Glossary: C

Definitions in this project whose names start with C. The whole glossary has the other sections.

A · B · C · D · E · F · G · H · I · J · K · L · M · N · O · P · Q · R · S · T · U · V · W · X · Y · Z · Other

C def

Entire real continuation of cosh(t sqrt x), also at negative x. · OAI.CriticalHoneycomb.FiniteFunctional.C

C def

Formal Hermitian adjoint. · OAI.ElasticityPlanar.C

C def

The evaluated one-parameter matrix C'. · OAI.EuclideanRamsey.NineGrid.C

C def

The x-chart of the blowup of the original plane at (0,0). Its coordinates s,t map to x,y/x, and the blowdown is x=s,y=st. · OAI.KummerTriple.C

C def

The x-chart of the blowup of the original plane at (0,0). Its coordinates s,t map to x,(x−y)/x, and the blowdown is x=s,y=s−st. · OAI.KummerTripleDiagonal.C

C def

C_K, computed without recursion in the upper argument of choose. · OAI.PixtonChow.ColorBudget.C

C def

The subgroup C is not replaced by its closure. · OAI.PlaneFiveColor.Spectral.C

c abbrev

The exponent in the exceptional-set estimate. · OAI.Problem337.RawDensityParameters.c

C1 def

The supremum of the first reciprocal degree moments over positive sizes. Boundedness is proved in reciprocal_degrees. · OAI.Thorp.Current.C1

c1 def

The infimum over every injective map into the full real sequence space ℓ₁. · OAI.TreeEdit.BinaryLower.c1

c1 def

Infimum over every injective map into the full real sequence space ℓ₁; there is no dimension or computability restriction. · OAI.TreeEdit.BinaryLower.c1

c1 def

Infimum over all injective ℓ₁ maps, used on finite sets with at least two points. · OAI.TreeEdit.c1

C1Bound def

The value and derivative bounds used to preserve immersion under smoothing. · OAI.ClosedSurfaceR4.FiniteOrderSmoothing.SmoothingAtlas.C1Bound

c1L2Submodule def

C¹ representatives of real H¹, with the L² inner product (not the H¹ norm). · OAI.SharpLiebThirring.SobolevProof.c1L2Submodule

c7 def

The square of the normal derivative of the unscaled conical function. · OAI.Bernoulli.DeSilvaJerison.c7

CacheControl structure

Numerical interface to the actual cached ensemble. The three tests below are proved from that ensemble, rather than postulated in the embedding theorem. · OAI.BoundedTreewidthL1.CacheControl

Cage structure

main (b=false) weight argument o. b=true witness uses score I, not the upper guarding score. · OAI.ExactDerandomization.Stem.Cage

CageQ structure

reusable request-origin signature. all indices in the allotted range, incl holes. · OAI.ExactDerandomization.SiteX.Line.CageQ

callback def

The option-valued emitter applies local overlap and prefix checks before an entry is retained. A rejected tuple consumes no dictionary space. · OAI.LowSpaceSubsetSum.RecordedDictionary.callback

cameo def

The commutator construction. Disjoint x->z, z->y adds. · OAI.PowerSaving.Slant.cameo

Camp structure

transparent query/list iteration · OAI.ExactDerandomization.Fret.Camp

Can abbrev

Ordinary closed computation guarantee. · OAI.PowerSaving.RAM.Can

cancel def

The cancellation homotopy is continuous jointly in both parameters and in the loop itself. · OAI.HyperbolicObstruction.LoopSpaceReversal.cancel

cancellationTowerModel def

The three transported components, keeping both shifted indices. · OAI.WeakPureInfiniteness.cancellationTowerModel

cancelledGreen def

The center is subtracted only off the near region, so this measurable kernel retains integrability at the center and realizes zero-mass cancellation. · OAI.PlanePotential.cancelledGreen

cancelledGreen def

The center is subtracted only off the near region, so this measurable kernel retains integrability at the center and realizes zero-mass cancellation. · OAI.SignedDisk.PlanePotential.cancelledGreen

cand def

candidate replacements for each repeated index · OAI.GeneralMahler.Scarf.cand

candidate def

The singleton interval of an inverse candidate entry. · OAI.AtomicTriangular.FiniteCertificate.AtomicMatrix.candidate

candidate def

Codes arising from actual boundaries are the only candidates needed. · OAI.PlanarBoundary.candidate

candidateAncestors def

Proper ancestors of a new child include its current parent map. · OAI.ReflexiveFixedPoints.CandidateContext.candidateAncestors

candidateBudget def

The dyadic budget of the candidate with zero-based ordinal n. · OAI.ReflexiveFixedPoints.candidateBudget

CandidateContext structure

Data already fixed when construction of a new finite level starts. · OAI.ReflexiveFixedPoints.CandidateContext

candidateCount def

Candidate counts include every map chosen before zero coefficients are removed. · OAI.ReflexiveFixedPoints.CandidateContext.candidateCount

candidateFunctional def

The detector functional is the one chosen for this candidate record. · OAI.ReflexiveFixedPoints.AnchoredCandidateContext.candidateFunctional

candidateLabel def

The original global candidate ordinal, with no renumbering after retention. · OAI.ReflexiveFixedPoints.AnchoredCandidateContext.candidateLabel

candidateLaw def

The left rule is uniform on its own chosen affine-answer candidates. · OAI.MaxCutGames.Decoder.PrivateStrategy.candidateLaw

candidateLaw def

The left rule is uniform on its own chosen affine-answer candidates. · OAI.UniqueGamesTheorem.Decoder.PrivateStrategy.candidateLaw

CandidateLedger structure

A completed candidate level, including every discarded candidate. · OAI.ReflexiveFixedPoints.CandidateLedger

candidateLedger def

The concrete ledger retains every candidate chosen at this level. · OAI.ReflexiveFixedPoints.AnchoredCandidateContext.candidateLedger

candidateLow def

Lower and upper endpoint subsets; the coefficient c is determined by these two subset products and the positive lag. · OAI.JointDickman.candidateLow

candidateMark def

An attempted mark on the geometric line joining consecutive point vertices. · OAI.Release075.candidateMark

CandidateMass def

The complete mathematical assertion of Lemma lem:filter, stated solely using the concrete source definitions and the finite independent fair-mask law. · OAI.MatroidProphet.CandidateSpecification.CandidateMass

candidateNumerator def

Numerators for the rational inverse candidates of I - R and I + R. · OAI.AtomicTriangular.FiniteCertificate.AtomicMatrix.candidateNumerator

candidatePointLedger def

The same actual ledger, evaluated at the single fixed anchor. · OAI.ReflexiveFixedPoints.AnchoredCandidateContext.candidatePointLedger

candidatePredictionLimit def

The same point expressed through the final actual ideal-cutoff evaluator. · OAI.ReflexiveFixedPoints.AnchoredCandidateContext.candidatePredictionLimit

candidatePrefix def

The actual history at the instant this retained candidate was chosen. · OAI.ReflexiveFixedPoints.AnchoredCandidateContext.candidatePrefix

candidatePrefixIndex def

This equivalence only changes the proof of the same prefix length. · OAI.ReflexiveFixedPoints.AnchoredCandidateContext.candidatePrefixIndex

candidatePrefixParameter def

The final retained parameter transported to the exact pre-map prefix. · OAI.ReflexiveFixedPoints.AnchoredCandidateContext.candidatePrefixParameter

candidatePrefixPrediction def

This is a member of the prediction family fixed before the map at v. · OAI.ReflexiveFixedPoints.AnchoredCandidateContext.candidatePrefixPrediction

candidateProfile def

The candidate unit-torus isoperimetric profile. · OAI.CubicTorus.candidateProfile

CandidateRecord structure

The auxiliary functional is chosen after this actual map has been selected. The budget and detector set are derived from the record's prefix. · OAI.ReflexiveFixedPoints.CandidateRecord

candidateRootWeight def

The five endpoint/coefficient factors and the sixth, auxiliary-root factor in the manuscript's kernel, including its normalization. · OAI.JointDickman.candidateRootWeight

candidates def

Exactly the bounded tested-class candidate set in the self-reference construction. · OAI.FiniteAlgebraicEnvelopes.Bootstrap.Canonical.candidates

candidates def

Trails with fewer than R steps connecting the specified endpoints. · OAI.GammaFPP.FiniteLocal.candidates

candidates def

Source candidate set: positive, unobserved, and not spanned by higher-priority observed labels. The full candidate set is analytical, not given to the algorithm. · OAI.MatroidProphet.CandidateSpecification.candidates

candidateTheta def

The monomial Θ indexed by all triples in lexicographic order, in the exterior algebra from Section 3. · OAI.PixtonChow.candidateTheta

candidateValue def

The exact real value of an inverse candidate entry. · OAI.AtomicTriangular.FiniteCertificate.AtomicMatrix.candidateValue

canonicalAffineLabel def

The chosen label depends only on the law and the actual profile, so different sequences use the same choice at any shared parameter. · OAI.PolycyclicRecognition.canonicalAffineLabel

canonicalAmplifiedAlgebra def

The countable amplification formed from the actual coordinate matrix units. · OAI.KadisonKastler.canonicalAmplifiedAlgebra

canonicalBall def

Canonical centered ball. Its radius on the listed branch is below 1/2. · OAI.CubicTorus.canonicalBall

canonicalBundleChartFiberIso def

Normalization of the genuine target over the canonical contractible chart. It is obtained from actual bundle evaluation and product projection. · OAI.Kaplansky.canonicalBundleChartFiberIso

canonicalBundleChartHomeomorph def

Actual bundle coordinates on the inverse image of the canonical base chart. This is a homeomorphism of the original total-space subspace. · OAI.Kaplansky.canonicalBundleChartHomeomorph

canonicalBundleChartRelativeInclusion def

Actual inclusion of the bundle over the canonical chart into the whole ordinary bundle, always removing the entire respective zero section. · OAI.Kaplansky.canonicalBundleChartRelativeInclusion

canonicalBundleChartTopHomologyIso def

Top relative homology over the actual bundle chart is ℤ, normalized using precisely the incoming recursive local generator. · OAI.Kaplansky.canonicalBundleChartTopHomologyIso

canonicalBundleCoverChartIso def

The removed-subspace convention in the general gluing theorem is identified with the preserved actual chart pair, with identity ambient map. · OAI.Kaplansky.canonicalBundleCoverChartIso

canonicalBundleCoverGenerator def

The incoming normalized generator, merely transported to the general relative-cover convention. No new generator or orientation is chosen. · OAI.Kaplansky.canonicalBundleCoverGenerator

canonicalBundleOverlapHomeomorph def

Actual coordinates on the overlap with the punctured bundle. Its base is punctured Euclidean space, not a contractible base. · OAI.Kaplansky.canonicalBundleOverlapHomeomorph

canonicalBundleOverlapRelativeIso def

A pair homeomorphism, retaining both the punctured base and the punctured fiber. This supplies no unjustified projection isomorphism. · OAI.Kaplansky.canonicalBundleOverlapRelativeIso

canonicalChartExcisionIso def

Actual local singular homology equivalence for the canonical chart. · OAI.Kaplansky.canonicalChartExcisionIso

canonicalChartFiberEmbedding def

The actual fiber over the canonical point, embedded using the proved bundle coordinates. Only the base coordinate is fixed. · OAI.Kaplansky.canonicalChartFiberEmbedding

canonicalChartRelativeInclusion def

Inclusion of the punctured chart into the punctured whole base. Its induced map is constructed; no excision isomorphism is asserted here. · OAI.Kaplansky.canonicalChartRelativeInclusion

canonicalChartSphereHomologyIso def

The local relative homology of the actual projective chart at its canonical zero is now compared with ordinary sphere homology. · OAI.Kaplansky.canonicalChartSphereHomologyIso

canonicalColumnName def

Perfect classes keep their forest/omitted name. Classes with no perfect occurrence use the index of their first occurrence. · OAI.TwoPointCorrelations.canonicalColumnName

CanonicalConvergence def

Literal filter version of the manuscript's canonical convergence formula. Every directed net is covered by its tail filter. · OAI.TraceConeClassification.CanonicalConvergence

canonicalCornerAlgebra def

The original algebra in the first coordinate of its countable sum. · OAI.KadisonKastler.canonicalCornerAlgebra

canonicalCoverOpen def

The five principal-open coefficient algebras used to compute the determinant lattice. · OAI.ExplicitCone.canonicalCoverOpen

CanonicalCyclic def

Canonical representatives permit exact word comparison of arrays. · OAI.ExactQuantumFactoring.BitArithmetic.CanonicalCyclic

canonicalDensity def

The exact rational density of a frame of K on the chart indexed by t. Its inverse fifth section factor is the same rational form on every chart. · OAI.SourceSymmetry.canonicalDensity

canonicalEquiv def

No route-existence hypothesis is needed for a connected realization. · OAI.HyperbolicObstruction.FinitePolyhedron.EdgePresentation.canonicalEquiv

canonicalExtension def

The canonical extension map, obtained from the actual last-vertex map. · OAI.Thomason.Construction.canonicalExtension

canonicalExtensionCounitHomotopy def

A double-subdivision homotopy gives the precise counit composite needed by the weak-equivalence argument. · OAI.Thomason.Construction.canonicalExtensionCounitHomotopy

canonicalFaceWeight def

Canonical normalized intrinsic area of the common closed cell face. · OAI.Problem358.PositiveFacet.canonicalFaceWeight

canonicalFiberGenerator def

The preserved recursive local class, now expressed in the actual fiber coordinate group. Its normalization is transported, never independently chosen. · OAI.Kaplansky.canonicalFiberGenerator

canonicalFields def

The fourth canonical record field is the saved occurrence ID. The equation RHS remains separately available on rhsField j. · OAI.MaxCutGames.Reduction.MachineSourceTuple.canonicalFields

canonicalFields def

The fourth canonical record field is the saved occurrence ID. The equation RHS remains separately available on rhsField j. · OAI.UniqueGamesTheorem.Reduction.MachineSourceTuple.canonicalFields

canonicalForm def

The source's rational 2-form dx∧dy/(y₀y₁y₂y₃y₄)^6. · OAI.KummerTriple.canonicalForm

canonicalFrame def

The regular frame expressed rationally as c₀₀⁵ dx∧dy/(∏yᵢ)^6. It is constructed in the determinant module. · OAI.KummerTriple.canonicalFrame

canonicalFrame def

A fixed, canonical choice from the positive square root of the inverse metric. No choice of curvature eigenvectors is needed for this frame. · OAI.PositiveEinsteinFour.Geometry.canonicalFrame

canonicalFrameSection def

The canonical generator, as a section of the intrinsic ∧² Ω. · OAI.SourceSymmetry.canonicalFrameSection

CanonicalFreeCornerCoreInput abbrev

The exact local structural input, with the native trace and faithfulness proofs supplied by the canonical generated model. · OAI.KadisonSimilarity.CanonicalFreeCornerCoreInput

CanonicalFreeCornerInput abbrev

Published local data on the exact canonical free-product model. · OAI.KadisonSimilarity.CanonicalFreeCornerInput

canonicalGeneratedMatrices def

The source matrix tower is transported into this exact generated model. · OAI.KadisonSimilarity.canonicalGeneratedMatrices

canonicalGeneratedSmallFrame def

The fixed small frame transported into the actual free second factor. · OAI.KadisonSimilarity.canonicalGeneratedSmallFrame

canonicalInitialEquiv def

The initial coordinate has exactly the original Hilbert norm. · OAI.KadisonKastler.canonicalInitialEquiv

canonicalInitialSpace def

The closed subspace supported at coordinate zero. · OAI.KadisonKastler.canonicalInitialSpace

canonicalIso def

Normalize a single-degree identification to induce the identity on its actual cohomology. · OAI.LittleFinitistic.DerivedTruncation.canonicalIso

canonicalize def

Replace the values outside the legal coordinates by false. · OAI.MaxCutGames.Foundations.PCP.UniformRestriction.canonicalize

canonicalize def

Replace the values outside the legal coordinates by false. · OAI.UniqueGamesTheorem.Foundations.PCP.UniformRestriction.canonicalize

canonicalizeTree def

Actual tree tours put every map/tree pair into the rooted normal form. · OAI.FKCRT.canonicalizeTree

canonicalLaw def

Explicit prepared law: the pushforward of the normalized tree density, using the actual overlap couplings, with the convention 0/0 = 0. · OAI.BoundedTreewidthL1.BagLabels.canonicalLaw

canonicalLine def

Invertibility is proved for the real intrinsic canonical sheaf by the chart cover of the source. · OAI.SourceSymmetry.canonicalLine

canonicalLinear def

Re-express a supplied k-linear map using the canonical scalars of ModuleCat D. · OAI.DerivedBar.ParityBridge.canonicalLinear

canonicalLocalRelativeHomologyIso def

Local section-coordinate isomorphism for the actual punctured chart, not just an abstract vector-space model. Excision into the full base is separate. · OAI.Kaplansky.canonicalLocalRelativeHomologyIso

CanonicallyReflexive def

Surjectivity of the canonical map into the continuous bidual. · OAI.ReflexiveFixedPoints.CanonicallyReflexive

canonicalMatrixCorner def

The standard countable matrix units on the Hilbert direct sum. · OAI.KadisonKastler.canonicalMatrixCorner

canonicalMatrixTail def

The literal tail of the universal dictionary's reversed insertion list. · OAI.DilutedSpinGlass.PrescribedTree.canonicalMatrixTail

canonicalNegativeProjection def

The negative-unit corner used in the direct-sum stabilization. · OAI.WeakPureInfiniteness.canonicalNegativeProjection

canonicalNegativeRow def

Transport of sᵢ ⊕ (sᵢ₊₂ p) along the rectangular row [s₁,p]. · OAI.WeakPureInfiniteness.canonicalNegativeRow

canonicalNormalizedEquiv def

The canonical normalized geometric cell Laplacian is a continuous linear isomorphism, with all positivity and connectivity consequences proved. · OAI.Problem358.PositiveFacet.canonicalNormalizedEquiv

canonicalNormalizedOverlapGenerator def

The unique overlap class with preserved normalization one. · OAI.Kaplansky.canonicalNormalizedOverlapGenerator

canonicalOracle def

Metadata-first factorization of exactly the same law, not a new oracle. · OAI.TraceReconstruction.PairedMechanisms.canonicalOracle

canonicalOverlapNormalization def

The preserved integer normalization measured through the actual overlap-to-chart map. It is not a global Thom class. · OAI.Kaplansky.canonicalOverlapNormalization

canonicalOverlapRatio def

The transition value is an element of the intersection ring. · OAI.SourceSymmetry.canonicalOverlapRatio

canonicalOverlapSection def

The intrinsic sheaf interpretation of an algebraic top form on the intersection. · OAI.SourceSymmetry.canonicalOverlapSection

canonicalProductChartHomeomorph def

The actual open neighborhood of the unique zero is homeomorphic to the full complex graph model, with the inherited topology on the neighborhood. · OAI.Kaplansky.canonicalProductChartHomeomorph

canonicalProductCoordinateMap def

Actual section coordinate map as a continuous map on the full graph model. · OAI.Kaplansky.canonicalProductCoordinateMap

canonicalProductCoordinateRelativeIso def

The local coordinate map induces an isomorphism on genuine relative singular homology. This does not assert that these homology groups are nonzero. · OAI.Kaplansky.canonicalProductCoordinateRelativeIso

canonicalProductGraphChart def

The genuine product graph chart used for the canonical zero. · OAI.Kaplansky.canonicalProductGraphChart

canonicalProductHomologyMap def

The actual canonical hyperplane-product section induces this integral homology morphism. Its nonvanishing still requires a global calculation. · OAI.Kaplansky.canonicalProductHomologyMap

canonicalProductLocalGenerator def

A specific primitive integral local class, obtained from 1 under the proved local homology isomorphism. Its orientation is the chosen recursive coordinate orientation; comparison with the standard complex orientation is separate. · OAI.Kaplansky.canonicalProductLocalGenerator

canonicalProductLocalHomologyIso def

Local relative integral homology at the canonical point of the full projective product is ℤ in its real dimension. · OAI.Kaplansky.canonicalProductLocalHomologyIso

canonicalProductRelativeHomologyMap def

Relative singular homology map on the genuinely punctured projective product. · OAI.Kaplansky.canonicalProductRelativeHomologyMap

canonicalProductSectionCoordinates def

The actual global product section expressed componentwise in the constructed graph evaluation trivializations. · OAI.Kaplansky.canonicalProductSectionCoordinates

canonicalProductSignedZeroCount def

The signed zero count of this specific section is one. Invariance under changing the section, and identification with an Euler/Chern invariant, remain separate obligations; neither is asserted by this definition. · OAI.Kaplansky.canonicalProductSignedZeroCount

canonicalProductZeroSet def

The finite set here is proved to be the zero set of the actual global section, and to lie inside the chart where the local sign is computed. · OAI.Kaplansky.canonicalProductZeroSet

canonicalPublishedOrientedData def

The canonical frame probability supplies the averaging and support fields, while the initial condition supplies orientation preservation. · OAI.PositiveEinsteinFour.Geometry.canonicalPublishedOrientedData

canonicalPuncturedBundle def

The complement of the entire canonical fiber, an open subset of the actual hyperplane bundle total space. · OAI.Kaplansky.canonicalPuncturedBundle

CanonicalRegularEntropyTests def

The finite family of entropy identities used by the prime-degree application. Both time directions are retained explicitly. · OAI.DukePrimeDegree.CanonicalRegularEntropyTests

CanonicalRegularEntropyUpperBounds def

The finite regular upper estimates, expressed without entropy identities. · OAI.DukePrimeDegree.CanonicalRegularEntropyUpperBounds

canonicalRepresentationSetup def

The solution is supplied only to the analysis of the fixed runtime data. · OAI.LowSpaceSubsetSum.canonicalRepresentationSetup

canonicalRestrictedSection def

The restriction of the actual section, before applying bundle coordinates. · OAI.Kaplansky.canonicalRestrictedSection

canonicalRootCoefficient def

The actual mean information at unit scale, with zero diagonal. · OAI.DukePrimeDegree.canonicalRootCoefficient

canonicalRootEntropyCoefficients def

The canonical entropy formula, together with the elementary coefficient properties, gives root entropy coefficients. · OAI.DukePrimeDegree.canonicalRootEntropyCoefficients

CanonicalRootEntropyFormula def

The canonical root entropy identity of EKL (2006), Proposition 3.1(3). · OAI.DukePrimeDegree.CanonicalRootEntropyFormula

CanonicalSecondFrame def

The chosen closed-range proof is included inside this data type. · OAI.KadisonSimilarity.CanonicalSecondFrame

canonicalSheaf def

The intrinsic top exterior power of the cotangent sheaf over the complex field on the source Proj. · OAI.SourceSymmetry.canonicalSheaf

canonicalShiftCycle def

The right cycle on all words of length n+1. · OAI.WeakPureInfiniteness.canonicalShiftCycle

canonicalSlab def

Canonical slab of width V between two parallel coordinate tori. · OAI.CubicTorus.canonicalSlab

canonicalSlice def

The actual completely positive coordinate-zero slice. · OAI.KadisonKastler.canonicalSlice

canonicalSlot def

First matching visible occurrence, with the third slot as an off-support default. Its inputs contain no hidden incidence choice. · OAI.MaxCutGames.Foundations.Hastad.SourceContexts.canonicalSlot

canonicalSlot def

Search the fixed six positions for the least occurrence of an address. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Relations.canonicalSlot

canonicalSlot def

First matching visible occurrence, with the third slot as an off-support default. Its inputs contain no hidden incidence choice. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceContexts.canonicalSlot

canonicalSlot def

Search the fixed six positions for the least occurrence of an address. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Relations.canonicalSlot

canonicalSmallFrameCore def

Transport all small-frame data into the actual second range. · OAI.KadisonSimilarity.canonicalSmallFrameCore

canonicalTableau def

One fixed tableau; changing its enumeration gives an isomorphic Specht module. · OAI.Saxl.canonicalTableau

canonicalTableIndex def

The lookup index uses exactly the same two swaps as the sampler. · OAI.ContingencyTables.DensePrograms.canonicalTableIndex

canonicalTailProjection def

The remaining projection after the first n canonical ranges. · OAI.WeakPureInfiniteness.canonicalTailProjection

canonicalTheta def

The first isomorphism theorem identifies the quotient of Q by the kernel with T. · OAI.FiniteLattice.CommonCore.canonicalTheta

canonicalTrace def

The canonical trace is the coefficient of the identity. · OAI.TorsionFreeBass.canonicalTrace

CanonicalTreeMap structure

Literal rooted-tour normal form, not a map with an assumed word inverse. · OAI.FKCRT.CanonicalTreeMap

canonicalTube def

Canonical tube around the shortest closed geodesic in coordinate direction 0. · OAI.CubicTorus.canonicalTube

canonicalUnit def

The distinguished CRT residue used in the list-completion branch. The parameter p₀ is subsequently chosen as the least prime component. · OAI.ExactQuantumFactoring.canonicalUnit

canonicalUnit def

Unit multiplying du∧dv in the canonical frame c₀₀⁵ ω at n=7. · OAI.KummerTriple.canonicalUnit

canonicalWholeBundleFiberMap def

Actual inclusion of this one fiber into the ordinary whole-bundle pair. · OAI.Kaplansky.canonicalWholeBundleFiberMap

canonicalWordEquiv def

Exact finite bimapping of the genuine canonical rooted planar tree maps and successful typed words; no symmetry divisor and no assumed inverse. · OAI.FKCRT.canonicalWordEquiv

CanSample def

A uniform sampler with a deterministic query budget on every execution and total-variation error at most one tenth. · OAI.LogConcaveSampling.CanSample

capacity def

Operational unassisted classical capacity, not a definition by regularization. · OAI.GAD.capacity

capacity def

The number of new elements on a level of the hierarchy. · OAI.LeanBlast.SecondKahnKalai.SetHierarchy.capacity

capacityBits def

The full product allowance is computed from the input capacities' binary lengths, rather than supplied as a counting hypothesis. · OAI.ContingencyTables.CountingTerminalSchedule.capacityBits

capacityPolynomial def

For fixed k,s,d, the entire direct-address space is a polynomial in n+m. · OAI.MaxCutGames.Reduction.CanonicalAddress.capacityPolynomial

capacityPolynomial def

For fixed k,s,d, the entire direct-address space is a polynomial in n+m. · OAI.UniqueGamesTheorem.Reduction.CanonicalAddress.capacityPolynomial

capacityReserve def

A history-free allowance for the spread parameter at capacity m. · OAI.LeanBlast.SecondKahnKalai.capacityReserve

capAmplitude def

The transverse unit-cylinder profile of the concrete upper barrier. · OAI.Bernoulli.DeSilvaJerison.capAmplitude

CapAvailable def

A radial cap exponent available after a power-small deletion of original pairs. Constants and the starting scale may depend on the desired loss. · OAI.Falconer.CapAvailable

capQuadratic def

The exact quadratics used in the positive-cap comparison. · OAI.AffineBernstein.capQuadratic

capTerm def

One rank's core-plus-scaled-flex contribution to the actual cap. · OAI.KServer.FineCaps.capTerm

capTiltedCovector def

Every coefficient of these covectors is positive, so their affine caps are uniformly bounded in the orthant model. · OAI.AffineBernstein.capTiltedCovector

capTiltedForm def

The actual continuous linear forms chosen in bounds.tex256. · OAI.AffineBernstein.capTiltedForm

capTransport def

A cap supplies a literal potential at every actual boundary occurrence. · OAI.HyperbolicObstruction.FinitePolygonModel.Data.capTransport

Captured def

An actual bad original coloring, its selected endpoint, and its full profile. · OAI.TalagrandThreshold.Selector.Captured

carClock def

Ordered finite product of the commuting occupation clocks. · OAI.WeakPureInfiniteness.carClock

carClockFactor def

The two occupation projections receive phases one and ζ. · OAI.WeakPureInfiniteness.carClockFactor

cardSplit def

Separate the first r coordinates from the remaining s coordinates. The sum is written s+r so its recursion agrees literally with the network. · OAI.CubeShuffle.cardSplit

carDyadicPhase def

The phase with half-period 2^i. · OAI.WeakPureInfiniteness.carDyadicPhase

carModeParity def

Twice the occupation projection is subtracted from the corner unit. · OAI.WeakPureInfiniteness.carModeParity

carried def

All positive mass is supported on p. We use the equivalent contrapositive with zero mass (and supply nonnegativity wherever we speak of a law). · OAI.HadwigerCounterexample.carried

carried def

easier lift over a scan on coordinates. · OAI.PowerSaving.RAM.Bench.carried

carrier def

Ambient open carrier of an arbitrary open subset of a chart source. It does not assert a smooth extension across a manifold boundary. · OAI.SmoothObstruction.CoordinateTubes.carrier

carrier def

Every actual realization point has a well-defined supported simplex subcomplex. · OAI.Thomason.Construction.RealizationPrism.carrier

carrier def

Adding an integral carrier changes no temporal eigenfrequency modulo one. · OAI.ThreeTorus.PassagePackets.carrier

carrier def

The entire cylinder, with all boundary points included. · OAI.TriangularCovering.TriCylinder.carrier

carrierCocone def

The supported carriers are compatible with every simplex arrow. · OAI.Thomason.Construction.RealizationPrism.carrierCocone

carrierHomeomorph def

The realization of a subcomplex is its intrinsic carrier subspace, homeomorphically. · OAI.Thomason.Construction.RealizationPrism.carrierHomeomorph

carrierTriangle def

Three paths in one carrier satisfy the endpoint-fixed triangle coherence needed by a relative simplicial approximation. · OAI.Thomason.Construction.PosetGeometry.carrierTriangle

Carries def

The converter-routing construction retains its finite admissible context, its exact renaming of the original input, and the normal positive carried type. · OAI.PureTypeSystem.Labelled.Carries

Carry def

Each carry case fixes just the low prefix, including its first noncarry bit. · OAI.WLIdentification.BinaryAddress.Carry

carry def

update of accumulated same-parity output. · OAI.FourierLLogL.Dyadic.Procedure.Paths.Beam.Lamp.carry

carryCoins def

Splitting a fair coin layer according to an inserted carried bit is a bijection. · OAI.CubeShuffle.carryCoins

cartanConjugation def

The conjugation induced on the actual complexified Cartan subalgebra. · OAI.PolycyclicRecognition.cartanConjugation

CartanFactorModel structure

The concrete group part of the manuscript's Cartan model. The ambient overlap is retained, and all lifting conclusions refer to the actual map. · OAI.PolycyclicRecognition.CartanFactorModel

cartanFactorToRotation def

The Cartan factor also retains its original multiplication. · OAI.PolycyclicRecognition.ComplexWeightFactorModel.cartanFactorToRotation

cartanHeightSection def

Choose a linear Cartan representative of each height. · OAI.PolycyclicRecognition.cartanHeightSection

cartanHeightWeight def

A Cartan root as a linear form on the canonical height quotient. · OAI.PolycyclicRecognition.cartanHeightWeight

cartanIdealAdjointAutomorphism def

The full Cartan action, including its nonsemisimple part. · OAI.PolycyclicRecognition.cartanIdealAdjointAutomorphism

cartanIdealResidualAutomorphism def

The residual automorphism of the nilpotent factor used in box estimates. · OAI.PolycyclicRecognition.cartanIdealResidualAutomorphism

cartanIdealResidualDerivation def

Restrict the residual derivation before taking its finite exponential. · OAI.PolycyclicRecognition.cartanIdealResidualDerivation

cartanIdealWeightDilation def

Restriction to the actual nilpotent Cartan ideal. · OAI.PolycyclicRecognition.cartanIdealWeightDilation

cartanLog def

The Cartan logarithm in the actual nilpotent exponential coordinates. · OAI.PolycyclicRecognition.cartanLog

cartanMultiplication def

Actual factor multiplication as a group homomorphism. · OAI.PolycyclicRecognition.cartanMultiplication

cartanRealShadowEquiv def

Only multiplication changes: the real vector space is unchanged. · OAI.PolycyclicRecognition.cartanRealShadowEquiv

cartanRealShadowSubalgebra def

The graph of minus the imaginary semisimple derivation. · OAI.PolycyclicRecognition.cartanRealShadowSubalgebra

cartanResidualAutomorphism def

The residual action exponentiates by a finite polynomial to an actual Lie automorphism. · OAI.PolycyclicRecognition.cartanResidualAutomorphism

cartanResidualDerivation def

The globally defined residual action. The library's inner h sends x to [x,h], hence the minus sign. · OAI.PolycyclicRecognition.cartanResidualDerivation

cartanRootIdeal def

The ideal generated by the nonzero generalized root spaces. · OAI.PolycyclicRecognition.cartanRootIdeal

cartanRootSubmodule def

Regard the generated subalgebra as a module for the Cartan subalgebra. · OAI.PolycyclicRecognition.cartanRootSubmodule

cartanScalarDerivation def

The scalar part as a genuine derivation of the ambient Lie algebra. · OAI.PolycyclicRecognition.cartanScalarDerivation

cartanScalarOperator def

The scalar part of the Cartan adjoint operator on the generalized-root sum, before exponentiation. · OAI.PolycyclicRecognition.cartanScalarOperator

cartanSupportDatum def

Support data for any finite collection of actual word bases. · OAI.PolycyclicRecognition.cartanSupportDatum

cartanWeightDilation def

Exact dilation as a Lie algebra automorphism, with inverse parameter -h. · OAI.PolycyclicRecognition.cartanWeightDilation

cartanWeightScale def

Scalar dilation on the actual generalized-root direct sum. · OAI.PolycyclicRecognition.cartanWeightScale

cartanWeightSection def

A linear choice of real Cartan representative for each weight vector. · OAI.PolycyclicRecognition.ComplexTriangularAdjoint.cartanWeightSection

CartanWordModel structure

Finite homogeneous generators, their abelianization basis, and an actual lower-central-series bound for the Cartan ideal. · OAI.PolycyclicRecognition.CartanWordModel

cartesianChange def

The literal Cartesian coefficient substitution z=x+iy,w=x-iy, including the index identification with the ambient Euclidean R^16. · OAI.HarmonicCounterexample.Control.cartesianChange

cartesianGenerator def

Actual Cartesian generators share the original real-form simultaneous control. · OAI.HarmonicCounterexample.Control.cartesianGenerator

cartierChart def

The actual polynomial Cartier chart above the fixed section z₀z₁z₄, inside the SAME rational function field used for the source cone. · OAI.ExplicitCone.cartierChart

casimir def

The spherical Laplacian expressed by its rotation generators. · OAI.PettyProjection.Spherical.casimir

casimirCurvatureForm def

The quadratic form after the two exact Casimir normalizations. · OAI.PositiveEinsteinFour.HighestHessian.casimirCurvatureForm

Cast def

Path additionally follows birth rules. · OAI.FourierLLogL.Dyadic.Procedure.Paths.Troupe.Cast

castCoeffs def

The canonical coefficient map, with no change to the index. · OAI.CirculantHadamard.BinaryCoefficients.castCoeffs

castDepth def

Reindex a tree along equality of depths, without changing its data. · OAI.LeanBlast.SecondKahnKalai.ProbabilityTree.castDepth

cat def

Disjoint coordinate intervals. · OAI.PowerSaving.Layout.cat

CAT0 def

Geodesic Euclidean triangle comparison. For points of parameters t and u on the sides xy and xz, the right hand side is the squared Euclidean comparison distance, expanded by the law of cosines. All geodesic sides, also degenerate ones, … · OAI.ArtinCAT0.CAT0

CATActionWitness structure

A genuine action on an arbitrary chosen space in the specified universe. · OAI.HyperbolicObstruction.CATActionWitness

Catalog def

Every stored edge is valid, and every graph edge is stored in at least one orientation with its exact capacity. · OAI.ContingencyTables.ForestGreedySemantics.Catalog

categoricalProductMass def

Mutually exclusive site hits, independent between coordinates. · OAI.JointDickman.categoricalProductMass

categoricalSubsetMass def

Empty hit set with mass 1-sum q, singleton {i} with mass q i, and zero on every other subset. The delta-sum form avoids any choices. · OAI.JointDickman.categoricalSubsetMass

categorification def

Categorification is the colimit of orientals indexed by all simplices. · OAI.Thomason.Construction.NormalizedSimplexIdentities.categorification

categorificationAdjunction def

The defining adjunction, derived from the colimit universal property. · OAI.Thomason.Construction.NormalizedSimplexIdentities.categorificationAdjunction

categorificationObjectsEquiv def

The objects of finite-dimensional categorification are the original vertices. · OAI.Thomason.Construction.FiniteReflectionInput.categorificationObjectsEquiv

categorificationObjectsEquiv def

Objects of categorification are exactly the original vertices. · OAI.Thomason.Construction.NormalizedSimplexIdentities.categorificationObjectsEquiv

categorificationObjIsoColimit def

An explicit identification with the simplex colimit, including degeneracies. · OAI.Thomason.Construction.NormalizedSimplexIdentities.categorificationObjIsoColimit

categorificationSimplexIso def

On standard simplices, categorification is naturally the given oriental. · OAI.Thomason.Construction.NormalizedSimplexIdentities.categorificationSimplexIso

category abbrev

Every strict law is detected by the jointly injective cocone evaluations. · OAI.Thomason.Construction.StrictOmegaCategory.ColimitConstruction.category

category def

The ordinary filtered cell colimit, equipped with its constructed strict operations and all the original strict laws. · OAI.Thomason.Construction.StrictOmegaCategory.FilteredColimit.category

category def

The strict n-globe, with its actual finite boundary cells. · OAI.Thomason.Construction.StrictOmegaCategory.Globe.category

category def

The compatible bounded families satisfy all strict laws componentwise. · OAI.Thomason.Construction.StrictOmegaCategory.LimitConstruction.category

category def

Adjoin a terminal vertex and one arrow from each original object. · OAI.Thomason.Construction.StrictOmegaCategory.TerminalCone.category

category def

The strict omega-category associated to a poset, truncated in dimension one. · OAI.Thomason.Construction.StrictPoset.category

CategoryDomain abbrev

The domain is the disjoint union of the three categories in the paper. · OAI.LowSpaceSubsetSum.CategoryDomain

categoryPrimeProduct def

Prime product corresponding to one of the common/exclusive categories. · OAI.JointDickman.categoryPrimeProduct

categoryTripleEquiv def

Three category bits at each site are exactly three Boolean vectors. · OAI.JointDickman.categoryTripleEquiv

CATNegOne def

The hyperbolic-cosh midpoint characterization of CAT(-1). · OAI.HyperbolicObstruction.CATNegOne

CATZero def

The CN characterization of CAT(0), with whole metric segments. · OAI.HyperbolicObstruction.CATZero

causalStep def

The sampled outcome history, without a bundled measurability certificate. · OAI.ZeroKey.AdaptiveOutcomeSampler.causalStep

caveat def

A local bit-Fisher budget · OAI.FourierLLogL.Dyadic.Procedure.Paths.caveat

cavityBackwardQuadratic def

The coefficient K_i = K (1 - H_i K)⁻¹ of the backward quadratic. · OAI.InvariantIsing.cavityBackwardQuadratic

cavityCoordinateEquiv def

Move the singled-out interaction coordinate to the distinguished last slot, preserving the relative order of all other slots. · OAI.DilutedSpinGlass.cavityCoordinateEquiv

cavityFactorMean def

The base Gibbs law times the independent uniform probability on the cavity cube, evaluated on the positive cavity factor. · OAI.InvariantIsing.cavityFactorMean

CavityGood def

Keep precisely the bonds meeting the new site at most once. · OAI.DilutedSpinGlass.CavityGood

cavityInnovationTree def

Retain and rescale each Gaussian mark while updating the running sum by the original increment. · OAI.InvariantIsing.cavityInnovationTree

cavityJoin def

Duplicate every block at every site before selecting its actual allocation. The unused duplicates disappear by prior-law projection, not by independence of tilted physical sites. · OAI.DilutedSpinGlass.PrescribedTree.cavityJoin

cavityResidualPartition def

The final Gaussian integral is an ordinary residual at each leaf, not an additional Poisson level. · OAI.InvariantIsing.cavityResidualPartition

CavityState def

Each cavity interaction has its own independent p-tuple of trial trees. · OAI.DilutedSpinGlass.PrescribedTree.CavityState

cavityStepTransition def

The conditional-mean matrix at a step from covariance P to C. · OAI.InvariantIsing.cavityStepTransition

cayleyAverage def

Average along the actual outgoing ports of the Cayley graph. · OAI.MaxCutGames.Foundations.PCP.CayleySpectral.cayleyAverage

cayleyAverage def

Average along the actual outgoing ports of the Cayley graph. · OAI.UniqueGamesTheorem.Foundations.PCP.CayleySpectral.cayleyAverage

cayleyGraph def

The undirected regular port graph generated by the indexed cube vectors. · OAI.MaxCutGames.Foundations.PCP.CayleySpectral.cayleyGraph

cayleyGraph def

The undirected regular port graph generated by the indexed cube vectors. · OAI.UniqueGamesTheorem.Foundations.PCP.CayleySpectral.cayleyGraph

cayleyLeft def

Left translation is an automorphism of the right-generator Cayley graph. · OAI.Release075.cayleyLeft

cdf def

The cumulative integral starting at the lower endpoint. · OAI.Problem358.PositiveIntervalDensity.cdf

cdfOrderIso def

Cumulative integration is an order isomorphism between the two compact intervals. · OAI.Problem358.PositiveIntervalDensity.cdfOrderIso

CEAlphaTwo abbrev

The Hamiltonicity specialization for graphs with no independent triple. · OAI.CycleClique.CEAlphaTwo

CEAlphaTwo def

The precise Chvátal–Erdős specialization used in the size test. Connectivity after every one-vertex deletion is the required two-connectivity condition at this order. · OAI.CycleClique.Construction.CEAlphaTwo

ceilingInTime def

Actual geometric search, placed onto the eventual family-level tape. · OAI.MaxCutGames.Foundations.Complexity.MachinePaddedExpanderFamily.ceilingInTime

ceilingInTime def

Actual geometric search, placed onto the eventual family-level tape. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachinePaddedExpanderFamily.ceilingInTime

ceilingSource def

The ceiling program preserves the family state as its ambient component. · OAI.MaxCutGames.Foundations.Complexity.MachinePaddedExpanderFamily.ceilingSource

ceilingSource def

The ceiling program preserves the family state as its ambient component. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachinePaddedExpanderFamily.ceilingSource

ceilingTape def

The only shared tape is the level: the first phase writes the exact word that the second phase's unary guard consumes. · OAI.MaxCutGames.Foundations.Complexity.MachinePaddedExpanderFamily.ceilingTape

ceilingTape def

The only shared tape is the level: the first phase writes the exact word that the second phase's unary guard consumes. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachinePaddedExpanderFamily.ceilingTape

ceilQuotient def

Integer ceiling; valid for positive a. · OAI.MaxCutGames.Reduction.WeightRounding.ceilQuotient

ceilQuotient def

Integer ceiling; valid for positive a. · OAI.UniqueGamesTheorem.Reduction.WeightRounding.ceilQuotient

Cell structure

Positive closed cubes of the actual Cayley cubical model. · OAI.EilenbergGanea.CubeRealization.Cell

Cell abbrev

A cell includes every tag field, even unused ones. · OAI.FiniteAlgebraicEnvelopes.Compiler.Validator.Cell

Cell abbrev

The three actual kinds of faces AFTER radial top triangles have been assembled and the closed macro walks split into simple cycles. Such assembly and its genus preservation are separate obligations, not fields here. · OAI.HyperbolicObstruction.ConeSkeleton.Cell

Cell def

Precisely the anchor and signature cells of one selected frame. · OAI.Problem348.GuardedPath.AnchorFrame.Cell

Cell structure

The finite positive chain table defining a cell of nu K. · OAI.Thomason.Construction.DirectedComplex.Cell

Cell abbrev

A small quotient: large cocones occur only in its defining proposition. · OAI.Thomason.Construction.StrictOmegaCategory.ColimitConstruction.Cell

Cell abbrev

Carrier of the intelligent truncation, before equipping its operations. · OAI.Thomason.Construction.StrictOmegaCategory.IntelligentTruncation.Cell

Cell structure

Compatible diagram cells sharing a finite dimension bound. · OAI.Thomason.Construction.StrictOmegaCategory.LimitConstruction.Cell

cell def

The extrusion of the upper facet normal to coordinate i. · OAI.DiagonalZonotope.cell

cell def

The cell table is finite when the alphabets and state set are finite. All tests and assignments are precisely the source's scan expansion. · OAI.FiniteAlgebraicEnvelopes.Compiler.Simulator.cell

cell def

Partition the unit interval into N half-open equal cells. · OAI.TorsionFreeZeroDivisors.BlockGrid.cell

cell def

Actual oriented physical two-cell with its group translate. The translation is measured from the common cone apex, not from the endpoint of the edge. · OAI.TorsionFreeZeroDivisors.ConeChains.cell

cellApp def

Extend the table entries of a cell linearly on the globe basis. · OAI.Thomason.Construction.DirectedGlobeComplex.cellApp

cellBoundaryMap def

The actual attaching map is the realization of the simplicial boundary followed by the given characteristic simplex. · OAI.Thomason.Construction.RealizationPrism.cellBoundaryMap

cellDiagram def

Single-cell diagram from stay, rightward, and leftward augmented transitions. · OAI.TwoWayComplementation.cellDiagram

cellDifferences def

The nonnegative differences of a set of cell indices from its first index. · OAI.Ostmann.cellDifferences

cellDirection def

Directed sides, retaining the literal nine labels, not just their group values. · OAI.ArtinCAT0.cellDirection

cellDisk def

The literal indicated cell, not a reselected word-equivalent relator. · OAI.HyperbolicObstruction.WordPictures.WordPicture.cellDisk

cellIndex def

An absent large cell receives the out-of-range code n. · OAI.ContingencyTables.SmallGraphProfiles.cellIndex

cellInteriorHomeomorph def

The open barycentric simplex is homeomorphic to the corresponding nondegenerate cell in the actual realization. · OAI.Thomason.Construction.RealizationPrism.cellInteriorHomeomorph

cellLabel def

The exact label of a point in a disjoint family; points outside all cells are assigned none, without choosing an arbitrary distinguished cell. · OAI.Problem356.FiniteCells.cellLabel

cellLinearMap def

Face derivatives assembled into one continuous linear cell-mass derivative. The index set can exclude the diagonal or any inactive faces. · OAI.Problem358.FaceDerivative.cellLinearMap

cellMap def

The canonical positive globe map associated to a cell table. Its top cell is the n-source of the table, hence the table itself when bounded. · OAI.Thomason.Construction.DirectedGlobeComplex.cellMap

cellNetwork def

A single cell retains stays internally and exposes its two moving transitions. · OAI.TwoWayComplementation.cellNetwork

cellOrbitEquiv def

Every lifted ordinary cell is uniquely a deck translate of the chosen lift of its base cell. This is the actual free-module coefficient indexing. · OAI.EilenbergGanea.CWCover.cellOrbitEquiv

CellPicture structure

A finite genus-zero side pairing with actual indexed cellular face maps. This describes a raw picture, without any reducedness or link bounds. · OAI.HyperbolicObstruction.PictureSwitch.CellPicture

cellPoint def

The positive preorder point lies inside its equal-mass time cell. · OAI.FKCRT.cellPoint

cellPoint def

The actual point in the broad component underlying a cell-domain point. · OAI.PlanarL1.Columns.Model.MainCell.cellPoint

cellProbabilityMeasure def

Uniform sampling from an actual measurable cell of positive finite mass. · OAI.PolycyclicRecognition.cellProbabilityMeasure

cellRoleCountBound def

Upper bound for all selected cell roles, including the anchors. · OAI.Ostmann.cellRoleCountBound

cellRoleEpsilon def

The fixed small parameter used in the character selection. · OAI.Ostmann.cellRoleEpsilon

cellRoleScale def

The larger exponential scale which absorbs the cell-role count. · OAI.Ostmann.cellRoleScale

Cells def

A concrete ordered disjoint union of consecutive columns. · OAI.Saxl.Columns.Cells

cells def

Masks and an order pointer can use fewer cells. Counting every mask bit and every remaining order element still gives this conservative bound. · OAI.LowSpaceSubsetSum.PrefixMachine.Control.cells

cells def

One scalar and the mask list, including its pointer/header. · OAI.LowSpaceSubsetSum.StoredEntry.cells

cells abbrev

The underlying cells, with every strict functor acting by its actual function. · OAI.Thomason.Construction.StrictOmegaCategory.FilteredColimit.cells

cellSheet def

The sheet-coordinate of a point on a lifted characteristic disk is continuous, including along the disk's nonembedded attaching boundary. · OAI.EilenbergGanea.CWCover.cellSheet

cellsIsColimit def

Forgetting to cells preserves the constructed filtered colimit exactly. · OAI.Thomason.Construction.StrictOmegaCategory.FilteredColimit.cellsIsColimit

cellSubtypeProbability def

The original normalized cell law, viewed on its carrying subtype. · OAI.PolycyclicRecognition.cellSubtypeProbability

cellTable def

The joint finite-cell mass table of a triple measure. · OAI.Problem356.cellTable

CellTarget structure

Oriented, individually indexed polygon boundaries in a target cell complex. In a lifted target, S includes the lift of the cell. Opposite orientations have disjoint side cycles; the same indexed side is retained when reversing. These … · OAI.HyperbolicObstruction.PictureSwitch.CellTarget

cellTops def

The tops in an actual translated angular cell, with the scale cutoff included in membership rather than imposed after assigning the cell. · OAI.LipschitzHilbert.AngularGrid.cellTops

cellToWalk def

Raw mixed pictures are now produced from actual indexed cellular pictures; no simple/reduced/fan-link condition is assumed at this bridge. · OAI.HyperbolicObstruction.ConeSkeleton.cellToWalk

CellTower structure

The original construction as a continuous tower of literal cells. Identity steps are allowed, so a selected or an omitted subconstruction can keep the same index order without compressing its limit cuts. · OAI.PartitionConsistency.ConservativeMonoids.CellTower

cellUpdate def

Assigned entry, remaining row margin, and remaining column margin. · OAI.ContingencyTables.GreedyFeasibleTable.cellUpdate

cellUpdateRealizer def

A fixed finite program over binary-encoded natural numbers. The compiler's correct field proves its literal execution and step bound. · OAI.ContingencyTables.GreedyFeasibleTable.cellUpdateRealizer

cellWalkStep def

Forget only the choice of simple circuit surrounding a top step. Frozen fan/decorative cell identities and all their positions are retained. · OAI.HyperbolicObstruction.ConeSkeleton.cellWalkStep

center def

A fixed actual opposite pair defines the common center. · OAI.HyperbolicObstruction.CoreGeometry.MetricCore.center

center def

The center is a function of the unconditional probability law alone. · OAI.MetricMeasureLimits.DiameterCenter.center

center def

The common final vertex is the cube center. · OAI.SharpLiebThirring.CubeFlags.Flag.center

centerAtom def

The actual full-outcome fiber of the center selector exposure. · OAI.Problem310.CenterFailureBound.centerAtom

centerChoice def

The actual first-success choice made by a selector table at a fixed center. · OAI.Problem310.CenterRouteExposure.centerChoice

centerChoice def

Canonical child choice from a family of selector bits at tree addresses. · OAI.Problem310.FirstDefault.centerChoice

centerCommutantInclusion def

The original algebra as a subalgebra of the decomposable operators. · OAI.KadisonSimilarity.centerCommutantInclusion

centerCompletion def

Complete any exposure assignment by setting all unexposed bits to false. This is only a deterministic representative of an atom, not a probability law. · OAI.Problem310.CenterRouteExposure.centerCompletion

centered def

The translate of the diagonal zonotope centered at the origin. · OAI.DiagonalZonotope.centered

centeredArc def

A representative of the centered arc indicator; endpoints have measure zero. · OAI.KadisonSimilarity.centeredArc

centeredBase def

The centered linear subspace of a base star algebra. · OAI.KadisonKastler.centeredBase

centeredBaseAlgebraVector def

The trace-zero algebra elements, with their actual centered Hilbert vectors. · OAI.KadisonSimilarity.TracialCornerGeometry.centeredBaseAlgebraVector

centeredChartHomeomorph def

Center an arbitrary full vector-space chart at the point under study. · OAI.Kaplansky.centeredChartHomeomorph

centeredCoefficient def

Coefficients of D^15 P((c + X)/D); all denominators are cleared. · OAI.DefocusingNLS.SeparatorArithmetic.centeredCoefficient

centeredCombination def

The centered linear combination appearing after interpolation. · OAI.JointDickman.PublishedInputs.centeredCombination

centeredCornerProjection def

The bounded map that compresses to the corner and removes its scalar part. · OAI.KadisonSimilarity.TracialCornerGeometry.centeredCornerProjection

centeredCount def

The centered Bernoulli count in a specified trial set. Trials belonging to other table entries are integrated out, not made dependent. · OAI.HyperbolicObstruction.FiniteLaw.centeredCount

centeredDigit def

The representative at exponent zero is included as an auxiliary coordinate. · OAI.QuantitativeVanDerWaerden.centeredDigit

centeredDihedralArc def

The actual centered operator used for the Haar arc. · OAI.KadisonSimilarity.centeredDihedralArc

CenteredGood def

The actual centered good set, with the left endpoint at time −2^k. · OAI.SmoothObstruction.GoodStationary.CenteredGood

centeredGrading def

The degree grading centered at n / 2. · OAI.LeanBlast.GotsmanLinial.centeredGrading

centeredHaarLaurentSpan def

Laurent polynomials with no constant Fourier term. · OAI.KadisonKastler.centeredHaarLaurentSpan

centeredHaarWord def

An alternating word, with a base coefficient followed by each Haar power. · OAI.KadisonKastler.centeredHaarWord

centeredImagePath def

Center the image path at its actual initial image. · OAI.PolycyclicRecognition.centeredImagePath

centeredL2Cocycle def

Subtracting the actual averaged slope produces a zero-mean cocycle. · OAI.PolycyclicRecognition.centeredL2Cocycle

centeredLabelMatrix def

The label map has the d⁻¹/² scale used at every free-word cut. · OAI.KadisonKastler.centeredLabelMatrix

CenteredPairWord inductive

A sequence of centered A,D pairs, including the empty sequence. · OAI.KadisonSimilarity.CenteredPairWord

CenteredProjectionMoments structure

The two centered mixed moments needed for the compressed projection. · OAI.KadisonKastler.CenteredProjectionMoments

centeredShortCoefficients def

The factor three accommodates a long average of norm at most two. · OAI.JointDickman.centeredShortCoefficients

centeredSobolevMap def

Centering preserves the linear dependence on the affine covector. · OAI.RegularPointCoordinates.centeredSobolevMap

centeredSubspace abbrev

The centered subspace is the orthogonal complement of the trace vector. · OAI.KadisonSimilarity.centeredSubspace

CenteredSymmetryWord inductive

Alternating words in centered other-factor letters and one fixed symmetry. true means that the next letter comes from the other factor. · OAI.KadisonSimilarity.CenteredSymmetryWord

centeredSymmetryWordProducts def

Ordinary products of the alternating reduced symmetry words. · OAI.KadisonSimilarity.centeredSymmetryWordProducts

centeredThetaSectionBasis def

The genuine centered source theta family is a basis for every real parameter 0 < tau < 1, with the literal nonintegral residue interval. · OAI.Nagata.Workers.W10.centeredThetaSectionBasis

centeredTracialConjugation def

Tracial conjugation restricted to the centered Hilbert space. · OAI.KadisonSimilarity.centeredTracialConjugation

centeredTracialPairing def

The continuous bilinear pairing used for matrix trace contractions is built from the actual centered tracial conjugation. · OAI.KadisonSimilarity.centeredTracialPairing

centeredTuplePath def

The common centering cancels between successive coordinates, including the first transition from the identity. · OAI.PolycyclicRecognition.centeredTuplePath

centeredWidth def

Exact half-width of the slab covered by centered indices from -q to q. · OAI.ContinuumCoulomb.SlabBoxSchedule.centeredWidth

centeredWordPairing def

The scalar product of two equally oriented centered words. Unequal lengths give zero, including a nonempty word paired with the vacuum. · OAI.KadisonSimilarity.centeredWordPairing

centerExposure def

The full deterministic center exposure: one bit for every selector table. · OAI.Problem310.CenterRouteExposure.centerExposure

centerNeighborEquiv def

Selected geometric sides and selected graph neighbors are in exact bijection; the hard-wall condition is used only for the missing exterior neighbors. Distinct ports may share an inside center. · OAI.CriticalHoneycomb.Transfer.FiniteConfigurations.centerNeighborEquiv

centerParameterDerivative def

The center derivative with respect to the moving positive outer point. · OAI.Problem356.GeometryJets.centerParameterDerivative

CenterPath structure

A literal simple honeycomb path, including its two exterior endpoint centers. It will be applied only when those endpoints lie outside the rectangle. · OAI.CriticalHoneycomb.Transfer.ChordOrientations.CenterPath

centerPosition def

The fixed Euclidean embedding in the unit triangular tiling. · OAI.CriticalHoneycomb.centerPosition

centers def

Initially empty. End-of-input states are frozen; requests before then use the genuine posterior at the new true prefix. · OAI.KServer.HeavyCenters.centers

centers def

The canonical finite indexing set of reduced centers, with duplicates removed. · OAI.Problem337.MinorArc.centers

centerSet def

Reduced centers in the unit interval, with bounded denominator. · OAI.Problem337.MinorArc.centerSet

centerStateHessian def

The center state Hessian, expressed using the single-pair unit-center Hessian. · OAI.Problem356.GeometryJets.centerStateHessian

centerTableAddress def

Center-table coordinate in the bounded-address convention used by the first-default probability theorem. · OAI.Problem310.FiniteTableModel.centerTableAddress

centerTensor def

The tensor sits at an actual honeycomb center, indexed by its actual three honeycomb wires. The order at an up and a down center is the source physical R factorization order, not a symmetrized substitute. · OAI.CriticalHoneycomb.Transfer.CenterPartition.centerTensor

centerTotalDerivative def

Total differential with the parameter block followed by the state block. · OAI.Problem356.GeometryJets.centerTotalDerivative

centralBlockPerm def

An arbitrary independent permutation on each fixed central size-2^L block. · OAI.CubeShuffle.centralBlockPerm

CentralBounds def

All entrywise comparisons in free:profile-enclosures, before taking roots. · OAI.DefocusingNLS.ProfileCertificate.CentralBounds

centralCell def

The central cell of the three-plane example. · OAI.Problem358.Sharpness.centralCell

centralCutPartialIso def

Restrict a partial isometry to a central summand. · OAI.BoundedHochschild.KadisonRingrose.centralCutPartialIso

CentralDecomposition structure

The structural input supplies one exceptional corner and tracial II₁ corners. · OAI.BoundedHochschild.KadisonRingrose.CentralDecomposition

CentralDecompositionData structure

The exact direct-integral identities needed for one separable algebra. No commutator estimate is a field of this structure. · OAI.KadisonSimilarity.CentralDecompositionData

CentralDecompositionInput def

Blackadar's central type decomposition, specialized to the application. · OAI.BoundedHochschild.KadisonRingrose.CentralDecompositionInput

centralDerivative def

First derivative of the central stationary branch for K = 20. · OAI.Problem356.LocalHessian.centralDerivative

centralEmbedding def

This is the concrete copy of the elementary abelian 2-group used by the paper's finite matrix quotient. Its realization as a quotient of the presented group remains separate from the verified embedding below. · OAI.LittleFinitistic.centralEmbedding

centralInclusion def

The concrete center included in the original algebra. · OAI.KadisonKastler.centralInclusion

centralInterval def

The one-dimensional central cell in the sharpness construction. · OAI.Problem358.centralInterval

centralizerSplitting def

The centralizer provides an actual homomorphic section of the extension. · OAI.PolycyclicRecognition.centralizerSplitting

centralLevelCost def

The actual alternating cycles at one expanded height, with the central permutations left arbitrary. No low-level cycles are counted. · OAI.CubeShuffle.centralLevelCost

centralMatrix def

The central coefficient embedding in the displayed 5-by-5 matrix model. · OAI.LittleFinitistic.centralMatrix

CentralMotifSupport def

A support criterion obtained from the actual side-count range. · OAI.TraceReconstruction.CentralMotifSupport

centralPalindromePerm def

A genuine palindrome with a specified permutation inserted in each central block. Fair coins below that boundary will later be absorbed or cancelled. · OAI.CubeShuffle.centralPalindromePerm

CentralProjection def

The projection index is unrestricted in cardinality. · OAI.KadisonSimilarity.CentralProjection

centralProjectionTwirl def

The finite average belonging to a finite family of central projections. · OAI.KadisonSimilarity.centralProjectionTwirl

centralSlice def

A central interval clipped to the source coordinate range. · OAI.Problem358.SharpSlice.centralSlice

centralSpaceConjugation def

The antiunitary on the actual central-vector subspace. · OAI.KadisonKastler.centralSpaceConjugation

centralThird def

Source central third, in the zero-based child label convention. The shift of the source one-based indices is recorded in center parameters. · OAI.TraceReconstruction.centralThird

centralVectorSpace def

Vectors on which left and conjugate-right multiplication agree. · OAI.KadisonKastler.centralVectorSpace

Certificate def

The source component certificate on literal active labels and an actual terminal link. Its labels, tip and positive mass are not extra input. · OAI.PlanarL1.Columns.Model.PairGeometry.Certificate

certificate def

Exact amplification with explicit prefix encodings and a finite polynomial-time machine. · OAI.ContinuumCoulomb.AmplificationProgram.certificate

certificate def

A concrete polynomial machine emits the canonical numerator and denominator of the certified dyadic approximation. · OAI.ContinuumCoulomb.DyadicRootProgram.certificate

certificate def

The actual finite machine, including rational normalization and square root search, has polynomial runtime in the explicit unary/binary inputs. · OAI.ContinuumCoulomb.MediatorProgram.certificate

certificate def

Execution with a polynomial clock gives a polynomial-time TM2 certificate. · OAI.Superstring.BitCode.Block.certificate

certificates def

.inv f here denotes the root augmentation pencil, not a new counted gate. · OAI.RationalHitting.Formula.certificates

Certified structure

certificate of a main form. n is the addition count and p the latest interval. · OAI.FourierLLogL.Dyadic.Thermal.Certified

certify def

Attach the already-proved property without executing anything or changing the input or output codec. · OAI.MaxCutGames.Explicit.MachineSubdivisionCompose.certify

certify def

Attach the already-proved property without executing anything or changing the input or output codec. · OAI.UniqueGamesTheorem.Explicit.MachineSubdivisionCompose.certify

cesaro def

Cesàro means of the powers of a bounded operator. · OAI.Nonhyperlinear.Diffuseness.cesaro

cesaro def

Cesaro averages indexed by n+1 to keep the total mass equal to one even at the first index. No ergodic theorem is used in their construction. · OAI.SmoothObstruction.CompactInvariant.cesaro

cfcSqrtRemainder def

The fourth-order remainder for the unitary path, still expressed in one continuous functional calculus. · OAI.KadisonSimilarity.cfcSqrtRemainder

CH def

The continuum hypothesis. · OAI.SeparableQuotient.CH

Chain structure

A concrete cofinal chain in a component. Its arrows contain actual translation indices, and the initial term is the chosen vertex. · OAI.PartitionConsistency.LocalDiagramSystem.Vertex.Chain

Chain def

A strict chain of p-subgroups with initial trivial subgroup, stabilized by J. The list records the nontrivial nodes, so its length is the chain length. · OAI.PCoreCancellation.Chain

Chain def

A chain of k+1 directed edges, including chains of zero probability. · OAI.TorsionFreeZeroDivisors.WeightedMoore.Chain

chain def

A finite affine Gaussian Markov chain, including a prescribed initial vector. · OAI.FKCRT.Radial.chain

chain def

The matrix product indexed by the actual consecutive position order. · OAI.Saxl.Columns.chain

chain def

The actual finite chain containing both endpoint vertex lists. · OAI.Thomason.Construction.RelativePrismImage.chain

chain def

The full two-sided, non-Zeno, specular chain, assembled from actual exits. · OAI.TriangularBilliards.TwoRays.chain

ChainAlgebra def

The radical-square-zero chain algebra, in vertex and arrow coordinates. Arrow i goes from i to i+1, with the manuscript's right-projective convention. · OAI.LittleFinitistic.OrdinarySimulation.ChainAlgebra

chainAtomIndices def

A finite integral positive chain specifies a finite list of atoms. · OAI.Thomason.Construction.SimplexComplex.chainAtomIndices

ChainBound def

A chain bound on the concrete forcing order. Compatibility here means existence of a common stronger condition, not agreement of two names. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.ChainBound

ChainCode def

The ground code set in the paper: at position i a literal local normalized name supported on the i-th chain diagram. · OAI.PartitionConsistency.ProductForcing.ChainCode

chainCosimplicial def

The normalized directed simplex complexes, including degenerate maps. · OAI.Thomason.Construction.NormalizedSimplexIdentities.chainCosimplicial

chainCycle def

An actual cycle represented by its underlying chain. · OAI.Kaplansky.chainCycle

chainCylinderRealizationIso def

The chain-cylinder cosimplicial object has precisely the earlier literal strict objects and maps. · OAI.Thomason.Construction.NormalizedSimplexIdentities.chainCylinderRealizationIso

chainDensity def

Joint density, in chronological order, for the radii of that chain. · OAI.FKCRT.Radial.chainDensity

chainEnumeration def

Increasing enumeration of the support, with the original partial order. · OAI.Thomason.Construction.PosetGeometry.chainEnumeration

chainEquiv def

Strict enumeration identifies actual subgroup chains with the corresponding strict chains of nontrivial p-subgroups normalized by J. · OAI.PCoreCancellation.chainEquiv

ChainIndex abbrev

Explicit positions in the ancestor chain. The name, not this occurrence, is the randomized identity; opposite portal occurrences may coincide. · OAI.PlanarL1.Columns.Model.Point.RadialTree.ChainIndex

chainIterate def

Iterates of an actual chain endomorphism. · OAI.Kaplansky.chainIterate

chainLength def

Positive integral lengths count exactly the original directed edges along the chains; the two orientations have the same length. · OAI.TorsionFreeZeroDivisors.DegreeTwoSuppression.chainLength

chainMap def

The unique positive map presenting an arbitrary strict globe map. · OAI.Thomason.Construction.DirectedGlobeComplex.chainMap

chainMap def

The positive augmented chain map obtained from the explicit AW formula. · OAI.Thomason.Construction.SimplexComplex.AlexanderWhitneyInput.chainMap

chainMaximum def

The greatest vertex of a finite nonempty chain in a linear order. · OAI.Thomason.Construction.chainMaximum

chainMaximumHom def

Greatest vertex as an order-preserving map. · OAI.Thomason.Construction.chainMaximumHom

chainModelComparison def

The actual chain-cylinder comparison, natural in the simplex. · OAI.Thomason.Construction.SimplexAtomsInput.chainModelComparison

chainNerveCylinder def

An actual simplicial cylinder map for the realization of every directed complex. Its simplices are positive chain maps, including degenerate ones. · OAI.Thomason.Construction.SimplexAtomsInput.chainNerveCylinder

chainNerveHomotopy def

Every strict map from the concrete chain cylinder determines an actual homotopy of Street nerves, with no Gray model-structure premise. · OAI.Thomason.Construction.SimplexAtomsInput.chainNerveHomotopy

chainNetwork def

Join a list of cells to a distinguished final cell. · OAI.TwoWayComplementation.chainNetwork

ChainNode def

Internal vertices of a chain with one distinguished final cell. · OAI.TwoWayComplementation.ChainNode

chainOfSimplex def

The positive chain map represented by an actual Street simplex. · OAI.Thomason.Construction.SimplexAtomsInput.chainOfSimplex

ChainOrdinal structure

An ordinal enumeration of a nonempty finite chain in a poset. · OAI.Thomason.Construction.ChainOrdinal

chainPackaging def

Reindex the actual integral edge chains as a free left group-ring module. · OAI.EilenbergGanea.chainPackaging

chainPiece def

The finite local chain prism, now with values in the categorified nerve. · OAI.Thomason.Construction.RelativePrismData.chainPiece

chainRealizationIso def

The comparison is natural for every strict map between globes. · OAI.Thomason.Construction.DirectedGlobeComplex.chainRealizationIso

chainRingMap def

Equivariance on integral chains is exactly full group-ring linearity. · OAI.EilenbergGanea.chainRingMap

chains def

The finite set of admissible chains. · OAI.FiniteChain.chains

chains def

Prefixes of a shortest representing word form a unit-step chain whose length is the word distance. · OAI.PolycyclicRecognition.WordMetric.chains

ChainsEquivariant def

The elementary equivariance expressing a left integral group-ring map. · OAI.EilenbergGanea.ChainsEquivariant

chainsSupported def

Chains whose original vertices all belong to a specified set. · OAI.Thomason.Construction.chainsSupported

chainStep def

Local moves on the bounded tape, with a final terminal position beyond the last cell. · OAI.TwoWayComplementation.chainStep

ChainSubdivision structure

A subdivision whose indexing agrees with an actual ordered cell list. · OAI.ThompsonNonamenability.ChainSubdivision

chainSymbol def

The symbol at a vertex in a nonempty cell chain. · OAI.TwoWayComplementation.chainSymbol

chainTemplates def

The same prefix-chain clauses used by splitLong, in their output order. · OAI.MaxCutGames.Foundations.Complexity.FinalCNFTemplate.chainTemplates

chainTemplates def

The same prefix-chain clauses used by splitLong, in their output order. · OAI.UniqueGamesTheorem.Foundations.Complexity.FinalCNFTemplate.chainTemplates

chainToIntegerDerived def

Comparison from the nonnegative-chain localization into the genuine integer cochain derived category, obtained by its localization universal property. · OAI.DerivedBar.chainToIntegerDerived

chainWalk def

Literal expansion of one suppressed edge into the original graph. Its endpoints are the two marked ports, and every internal vertex is degree two. · OAI.TorsionFreeZeroDivisors.SuppressedMoore.chainWalk

chainWeight def

The signed sum over chains of nontrivial p-subgroups normalized by J, with the trivial subgroup understood as the first term of each chain. · OAI.PCoreCancellation.chainWeight

Chamber def

A functional in the chamber indexed by y has exactly Inv(y) as its negative roots. This is stronger data used only to construct the order. · OAI.KLInvariance.TitsSpace.RatioOrder.Chamber

change def

Precomposition by the free-group automorphism, with its inverse. This works for full-ring matrix groups, not only for complex points. · OAI.IntegralCharacterVarieties.SurfaceSurgery.Hurwitz.change

change def

The additive change on the actual, uniquely parameterized open tube. · OAI.ThompsonNonamenability.Analytic.CorrectionData.change

changedCoordinatesEquiv def

Changed perturbations correspond exactly to an arbitrary line coordinate and a nonzero hyperplane coordinate. In particular this is uniform in p. · OAI.MaxCutGames.Quadratic.changedCoordinatesEquiv

changedCoordinatesEquiv def

Changed perturbations correspond exactly to an arbitrary line coordinate and a nonzero hyperplane coordinate. In particular this is uniform in p. · OAI.UniqueGamesTheorem.Quadratic.changedCoordinatesEquiv

changeHidden def

Replace the entire hidden matrix once, consistently across all overlapping column events; the row advice remains fixed. · OAI.MaxCutGames.Decoder.AdviceExperiment.changeHidden

changeHidden def

Replace the entire hidden matrix once, consistently across all overlapping column events; the row advice remains fixed. · OAI.UniqueGamesTheorem.Decoder.AdviceExperiment.changeHidden

changeMatrix def

Coordinates of the second frame in the first. · OAI.PositiveEinsteinFour.Geometry.OrthonormalFrame.changeMatrix

changeModelCore def

A fixed linear change of model coordinates; the gluing is conjugated. · OAI.Kaplansky.changeModelCore

changeModelCoreTotalHomeomorph def

The coordinate change is a homeomorphism of actual total spaces. · OAI.Kaplansky.changeModelCoreTotalHomeomorph

changePassageBases def

Rebase every named fiber continued through a passage, with the exact permutation and rank equalities of the original vertex. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.LocalRanks.changePassageBases

changeScalarsEquiv def

Exterior powers commute with transport of the coefficient ring. · OAI.ActualExterior.changeScalarsEquiv

changeSplitParent def

Change the common parent frame, keeping the named third-seam frame. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.LocalRanks.changeSplitParent

channel def

Observation type-A. · OAI.FourierLLogL.Dyadic.Thermal.channel

channel def

The memoryless n-fold generalized amplitude-damping channel. · OAI.GAD.channel

ChannelCode structure

The fixed odd paths of the first (and later probe) observation channel. · OAI.PureTypeSystem.Labelled.ChannelCode

ChannelReturn structure

The parity-matched return is retained alongside, not substituted for, the fixed channel paths. · OAI.PureTypeSystem.Labelled.ChannelReturn

ChannelSetup structure

All roles use fixed path syntax selected once from the forbidden configuration. In particular the shared raw wrapper is not reselected when constructing a query, bundle, or the second channel. · OAI.PureTypeSystem.Labelled.ChannelSetup

char def

The exact evaluation character, as a bounded measurable function. · OAI.PlaneFiveColor.Spectral.char

Character structure

A normalized, additive, multiplicative, restriction-monotone tensor valuation. · OAI.MatrixMultiplication.AuxiliarySeparation.Character

character def

Probability-normalized finite Walsh transform, expressed in the same sign-bit coordinates as the tree. · OAI.OptimalMaxCut.TreeCode.character

character def

The diagonal entries of the adjoint representation. · OAI.PolycyclicRecognition.TriangularAdjoint.character

character def

The diagonal entries of the adjoint representation are linear forms. · OAI.PolycyclicRecognition.TriangularAdjoint.character

characterAverage def

The same proved average, expressed with the spectral-order instances used by the descent criterion. Its matrix positivity is unchanged. · OAI.KadisonKastler.characterAverage

characterBulk def

The top prime is position zero; the bulk positions are its successors. · OAI.Ostmann.characterBulk

characterCellIndex def

A selected ordered list is used once in each signed copy. · OAI.Ostmann.characterCellIndex

characterCommutant def

The closed C-star algebra commuting with all real character multipliers. · OAI.KadisonKastler.characterCommutant

characterDetection def

Detection by any member of a character family. Classical decidability only selects an event indicator on the finite noise sample space. · OAI.MaxCutGames.Gadget.DetectionBridge.characterDetection

characterDetection def

Detection by any member of a character family. Classical decidability only selects an event indicator on the finite noise sample space. · OAI.UniqueGamesTheorem.Gadget.DetectionBridge.characterDetection

characterEndpointExtension def

The endpoint tests can be extended outside their selected finite set without strengthening any premise about actual endpoints. · OAI.Ostmann.characterEndpointExtension

characterFieldLine def

A total projective-line map; the default is irrelevant on nonzero characters. · OAI.MaxCutGames.Quadratic.characterFieldLine

characterFieldLine def

A total projective-line map; the default is irrelevant on nonzero characters. · OAI.UniqueGamesTheorem.Quadratic.characterFieldLine

characterFullPrimeCells def

Both signs use the same marginal. Signs change the characters, never the original positive harmonic measure. · OAI.Ostmann.characterFullPrimeCells

characteristic def

The spatial position at time t of a characteristic starting from z. · OAI.BoltzmannNonuniqueness.AxialFlow.characteristic

characteristic def

Boundary coordinates equal to1 translate the initial vertex; coordinates0 or1 are dropped from the unique open face. · OAI.EilenbergGanea.CubeRealization.characteristic

characteristic def

Crossing a far face removes a descent and strictly lowers the top vertex. · OAI.EilenbergGanea.DescendingCube.Data.characteristic

characteristic def

The characteristic map is defined on the entire parameter space, as required by Mathlib's classical CW definition, but only its closed-disk values matter. · OAI.GroupRingTopology.Attachment.characteristic

characteristic def

A characteristic map on its closed disk. · OAI.GroupRingTopology.Cellular.characteristic

characterPivot def

Extend the finitely many real pivots by an arbitrary existing word slot. All transfer theorems use only the genuine prefix. · OAI.Ostmann.characterPivot

characterPivotGap def

The index is zero based here; this is the manuscript's Delta_(i+1). · OAI.Ostmann.characterPivotGap

characterSparse def

Character formulation of the sparse-node predicate. · OAI.TreeEdit.BinaryLower.FiniteActivity.characterSparse

CharacterTargetWord structure

A word consists of cells from one whole, previously tested rich shell. · OAI.Ostmann.CharacterTargetWord

characterTwist def

The multiplicative twist appearing in the autocorrelation formula. · OAI.Ostmann.characterTwist

characterTwist def

The sequence χ(n) n^(it), with the value at zero immaterial to the prime sum. · OAI.TwoPointCorrelations.characterTwist

characterWordEquiv def

The actual positive word, including its one top label. · OAI.Ostmann.characterWordEquiv

charFunctional def

Bilinear character scalar product used by the finite-group orthogonality theorem. · OAI.Saxl.charFunctional

charge def

Number of successful forced fresh-bit queries along the realized trace. · OAI.CubeShuffle.QueryTree.charge

charge abbrev

Final pointwise estimate after starting before group j. · OAI.ExactDerandomization.Buf.Run.charge

charge def

Exact annular charge made by a real center edit, zero on non-edit steps. · OAI.KServer.HeavyCenters.charge

charge def

The genuine insertion mass W of the manuscript, zero on noninsertions. · OAI.KServer.TierProcess.charge

charged def

A charged step is precisely the first branching at a designated depth. · OAI.DilutedSpinGlass.PrescribedTree.charged

chargedCoarseQuery def

The flattened program receives each coarse answer from an explicit external read. The constant zero here is never used as a seed answer. · OAI.EditApproximation.chargedCoarseQuery

chargedRandomCoarseQuery def

The flattened program receives each coarse answer from an explicit external read. The constant zero here is never used as a seed answer. · OAI.EditApproximation.chargedRandomCoarseQuery

ChargeHighBasis abbrev

High occupations exclude the singly occupied spin subspace. · OAI.ContinuumCoulomb.ChargeHighBasis

chargeMass def

Histories having at least b charged steps, preserving their multiplicities. · OAI.DilutedSpinGlass.PrescribedTree.chargeMass

Chart structure

A transitive action with actual chosen representatives. For the application, Ω is the set of ordered placements; the fiber group fixes the base placement. · OAI.CoordinateSweeps.PlacementInduction.Chart

Chart structure

Concrete frame equations defining one splitting chart. The domains are free modules in the matrix instantiation below; no affine-bundle hypothesis. · OAI.IntegralCharacterVarieties.ExactSequenceChart.Chart

chart def

The same chart with the literal cardinal as ambient length. · OAI.CoordinateSweeps.FiberBlocks.chart

chart def

Every ordered placement is reached by an actual permutation extending the bijection on its marked positions. No cardinal or transitivity axiom is used. · OAI.CoordinateSweeps.OrderedPlacements.chart

chart def

The actual positive-cube open chart. · OAI.EilenbergGanea.CubeRealization.chart

chart def

The literal fraction algebra of an ordinary blowup chart. · OAI.FractionChartComparison.chart

chart def

Convert back to the old ordinary orientation, as in /main-packing. The -4E proposal shift is part of the literal atom constant. · OAI.PlanarL1.Columns.Model.Point.RadialTree.AtomSource.chart

chart def

The polynomial line chart is an actual subalgebra of the ORIGINAL cone function field, with fibre coordinate cX, not an abstract replacement. · OAI.PolynomialFieldChart.chart

chart def

Projection onto an affine hyperplane along a fixed transverse direction. · OAI.Problem358.MovingHyperplaneChart.chart

chart def

The chart on the jth closed region, in the corresponding (n+1)-simplex. · OAI.Thomason.Construction.PrismBarycentric.chart

ChartA abbrev

The base chart D(z+1). · OAI.StableCoordinate.BundleCharts.ChartA

chartBoundaryFlux def

Actual oriented boundary-chart integral in the level coordinates. · OAI.MahlerStokes.chartBoundaryFlux

chartBoundaryWeightIdeal def

The monomial ideal of boundary exponents above a given weight in the local chart. · OAI.CartierSections.chartBoundaryWeightIdeal

chartCompletionMap def

The completion map induced by an algebra map. · OAI.CartierSections.chartCompletionMap

chartCoordinate def

The affine coordinate functions on their image in Y. · OAI.ActualCotangent.chartCoordinate

chartCoordinates def

Six functions suffice: the fifth original root is reconstructed. · OAI.ExplicitCone.chartCoordinates

chartCoordinates def

Coordinates of a projective point in chart c. This map is used only when the chart coordinate is nonzero. · OAI.Nagata.ProjectiveGeometry.chartCoordinates

ChartD abbrev

The base chart D(z). · OAI.StableCoordinate.BundleCharts.ChartD

chartDehomogenize₂ def

The same genuine affine chart equation with its two variables enumerated by Fin 2, for the finite jet matrix interface. · OAI.Nagata.ProjectiveGeometry.chartDehomogenize₂

chartDenominator def

The denominator defining one principal open. · OAI.IntegralCharacterVarieties.ExactSequenceChart.chartDenominator

chartDerivativeEquiv def

The derivative of a smooth open boundary chart is continuously invertible. · OAI.Elasticity.chartDerivativeEquiv

chartEquiv def

The modules on any open-immersion chart are exactly the modules on the target's slice site at its image open. · OAI.ActualOpenModules.chartEquiv

chartEquiv def

A domain generated by actual section ratios is the actual fraction chart, not merely a ring agreeing with it modulo the exceptional equation. · OAI.FractionChartComparison.chartEquiv

chartEquivA def

The actual algebra equivalence C[a⁻¹] ≃ k[a,a⁻¹,d,u][b]. · OAI.StableCoordinate.DeterminantRing.chartEquivA

chartEquivAOverCoefficients def

The a-chart equivalence, over its full localized coefficient ring. · OAI.StableCoordinate.DeterminantRing.chartEquivAOverCoefficients

chartEquivD def

The actual algebra equivalence C[d⁻¹] ≃ k[a,d,d⁻¹,u][c]. · OAI.StableCoordinate.DeterminantRing.chartEquivD

chartEquivDOverCoefficients def

The d-chart equivalence, over its full localized coefficient ring. · OAI.StableCoordinate.DeterminantRing.chartEquivDOverCoefficients

chartEvaluation def

The line-bundle-chart evaluation of the section ring, rescaling each pure section by c^degree and retaining the fiber parameter X. · OAI.SectionCompletion.chartEvaluation

chartExponentialFlow def

Right multiplication is the flow of the left-invariant field. · OAI.PolycyclicRecognition.chartExponentialFlow

ChartFiberRetention def

Finite sampling bridge, stated on the actual equality test, actual lifted output fibers, and actual Grassmann retention. The positive dimensions exclude the degenerate empty-neighbor cases. No KMS or inverse principle is part of this … · OAI.MaxCutGames.Inverse.ShortcodeFromGrassmann.ChartFiberRetention

ChartFiberRetention def

OPEN finite sampling bridge, stated on the actual equality test, actual lifted output fibers, and actual Grassmann retention. The positive dimensions exclude the degenerate empty-neighbor cases. No KMS or inverse principle is part of this … · OAI.UniqueGamesTheorem.Inverse.ShortcodeFromGrassmann.ChartFiberRetention

chartGain def

Carleman gain against a possibly singular incoming velocity measure. The transverse degeneracy set has measure zero for the sheet laws used here. · OAI.BoltzmannNonuniqueness.chartGain

chartGramBlock def

The framed coordinate curvature, expressed entirely in one fixed chart. · OAI.PositiveEinsteinFour.Geometry.chartGramBlock

chartIdealSectionMap def

The kernel ideal on the chart, in its coefficient module. · OAI.ActualCartier.chartIdealSectionMap

chartIndexEquiv def

Ordered enumeration of the two coordinates different from c. · OAI.Nagata.ProjectiveGeometry.chartIndexEquiv

chartInverse def

Real inverse coordinates of an open cube, without any assumed chart. · OAI.EilenbergGanea.CubeRealization.chartInverse

chartIso def

Functoriality connects the global sheaf pullback with the actual regular quotient on each surface affine chart. · OAI.SourceExceptionalRestriction.chartIso

chartJetPolynomialMap def

Polynomial jet of one actual global homogeneous form at one point chart. · OAI.Nagata.W27.chartJetPolynomialMap

chartMap def

Transfer a chart morphism to the target slice site by the fully faithful inverse. · OAI.ActualOpenModules.chartMap

chartMapAOverBase def

Compatibility with the actual polynomial coefficient algebra. · OAI.StableCoordinate.DeterminantRing.chartMapAOverBase

chartMapDOverBase def

Compatibility with the actual polynomial coefficient algebra. · OAI.StableCoordinate.DeterminantRing.chartMapDOverBase

chartMatrixFrame def

A matrix normalized by the chart metric gives an actual orthonormal frame, after applying the inverse-chart tangent map. · OAI.PositiveEinsteinFour.Geometry.chartMatrixFrame

chartModelIso def

The affine double-dual restriction model for P on each surface chart. · OAI.SourceExceptionalRestriction.chartModelIso

chartOccupationPoint def

Normalization at the actual chart point, including the deterministic suffix between the two sampled factors. · OAI.PolycyclicRecognition.chartOccupationPoint

chartPatch def

A literal common-label scalar chart replacement yields local patch data. P already contains the source κ and comparison-orientation factors. · OAI.PlanarL1.Threshold.chartPatch

chartPlaneJetMatrix def

The matrix uses a single basis of the global homogeneous form space, regardless of which affine chart is selected at each individual point. · OAI.Nagata.W27.chartPlaneJetMatrix

chartPoint def

The actual covering-space chart x ↦ tau^x0 exp x. · OAI.Nagata.Workers.W05.chartPoint

chartPolarLowering def

Base point, direction and positive radius as actual cotangent coordinates, using the smooth Gram frame on the chart. · OAI.PositiveEinsteinFour.Geometry.chartPolarLowering

chartPolynomialRepresentatives def

Polynomial homogeneous representatives with fixed distinguished coordinate one in a separately chosen affine chart at every marked point. · OAI.Nagata.Workers.W14.chartPolynomialRepresentatives

chartPunctureIso def

The geometric puncture lies above the affine Cartier nonvanishing open, so its fixed isomorphism restricts to the affine principal puncture. · OAI.ActualCartier.chartPunctureIso

chartRatio def

The section ratio determining the overlap in chart s. · OAI.SourceSymmetry.chartRatio

chartRegionGain def

Restriction in the actual incoming velocity and the two chart coordinates. · OAI.BoltzmannNonuniqueness.chartRegionGain

chartRestrictionEquiv def

This same normalization is the actual restriction of the original normalized five-root affine cover, not a newly chosen function field. · OAI.KummerAffineExtra.chartRestrictionEquiv

chartRestrictionEquiv def

This normalization is the actual restriction of the normalization of the blowup chart in the original five-root field. · OAI.KummerShifted.chartRestrictionEquiv

chartRestrictionEquiv def

This normalization is the actual restriction of the normalization of the blowup chart in the original five-root field. · OAI.KummerShiftedDiagonal.chartRestrictionEquiv

chartRestrictionEquiv def

This normalization is the actual restriction of the normalization of the blowup chart in the original five-root field. · OAI.KummerTriple.chartRestrictionEquiv

chartRestrictionEquiv def

This normalization is the actual restriction of the normalization of the blowup chart in the original five-root field. · OAI.KummerTripleDiagonal.chartRestrictionEquiv

chartRestrictionIso def

The chart equivalence intertwines global restriction with slice-site restriction. · OAI.ActualOpenModules.chartRestrictionIso

chartScaling def

Local normalization with harmless zero values outside a chart. · OAI.Kaplansky.chartScaling

chartSectionTrivialization def

An affine-scheme line trivialization gives the section trivialization on its image open, accounting for the coefficient-ring iso. · OAI.ActualCartier.chartSectionTrivialization

chartSeriesExpansion def

Expansion in completion coordinates as an algebra map over the coefficient ring. · OAI.CartierSections.chartSeriesExpansion

chartSmallRelativeChainsIso def

Local relative chains are isomorphic to the small relative chains of the actual cover by the chart and the punctured ambient space. · OAI.Kaplansky.chartSmallRelativeChainsIso

chartSupport def

The image compact set for the local coordinate system. · OAI.ClosedSurfaceR4.JetPolynomial.chartSupport

chartTransitionToLocalRing def

The concrete rational substitution on an overlap of affine projective charts, with values in the actual local ring at the target point. · OAI.Nagata.ProjectiveGeometry.chartTransitionToLocalRing

chartTrivialization def

Each actual projective affine chart trivializes the quotient fiber. · OAI.Nagata.W16.AssociatedBundle.chartTrivialization

ChartVariables abbrev

Each affine projective chart has exactly the two coordinates other than its distinguished nonzero homogeneous coordinate. · OAI.Nagata.ProjectiveGeometry.ChartVariables

cheapTrailEvent def

Cheap trails at one exact length and one deterministic starting vertex. · OAI.GammaFPP.cheapTrailEvent

ChebyshevPrimeCountingInput def

The large-argument Chebyshev upper bound; Montgomery–Vaughan, Multiplicative Number Theory I, Corollary 2.6. · OAI.JointDickman.PublishedInputs.ChebyshevPrimeCountingInput

check def

Canonical names in the actual condition system. · OAI.PartitionConsistency.SetForcingNames.ConditionAction.Name.check

checkName def

The canonical name for a ground set, using the condition t. In applications t is the largest condition. · OAI.PartitionConsistency.SetForcingNames.checkName

cheegerEnergy def

Cheeger energy is the L² relaxation of one half the squared local slope. · OAI.WeakHessian.cheegerEnergy

Chess structure

canonical path selection supplies a coherent and costed path, not just a validity bit. · OAI.ExactDerandomization.Sub.Quill.Chess

chi def

Paper §01 χ_C(V). Used with V interior to the dual. · OAI.GeneralMahler.chi

chi def

The cutoff supported inside the full-radius ball. · OAI.ThompsonNonamenability.Analytic.chi

chi def

The Kuratowski measure of noncompactness, on bounded sets. · OAI.Tingley.chi

chiExt def

For limiting arguments keep track of infinity. · OAI.GeneralMahler.chiExt

Child def

The subname relation on literal set codes. The second component of an edge is a forcing condition; the first component is a name for a potential member. · OAI.PartitionConsistency.SetForcingNames.Child

Child abbrev

Child M is default and the other M children have selectors. · OAI.Problem310.FiniteTableModel.Child

Child inductive

Only literal syntax descent, including both annotations. · OAI.PureTypeSystem.Expr.Child

child def

One child on the sphere having the two incident velocities as diameter endpoints. · OAI.BoltzmannNonuniqueness.ReferenceGeometry.child

child def

Natural enumeration of the positive-integer children. · OAI.IndependentProducts.child

child def

An edge recovers a name, with hereditary well-formedness proved from its parent rather than imposed as an additional hypothesis. · OAI.PartitionConsistency.SetForcingNames.ConditionAction.Name.child

child def

An active entry is automatically HS by hereditary symmetry. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.Name.child

childAngularEnvelopeMaximum def

The finite maximum uses the same original measures and the actual child good sets. · OAI.Falconer.childAngularEnvelopeMaximum

childAverage def

Actual parentwise convex average of the new cutoff outputs. · OAI.ReflexiveFixedPoints.FrontierExtension.childAverage

childAverage def

The actual convex average of the child outputs. · OAI.ReflexiveFixedPoints.TreeEvaluation.childAverage

childCharacter def

The logical map of a child, obtained from the common parent first-coordinate map and the prescribed logical character. · OAI.MaxCutGames.Gadget.childCharacter

childCharacter def

The logical map of a child, obtained from the common parent first-coordinate map and the prescribed logical character. · OAI.UniqueGamesTheorem.Gadget.childCharacter

childCoordinates def

A proper child retains the first branching coordinate as its evaluation coordinate, and precisely its own internal branching coordinates. · OAI.DilutedSpinGlass.ReducedTopology.childCoordinates

childDiagonal def

Inverse CHILD transport gives the correct oriented seam convention. · OAI.IntegralCharacterVarieties.IntrinsicInterval.Transport.childDiagonal

childEdge def

The new coordinate on the edge from a vertex to one of its children. · OAI.IndependentProducts.childEdge

childEdge def

Edges corresponding to appending a child at a strict internal node. · OAI.Problem310.TreePreorder.childEdge

childEnergy def

Covariance energy of one actual child, including its incoming transition. The rest of the tree has not been sampled or conditioned upon. · OAI.DilutedSpinGlass.PrescribedTree.childEnergy

childEnumeration def

The source order on each child list: prefix, split block/subblocks, suffix. This is not an arbitrary finite enumeration or an unconstrained germ bijection. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.Kind.childEnumeration

childEquiv def

Every nonroot vertex occurs in exactly one child list. · OAI.KServer.TreeRounding.Tree.childEquiv

childIncrementL2 def

The child jump in the Hilbert space L2. · OAI.IndependentProducts.childIncrementL2

ChildLabels abbrev

Reference labels in a particular child, preserving their identities. · OAI.DilutedSpinGlass.PrescribedTree.ChildLabels

childMarks def

The marked input paths assigned to one of the two recursive children. · OAI.CubeShuffle.childMarks

children def

Literal nine-child support of an adjacent dyadic refinement row. · OAI.DyadicHats.children

children def

Literal nine-child support of an adjacent dyadic refinement row. · OAI.SignedDisk.DyadicHats.children

childScheduleProjection def

Product of the actual proper-child schedule projectors, restricted from the parent depth-assignment cube. Coordinates may be independently translated. No independent sampling of aligned copies occurs in this definition. · OAI.DilutedSpinGlass.ReducedTopology.childScheduleProjection

childSpan def

Span of one incoming-edge window and the subtree beneath that edge. · OAI.Problem310.WindowRecursion.childSpan

childWindowIndex def

A displaced child window as an actual interval of the deterministic parent template. t is the realized count of atoms before the window. · OAI.TraceReconstruction.childWindowIndex

chimera def

Per-position beta status overlap at fixed orientation pair. · OAI.HadwigerCounterexample.StatusAnalysis.Typus.chimera

choiceFailureFamily def

The internal family P(A) minus the empty set, formed by the actual Power Set and Separation constructions. Quantified subsets have no rank bound. · OAI.PartitionConsistency.ProductForcing.choiceFailureFamily

choiceForFamily def

c(ξ) ∈ F(ξ) for every ξ ∈ η, with c an internal function. · OAI.PartitionConsistency.SetLanguage.choiceForFamily

choiceLaw def

All stage-lines randomize independently, exactly as in 03-paths:5-17. · OAI.CoordinateSweeps.Grid.choiceLaw

choiceOn def

A choice function on the set A of nonempty sets. · OAI.PartitionConsistency.SetLanguage.choiceOn

choices def

Right-associated finite disjunction used by the source query object. · OAI.PureTypeSystem.Labelled.LogicCode.choices

choicesSplit def

Exact two-part choice-space equivalence, not just an equality in law. · OAI.CoordinateSweeps.Grid.choicesSplit

ChoiceValue def

The actual coefficient type after a choice has been fixed. · OAI.MaxCutGames.Decoder.PaddedLaw.ChoiceValue

ChoiceValue def

The actual coefficient type after a choice has been fixed. · OAI.UniqueGamesTheorem.Decoder.PaddedLaw.ChoiceValue

choiceValuesRawTupleEquiv def

Choice fibers and the already-verified RawTuple model contain exactly the same coefficients, with no unused coordinate and no multiplicity. · OAI.MaxCutGames.Decoder.PaddedLaw.choiceValuesRawTupleEquiv

choiceValuesRawTupleEquiv def

Choice fibers and the RawTuple model contain exactly the same coefficients, with no unused coordinate and no multiplicity. · OAI.UniqueGamesTheorem.Decoder.PaddedLaw.choiceValuesRawTupleEquiv

choiceWeight def

Independent line permutations with law (1-z) U + z B. · OAI.CoordinateSweeps.Grid.choiceWeight

choiceWeight def

Joint mass of all independent stage-line choices. · OAI.CoordinateSweeps.Grid.choiceWeight

choose def

true is rule II. Ties select I at a restart. · OAI.KServer.RuleSchedule.choose

choose def

A local winner other than the old field receives exactly one new name. The height-only branch is zero in the fresh block, and spatial atom a is a+1. · OAI.PlanarL1.Columns.Model.MainCell.choose

choose def

Choose the arithmetic data whose existence is proved by the prime and prime-power scale construction. No coloring or covering data is assumed. · OAI.QuantitativeVanDerWaerden.ConstructionModel.choose

choose def

A foreign-text filter is deliberately separate: its equality test uses only a common product period, not a potentially unbounded infinite stream. · OAI.Superstring.UpperCode.choose

choose def

Every nonempty finite chain has an increasing enumeration. · OAI.Thomason.Construction.ChainOrdinal.choose

chooseChild def

The first successful nondefault selector, or the final (default) child. · OAI.Problem310.OrderedRouting.chooseChild

chooseInt def

Binomial coefficients extended by zero to negative lower arguments. · OAI.FKCRT.SimpleWalk.chooseInt

choosePath def

A fixed deterministic full path, used only to resolve finite choices. · OAI.LeanBlast.SecondKahnKalai.ProbabilityTree.choosePath

choosePivot def

An active greedy pivot, or a fixed center when every residual is small. · OAI.MetricEntropyDuality.GreedyPivots.choosePivot

chosen def

The raw machine has thirteen tapes. These are precisely the twelve non-output tapes, including the original input and retained archive. · OAI.MaxCutGames.Foundations.Complexity.FinalCNFCleanup.chosen

chosen def

The raw machine has thirteen tapes. These are precisely the twelve non-output tapes, including the original input and retained archive. · OAI.UniqueGamesTheorem.Foundations.Complexity.FinalCNFCleanup.chosen

chosenBlockOrientation def

Existence uses the proved dimension of the actual quadratic block. · OAI.MaxCutGames.Quadratic.chosenBlockOrientation

chosenBlockOrientation def

Existence uses the proved dimension of the actual quadratic block. · OAI.UniqueGamesTheorem.Quadratic.chosenBlockOrientation

chosenBucketData def

This is one bucket visited by the real residue loop, with the designated normalized residue used only to name that iteration in the analysis. · OAI.LowSpaceSubsetSum.RepresentationSetup.chosenBucketData

chosenCell def

An arbitrary choice of one disk lift per base cell; no finiteness is needed. · OAI.EilenbergGanea.CWCover.chosenCell

chosenIntegralPresentation def

A projective presentation chosen with finite source whenever the target is finitely generated. The finiteness proof stays inside the chosen object. · OAI.PolycyclicRecognition.chosenIntegralPresentation

chosenLowEntries def

The actual distinct-entry universe after only the parent low filter. · OAI.LowSpaceSubsetSum.RepresentationSetup.chosenLowEntries

chosenMatrixExtension def

A choice of the published algebraic extension, with no unitary or tensor conclusion included in that input. · OAI.KadisonSimilarity.TracialCornerGeometry.chosenMatrixExtension

chosenOtherBucket def

Different-weight entries in the designated full parent bucket. · OAI.LowSpaceSubsetSum.RepresentationSetup.chosenOtherBucket

chosenParentBucket def

The actual distinct-entry bucket containing the designated representation. · OAI.LowSpaceSubsetSum.RepresentationSetup.chosenParentBucket

chosenParentData def

The designated analysis task uses one of the actual normalized parent residues. The program itself receives only the primitive data of this record. · OAI.LowSpaceSubsetSum.RepresentationSetup.chosenParentData

chosenPartition def

The finite prescribed-boundary orientation coefficient of one exact unoriented honeycomb configuration. · OAI.CriticalHoneycomb.Transfer.RasterGrid.chosenPartition

chosenTopRepresentative def

The top witness is fixed before either child residue is inspected. · OAI.LowSpaceSubsetSum.chosenTopRepresentative

chosenWeakLimit def

A chosen actual limit, obtained from compactness rather than a limit-existence field. · OAI.ReflexiveFixedPoints.Prediction.chosenWeakLimit

chosenWitness def

The witness choice has only the left observation as its variable input. · OAI.MaxCutGames.Decoder.VisiblePolicies.chosenWitness

chosenWitness def

The witness choice has only the left observation as its variable input. · OAI.UniqueGamesTheorem.Decoder.VisiblePolicies.chosenWitness

ChristensenMutual def

Precise published input: the mutual case of CCSSWW (2014), Theorem 3.3(i). The location of the unitary in (M ∪ N)'' is omitted because this application does not use it. The equality conclusion applies for γ < 1/101. · OAI.KadisonKastler.ChristensenMutual

christoffel def

Levi-Civita connection in the single global Cartesian chart. · OAI.HarmonicCounterexample.christoffel

chunk def

true denotes a V layer and false denotes a U layer. · OAI.Problem335.BalancedSchedule.chunk

Chunking structure

A literal partition into signed integer chunks, not a count of formal pieces. · OAI.TorsionFreeZeroDivisors.ChunkComparisons.Chunking

chunkLength def

Number of U layers following the ith V layer. · OAI.Problem335.BalancedSchedule.chunkLength

CircleBases abbrev

A basis at each old boundary circle; repeated facet labels do not identify these independent basing paths. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.CircleBases

CircleBounded def

A bounded measurable coefficient on the temporal circle. · OAI.ThreeTorus.CircleBounded

CircleConfiguration structure

The source's generic circle placement, stated only in geometric terms. · OAI.Paper170.CircleConfiguration

circleCutoff def

A unit-additive real modification descends pointwise to the circle. · OAI.CircleDomainRigidity.FiniteTransfer.circleCutoff

circleDegree def

The degree-n self-map of the additive circle. · OAI.CoarseAssembly.circleDegree

circleDensityLift def

The usual interval representative of a circle weight, with zero extension. · OAI.ThreeTorus.circleDensityLift

CircleDomainRemovabilityRigidity def

Paper 047-01, Introduction, thm:main: removability implies rigidity. · OAI.CircleDomainRigidity.CircleDomainRemovabilityRigidity

circleInputOfIndexOne def

All the published circle fields follow from a single coordinate-index-one class. · OAI.CoarseAssembly.circleInputOfIndexOne

circleIntegral def

Normalized Haar integral on the ordinary circle. · OAI.PlaneFiveColor.InvariantMean.circleIntegral

CircleLiftCover def

The literal pullback of the universal real cover by a circle map. · OAI.CircleDomainRigidity.FiniteTransfer.CircleLiftCover

circleMap def

Barycentric interpolation of the cocycle, with values in the unit circle. · OAI.CoarseAssembly.Rips.IntegralCocycle.circleMap

circleMul def

Circle coordinate multiplication, constructed on L² representatives. · OAI.ThreeTorus.circleMul

CirclePartition structure

Literal half-open cells on a single fundamental interval. A centered partition is represented by translating both its endpoints by 1/2. · OAI.QuantitativeVanDerWaerden.CirclePartition

circlePathIncrement def

Winding measured in turns; on a loop it is an integer. · OAI.CircleDomainRigidity.FiniteTransfer.circlePathIncrement

circlePrimitive def

The precise globally defined circle-valued primitive used by a passage. · OAI.ThreeTorus.circlePrimitive

circlePull def

Pullback along the dense algebraic circle embedding. · OAI.PlaneFiveColor.InvariantMean.circlePull

circleRep def

The representative in the old good-arc coordinate chart. · OAI.ThreeTorus.circleRep

circleRepresentation def

Continuous functional calculus of the actual bilateral shift. · OAI.CoarseAssembly.Analytic.circleRepresentation

circleTraceMap def

The actual continuous restriction to the circle, with continuity proved locally. · OAI.Nagata.W01.circleTraceMap

circleUnitary def

The scalar unitary represented by a circle-valued map. · OAI.CoarseAssembly.circleUnitary

Circuit structure

Outputs name available values, so permutations and arbitrary fanout are uncharged. · OAI.ExactFourier.Circuit

Circuit abbrev

A simple loop with its actual base vertex and edge traversals. · OAI.HyperbolicObstruction.ConeWalks.Circuit

circuit def

The clean finite-gate controlled-shift implementation derived from its forward and inverse verified combinational networks. · OAI.ExactQuantumFactoring.ControlledTranslation.circuit

circuit def

A single family (depending only on n) prepares the manuscript's exact runtime-b(n,m) fair sample. No Hadamard is chosen by a metalevel value of m. · OAI.ExactQuantumFactoring.ListSampler.circuit

circuit def

The qualified Fin-indexed circuit produced by block expansion. · OAI.Problem335.BlockData.circuit

circuitCoefficientField def

The subfield generated by the finitely many coefficients of a circuit. · OAI.Problem335.circuitCoefficientField

circuitCoefficientSet def

All coefficients occurring in a circuit, including weighted wires. · OAI.Problem335.circuitCoefficientSet

circuitProductEnvelope def

A common bound for a small product and for one expanded high product. · OAI.Problem335.circuitProductEnvelope

circuitValue def

Polynomial computed by a circuit indexed by arbitrary finite types. · OAI.Problem335.IndexedDepth5Circuit.circuitValue

CircularTipSeparates def

Strictly opposite sides of an unoriented circular chord. · OAI.PlanarL1.Columns.Model.Field.CircularTipSeparates

clamp def

Projection to a closed scalar interval. It is used only away from the source singularities, and is proved inactive by exact comparison barriers. · OAI.SepticProfile.RegularContinuation.clamp

clamp def

Clamp all later vertices to the last sieve vertex. · OAI.Thomason.Construction.SieveSimplex.Cut.clamp

clampedPath def

Freeze a continuous path outside a finite time interval. · OAI.SLEExactGauge.FrontSuccessMeasurability.clampedPath

classAt def

The plus and minus names coincide precisely at the leaf level. · OAI.Problem348.SampledPath.classAt

classAvailability def

The availability set of an eligible E-class is the source's actual ideal: it records existence of representatives at subdiagrams, not an ideal supplied as extra structure. This specializes the proved intersection calculus. · OAI.PartitionConsistency.ProductForcing.classAvailability

classBody def

Source class callback w ↦ (w ∼ y); G retains its predicate binder. · OAI.PureTypeSystem.Labelled.ChannelCode.classBody

classChartIdeal def

The full principal ideal of the local equation, without radicalization. In particular, powers of equations retain their nonreduced structure. · OAI.Nagata.W16.classChartIdeal

classes def

Signed recurrent equivalence classes of an idempotent diagram. · OAI.TwoWayComplementation.Diagram.classes

classEvaluation def

Select one representative where a cycle partition is realized; unused coordinates are zero. This gives an injection even without surjectivity of cycle-type realization. · OAI.Saxl.classEvaluation

classFiberEmbedding def

Distinct hit probes yield distinct symbolic classes, all in the exact specified fiber. This map uses actual names, actual forcing eligibility, and the proved intersection theorem. · OAI.PartitionConsistency.ProductForcing.classFiberEmbedding

classFromLocal def

A supported eligible class at any subdiagram gives a genuine element of D_F(r), even when that subdiagram is not itself below q. · OAI.PartitionConsistency.ProductForcing.classFromLocal

classFunctions def

Class functions on the symmetric group, indexed by cycle type. · OAI.Saxl.classFunctions

ClassificationOn def

The family of main claims for a fixed smooth four-manifold. Compactness encodes "closed" since the model has no boundary. · OAI.PositiveEinsteinFour.ClassificationOn

ClassificationPublishedInputs structure

Precisely the classical oriented metric identities and the classical orientation-cover/Euler inputs required by the full proof. · OAI.PositiveEinsteinFour.ClassificationPublishedInputs

Classified def

Positive rescaling identifies the metric with one of the three standard models. · OAI.PositiveEinsteinFour.Classified

Classified def

The three metric models in the manuscript's unoriented classification. a is the positive factor multiplying the original Riemannian metric. · OAI.PositiveEinsteinFour.Classified

classify def

Assign the all-equal case to normal pairing, as required for disjoint classes. · OAI.Problem335.Pairings.classify

classifyLayer def

Assign the zero difference to the normal class; all other compatible coordinate differences belong to the diagonal class. · OAI.Problem335.MomentPairing.classifyLayer

classifyWord def

The unique exact pairing word of a compatible quadruple. · OAI.Problem335.MomentPairing.classifyWord

classInfluence def

Traversals with one designated occurrence of a recurrent rectangle in the middle factor. · OAI.TwoWayComplementation.Diagram.classInfluence

classMultiplicityAtLeast def

The local-order lower bound of an equation class, using the actual chart coordinate rings and their point maximal ideals. · OAI.Nagata.W16.classMultiplicityAtLeast

classOf def

The quotient class of a tensor on any three finite coordinate sets. · OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.classOf

classOf def

Projection to the zero-ultranorm quotient. · OAI.Tingley.classOf

classOfFormula def

τ is precisely the E-class of x relative to A, expanded in pure first-order membership syntax. Slots are τ,A,E,x. · OAI.PartitionConsistency.SetLanguage.classOfFormula

classRectangle def

The rectangular port edges belonging to a signed recurrent class. · OAI.TwoWayComplementation.Diagram.classRectangle

classRepresentationDenseSet def

Padding the preceding local witness set with incompatible conditions is dense in the entire Cohen poset, as required for ordinary genericity. · OAI.PartitionConsistency.ProductForcing.classRepresentationDenseSet

classRepresentationSet def

A genuinely dense set of Cohen conditions giving literal local records. The diagram may be strengthened and the condition's support put in it. · OAI.PartitionConsistency.ProductForcing.classRepresentationSet

classSum def

Sum of a weight ONCE per actual equivalence class. · OAI.HyperbolicObstruction.PictureSwitch.classSum

classTag def

Class and edge query tags in the source's fixed seven structural tags. · OAI.PureTypeSystem.Labelled.ChannelCode.classTag

ClassTestCochainUpper def

Unlike ClassTestUpper, this bounds ordinary cochain degree, uniformly in total degree. · OAI.HarmonicArtin.ParabolicIntersections.ClassTestCochainUpper

classZeroLocus def

The underlying projective zero locus is kept separately from the equation class: it is not the equivalence relation defining the class. · OAI.Nagata.W16.classZeroLocus

Clause abbrev

`Clause` abbreviates `Vector Literal 3`, so its arity is fixed by the type rather than checked at runtime. · OAI.UniqueGamesTheorem.BinaryFormula.Clause

clauseAnswerEquiv def

Executable tuple encoding of the eight clause answers. · OAI.MaxCutGames.Foundations.Hastad.SourceContexts.clauseAnswerEquiv

clauseAnswerEquiv def

Executable tuple encoding of the eight clause answers. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceContexts.clauseAnswerEquiv

ClauseControlled def

Finite-memory invariant: stored representatives, not just their sets. · OAI.ParityWL.Complexity.ClauseControlled

ClauseGap def

A clause gap includes nonemptiness: a cross-multiplied inequality alone would hold vacuously for a formula with no clauses. · OAI.MaxCutGames.Foundations.Hastad.SourceGap.ClauseGap

ClauseGap def

A clause gap includes nonemptiness: a cross-multiplied inequality alone would hold vacuously for a formula with no clauses. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceGap.ClauseGap

clauseInstruction def

Discard a clause marker, or finish at the final formula terminator. · OAI.MaxCutGames.BinaryTokenMachine.clauseInstruction

clauseInstruction def

Discard a clause marker, or finish at the final formula terminator. · OAI.UniqueGamesTheorem.BinaryTokenMachine.clauseInstruction

clauseNames def

An ordinary conjunction of three-slot clauses with sparse variable names. Repeated names and repeated literals are permitted in every clause. · OAI.MaxCutGames.BinaryFormula.clauseNames

clauseNames def

An ordinary conjunction of three-slot clauses with sparse variable names. Repeated names and repeated literals are permitted in every clause. · OAI.UniqueGamesTheorem.BinaryFormula.clauseNames

clauseSlots def

Control roles may include preserved headers and dedicated protected extras. · OAI.MaxCutGames.Foundations.Hastad.SourceRankPhase.clauseSlots

clauseSlots def

Control roles may include preserved headers and dedicated protected extras. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceRankPhase.clauseSlots

Clay structure

Root kernel proof data. hand maps addresses to the shared register carrying that address and a catalyst, not to local pc states. Different hand embeddings of the same addresses during different invocations are allowed. · OAI.ExactDerandomization.Swirl.Clay

Clean def

Unused higher counters may contain arbitrary data, for compositional use. · OAI.Superstring.BitCode.Program.Clean

clean def

The same clean tag is used on every base, including its unused fields. · OAI.FiniteAlgebraicEnvelopes.Compiler.clean

clean def

a component uses at most R nonattack labels. · OAI.HadwigerCounterexample.Volley.clean

cleanBit def

origin is instantiated by the zero advice slope. Keeping it explicit also shows that every specified slope has exactly the same chance. · OAI.MaxCutGames.Clean.cleanBit

cleanBit def

origin is instantiated by the zero advice slope. Keeping it explicit also shows that every specified slope has exactly the same chance. · OAI.UniqueGamesTheorem.Clean.cleanBit

cleanEntry def

The literal entry and handoff of compiler-clean-entry. · OAI.FiniteAlgebraicEnvelopes.Compiler.Compiled.cleanEntry

CleanGallery def

A refinement/coarsening containing no crossings, with actual fresh names. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.CleanGallery

cleanPermutation def

The standard clean implementation of a bijection uses a verified forward and inverse Boolean network, two work registers, and only the named gates. · OAI.ExactQuantumFactoring.BooleanNetwork.cleanPermutation

cleanup def

Wipe stacks and rewind public input, yielding the initial empty stack configuration (no arg tapes). · OAI.Fifths.SM.cleanup

cleanupExecution def

Actual drains followed by a real state-reset instruction reach canonical halt. · OAI.MaxCutGames.Foundations.Complexity.FinalCNFCleanup.cleanupExecution

cleanupExecution def

The cleanup path consists of actual drains and one physical state reset. · OAI.MaxCutGames.Foundations.Complexity.MachineCanonicalOutput.cleanupExecution

cleanupExecution def

Actual drains followed by a real state-reset instruction reach canonical halt. · OAI.UniqueGamesTheorem.Foundations.Complexity.FinalCNFCleanup.cleanupExecution

cleanupExecution def

The cleanup path consists of actual drains and one physical state reset. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineCanonicalOutput.cleanupExecution

cleanupInTime def

Actual cleanup execution, bounded using the original selected-stack lengths. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Cleanup.cleanupInTime

cleanupInTime def

All remaining driver stacks are actually popped, and the exact initial finite state is restored before the halting configuration is reached. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Finish.cleanupInTime

cleanupInTime def

Actual cleanup execution, bounded using the original selected-stack lengths. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Cleanup.cleanupInTime

cleanupInTime def

All remaining driver stacks are actually popped, and the exact initial finite state is restored before the halting configuration is reached. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Finish.cleanupInTime

cleanupTapes def

A fixed explicit list of every non-output tape, independent of input data. · OAI.MaxCutGames.Foundations.Complexity.MachineLazyTableRuntime.cleanupTapes

cleanupTapes def

A fixed finite list of every non-output stack. This includes both retained archives, the original input, and the physically consumed level. · OAI.MaxCutGames.Foundations.Complexity.MachinePaddedOverlayRuntime.cleanupTapes

cleanupTapes def

Every name is distinct, so the exact drain cost uses the entry lengths. · OAI.MaxCutGames.Foundations.Complexity.MachineRegularOwnerCleanup.cleanupTapes

cleanupTapes def

A fixed explicit list of every non-output tape, independent of input data. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineLazyTableRuntime.cleanupTapes

cleanupTapes def

A fixed finite list of every non-output stack. This includes both retained archives, the original input, and the physically consumed level. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachinePaddedOverlayRuntime.cleanupTapes

cleanupTapes def

Every name is distinct, so the exact drain cost uses the entry lengths. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRegularOwnerCleanup.cleanupTapes

clear def

Clearing and transfers are explicit bit programs. In particular, reversing or copying a variable-width label is never given unit cost. · OAI.Superstring.BitCode.Block.clear

clearance def

Exact distance to the three cone tips. · OAI.TriangularBilliards.Triangle.clearance

clearBuffer def

The row primitive clears only its own buffer; other finite registers survive. · OAI.MaxCutGames.Foundations.Complexity.PoweringMasterState.clearBuffer

clearBuffer def

The row primitive clears only its own buffer; other finite registers survive. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringMasterState.clearBuffer

clearedLattice def

The coherent sheaf obtained by twisting a coherent lattice by a nonnegative Cartier ideal power. · OAI.ActualCartier.clearedLattice

clearFields def

The four explicit updates made by the successive cleanup labels. · OAI.MaxCutGames.Explicit.MachineClone100Cleanup.clearFields

clearFields def

The four explicit updates made by the successive cleanup labels. · OAI.UniqueGamesTheorem.Explicit.MachineClone100Cleanup.clearFields

clearFieldsInTime def

The three cleanup phases are actual drains in the concrete program. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Body.clearFieldsInTime

clearFieldsInTime def

The three cleanup phases are actual drains in the concrete program. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Body.clearFieldsInTime

clearKeys def

The finalizer visits every other tape in a fixed finite order. · OAI.MaxCutGames.Reduction.AddressMachineProgram.clearKeys

clearKeys def

The finalizer visits every other tape in a fixed finite order. · OAI.UniqueGamesTheorem.Reduction.AddressMachineProgram.clearKeys

clearLoadedSnapshotInTime def

Apply the checked actual field cleanup directly to an overlaid loaded context. · OAI.MaxCutGames.Foundations.Hastad.SourceTupleTape.clearLoadedSnapshotInTime

clearLoadedSnapshotInTime def

Apply the checked actual field cleanup directly to an overlaid loaded context. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceTupleTape.clearLoadedSnapshotInTime

clearLoadedToBaseInTime def

With initially clean fields and lookup work tapes, loading followed by this cleanup restores the caller's original frame exactly. · OAI.MaxCutGames.Foundations.Hastad.SourceContextClear.clearLoadedToBaseInTime

clearLoadedToBaseInTime def

With initially clean fields and lookup work tapes, loading followed by this cleanup restores the caller's original frame exactly. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceContextClear.clearLoadedToBaseInTime

clearMemory def

Seven real drain loops, including the complete materialized rotor. · OAI.MaxCutGames.Foundations.Complexity.MachineRegularOriginalBody.clearMemory

clearMemory def

Seven real drain loops, including the complete materialized rotor. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRegularOriginalBody.clearMemory

clearP def

check nondivisibility for j = 2 to n-1 · OAI.PowerSaving.RAM.Bench.clearP

clearTapes def

Actual tape family after the selected stacks have been drained in order. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Cleanup.clearTapes

clearTapes def

Actual tape family after the selected stacks have been drained in order. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Cleanup.clearTapes

cleMatrix def

The actual square matrix in the frozen Euclidean coordinates. · OAI.AffineBernstein.cleMatrix

clim def

choose incoming cap after pilot column thresholds. Use ranking with same convention (zero-based; cap at c, c+1st score gate). · OAI.ExactDerandomization.At.clim

climb def

Run state updates in order, with natural prefix addressing. For step: parameters (axis, old prefix, extension digit, previous state). · OAI.PowerSaving.Grove.climb

clip def

Clip at a nonnegative L1 height. Clips are not required to belong to E. · OAI.ExactModuli.clip

clip def

Natural stages saturate if applied in arbitrary intermediate phases at an out-of-range endpoint. On real path proofs will show bounds. · OAI.Fifths.Circuit.clip

clip def

The literal scalar interval used by the source's preliminary clipping. · OAI.KServer.Rounding.clip

clip def

Single kept-cell clipping, before chart identifiers/rebasing. · OAI.PlanarL1.Clipping.clip

clip01 def

Exterior extension is by constant coefficients, never by altering the actual solution on the pulse. · OAI.HarmonicCounterexample.LinearODE.clip01

clipped def

Values are clipped at the exact cost only for the regret analysis. · OAI.EditApproximation.clipped

clipped def

Multiply the literal torus amplitudes by smooth disjoint cylinder cutoffs and retain their actual lifts. All passage regularity is derived. · OAI.ThreeTorus.PassagePackets.clipped

clippedCDF def

The cumulative integral is clamped to its genuine support interval. · OAI.Problem358.PositiveIntervalDensity.clippedCDF

clippedGraphRow def

Spectral clipping of a rectangular row on its left square. · OAI.KadisonKastler.clippedGraphRow

ClippedOracle structure

A pair produced by genuine common-event normalized conditioning. The outcome space and public decoder remain unchanged. · OAI.TraceReconstruction.ClippedOracle

clippedPulse def

Constant exterior extension of a pulse coefficient. Its boundedness follows from genuine continuity on the compact pulse, rather than global invertibility of an irrelevant scalar comparison. · OAI.HarmonicCounterexample.LinearODE.clippedPulse

clipReal def

A bounded representative, used only with a proof of AE equality. It does not impose a pointwise bound on all values of an unrestricted sample space. · OAI.DilutedSpinGlass.clipReal

cliqueCell def

Every point with prescribed initial vertex belongs to some closed positive cube at that vertex. · OAI.EilenbergGanea.CubeRealization.cliqueCell

cliqueEnergy def

The unordered complete exchange energy, expressed as half the ordered sum, with the manuscript's minimum-endpoint conductance. · OAI.ContingencyTables.SignedTransport.cliqueEnergy

CliqueIndex abbrev

Actual finite cliques, including the empty clique. · OAI.EilenbergGanea.CubeRealization.CliqueIndex

clock def

Uniform bit clock; the polynomial work counts syntactic bounded iterations. · OAI.Superstring.BitCode.Program.clock

clockCancellationCross def

The positive-degree off-diagonal coefficient. · OAI.WeakPureInfiniteness.clockCancellationCross

clockCancellationDegrees def

The three actual homogeneous degrees and coefficients. · OAI.WeakPureInfiniteness.clockCancellationDegrees

clockCancellationDiagonal def

The constant coefficient of the cancellation polynomial. · OAI.WeakPureInfiniteness.clockCancellationDiagonal

clockCancellationIncrement def

The sum of the two diagonal corner increments. · OAI.WeakPureInfiniteness.clockCancellationIncrement

clockCoefficientContraction def

Coefficient contraction using the two paths on the same partition. · OAI.WeakPureInfiniteness.clockCoefficientContraction

clockCorrectedCancellation def

The explicit correction of the degree 0,1,-1 cancellation formula. · OAI.WeakPureInfiniteness.clockCorrectedCancellation

clockDegreeCorrection def

Left correction of an integer homogeneous degree by a unitary clock. · OAI.WeakPureInfiniteness.clockDegreeCorrection

ClockPathControl structure

The two estimates needed when a finite path is sampled on the cyclic tower. · OAI.WeakPureInfiniteness.ClockPathControl

clockPolarValue def

The explicit polar correction, on the open set used below. · OAI.WeakPureInfiniteness.clockPolarValue

clockStatement def

Homogeneous Boolean embedding of the fixed logarithm program. · OAI.MaxCutGames.Foundations.Complexity.GraphCounterModel.clockStatement

clockStatement def

Homogeneous Boolean embedding of the fixed logarithm program. · OAI.UniqueGamesTheorem.Foundations.Complexity.GraphCounterModel.clockStatement

clockTailTime def

Move from a chosen sampling time to the terminal time. · OAI.WeakPureInfiniteness.clockTailTime

clone def

Copy stack k onto empty y using empty tmp t. · OAI.Fifths.SM.clone

cloneName def

A total bounded-name helper. Every index used by the table is below 48, and cloneName_value removes the remainder operation on those indices. · OAI.MaxCutGames.Reduction.CloneTable.cloneName

cloneName def

A total bounded-name helper. Every index used by the table is below 48, and cloneName_value removes the remainder operation on those indices. · OAI.UniqueGamesTheorem.Reduction.CloneTable.cloneName

close def

Close all internal tips with a matching of local joins. · OAI.OneWayLiveness.Pairing.close

close def

Universal closure of all the free variables. · OAI.PartitionConsistency.Formula.close

close def

Deterministic closeness of each forced split; it is a restriction on labels, never an additional conditioning event for the random bits. · OAI.TraceReconstruction.BinaryWindowLabels.close

closed def

The actual balanced and short-closed system, with all cut comparisons and all closure positions left unpaired. · OAI.TorsionFreeZeroDivisors.SampleGraph.FullPairing.closed

closed_split def

Transfer the SAME separator sets/parts/cost to the actual short-closed paths. In particular no extra charge for the appended tails is introduced. · OAI.TorsionFreeZeroDivisors.SampleGraph.FullPairing.closed_split

closedAnnulus def

The genuine closed annulus in the complex plane. · OAI.Nagata.Workers.W05.closedAnnulus

closedBallLinearSelf def

A linear contraction restricts continuously to the same ball. · OAI.PolycyclicRecognition.closedBallLinearSelf

closedCanonicalRowAlgebra def

The closed unital algebra generated by the complete canonical row. · OAI.WeakPureInfiniteness.closedCanonicalRowAlgebra

closedCell def

The closed cell of one affine branch, including ties. · OAI.Problem358.CellGeometry.closedCell

closedCentralInterval def

Closed central cells have the same slice law as open central cells. · OAI.Problem358.closedCentralInterval

closedChartEquiv def

The completed coordinate isomorphism at a closed étale point. · OAI.CartierSections.closedChartEquiv

closedChartFieldRetraction def

The retraction is a valuation of the chart's actual rational function field. · OAI.CartierSections.closedChartFieldRetraction

closedChartRetraction def

The monomial valuation pulled back by the formal coordinate expansion. · OAI.CartierSections.closedChartRetraction

closedCoverGraph def

Distinct cover members are adjacent when they meet. · OAI.Problem358.closedCoverGraph

closedCoverMap def

Finite closed-cover pasting, with the actual glued continuous map. · OAI.EuclideanFiveColor.closedCoverMap

closedDescent_localIso def

Canonical local comparison of the descended module with its given closed-support representative. · OAI.CoherentGlobal.closedDescent_localIso

ClosedFiniteCodim structure

Closed finite-codimensional subspaces for asymptotic moduli. · OAI.BoundedTreePotentials.ClosedFiniteCodim

ClosedIdeal def

A literal closed two-sided star ideal of the coefficient algebra. · OAI.TraceConeClassification.ClosedIdeal

closedIdealEquivIdempotent def

The bijection in the first paragraph of the proof of st:ideals. · OAI.TraceConeClassification.closedIdealEquivIdempotent

closedIdealEquivOfAdditive def

Recovering the closed ideals from their idempotent trace weights. · OAI.TraceConeClassification.closedIdealEquivOfAdditive

closedIdealOrderEquivOfAdditive def

The order isomorphism of closed ideals induced by an additive trace-cone bijection. · OAI.TraceConeClassification.closedIdealOrderEquivOfAdditive

closedLevelNet def

An exact level count, selected by the bounded divisibility predicate. The branch at t≥n is zero since a is a nonzero n-bit word. · OAI.ExactQuantumFactoring.BitArithmetic.closedLevelNet

closedLiftFunctor def

The open-lift functor for a closed immersion. · OAI.CoherentGlobal.closedLiftFunctor

closedLiftOpen def

The largest target open whose inverse image lies in a given source open. · OAI.CoherentGlobal.closedLiftOpen

ClosedMarking def

A Boolean relation is closed if it is an equivalence containing every marked edge. · OAI.FiniteCongruence.ClosedMarking

closedNeighborhood def

The union of an independent set and all of its neighbours. · OAI.CycleClique.Construction.closedNeighborhood

closedObject def

Closure under factors and ambient joins supplies a genuine object. · OAI.PartitionConsistency.ConservativeMonoids.closedObject

closedOpenAdjunction def

Inverse image of opens is a reflective localization for a closed embedding. · OAI.CoherentGlobal.closedOpenAdjunction

ClosedOrientedMetricFour structure

A connected closed oriented smooth Riemannian four-manifold. · OAI.PositiveEinsteinFour.ClosedOrientedMetricFour

ClosedOrientedMetricFour structure

A closed connected oriented Riemannian four-manifold, with its genuine smooth manifold structure. The measurable structure can be installed as the Borel structure when applying the analytic theorem. · OAI.PositiveEinsteinFour.ClosedOrientedMetricFour

closedPresheafFullyFaithful def

Additive presheaves are recovered fully faithfully from their direct images. · OAI.CoherentGlobal.closedPresheafFullyFaithful

closedPushforwardPreimageAdd def

Underlying additive natural transformation of the descended map. · OAI.CoherentGlobal.closedPushforwardPreimageAdd

closedRuns def

Completed runs, in reverse chronological order. · OAI.SignedDisk.SignedRuns.closedRuns

closedRuns def

Completed runs, in reverse chronological order. · OAI.SignedRuns.closedRuns

closedShell def

The closed annulus is used only for local differentiability and continuity. · OAI.MahlerStokes.closedShell

closedSimplexHomeomorph def

The closed simplex in the carrier and its ambient barycentric subtype carry exactly the same topology. · OAI.HyperbolicObstruction.FinitePolyhedron.closedSimplexHomeomorph

closedSpan def

The concrete Banach completion. Its norm is literally the supremum of absolute evaluations over K. · OAI.SeparableQuotient.NormConstruction.NormingSet.closedSpan

closedSpan def

The closed linear span of the sequence. · OAI.SeparableQuotient.PrefixNormers.closedSpan

closedStructureBaseChangeIso def

The composite of base change with the pullback unit isomorphism. · OAI.SourcePullbackChart.closedStructureBaseChangeIso

ClosedUnder def

Closure under the edges whose modules survive, strictly weaker than being an ambient upper set. · OAI.KLInvariance.MomentGraph.Sheaf.ClosedUnder

closeLabels def

Admissible close child labels, without relaxing any plateau restrictions on the anchor. An arbitrary finite label domain is retained. · OAI.TraceReconstruction.closeLabels

closePairs def

Ordered distinct phase pairs within a fixed distance. · OAI.EuclideanFiveColor.PhaseCollision.closePairs

closePath def

A closing path has a specified endpoint rather than an adaptively chosen one. It issues exactly the route's coin requirements. · OAI.CubeShuffle.QueryTree.closePath

closePath def

A single path with its two external vertices fixed at a. · OAI.Problem335.Pairings.closePath

closePaths def

Add the two fixed external vertices to four paths specified by their internal labels. · OAI.Problem335.Pairings.closePaths

closureLift def

Lift a complex-linear automorphism of the rational function field to its algebraic closure. · OAI.SourceSymmetry.closureLift

Cloud abbrev

The darts departing a specified vertex. · OAI.MaxCutGames.Foundations.PCP.DegreeReplacement.Cloud

Cloud abbrev

The darts departing a specified vertex. · OAI.UniqueGamesTheorem.Foundations.PCP.DegreeReplacement.Cloud

cloud def

Envelope for propagation coefficient. · OAI.ExactDerandomization.Buf.Run.cloud

cloudCountInTime def

A linear transition budget for this concrete cloud-count subroutine. · OAI.MaxCutGames.Foundations.Complexity.MachineCloudCount.cloudCountInTime

cloudCountInTime def

A linear transition budget for this concrete cloud-count subroutine. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineCloudCount.cloudCountInTime

cloudDarts def

Retain the input dart order and test only the stored tail index. · OAI.MaxCutGames.Foundations.PCP.PreprocessingCloudIndex.cloudDarts

cloudDarts def

Retain the input dart order and test only the stored tail index. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingCloudIndex.cloudDarts

cloudEquiv def

This equivalence contains executable rank and select functions, including when both its domain and codomain are empty. · OAI.MaxCutGames.Foundations.PCP.PreprocessingCloudIndex.cloudEquiv

cloudEquiv def

A positive padded cloud has exactly the size of one explicit family member. · OAI.MaxCutGames.Foundations.PCP.Regularization.cloudEquiv

cloudEquiv def

This equivalence contains executable rank and select functions, including when both its domain and codomain are empty. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingCloudIndex.cloudEquiv

cloudEquiv def

A positive padded cloud has exactly the size of one explicit family member. · OAI.UniqueGamesTheorem.Foundations.PCP.Regularization.cloudEquiv

cloudFirst def

One inner step in a cloud, retaining its return port and the unused port. · OAI.MaxCutGames.Foundations.PCP.ZigzagGraphs.cloudFirst

cloudFirst def

One inner step in a cloud, retaining its return port and the unused port. · OAI.UniqueGamesTheorem.Foundations.PCP.ZigzagGraphs.cloudFirst

cloudGraph def

Empty clouds use the empty rotation; every other cloud uses the proved family. · OAI.MaxCutGames.Foundations.PCP.Regularization.cloudGraph

cloudGraph def

Empty clouds use the empty rotation; every other cloud uses the proved family. · OAI.UniqueGamesTheorem.Foundations.PCP.Regularization.cloudGraph

cloudGraphs def

A supplied finite cloud rotation is transported by the explicit cloud list. · OAI.MaxCutGames.Foundations.PCP.PreprocessingRegularTables.cloudGraphs

cloudGraphs def

A supplied finite cloud rotation is transported by the explicit cloud list. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingRegularTables.cloudGraphs

cloudInTime def

The common dummy-owner path also carries the checked physical scan's polynomial certificate; the caller's existing rank, prefix, and rotor survive. · OAI.MaxCutGames.Foundations.Complexity.MachineRegularMetadata.cloudInTime

cloudInTime def

The common dummy-owner path also carries the checked physical scan's polynomial certificate; the caller's existing rank, prefix, and rotor survive. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRegularMetadata.cloudInTime

cloudOperator def

Average only the inner cloud coordinate, keeping the outer vertex fixed. · OAI.MaxCutGames.Foundations.PCP.ZigzagGraphs.cloudOperator

cloudOperator def

Average only the inner cloud coordinate, keeping the outer vertex fixed. · OAI.UniqueGamesTheorem.Foundations.PCP.ZigzagGraphs.cloudOperator

cloudPaddedSize def

Empty clouds remain empty; nonempty clouds use the geometric family. · OAI.MaxCutGames.Foundations.PCP.PreprocessingLevels.cloudPaddedSize

cloudPaddedSize def

Empty clouds remain empty; nonempty clouds use the geometric family. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingLevels.cloudPaddedSize

cloudPaddingInTime def

The bound is quadratic in the combined actual graph/query tape length. All transitions are supplied by the concrete program and its phase traces. · OAI.MaxCutGames.Foundations.Complexity.MachineCloudPadding.cloudPaddingInTime

cloudPaddingInTime def

The bound is quadratic in the combined actual graph/query tape length. All transitions are supplied by the concrete program and its phase traces. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineCloudPadding.cloudPaddingInTime

cloudRank def

Find the local index of an original dart belonging to this cloud. · OAI.MaxCutGames.Foundations.PCP.PreprocessingCloudIndex.cloudRank

cloudRank def

Find the local index of an original dart belonging to this cloud. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingCloudIndex.cloudRank

cloudRankInTime def

A genuine timed execution witness with a polynomial (indeed linear) transition budget. Neither the result nor a body trace is a premise. · OAI.MaxCutGames.Foundations.Complexity.MachineCloudRank.cloudRankInTime

cloudRankInTime def

A genuine timed execution witness with a polynomial (indeed linear) transition budget. Neither the result nor a body trace is a premise. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineCloudRank.cloudRankInTime

cloudSelect def

Read the original dart stored at a valid local cloud index. · OAI.MaxCutGames.Foundations.PCP.PreprocessingCloudIndex.cloudSelect

cloudSelect def

Read the original dart stored at a valid local cloud index. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingCloudIndex.cloudSelect

cluster def

Finite open paths, including the path of length zero, define the open cluster. · OAI.CriticalPercolation.BondGraph.cluster

cluster def

The actual depth-d vertex is represented by its set of descendant leaf positions. Depths range from the root through the terminal leaf positions. · OAI.DilutedSpinGlass.PrescribedTree.cluster

clusterCount def

The source k_M(A), including isolated vertices. · OAI.FKCRT.clusterCount

clusterCutCount def

A cut expressed in cluster coordinates. · OAI.OptimalMaxCut.Unweighted.clusterCutCount

cmean def

Complexification of the same (not a separately chosen) real mean. · OAI.PlaneFiveColor.InvariantMean.cmean

coaction def

The Laurent weight coaction on the principal quotient by a homogeneous relation. · OAI.StableCoordinate.HomogeneousQuotientWeight.coaction

CoarseAccuracyEvent def

The actual finite-tree event. At every node it uses only deterministic true-cost minima and the node's current child threshold labels. · OAI.EditApproximation.CoarseAccuracyEvent

coarseArrayEval def

Actual shared-array recursion. Pruning creates a zero marker and does not traverse any descendants. · OAI.EditApproximation.coarseArrayEval

coarseArrayEvalChildren def

Each list member is evaluated once and its whole array is reused. · OAI.EditApproximation.coarseArrayEvalChildren

CoarseArrayRep def

The array relation ignores proof fields and keeps exact stored values. · OAI.EditApproximation.BinaryFraction.CoarseArrayRep

coarseArrayStorage def

The stored numerator bits, denominator bits, sign and list links. · OAI.EditApproximation.BinaryFraction.coarseArrayStorage

CoarseBinaryBox def

A zero marker or one full shift array on the execution's common grid. · OAI.EditApproximation.BinaryFraction.CoarseBinaryBox

coarseBinaryChildren def

Shared evaluation of each child, followed by its exact two-pass penalty transform, threshold contribution and fixed-grid addition. · OAI.EditApproximation.BinaryFraction.coarseBinaryChildren

coarseBinaryChildrenWeight def

Sum the weights of exactly those child calls made by the binary loop. · OAI.EditApproximation.BinaryFraction.coarseBinaryChildrenWeight

coarseBinaryEval def

Binary implementation of the pruning-stopped shared-array evaluator. The complete tree is a semantic indexed source; visited child descriptors carry their source offset and length, charged in coarseBinaryChildren. · OAI.EditApproximation.BinaryFraction.coarseBinaryEval

coarseBinaryWeight def

One array unit per expanded child shift and one unit per visited node. This counter follows the executed pruning and threshold branches. · OAI.EditApproximation.BinaryFraction.coarseBinaryWeight

coarseCandidateWithWork def

A candidate executes the whole root recurrence, then adds its literal guess and doubles the result using Boolean arithmetic. · OAI.EditApproximation.BinaryFraction.coarseCandidateWithWork

CoarseChildAccuracy def

Child accuracy at precisely the source positions visited by the actual traversal, with its actual per-child allowance. · OAI.EditApproximation.CoarseChildAccuracy

coarseChildListStart def

Actual source starts of all children of a literal list traversal. · OAI.EditApproximation.coarseChildListStart

coarseChildrenArrayHistory def

Sum the stored bits of the four array results at each child step, including every recursive child's array history. This allows all these arrays to remain allocated instead of assuming prompt reclamation. · OAI.EditApproximation.BinaryFraction.coarseChildrenArrayHistory

coarseChildThresholdAllocation def

Includes all three natural literal numerators and their unit denominators. · OAI.EditApproximation.BinaryFraction.coarseChildThresholdAllocation

coarseChildThresholdInvocations def

Includes a conservative allowance for the three literal constructors. · OAI.EditApproximation.BinaryFraction.coarseChildThresholdInvocations

CoarseCompact def

Leading-zero removal is enough: this property does not assert that a stored rational is reduced by a gcd. · OAI.EditApproximation.BinaryFraction.CoarseCompact

coarseCostedPenaltyTransform def

Instrumented actual sweeps plus the final cellwise comparisons. · OAI.EditApproximation.coarseCostedPenaltyTransform

coarseCostedScan def

A costed scan records the addition and comparison actually performed at each transition; list traversal is charged separately by the RAM model. · OAI.EditApproximation.coarseCostedScan

coarseDataListStart def

Source starts attached to the full child metadata list. · OAI.EditApproximation.coarseDataListStart

coarseDefect def

The literal triangle defect over twelve positive refinements. · OAI.DirectionNeighbors.coarseDefect

coarseDefect def

The literal triangle defect over twelve positive refinements. · OAI.SignedDisk.DirectionNeighbors.coarseDefect

CoarseDescendantEvents def

Events for all children, including children below a pruned node. · OAI.EditApproximation.CoarseDescendantEvents

coarseDescriptorLength def

Cached length of a descriptor into a virtually padded source. · OAI.EditApproximation.coarseDescriptorLength

coarseEntryBinaryCost def

Keep the branch dimensions symbolic when forming the finite source sum. · OAI.EditApproximation.coarseEntryBinaryCost

coarseEntryTailCount def

Multiplicity of all local Bernstein tails for one complete entry. · OAI.EditApproximation.coarseEntryTailCount

coarseExpectedArrayWork def

Finite conditional expectation recurrence for the same active work. The root labels are independent uniforms, and each child uses its own independent recursive label tree. · OAI.EditApproximation.coarseExpectedArrayWork

CoarseGridCell def

Invariant of a stored coarse cell on the common denominator. · OAI.EditApproximation.BinaryFraction.CoarseGridCell

coarseGridNatAllocation def

Arithmetic-only allocation; the caller owns the input words. · OAI.EditApproximation.BinaryFraction.coarseGridNatAllocation

coarseGridNatInvocations def

Control invocations of the shared-denominator scalar construction. · OAI.EditApproximation.BinaryFraction.coarseGridNatInvocations

coarseGridPenaltyAllocation def

Both reverse operations allocate their list spines; the element words are shared. Includes both sweeps, their discarded arrays and the final zip. · OAI.EditApproximation.BinaryFraction.coarseGridPenaltyAllocation

coarseGridPenaltyInvocations def

Includes one invocation per reverse-list step, even if implemented with an iterative constant-stack loop. · OAI.EditApproximation.BinaryFraction.coarseGridPenaltyInvocations

coarseGroupRingCohomologyIso def

The group-ring case of coarse cohomology invariance, with the actual left regular representations and standard group cohomology. · OAI.PolycyclicRecognition.coarseGroupRingCohomologyIso

coarseIndexChildrenAllocation def

Allocation follows the same child sequence, thresholds, recursive calls, penalty sweeps and accumulated array as the indexed evaluator. · OAI.EditApproximation.BinaryFraction.coarseIndexChildrenAllocation

coarseIndexChildrenInvocations def

Invocation count follows the same child sequence, thresholds, recursive calls, penalty sweeps and accumulated array as the indexed evaluator. · OAI.EditApproximation.BinaryFraction.coarseIndexChildrenInvocations

coarseIndexedAllocation def

Fresh bit and list cells in the actual indexed coarse evaluator. Input arrays and random draws are shared; virtual source trees are never materialized. Child offsets/descriptors follow the repeated reads charged by the executable evaluator. · OAI.EditApproximation.BinaryFraction.coarseIndexedAllocation

coarseIndexedArrayHistory def

Every array footprint of the literal indexed recursion. Descriptor and address words are charged separately by the indexed bit counter. · OAI.EditApproximation.BinaryFraction.coarseIndexedArrayHistory

coarseIndexedExtra def

All descriptor and original-input operations alongside the core arithmetic. Threshold decisions are exactly those of the evaluated call. · OAI.EditApproximation.BinaryFraction.coarseIndexedExtra

coarseIndexedInitialAllocation def

Fresh arithmetic cells and root/index list spines. Constant words and preexisting input/cached words are shared. The retained input and output words are counted separately by the global storage capacity. · OAI.EditApproximation.BinaryFraction.coarseIndexedInitialAllocation

coarseIndexedInvocations def

Conservative recursive invocation count for the indexed evaluator, including scalar callees, index traversals and repeated descriptor reads. · OAI.EditApproximation.BinaryFraction.coarseIndexedInvocations

coarseInitialWithWork def

Entire dyadic initial estimator, including direct cases, every root execution, binary minimum, ceiling, and final integer clamp. · OAI.EditApproximation.BinaryFraction.coarseInitialWithWork

coarseInternalNodes def

Symbolic number of internal nodes in a complete coarse tree. · OAI.EditApproximation.coarseInternalNodes

coarseLeftSweep def

The executable left sweep; Lean's scanl traverses its input once. · OAI.EditApproximation.coarseLeftSweep

coarseLevelMagnitude def

Loose magnitude bound for all partial child sums. Its bit length is linear in the depth and parameter bit lengths. · OAI.EditApproximation.BinaryFraction.coarseLevelMagnitude

coarseNodeSamplingValue def

The two deterministic cost lists used at a node, simultaneously for all shifts. The Boolean coordinate selects the lower or upper list. · OAI.EditApproximation.coarseNodeSamplingValue

coarsePaddedRead def

Padding and framing are index calculations over the original stored input. These views do not allocate a padded string. · OAI.EditApproximation.coarsePaddedRead

CoarsePair abbrev

All pairs with the fixed coarse bounds, without basepoint normalization. · OAI.PolycyclicRecognition.CoarsePair

coarsePenaltyTransform def

Literal array output of the two linear sweeps. · OAI.EditApproximation.coarsePenaltyTransform

coarsePeriodGap def

Twice the coarse period gaps (endpoints are half-integral before doubling). · OAI.TraceReconstruction.coarsePeriodGap

CoarseReadBudget structure

A finite uniform bound on the actually executed source-index operations. The depth and offset ranges are the original padded source capacity. · OAI.EditApproximation.CoarseReadBudget

coarseReadCore def

Core arithmetic over source descriptors. The separate descriptor/read counter charges address arithmetic and original-array accesses. · OAI.EditApproximation.BinaryFraction.coarseReadCore

coarseReadTree def

A source descriptor is expanded by index arithmetic. No sliced or padded list is allocated. The function is the original stored input's read operation. · OAI.EditApproximation.coarseReadTree

coarseRootArray def

The actual array immediately before the root lookup. · OAI.EditApproximation.BinaryFraction.coarseRootArray

coarseRootWithWork def

Counted root setup and array lookup around the complete binary coarse recursion. The common denominator is built from bits, not a free conversion. · OAI.EditApproximation.BinaryFraction.coarseRootWithWork

coarseSaturatingSubtractAllocation def

Comparisons and minimum reuse their input bit arrays. · OAI.EditApproximation.coarseSaturatingSubtractAllocation

CoarseStoredArray abbrev

Stored cells have the complete shift-domain length; none is zero. · OAI.EditApproximation.CoarseStoredArray

coarseStoredPenalty def

The costed linear sweeps operate on the stored cells. · OAI.EditApproximation.coarseStoredPenalty

coarseStructuralFactor def

The exact depth factor in Section 3. · OAI.EditApproximation.coarseStructuralFactor

coarseThresholdBitWork def

Power-of-two threshold sampling consumes exactly the fixed word length per used label. binaryWord_uniform proves the output law of every draw. · OAI.EditApproximation.coarseThresholdBitWork

coarseTreeValue def

The zero root shift is the middle integer code. · OAI.EditApproximation.coarseTreeValue

coarseTrueThresholdSum def

The true-cost threshold sum appearing in the simultaneous event. · OAI.EditApproximation.coarseTrueThresholdSum

CoassociativeAt def

Coassociativity at one particular ring element. · OAI.StableCoordinate.FrameLift.CoassociativeAt

coast def

mask kind2 and kind1 use lit-only pricing. · OAI.ExactDerandomization.Cambria.Wear.coast

coastline def

Total certificate action. · OAI.PowerSaving.RAM.coastline

coaugmentation def

Its canonical coaugmentation is the inclusion of the original stage. · OAI.Thomason.Construction.InfiniteExtension.coaugmentation

cobaseWeak abbrev

A map remains weak after every individual cobase change. · OAI.Thomason.Construction.FiniteSubsetAttachments.cobaseWeak

coboundaries def

Actual coboundaries, with constant cochains in degree zero. · OAI.CoboundaryExpanders.Complex.coboundaries

coboundaries def

Actual coboundaries; in degree zero use constants, as in the manuscript. · OAI.CoboundaryExpanders.Complex.coboundaries

coboundary def

Simplicial coboundary over F₂. · OAI.CoboundaryExpanders.Complex.coboundary

coboundary def

Simplicial coboundary with coefficients in F₂. · OAI.CoboundaryExpanders.Complex.coboundary

CochainBound def

A linear bound on the integral, for every clipped distance cochain. · OAI.Release075.Discrete.CochainBound

cochainDifferential def

The scalar Hochschild differential of a bilinear cochain. · OAI.KadisonKastler.cochainDifferential

CochainLinear def

A quantitative consequence of actual linear word filling, tested by Lipschitz cochains. This property makes no geometric conclusion an assumption. · OAI.HyperbolicObstruction.CochainLinear

cochainModule def

The ordinary C-module O formed from the actual terms and differential of P. · OAI.LittleFinitistic.OrdinarySimulation.cochainModule

cocone abbrev

The colimit point is the original categorification with its membership label. · OAI.Thomason.Construction.CosieveSimplex.cocone

cocone abbrev

The canonical cocone, using Mathlib's realization comparison on simplices. · OAI.Thomason.Construction.RealizationPrism.cocone

cocone abbrev

The colimit point is the original categorification with its membership label. · OAI.Thomason.Construction.SieveSimplex.cocone

cocone def

All the original diagram relations hold in the quotient. · OAI.Thomason.Construction.StrictOmegaCategory.ColimitConstruction.cocone

coconeIsColimit def

Forgetting the labels recovers exactly the original simplex colimit. · OAI.Thomason.Construction.CosieveSimplex.coconeIsColimit

coconeIsColimit def

Forgetting the labels recovers exactly the original simplex colimit. · OAI.Thomason.Construction.SieveSimplex.coconeIsColimit

cocycleGridTime def

The actual finite mesh. Its endpoint belongs to the same unit interval. · OAI.WeakPureInfiniteness.cocycleGridTime

Code structure

An unrestricted n-use quantum code with one collective output POVM. · OAI.GAD.Code

Code abbrev

A pair of members of the positive approximation list. · OAI.MetricEntropyDuality.SignedCompression.Code

Code structure

A finite integer mixed-moment descriptor, retaining every index size. · OAI.MUB6.Certificate.MixedSemantics.Code

Code inductive

call is internal for bounded recursion; a program has Port = none. loop, tab and sow retain the input to access parameters in the body. descend sets up bounded recursion, reading fuel and initial argument. · OAI.PowerSaving.RAM.Code

Code abbrev

Bounded path sequences, marked graph incidence/reversal, lengths, sides, root designation and interval data. Unused graph indices may be padded. The use of oriented darts avoids choosing orientations of loop chains. · OAI.TorsionFreeZeroDivisors.PatternCoding.Code

code def

A finite code for all chart data, not a code for arbitrary cube points. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Table.code

code def

The callback is executed once for every coordinate. · OAI.LowSpaceSubsetSum.StoredSeparatorQueries.code

code def

The actual set code, not a semantic truth value. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.code

code def

The result register is separate from immutable inputs; all temporary register numbers are at least base and are cleared before return. · OAI.WLIdentification.WordExpressions.Expr.code

codeArray def

Dense serialized arrays of variable-width binary records. · OAI.RationalHitting.StackLanguage.codeArray

codedAction def

Conjugating an honest order action by the coding equivalence. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.codedAction

CodedChain abbrev

Standard-Borel coding of the exact complete trajectories. · OAI.TriangularBilliards.CodedChain

codeOrder def

The ordering on literal condition codes, false outside those codes. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.codeOrder

codeOrderBound def

The order bound is the order type of the one fixed ground-code wellordering, computed on a small copy of the code set. · OAI.PartitionConsistency.ProductForcing.ChainCode.codeOrderBound

codeSpineSum def

The exact deterministic ancestral statistic at preorder position v. · OAI.FKCRT.TreeCoding.codeSpineSum

codeWord def

Self-delimiting word records (distinct from the final output encoding). · OAI.RationalHitting.StackLanguage.codeWord

codeword def

The encoding is a literal coordinate read from a finite Boolean truth table. · OAI.MaxCutGames.Foundations.PCP.CodeComposition.codeword

codeword def

The encoding is a literal coordinate read from a finite Boolean truth table. · OAI.UniqueGamesTheorem.Foundations.PCP.CodeComposition.codeword

Coding class

Canonical storage format for the compositional finite-program proofs. · OAI.RationalHitting.StackLanguage.Coding

codingPoint def

All infinite addresses, not an almost-everywhere image. · OAI.CorSetReference.codingPoint

codiscrete def

There is a unique cell between each pair of labels, and only identities above dimension one. · OAI.Thomason.Construction.StrictOmegaCategory.codiscrete

codiscreteFunctor def

Relabelling codiscrete categories is functorial. · OAI.Thomason.Construction.StrictOmegaCategory.codiscreteFunctor

codiscreteLabel def

Read the label of a zero-cell in a codiscrete target. · OAI.Thomason.Construction.StrictOmegaCategory.codiscreteLabel

codiscreteLift def

Any assignment of object labels extends to the codiscrete strict category. · OAI.Thomason.Construction.StrictOmegaCategory.codiscreteLift

codomainEquiv def

Codomain coordinates split the range off from a chosen complement. · OAI.MaxCutGames.Appendix.CoordinateTransport.codomainEquiv

codomainEquiv def

Codomain coordinates split the range off from a chosen complement. · OAI.UniqueGamesTheorem.Appendix.CoordinateTransport.codomainEquiv

codomainFrequencyEquiv def

Actual dual frequencies are reindexed by precomposition with the inverse. · OAI.MaxCutGames.Inverse.KMSAnalytic.codomainFrequencyEquiv

codomainFrequencyEquiv def

Actual dual frequencies are reindexed by precomposition with the inverse. · OAI.UniqueGamesTheorem.Inverse.KMSAnalytic.codomainFrequencyEquiv

codomainSplit def

A question into the symbol/perpendicular product is exactly a pair of independent coordinate questions. · OAI.MaxCutGames.Fourier.SplitMaps.codomainSplit

codomainSplit def

A question into the symbol/perpendicular product is exactly a pair of independent coordinate questions. · OAI.UniqueGamesTheorem.Fourier.SplitMaps.codomainSplit

coeff def

Coherent refinement coefficients, evaluated by exact positive matrix composition on the finite box. These are not row-stochastic transitions. · OAI.DyadicHats.coeff

coeff def

Coordinates in the basis of unnormalized cluster columns. · OAI.HyperedgeComparison.coeff

coeff def

The exact coefficient at one key, including all repeated occurrences. · OAI.MUB6.Certificate.Sparse.coeff

coeff def

Coherent refinement coefficients, evaluated by exact positive matrix composition on the finite box. These are not row-stochastic transitions. · OAI.SignedDisk.DyadicHats.coeff

coeffA def

First coefficient in the global Bezout relation. · OAI.StableCoordinate.Bundle.coeffA

coeffB def

Second coefficient in the global Bezout relation. · OAI.StableCoordinate.Bundle.coeffB

coefficient def

Coefficients of the central isotypic projector, in unnormalized trace convention. They are derived from an actual irrep, not supplied as character data satisfying unproved orthogonality equations. · OAI.CoordinateSweeps.UnitaryIrrep.coefficient

coefficient def

Replace only the free leading algebra coefficient along f. · OAI.DerivedBar.FreeBaseChange.coefficient

coefficient def

Coefficients at nonzero monoid elements descend unchanged to the quotient. · OAI.FiniteAlgebraicEnvelopes.Contracted.coefficient

coefficient def

The bounded scalar coefficient to which the scalar mean is applied. · OAI.KadisonKastler.ScalarInvariantMean.coefficient

coefficient def

A scalar matrix coefficient is an actual L-infinity class. · OAI.KadisonKastler.UniformMeasurableOperators.coefficient

coefficient def

Matrix coefficients depend only on the almost-everywhere operator class. · OAI.KadisonKastler.WeakOperatorField.coefficient

coefficient def

Scalar matrix coefficients as L-infinity classes. · OAI.KadisonKastler.WeakOperatorRepresentative.coefficient

coefficient def

Extend a finite family of actual components by zero in every other degree. · OAI.LittleFinitistic.FiniteSupportReindex.coefficient

coefficient def

The coefficient of the unique possible arrow from j to i. · OAI.LittleFinitistic.OrdinarySimulation.ChainAlgebra.coefficient

coefficient def

Coefficient of the path b a in a vector of the actual relation subspace. · OAI.LittleFinitistic.RelationBasisCoefficients.coefficient

coefficient def

Fourier coefficient for an arbitrary finite weighted system. · OAI.Problem313.FiniteParseval.coefficient

coefficient def

The usual cubic Ramanujan coefficient, extended by zero at modulus zero. At positive modulus this is the actual unit-character sum weighted by the cube of the Möbius-to-totient ratio. · OAI.Problem337.RamanujanEuler.coefficient

coefficient def

Retained weights at retained candidates, and zero at discarded candidates. · OAI.ReflexiveFixedPoints.CandidateLedger.coefficient

coefficient def

The source's common coefficient in the two remaining generator values. · OAI.StableCoordinate.VerticalLND.coefficient

coefficient def

Coefficients in the actual tuple-of-Finsupp chain group. · OAI.Thomason.Construction.PastingGlobeCylinderBasis.coefficient

coefficient def

The constant in the polynomial pattern bound depends only on fixed bounds. · OAI.TorsionFreeZeroDivisors.PatternCoding.coefficient

coefficient def

The normalized Fourier coefficient, with conjugation on the character. · OAI.TreeEdit.BinaryLower.FiniteFourier.coefficient

coefficientAction def

A right-regular coefficient acts on the left tensor factor by its unop. · OAI.LittleFinitistic.BalancedTensor.Free.coefficientAction

coefficientAction def

A coefficient of a free left module acts on the right tensor factor. · OAI.LittleFinitistic.BalancedTensor.FreeLeft.coefficientAction

coefficientAlgebra def

Smooth temperate coefficients act on actual tempered distributions. · OAI.ElasticityDistribution.coefficientAlgebra

coefficientCanonicalFrame def

A frame of the top cotangent module on every good affine open. · OAI.SourceSymmetry.coefficientCanonicalFrame

coefficientCanonicalGenerator def

A generator of the top differential module. · OAI.SourceSymmetry.coefficientCanonicalGenerator

coefficientCanonicalIso def

The sheaf/module comparison on the twelve smooth coefficient charts. · OAI.SourceSymmetry.coefficientCanonicalIso

coefficientCanonicalUnitIso def

The canonical line is intrinsically trivial on each coefficient chart by the explicitly computed top differential. · OAI.SourceSymmetry.coefficientCanonicalUnitIso

coefficientConvolution def

Frequency convolution, with counting measure on the frequency space. · OAI.MaxCutGames.Fourier.MatrixProducts.coefficientConvolution

coefficientConvolution def

Frequency convolution, with counting measure on the frequency space. · OAI.UniqueGamesTheorem.Fourier.MatrixProducts.coefficientConvolution

coefficientCotangentIso def

The intrinsic source cotangent sheaf on each good chart is the Kähler differential module already used in the canonical-frame calculation. · OAI.SourceSymmetry.coefficientCotangentIso

coefficientCurve def

A continuous coefficient jet curve induces a continuous curve of multiplication operators. · OAI.ForcedComputation.PlaneScalarMild.coefficientCurve

coefficientDerivation def

Coefficientwise extension of a derivation, fixing the Laurent parameter. · OAI.StableCoordinate.FilteredRees.coefficientDerivation

coefficientEnlargement def

Finite coefficient enlargement of a map, as in KR02, equation (7.10). · OAI.WeakPureInfiniteness.coefficientEnlargement

coefficientEquation def

A coefficient-linear equation whose entries are polynomial in the point coordinates. · OAI.Nagata.W27.coefficientEquation

coefficientEquiv def

The actual coefficient field is identified with the rational function field using exactly the given algebraic independence of the real parameters. · OAI.EuclideanRamsey.NineField.coefficientEquiv

coefficientEquiv def

Actual matrix-coefficient equivalence, rather than just a count model. · OAI.MaxCutGames.Reduction.ActualEnumeration.coefficientEquiv

coefficientEquiv def

Injectivity and checked delta coefficient values produce a linear equivalence. · OAI.Nagata.Workers.W10.coefficientEquiv

coefficientEquiv def

Actual matrix-coefficient equivalence, rather than just a count model. · OAI.UniqueGamesTheorem.Reduction.ActualEnumeration.coefficientEquiv

coefficientFamily def

All group coordinates of a coefficient form a uniformly bounded family, the precise source space of the scalar mean. · OAI.KadisonKastler.UniformMeasurableOperators.coefficientFamily

coefficientFamily def

The intrinsic scalar coefficient family. · OAI.KadisonKastler.WeakOperatorField.coefficientFamily

coefficientFamilyBoundary def

Boundary of a family after specified coefficient embeddings. · OAI.CoarseAssembly.DiscreteRoe.coefficientFamilyBoundary

CoefficientField abbrev

Exactly F₂[Z]/(g), with g the lexicographically first monic irreducible polynomial of the prescribed degree. · OAI.KaplanskyCounterexample.Prescription.CoefficientField

coefficientHom def

Every literal coefficient is an additive map. · OAI.Thomason.Construction.PastingGlobeCylinderBasis.coefficientHom

coefficientIndex def

Reindexing the genuine lifted-cell basis by monodromy gives the actual integral basis on G×(base2-cells). · OAI.EilenbergGanea.CWPresentation.coefficientIndex

coefficientIota def

The coefficient charts are open immersions into the source Proj. · OAI.SourceSymmetry.coefficientIota

coefficientJetMap def

The manuscript's actual finite row shape, with entries obtained by complex differentiation of the local basis functions. · OAI.Nagata.W20.coefficientJetMap

coefficientLinear def

Coefficientwise application of a linear map to Laurent polynomials. · OAI.StableCoordinate.FilteredRees.coefficientLinear

coefficientLinear def

Constants followed by a specified coefficient of the derivation. · OAI.StableCoordinate.LaurentDescent.coefficientLinear

coefficientLinearMap def

The coefficient linear map defining the jet conditions at fixed points. · OAI.Nagata.W27.coefficientLinearMap

coefficientMap def

Map coefficients and retain the same cyclic basis. · OAI.CirculantHadamard.CyclicRing.coefficientMap

coefficientMap def

Every additive multiplicative coefficient map induces a Steinberg map; it is not required to preserve one. · OAI.FiniteAlgebraicEnvelopes.Steinberg.coefficientMap

coefficientMatrix def

An elementary tensor in coordinates of the fixed second factor. · OAI.NuclearUltrapower.coefficientMatrix

coefficientMatrixEquiv def

Entrywise transport along a star-algebra equivalence. · OAI.KadisonKastler.coefficientMatrixEquiv

coefficientMatrixMap def

Entrywise application of a unital star homomorphism to square matrices. · OAI.KadisonKastler.coefficientMatrixMap

coefficientMeasure def

The weighted coefficient measure in the constant-ratio preparation. · OAI.ThreeTorus.coefficientMeasure

coefficientOperator def

Pointwise action of a continuous coefficient on a continuous path. · OAI.ForcedComputation.WeaklySingular.coefficientOperator

coefficientOperator def

The continuous operator associated to the finite coefficient vector. · OAI.PolycyclicRecognition.ComplexCartan.coefficientOperator

coefficientOverlap def

The common principal open inside the same fixed function field. · OAI.SourceSymmetry.coefficientOverlap

coefficientOverlapIota def

The common open inclusion into the projective surface. · OAI.SourceSymmetry.coefficientOverlapIota

coefficientOverlapIso def

The scheme overlap. This isomorphism is · OAI.SourceSymmetry.coefficientOverlapIso

CoefficientParameter abbrev

A coefficient-only representation of each fixed linear-map parameter. · OAI.MaxCutGames.Reduction.FixedOutcomes.CoefficientParameter

CoefficientParameter abbrev

A coefficient-only representation of each fixed linear-map parameter. · OAI.UniqueGamesTheorem.Reduction.FixedOutcomes.CoefficientParameter

coefficientPolynomial def

The polynomial with the first n entries of a coefficient function. · OAI.CirculantHadamard.DyadicRemainder.coefficientPolynomial

coefficientPolynomial def

Finite coefficient form, suitable for the source's homogenization rule. · OAI.LittleFinitistic.QuadraticPresentation.coefficientPolynomial

coefficientProductEquiv def

Splitting a raw coefficient tuple into its actual independent coordinates. · OAI.MaxCutGames.Clean.CoefficientSampling.coefficientProductEquiv

coefficientProductEquiv def

Splitting a raw coefficient tuple into its actual independent coordinates. · OAI.UniqueGamesTheorem.Clean.CoefficientSampling.coefficientProductEquiv

coefficientRepresentative def

An actual coefficient vector in the fiber of a list function, or zero when this fiber is empty. No inverse of the column-combination map is used. · OAI.MetricEntropyDuality.coefficientRepresentative

Coefficients abbrev

An l1-of-Euclidean-block ball gives a compact convex parameter space for all finite atomic combinations. No Caratheodory theorem is assumed. · OAI.DiamondDistortion.SegmentSystem.Coefficients

coefficients def

The coefficient matrix for h ↦ U h Q (column indices on Hom). · OAI.IntegralCharacterVarieties.HomTransport.coefficients

coefficients def

An actual finite coefficient list for every fixed natural polynomial. No evaluation of an input is included in this static representation choice. · OAI.MaxCutGames.Foundations.Complexity.CookLevin.PolynomialMachine.coefficients

coefficients def

The coefficient form of the conditional projection operator. · OAI.MaxCutGames.Repetition.ProjectionKernel.coefficients

coefficients def

Coefficients of the basis -I_i J_j, whose squared norms are four. · OAI.PositiveEinsteinFour.ScalarHessian.coefficients

coefficients def

An actual finite coefficient list for every fixed natural polynomial. No evaluation of an input is included in this static representation choice. · OAI.UniqueGamesTheorem.Foundations.Complexity.CookLevin.PolynomialMachine.coefficients

coefficients def

The coefficient form of the conditional projection operator. · OAI.UniqueGamesTheorem.Repetition.ProjectionKernel.coefficients

coefficientSet def

All nonnegative earlier-candidate coefficients with parent-group mass at most one. · OAI.ReflexiveFixedPoints.Prediction.coefficientSet

CoefficientShape inductive

An arbitrary occupied ordered subtree, with motif values at its actual child labels. Depth zero here means a bottom sibling group, so it already has one internal level. No binary-shape assumption is made. · OAI.TraceReconstruction.CoefficientShape

CoefficientSimplex def

Nonnegative coefficients with total mass one. · OAI.Tingley.CoefficientSimplex

coefficientSpace def

Coefficient block V_j of the actual polynomial space W_tau. Parameters d,m,j are integers, so tensor exponents remain integer powers as in the source. · OAI.Nagata.CoefficientSpaces.coefficientSpace

coefficientSpan def

The closed coefficient span Z. · OAI.SeparableQuotient.PrefixNormers.coefficientSpan

coefficientSubring def

Ring coefficients whose root elements belong to a subgroup form a subring. The rank-six third index is used in the multiplication step. · OAI.FiniteAlgebraicEnvelopes.Steinberg.coefficientSubring

coefficientSum def

Total coefficient mass belonging to one parent. · OAI.ReflexiveFixedPoints.Prediction.coefficientSum

CoefficientSupport def

The antichain model really requires finite support, and the global model really requires square summability. Neither condition is discarded. · OAI.BoundedTreePotentials.TreeCalculus.CoefficientSupport

coefficientTensorEquiv def

The tensor equivalence transports both copies of the actual field. · OAI.EuclideanRamsey.NineField.coefficientTensorEquiv

coefficientTopInclusion def

Restriction of top forms along an inclusion of coefficient rings. · OAI.SourceSymmetry.coefficientTopInclusion

coefficientTriples def

The finite coefficient table's three coefficients for each local record. · OAI.MaxCutGames.Reduction.CanonicalBodyTemplate.coefficientTriples

coefficientTriples def

The finite coefficient table's three coefficients for each local record. · OAI.UniqueGamesTheorem.Reduction.CanonicalBodyTemplate.coefficientTriples

coefficientWordMap def

Word evaluation with a fixed alphabet of coefficient letters and a finite alphabet of unknowns. Coefficients may vary smoothly with a real parameter. · OAI.EilenbergGanea.coefficientWordMap

coeffP def

Coefficient polynomials for the exact cyclic-diagonal Fourier transform. · OAI.RationalHitting.ThreeLift.coeffP

Cof def

Closed finite-codimensional subspaces, with the two conditions separately retained, not replaced by a coordinate-null convention. · OAI.DiamondDistortion.Midpoint.Cof

cofactorHessian def

The coefficient U^{ij} = det(D²u) (D²u)^{-1}_{ij}. · OAI.AffineBernstein.cofactorHessian

Cofibration structure

The specialized cofibration conditions for this span orientation. · OAI.Thomason.Construction.SimplicialSpan.Cofibration

Cofinite def

Closed linear subspaces of finite codimension. · OAI.ExactModuli.Cofinite

Cofinite def

The closed finite-codimensional subspaces. · OAI.ForestSpace.Cofinite

CofiniteSubspaces def

Closed subspaces of finite codimension, with respect to the inherited topology. · OAI.IndependentProducts.CofiniteSubspaces

Coh abbrev

The abelian category of coherent structure-sheaf modules. · OAI.CoherentGlobal.Coh

cohClosedPushforwardFullyFaithful def

Fully faithful exact closed pushforward on coherent sheaves. · OAI.CoherentGlobal.cohClosedPushforwardFullyFaithful

Coherent inductive

Syntactic coherence generated exclusively by commuting two different coordinate splits. No semantic equality is admitted as a generating relation. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Tree.Coherent

CoherentBases structure

Fiber bases compatible across every vertex corner. For a Diagram this property is derived from bases on its boundary circles. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.LocalRanks.CoherentBases

coherentCartierKernel def

The coherent first Tor object computed by the Cartier line resolution, viewed on the ambient scheme. · OAI.ActualSheafTensor.coherentCartierKernel

CoherentCore structure

Pointwise bounded rational sesquilinearity, including the two tests with multiplication by i that force complex sesquilinearity by continuity. · OAI.KadisonKastler.ScalarOperatorKernel.CoherentCore

coherentFree def

The finite free coherent module sheaf. · OAI.ActualCartier.coherentFree

coherentIdealAction def

The resolution morphism for an Cartier line ideal, inside the category of coherent sheaves. The source is the tensor twist. · OAI.ActualSheafTensor.coherentIdealAction

coherentTensorLine def

The Cartier line twist on the abelian category of coherent sheaves. · OAI.ActualSheafTensor.coherentTensorLine

cohGamma def

Sections of a coherent sheaf on an affine Noetherian scheme form a finite module. · OAI.CoherentGlobal.cohGamma

cohInclusion abbrev

Inclusion of the coherent category into all structure-sheaf modules. · OAI.CoherentGlobal.cohInclusion

cohomologicalDimension def

Projective dimension of the trivial integral left group-ring module. · OAI.EilenbergGanea.cohomologicalDimension

cohomologicalDimension def

Integral, left-module cohomological dimension; no rational replacement. · OAI.EilenbergGanea.cohomologicalDimension

cohomology def

The entire cohomology B-module is the manuscript's M(H(Y)), in both degrees. · OAI.LittleFinitistic.SelectedRealization.cohomology

cohPushforward def

Finite pushforward as a functor of coherent categories. · OAI.CoherentGlobal.cohPushforward

cohTilde def

The associated sheaf functor on finite modules. · OAI.CoherentGlobal.cohTilde

coincides def

Coordinate coincidence in the normal pairing. · OAI.Problem335.MomentPairing.coincides

coinProduct def

Separate coins from all other finite independent local data, including when the local data type depends on the cell/group. · OAI.PlanarL1.FiniteRandom.coinProduct

Coins def

One coin for each pair, indexed by the unchanged tail of a binary position. The one-card case is the deterministic extension of the physical definition. · OAI.CubeShuffle.Coins

coinsEasy def

meter values · OAI.Fifths.Tour.coinsEasy

coinStepEquiv def

Exact independent split of a butterfly's last layer and its two children. · OAI.CubeShuffle.coinStepEquiv

CoinTape abbrev

The formalization proves $\mathsf L=\mathsf{RL}=\mathsf{BPL}$ for binary languages in an explicit Turing-machine model with logarithmic workspace. The randomized classes use polynomial time: $\mathsf{RL}$ never accepts a no-instance and … · OAI.ExactDerandomization.CoinTape

cokernelBiprodIso def

The quotient of a direct sum is the direct sum of the actual quotients. · OAI.Kaplansky.cokernelBiprodIso

ColChosen def

A selected column after the independent column reindexing. · OAI.Problem348.Construction.HostSampling.ColChosen

coldDilationError def

S bounds the rescaled jet size, and U bounds rescaled time. · OAI.BoltzmannNonuniqueness.coldDilationError

colimitData def

A countable composite of stationary deformation inclusions is again one. · OAI.Thomason.Construction.TopologicalDeformation.Sequential.colimitData

colIncident def

Incidence with either column role of a mode. · OAI.Problem348.ModeIncidence.colIncident

colKey def

Coordinates for the column order: anchors, dummy, variables. · OAI.Problem348.AnchorFrames.colKey

collapse def

Evaluation of D ⊗_k (A ⊗_k W) on the two leading coefficients. · OAI.DerivedBar.Balanced.collapse

collapse def

Collapse the first two slots using the right D-action on X. · OAI.DerivedBar.BimoduleTensor.collapse

collapse def

Literal radial-path collapse of the actual cone one-skeleton. It fixes all letter edges, and does not identify two indexed radial positions. · OAI.HyperbolicObstruction.ConeSkeleton.collapse

collapse def

Collapse by genuine external well-founded recursion, not recursion over an arbitrary first-order model's internal membership relation. · OAI.PartitionConsistency.Mostowski.collapse

collapse def

The current maps act once, after the new child averages. · OAI.ReflexiveFixedPoints.FrontierExtension.collapse

collapse def

One genuine level of the nonlinear recursive evaluation. · OAI.ReflexiveFixedPoints.TreeEvaluation.collapse

collapse def

Collapse the universal m-cell to its chosen n-face, retaining all its lower faces by the universal property of the actual globe. · OAI.Thomason.Construction.StrictOmegaCategory.Globe.collapse

collapseTo def

Collapse a higher globe to the universal cell of a lower globe. · OAI.Thomason.Construction.StrictOmegaCategory.PastingGlobe.collapseTo

collapseVertex def

The cone on the greatest vertex collapses back to that vertex. · OAI.Thomason.Construction.ConePrism.collapseVertex

collarFace def

Two simplices in each mapping-cylinder rectangle and one cap simplex. · OAI.HyperbolicObstruction.FinitePolygonModel.Data.collarFace

collarHomotopy def

The radial collar is a genuine strong deformation neighborhood of the lower-dimensional CW skeleton. · OAI.EilenbergGanea.CWCollar.Attachment.collarHomotopy

collect def

The collected name is packaged with the proved hereditary symmetry. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.collect

collectedHessianForm def

The first-derivative form obtained after all four Green calculations. · OAI.PositiveEinsteinFour.HighestHessian.collectedHessianForm

collection def

The finite ordered collection is the actual sorted base decomposition, not an arbitrary family substituted for its multiplicities. · OAI.Superstring.Hierarchical.Threading.collection

collectiveMatrix def

Concatenation of all projected maps, whose range is their collective image. · OAI.Kaplansky.collectiveMatrix

collectivePlan def

A collective top block is valid INCLUDING k=1. Its ordinary link, free period internal alternative, and maximal-rank short-contact realization are all materialized, with every constituent base turn included once. · OAI.Superstring.Hierarchical.LayerCollection.collectivePlan

collectRaw def

A small family of HS names with any predicate on its entry conditions. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.collectRaw

Collision def

Collision of equal-length nonbacktracking paths, before any cancellation. · OAI.TorsionFreeZeroDivisors.ShortClosure.Collision

collisionCenters def

Distinct real points in the complex disk, scaled to zero. · OAI.Nagata.W20.collisionCenters

collisionCutoff def

The common relative-speed cutoff used for both pre- and post-collisional variables. · OAI.BoltzmannNonuniqueness.collisionCutoff

collisionDefect def

The three nonnegative terms in the collision stability estimate. · OAI.Ostmann.collisionDefect

collisionEnergy def

The sum of squared residue-class sizes. · OAI.LowSpaceSubsetSum.collisionEnergy

collisionExceptional def

A countable explicit null superset of all trajectories at a boundary at a prescribed time; it is defined without making trajectory choices. · OAI.TriangularBilliards.collisionExceptional

collisionFactor def

C=(x-y)(x-v)(u-y)(u-v), with coordinates x,u,y,v. · OAI.Paper170.collisionFactor

collisionGenerators def

Exact set of the exponent vectors of the standard generators of (X^q,Y)^m. · OAI.Nagata.Workers.W30.collisionGenerators

collisionIdeal def

(X^q,Y)^m in the genuine bivariate polynomial ring. · OAI.Nagata.Workers.W30.collisionIdeal

collisionKernel def

The contribution of a suffix at distance h from the end, after restoring its common prefix and integrating the dyadic suffix estimates. · OAI.TotientAsymptotic.collisionKernel

collisionPairs def

Ordered collisions include diagonal pairs. · OAI.LowSpaceSubsetSum.collisionPairs

collisionPairs def

Ordered pairs with equal images, including the diagonal. · OAI.Problem337.collisionPairs

collisionValue def

Maximum squared collision norm of an ordinary deterministic left strategy. · OAI.MaxCutGames.Repetition.ProjectionKernel.collisionValue

collisionValue def

Maximum squared collision norm of an ordinary deterministic left strategy. · OAI.UniqueGamesTheorem.Repetition.ProjectionKernel.collisionValue

color def

Permanent color of a chronological record; later stages never modify it. · OAI.KServer.OnlineLabels.color

color def

The index-seven coloring of triangular-lattice sites. · OAI.Problem160.SevenLattice.color

ColorBalanced def

Balance of the three non-outside color classes. · OAI.LowSpaceSubsetSum.ColorBalanced

colorBases def

Bases on named secondary labels satisfy corner coherence at every allowed vertex. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.LocalRanks.colorBases

ColorBlock structure

One full period, recording its integral horizontal and vertical origins. · OAI.DoublingHilbert.ColorBlock

ColorChain def

Lists must match in their order, including repetitions. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VariableGallery.ColorChain

ColorCompatible def

Entire ordered lists, not merely multisets of ranks, agree at opposite ends. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.PortWiring.ColorCompatible

colOrder abbrev

The full raw column order, suitable for local instance installation. · OAI.Problem348.AnchorFrames.colOrder

colored def

Local decorations with proved complete seam matching produce an actual allowed occurrence assembly, not only rank/port endpoint lists. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.PortWiring.colored

coloredAt def

The literal ordered colored extension calculus. Each color either uses one unused old leaf, which is then removed from the available list, or grows one fresh branch with its original signed gamma coefficient. · OAI.DilutedSpinGlass.PrescribedTree.coloredAt

ColoredPair structure

A pair retaining real passage geometry, not merely a density transition. · OAI.ThreeTorus.ColoredPair

ColoredPath inductive

A finite trace contains actual neutral--signal pairs, including their smooth geometry and exact intermediate speeds. · OAI.ThreeTorus.ColoredPath

coloredPositions def

Actual positions carrying a label of the specified color. · OAI.QuantitativeVanDerWaerden.RecordedSupport.coloredPositions

colorFiberEquiv def

Erasing the choice of a distinguished color in a fiber of incidenceColor leaves precisely a triple containing that color. · OAI.PixtonChow.colorFiberEquiv

colorFin def

The colors can be used directly as the Fin 7 target of a coloring. · OAI.Problem160.SevenLattice.colorFin

colorGauge def

Row and column contacts have separate named gauges, including when their facet labels coincide. · OAI.IntegralCharacterVarieties.NamedBandGrades.IdentifiedBand.colorGauge

colorIndicator def

Color indicator of one actual labeling. · OAI.MaxCutGames.Fourier.MatrixEnergy.colorIndicator

colorIndicator def

Color indicator of one actual labeling. · OAI.UniqueGamesTheorem.Fourier.MatrixEnergy.colorIndicator

Coloring abbrev

A coloring of the selected vertices constant along every retained edge of an oriented permutation graph. Off the selected set its value is fixed. · OAI.CubeShuffle.CycleColoring.Coloring

coloringLaw def

The independent product law on exactly the original vertex set. · OAI.TalagrandThreshold.coloringLaw

coloringOfPermutation def

Fix one block labeling; a uniformly permuted labeling has the exact uniform law on all labelings with these block sizes. · OAI.LowSpaceSubsetSum.coloringOfPermutation

colorInstruction def

The one-colour stack instruction attached to a typed inventory symbol. · OAI.FKCRT.Inventory.colorInstruction

colorMark def

Factor assigned to one color from a pair requirement. Pair endpoints are assumed distinct colors, as stipulated in the canceled-anchor expansion. · OAI.DilutedSpinGlass.MarkSelector.colorMark

colorMass def

Probability mass of a complete coloring, with one factor per vertex. · OAI.TalagrandThreshold.colorMass

ColorMatch def

Literal ordered color matching for a whole seam. Repeated names retain separate coordinates; child reindexing is forced by the seam's arity equality. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.PortWiring.ColorMatch

colorProb def

The literal single-coordinate probabilities from the selector experiment. · OAI.TalagrandThreshold.colorProb

ColorSlot abbrev

Slots indexed by a triple together with one of its colors. · OAI.PixtonChow.ColorSlot

ColorStabilizer abbrev

A permutation of slots preserving their colors. · OAI.PixtonChow.ColorStabilizer

colorWord def

The concatenation of the increasing triples, with colors numbered 0,...,g-1 instead of 1,...,g. · OAI.PixtonChow.colorWord

colourLeaves def

Leaf transport of a coordinate relabelling, with the same binary shape. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Tree.colourLeaves

Column abbrev

The induced column in Section 4, with its external actions. · OAI.ExternalTensor.Column

Column def

The actual column type supplied to interpolation is finite by construction. · OAI.Nagata.FiniteExponents.Column

column def

Actual derivative columns of global extensions. · OAI.DoublingHilbert.column

column def

The exact column vector x_v, indexed by the chosen holes. · OAI.KaplanskyCounterexample.PolynomialPairing.column

column def

The column at right degree p, including the standard total-complex sign (-1)^p. · OAI.LittleFinitistic.ComplexTensor.column

column def

The actual matrix, in the column-first convention of compression. · OAI.MetricEntropyDuality.FiniteFieldRows.column

column def

The graph-distance column function. · OAI.MetricEntropyDuality.GraphProfile.column

column def

A column of a bilinear exponential sum with its actual hyperbolic support. · OAI.Problem337.HyperbolaCorrelation.column

column def

This declaration is an exact comparator-selected statement. Read the Lean statement below for its mathematical content. · OAI.SpinAngle.TypeAngle.column

column def

The individual vertical column, including both its generating cells. · OAI.Thomason.Construction.StrictOmegaCategory.GridGlobe.column

columnAction def

Application of an operator-valued column to a single vector. · OAI.KadisonSimilarity.columnAction

columnAlt def

Actual column antisymmetrizer on the word-coordinate module. · OAI.Saxl.columnAlt

columnBasisMatrix def

The columns chosen from a finite list of vectors. · OAI.PolycyclicRecognition.columnBasisMatrix

ColumnBound def

Uniform column-amplification bound with the rectangular C-star matrix norm. · OAI.KadisonSimilarity.ColumnBound

columnChunkCuts def

An actual column is a sequence of perfect blocks and individual imperfect entries. · OAI.TwoPointCorrelations.columnChunkCuts

columnChunkPieces def

Apply the omitted-label cut to every actual compressed perfect block. · OAI.TwoPointCorrelations.columnChunkPieces

ColumnCode abbrev

Fixed-size slots can be padded once the nonempty index universe is fixed. · OAI.TwoPointCorrelations.ColumnCode

columnColumn def

A column, realized inside the algebra by orthogonal isometries. · OAI.BoundedHochschild.KadisonRingrose.columnColumn

columnComplexEquiv def

Explicit equivalence between the genuine integer columns and interpolation's complex finite set. Its map is the ordinary ℤ²→ℂ² inclusion. · OAI.Nagata.Workers.W12.columnComplexEquiv

columnCompression def

The matrix of an element with respect to unital isometry columns. · OAI.BoundedHochschild.KadisonRingrose.columnCompression

ColumnConnection def

A localized finite path between fixed-target lattice vertices. · OAI.GammaFPP.CrossingBridge.ColumnConnection

columnContraction def

Contract the vertical wires within a literal column and all right-side input wires; the left-side output and upper auxiliary are not summed. · OAI.CriticalHoneycomb.Transfer.PlanarNormalization.columnContraction

columnCoordinates def

Coordinates in an invertible selected column basis. The definition also exists for singular selections, which simplifies the finite uniform bound. · OAI.PolycyclicRecognition.columnCoordinates

columnEntry def

The coordinates of a bounded column operator. · OAI.KadisonSimilarity.columnEntry

columnEntry def

Column entry times in ribbon-width units: straight length 200m, then a semicircle of radius 50. The connector contributes A. · OAI.SLEExactGauge.Serpentine.columnEntry

columnFamily def

Each actual tracial matrix stage supplies its finite corner frame. · OAI.KadisonSimilarity.NestedTracialMatrices.columnFamily

columnFiberEquiv def

The finite coefficient-index family is equivalent to the exact literal columns. · OAI.Nagata.Workers.W10.columnFiberEquiv

columnFinalState def

The previous label after reading a finite part of a column. · OAI.TwoPointCorrelations.columnFinalState

columnFixedEquiv def

Identifies the original enveloping vertex action with its corner projector. · OAI.LittleFinitistic.BimoduleVertex.columnFixedEquiv

columnFlags def

Run boundaries reset at imperfect positions. · OAI.TwoPointCorrelations.columnFlags

columnFunctor def

Induction by the actual left ideal Be, on right modules and maps. · OAI.ExternalTensor.Section4.columnFunctor

columnGroup def

The column permutations of a tableau. · OAI.Saxl.columnGroup

columnHoles def

All holes in one junction column, with the inherited tail trajectories. · OAI.CoordinateSweeps.Grid.Holes.columnHoles

columnInclusion def

The right-linear insertion used in the vertex maps iota_i. · OAI.ExternalTensor.columnInclusion

columnInclusion def

The original first induced column includes into the actual zero column. · OAI.LittleFinitistic.FirstVertex.columnInclusion

columnIndex def

The increasing column enumeration, independent of the row enumeration. · OAI.Problem348.OrderedHost.columnIndex

columnLines def

Column line chart: the fixed head label and the other tail coordinates. · OAI.CoordinateSweeps.Grid.columnLines

columnMap def

The differential or relation-homotopy component c tensor n ↦ c tensor f(n). · OAI.ExternalTensor.columnMap

columnMap def

Input chain maps act on the signed columns by the actual shifted tensor functor. · OAI.LittleFinitistic.ComplexTensor.columnMap

columnMap def

Every fixed column is an isometric copy of its allowed height interval. · OAI.PlanarL1.Columns.Model.Point.columnMap

columnMatrixEquiv def

The compression/reconstruction identification preserves the C*-norm. · OAI.BoundedHochschild.KadisonRingrose.columnMatrixEquiv

columnOperator def

The column operator on any genuine representation of the same group. · OAI.Saxl.columnOperator

columnOrbitLinearMap def

The corresponding orbit map of the finite family. · OAI.KadisonSimilarity.columnOrbitLinearMap

columnOrder abbrev

Columns: anchors first, then dummy, then the plus/subtree/minus tree order. · OAI.Problem348.OrderedHost.columnOrder

columnPath def

The path-multiplication component c tensor n ↦ ca tensor n of the rectification maps. · OAI.ExternalTensor.columnPath

columnPattern def

A finite signed column pattern. · OAI.InfiniteMatroidCounterexample.columnPattern

columnPoint def

The closed width-one band over one straight pass, in physical scale. · OAI.SLEExactGauge.Serpentine.columnPoint

columnReconstruction def

Reconstruction from the actual matrix entries. · OAI.BoundedHochschild.KadisonRingrose.columnReconstruction

columnRightIso def

The restricted action is the right action already verified for the actual tensor. · OAI.ExternalTensor.Section4.columnRightIso

columnRow def

A row, realized inside the algebra by orthogonal isometries. · OAI.BoundedHochschild.KadisonRingrose.columnRow

columnRunHeads def

Only the first position of each perfect constant run consumes a label. · OAI.TwoPointCorrelations.columnRunHeads

Columns abbrev

Actual column indices, including the choice of partition graph. · OAI.MetricEntropyDuality.GraphProfile.Columns

ColumnSample abbrev

A sampled column sequence with a sign attached to each occurrence. · OAI.JointDickman.ColumnSample

columnSpan def

Column specifications fix the matrix on this entire subspace. · OAI.MaxCutGames.Inverse.Shortcode.Slice.columnSpan

columnSpan def

Column specifications fix the matrix on this entire subspace. · OAI.UniqueGamesTheorem.Inverse.Shortcode.Slice.columnSpan

ColumnSparse def

Variable-column sparseness on a selected prefix. · OAI.Problem348.HostAnchorRigidity.ColumnSparse

columnSpinEquiv def

Changing from vacant/arrow spins to the manuscript's +,0,- ordering. · OAI.CriticalHoneycomb.Transfer.RasterGrid.columnSpinEquiv

columnToComplex def

Inclusion of a genuine exponent column into the coordinatewise complex image. · OAI.Nagata.Workers.W12.columnToComplex

comap def

Pullback along an arbitrary group homomorphism, retaining the Hilbert space. · OAI.NuclearUltrapower.FiniteUnitaryRep.comap

comap def

Pulling a retained submonoid back to an original stage also preserves all the conservation requirements. · OAI.PartitionConsistency.ConservativeMonoids.Subobject.comap

comap def

Change of coordinate columns acts on the full signed measures. · OAI.RegularPointCoordinates.PositiveBilinearMeasure.comap

comapEquiv def

A change of the concrete model for the source changes neither the represented quotient nor its trace. · OAI.KadisonKastler.TraceQuotientMap.comapEquiv

comapEquiv def

Transport the trace-pairing characterization to the inverse image of the same concrete subalgebra. Only the actual trace identity is required. · OAI.KadisonSimilarity.TracialExpectation.comapEquiv

comapLieEquiv def

The same adjoint matrices in the pulled-back basis. · OAI.PolycyclicRecognition.TriangularAdjoint.comapLieEquiv

comapStarAlgEquiv def

Pull a faithful tracial state back along a star-algebra equivalence. · OAI.KadisonKastler.FaithfulTracialState.comapStarAlgEquiv

combFold def

Tensor a nonempty string of n+1 labels in the right comb. · OAI.FiniteAlgebraicEnvelopes.BinaryGroupCategory.combFold

combHead def

The first adjacent swap on a right-associated comb with at least three leaves. · OAI.FiniteAlgebraicEnvelopes.BinaryGroupCategory.combHead

Combination inductive

A balanced tree of linear combinations avoids a long flattened list during kernel reduction. Each leaf refers to one independently justified equation. · OAI.MUB6.Certificate.Combination

combine def

Coordinatewise nonnegative linear combination of weight vectors. · OAI.CartierSections.Monomial.combine

combine def

Finite represented vectors retain their segment families as data. · OAI.DiamondDistortion.SegmentSystem.combine

combine def

Assemble a proposed linear-combination witness. · OAI.MUB6.Certificate.Sparse.combine

combPerm def

Conjugate an ordinal permutation to the exact leaves of the comb. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Tree.combPerm

combPermutation def

The literal symmetric-group action on every nonempty right comb. · OAI.FiniteAlgebraicEnvelopes.BinaryGroupCategory.combPermutation

combSwap def

n adjacent generators acting on the n+1 leaf right comb. · OAI.FiniteAlgebraicEnvelopes.BinaryGroupCategory.combSwap

Command inductive

All varying literals are finite: one Boolean or one bounded offset. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Emitter.Command

Command inductive

A structured program still has only finitely many syntactic labels. loop r c tests/decrements its own unary counter, then runs c. It is a genuine while loop: no claim of termination is built into the syntax. · OAI.OptimalMaxCut.CounterMachine.Command

Command inductive

All varying literals are finite: one Boolean or one bounded offset. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Emitter.Command

Command inductive

Finite structured source code. Tests and registers have fixed finite range; all unbounded data remain in the explicitly named bit stacks. · OAI.WLIdentification.StackModel.Command

commands def

The complete fixed-alphabet row schedule: two endpoints followed by every forward-table entry in increasing numeric label order. · OAI.MaxCutGames.Explicit.ProductMachineSemantics.commands

commands def

The complete fixed-alphabet row schedule: two endpoints followed by every forward-table entry in increasing numeric label order. · OAI.UniqueGamesTheorem.Explicit.ProductMachineSemantics.commands

commandStatement def

A literal Boolean is emitted by a real finite branch. A bounded offset is dispatched to a finite label, whose push chain has a fixed length. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Emitter.commandStatement

commandStatement def

A literal Boolean is emitted by a real finite branch. A bounded offset is dispatched to a finite label, whose push chain has a fixed length. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Emitter.commandStatement

commGroup def

The group multiplication is the original fundamental-group multiplication. · OAI.HyperbolicObstruction.HSpaceFundamental.commGroup

Common def

Two vertices have a common successor. Directedness will identify this with connectedness in the undirected contraction graph. · OAI.PartitionConsistency.LocalDiagramSystem.Vertex.Common

Common def

A chart identifier is common on the actual unrestricted closed ball. The test includes every representative, not merely the original vertices. · OAI.PlanarL1.Columns.Model.Ambient.Common

common def

Source-intersection table, retaining the left-hand transported charts. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Table.common

commonCarrierHomotopyRel def

A homotopy of paths with exactly fixed ends and stationary side. All interpolation occurs in the given common carrier. · OAI.Thomason.Construction.PosetGeometry.commonCarrierHomotopyRel

CommonCenterSeparableStability def

Strong stability for algebras with a common center, uniform over separable Hilbert-space representations. · OAI.KadisonKastler.CommonCenterSeparableStability

commonCentralField def

All operators commuting with the center acquire fields in the single common spectral model. · OAI.KadisonSimilarity.commonCentralField

commonChainHomotopy def

Two maps with pointwise compatible positive supports are homotopic by straight-line interpolation in the proved barycentric image. · OAI.Thomason.Construction.PosetGeometry.commonChainHomotopy

commonChainHomotopyRel def

The interpolation fixes the full set where the endpoint maps agree. · OAI.Thomason.Construction.PosetGeometry.commonChainHomotopyRel

commonCountIndex def

An injective enumeration of m common integer counts; no asymptotic cardinality argument is left implicit. · OAI.TraceReconstruction.commonCountIndex

commonDegree def

Common degree of the first coefficient blocks. · OAI.Nagata.FiniteExponents.commonDegree

commonFieldCompression def

The original operator obtained by compressing the reconstructed field. · OAI.KadisonSimilarity.commonFieldCompression

commonFunction def

The common scalar function is read from the uniquely determined zeroth-order path. · OAI.ForcedComputation.PlaneScalarMild.commonFunction

CommonHeads def

Fixed across each level (stronger than predictability); required only by the deterministic leakage input. · OAI.DiamondDistortion.SignCube.Protocol.CommonHeads

CommonInterface def

One interface excludes both labels from the same straddling set and approaches its center in an actual unit direction. · OAI.EuclideanFiveColor.CommonInterface

commonPlacement def

The unary start vertex and outer counter occupy the same physical tape. · OAI.MaxCutGames.Foundations.Complexity.PoweringMachineInitialize.commonPlacement

commonPlacement def

The unary start vertex and outer counter occupy the same physical tape. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringMachineInitialize.commonPlacement

commonPositionError def

Explicit common-position error after returning to the original space. · OAI.KadisonKastler.commonPositionError

commonProjectionCentralData def

Central decomposition with all scalar/matrix norms and countable contraction tests derived from the common spectral embedding. · OAI.KadisonSimilarity.commonProjectionCentralData

commonSquareNullIdeal def

The common square nullspace is a genuine closed star ideal, including for nonunital algebras. No ideal-generation hypothesis is built into the definition. · OAI.WeakPureInfiniteness.commonSquareNullIdeal

commonTorusOpenPartialHomeomorph def

The fixed exponential disk is a genuine local homeomorphism into the torus. Its inverse is continuous on the actual open quotient image. · OAI.Nagata.Workers.W05.commonTorusOpenPartialHomeomorph

CommonTracialPositionData structure

Common-position data retaining only the conclusions used in the finite-factor application. · OAI.KadisonKastler.CommonTracialPositionData

CommonWitness structure

The same actual family member and cutoffs work for every original coloring in a fixed final-coloring/profile group. · OAI.TalagrandThreshold.Selector.CommonWitness

commutant def

The actual signed-action commutant as a linear subspace of matrices. · OAI.CoordinateSweeps.SignedTensor.commutant

commutant def

The ordinary commutant in all bounded operators, not only in a chosen factor. · OAI.Hyperinvariant.commutant

commutantData def

Record one matrix coefficient per joint-symbol contingency table. Zero coefficients in these representatives force every coefficient to be zero. · OAI.RowColumn.Signed.commutantData

CommutantProjection abbrev

The projections indexing the usual off-diagonal distance seminorm. · OAI.KadisonSimilarity.CommutantProjection

commutantReducingHom def

Restrict the original algebra to a subspace reduced by it. · OAI.KadisonKastler.commutantReducingHom

commutantTracialRepresentation def

The left commutant carries the faithful normal vacuum trace. · OAI.KadisonSimilarity.TracialCornerGeometry.commutantTracialRepresentation

commutativeContent def

A compact content constructed from the SAME ambient dimension state. · OAI.Kaplansky.CuntzDimensionState.commutativeContent

commutativeValueLinearMap def

Gelfand duality transfers the proved integral linearity to any unital commutative C*-algebra represented in a finite matrix algebra over A. · OAI.Kaplansky.CuntzDimensionState.commutativeValueLinearMap

commutator def

The commutator, with the sign convention of the manuscript. · OAI.KadisonSimilarity.commutator

commutatorAlong def

A spatial commutator along a star representation, as a bounded map. · OAI.KadisonKastler.commutatorAlong

CommutatorEstimate def

The fixed-algebra conclusion of the manuscript's uniform estimate. · OAI.KadisonSimilarity.CommutatorEstimate

commutatorFamily def

The equations for commuting with every member of a matrix family. · OAI.DukePrimeDegree.commutatorFamily

commutatorVector def

A represented commutator tested on one vector. · OAI.KadisonKastler.commutatorVector

Commutes def

Coordinate projections commute even when their norms are not uniformly bounded. · OAI.DiamondDistortion.Midpoint.Heads.Commutes

CommutesWithWeights def

A polynomial family commutes with a Laurent weight coaction if the two orders of applying their coefficient maps agree, after interchanging parameters. · OAI.StableCoordinate.ActionWeightDerivative.CommutesWithWeights

commutingEval def

Evaluation on one Hilbert space; used only when the two ranges commute. · OAI.NuclearUltrapower.commutingEval

commutingOperatorEval def

Evaluate commuting operators by passing through their commutative generated algebra. · OAI.PolycyclicRecognition.commutingOperatorEval

commutingProjectionRestriction def

Multiplication by a commuting projection is a genuine nonunital star hom. · OAI.WeakPureInfiniteness.commutingProjectionRestriction

commutingWord def

The commutation relator for the two different halves. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Generators.commutingWord

comp def

Exact composition: all previous values can be reused without being consumed. · OAI.ExactFourier.LinearDAG.comp

comp def

Checked composition for actual finite Boolean-stack procedures. This constructs the combined program and accounts for the intermediate-word copy; it does NOT invoke Mathlib's unproved TM2 polytime-composition declaration. · OAI.ExactQuantumFactoring.BitStackProgram.Procedure.comp

comp def

Composition gives the descendants 00 and 01 of the zero corner. · OAI.FiniteAlgebraicEnvelopes.Steinberg.SupportedPair.comp

comp def

Applying one finite average after another is again a finite average. · OAI.KadisonKastler.FiniteInnerAverage.comp

comp def

Composition is the regular existential relation, not an external composition of selected evaluations. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.BinaryRelation.comp

comp def

Composition uses disjoint scratch banks and an actual order-preserving stack transfer. · OAI.RationalHitting.StackLanguage.Procedure.comp

comp def

Substitution is again a polynomial with nonnegative coefficients. · OAI.RationalHitting.TimeBound.comp

comp def

Composition is actual composition of the underlying norm selfmaps. · OAI.ReflexiveFixedPoints.WeakSelfMap.comp

comp def

Linear coordinate changes pull back the full positive measure, including its singular part. The global extension bound pays only the operator norm. · OAI.RegularPointCoordinates.LocalHarmonicEnergy.comp

comp def

Actual composition on the mapping-family carrier. · OAI.Thomason.Construction.StrictOmegaCategory.GrayMapping.comp

comp def

Matching families can be composed componentwise. · OAI.Thomason.Construction.StrictOmegaCategory.LimitConstruction.Cell.comp

comp def

Composition of boundary diagrams in path order. · OAI.TwoWayComplementation.Diagram.comp

Compact def

No internal node is retained if both its branches are empty. · OAI.LowSpaceSubsetSum.BinaryTrie.Compact

compactCommutant def

Operators commuting with P modulo compacts, together with their adjoints. · OAI.CoarseAssembly.Analytic.compactCommutant

CompactContraction structure

Actual continuous contraction data; this contains no K-theory assertion. · OAI.CoarseAssembly.Analytic.CompactContraction

CompactFundamentalSet def

A compact fundamental set need not be disjoint or a strict domain. · OAI.GroupRingTopology.Milnor.CompactFundamentalSet

compactHeatEval def

Positive heat evaluation on compactly supported continuous functions. · OAI.RegularPointCoordinates.NoncollapsedRCD.compactHeatEval

compactificationHomologyIso def

A genuine homeomorphism of total spaces gives an isomorphism on the integral homology of their one-point compactifications. · OAI.Kaplansky.compactificationHomologyIso

compactifiedNonzeroBundle def

Complement of the actual zero section in the one-point total space. It includes infinity; it is not just the punctured total space. · OAI.Kaplansky.compactifiedNonzeroBundle

compactifiedSection def

The map of a section into the one-point compactification of its actual total space. · OAI.Kaplansky.compactifiedSection

compactifiedSectionHomotopy def

Global section independence at the level of homotopy, without a regularity or finiteness assumption on any intermediate zero set. · OAI.Kaplansky.compactifiedSectionHomotopy

compactifiedSectionNullHomotopy def

A nowhere-zero section is null-homotopic in the one-point total space. Continuity at infinity is proved from the actual continuous fiber norm. · OAI.Kaplansky.compactifiedSectionNullHomotopy

compactIndexPairing def

The index pairing on the actual locally finite K-homology group. · OAI.CoarseAssembly.Analytic.compactIndexPairing

compactLaplacianCore def

Compact Lipschitz Sobolev functions whose weak Laplacian belongs to L². · OAI.RegularPointCoordinates.compactLaplacianCore

CompactlyGeneratedLCModel structure

Only the locally compact group data used from an intermediate Lie group in the CT reduction; the smooth structure is unnecessary here. · OAI.PolycyclicRecognition.CompactlyGeneratedLCModel

compactMap def

Continuous maps from a compact source are cocompact. · OAI.CoarseAssembly.compactMap

compactMeasureTestOperator def

The actual averaging operator used in compact-family separation. · OAI.CircleDomainRigidity.compactMeasureTestOperator

compactnessSequence def

The radial potentials and coordinates form a compactness sequence; the accompanying harmonic function is chosen to be zero. · OAI.RegularPointCoordinates.SharpProjectionFamily.compactnessSequence

CompactOneManifoldClassification def

Published input: the compact connected boundaryless case of Milnor's classification of smooth one-manifolds. · OAI.CircleDomainRigidity.CompactOneManifoldClassification

compactPassageCertificate def

The three compact tests retain the full prefix, a middle interval, and a prefix ending before any two-thirds front crossing. · OAI.SLEExactGauge.PassageRetention.compactPassageCertificate

CompactPhasePartition structure

A finite smooth partition on a compact target set, with compact supports subordinate to the prescribed phase domains. · OAI.ClosedSurfaceR4.PhasePartitions.CompactPhasePartition

CompactProfileEstimates structure

Affine means, vanishing offsets, and deterministic suffix concentration for the same profiles and ordinary sampling laws. · OAI.PolycyclicRecognition.CompactProfileEstimates

compactRealTorus def

Rescale a compactly supported real frequency variable to one fundamental interval. Its endpoints vanish, so the quotient is genuinely continuous. · OAI.ThreeTorus.compactRealTorus

compactRotationAction def

The compact action on the original group, now with its invariant character. · OAI.PolycyclicRecognition.ComplexWeightFactorModel.compactRotationAction

compactSmoothFamily def

Multiplication of a jointly smooth family by a fixed compact cutoff. · OAI.DefocusingNLS.compactSmoothFamily

compactSpatialMass def

The physical measure carried by a fixed compact part of the half-plane. · OAI.SLEExactGauge.compactSpatialMass

compactSubcomplex def

The finite subcomplex generated by all carriers met by a compact subset. · OAI.Thomason.Construction.RealizationPrism.compactSubcomplex

compactTestEnergy def

A local distribution of the form energy pairing plus a constant density. · OAI.RegularPointCoordinates.compactTestEnergy

compactTestSobolev def

The lift is linear because its actual L² representative is the test. · OAI.RegularPointCoordinates.compactTestSobolev

compactTestToL2 def

Compact continuous tests as actual L² classes. · OAI.RegularPointCoordinates.compactTestToL2

compactUnitary def

A circle-valued map as a unitary in the compact continuous-function algebra. · OAI.CoarseAssembly.Analytic.compactUnitary

companionVelocity def

The unobserved outgoing velocity is fixed by momentum conservation. · OAI.BoltzmannNonuniqueness.companionVelocity

comparableStarHomeomorph def

Removing vertices incomparable with v does not alter its open star, including the topology inherited from the actual realization. · OAI.Thomason.Construction.PosetGeometry.comparableStarHomeomorph

compare def

The explicit global comparison determined by a parameter-level map. · OAI.Thomason.Construction.StrictOmegaCategory.GlobePasting.compare

compareEquiv def

This is definitionally the state already used by the equality-bit producer. · OAI.MaxCutGames.Foundations.Complexity.PoweringMasterState.compareEquiv

compareEquiv def

This is definitionally the state already used by the equality-bit producer. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringMasterState.compareEquiv

compareInTime def

Linear runtime in the extracted candidate and stored key lengths. · OAI.MaxCutGames.BinaryNameCompare.compareInTime

compareInTime def

An actual timed execution witness, with the exact stored transition count. · OAI.MaxCutGames.Foundations.Complexity.MachineCompare.compareInTime

compareInTime def

The stored flag is equality of the actual unary endpoint names. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.CompareLoop.compareInTime

compareInTime def

Linear runtime in the extracted candidate and stored key lengths. · OAI.UniqueGamesTheorem.BinaryNameCompare.compareInTime

compareInTime def

An actual timed execution witness, with the exact stored transition count. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineCompare.compareInTime

compareInTime def

The stored flag is equality of the actual unary endpoint names. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.CompareLoop.compareInTime

compareInvocations def

Actual fraction-comparison invocation count, including signed comparison. · OAI.EditApproximation.BinaryFraction.compareInvocations

compareStateEquiv def

Reassociate the actual three comparison registers into the shared state. · OAI.MaxCutGames.Foundations.Complexity.MachineUnaryEqualityBit.compareStateEquiv

compareStateEquiv def

Reassociate the actual three comparison registers into the shared state. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineUnaryEqualityBit.compareStateEquiv

compareWordsInTime def

Compare arbitrary words while restoring every tape, including work tapes. All program hypotheses specify only the statements at the seven labels. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.CompareLoop.compareWordsInTime

compareWordsInTime def

Compare arbitrary words while restoring every tape, including work tapes. All program hypotheses specify only the statements at the seven labels. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.CompareLoop.compareWordsInTime

compareWordsInTime_bounded def

A linear word-length budget, with the actual complete execution retained. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.CompareLoop.compareWordsInTime_bounded

compareWordsInTime_bounded def

A linear word-length budget, with the actual complete execution retained. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.CompareLoop.compareWordsInTime_bounded

Comparison abbrev

Two intervals and their comparison sign. Inactive comparison slots are absent. · OAI.TorsionFreeZeroDivisors.PatternCoding.Comparison

comparison def

All and only the permitted source vertex comparisons, including complement and quotient maps at a swap, and reverse refinement at a merge. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.LocalRanks.comparison

comparison def

The canonical comparison between faithful trace quotients of the same bounded-sequence algebra. · OAI.KadisonKastler.TraceQuotientMap.comparison

comparison def

Vertex extraction commutes naturally with the original balanced tensor functor. · OAI.LittleFinitistic.BalancedVertexFunctor.comparison

comparison def

All induced differential and module maps are retained by the comparison. · OAI.LittleFinitistic.BimoduleTensorVertexBridge.comparison

comparison def

All powers on the exact selected first-block derived family are the original bar-derived powers; every resolution, scalar, action and stalk bridge is discharged. · OAI.LittleFinitistic.SelectedBarPowers.comparison

comparison def

Its exact standard integer-shift identification with the genuine derived iterate. · OAI.LittleFinitistic.SelectedCountCohomology.comparison

comparison def

The previously explicit stalk-comparison premise is now discharged for the actual fixed selecting bimodule and its actual derived tensor endofunctor. · OAI.LittleFinitistic.SelectedDerivedFunctor.comparison

comparison def

Tensoring a selected stalk by the shifted fixed bimodule shifts the already verified full B-linear tensor complex. · OAI.LittleFinitistic.SelectedShiftedRealization.comparison

comparison def

Natural comparison for the full B-valued tensor functor. · OAI.LittleFinitistic.TensorArrowComparison.comparison

comparison def

From cohomology of an evaluated representative to the actual vertex of cohomology of the full B-module tensor complex. · OAI.LittleFinitistic.TensorCohomologyDiagram.comparison

comparison def

The canonical sheafification comparison for modules and any continuous cocontinuous site functor, including nontrivial scalar transport. · OAI.ModuleSheafification.comparison

comparison def

Concrete scalar comparing two frames over the same base. · OAI.StableCoordinate.FrameLift.comparison

comparison abbrev

The map between the two pushouts induced by a map of attaching targets. · OAI.Thomason.Construction.CobaseChangeCellularClosure.comparison

comparison def

The canonical continuous comparison from the topological cylinder to realization of the actual simplicial cylinder. · OAI.Thomason.Construction.RealizedCylinder.comparison

comparison def

The canonical comparison to the chain model on arbitrary strict categories. · OAI.Thomason.Construction.StrictOmegaCategory.GrayChainExtension.comparison

comparisonBad def

Bad comparisons include only valid windows longer than the cutoff. · OAI.TreeEdit.BinaryLower.SegmentCode.comparisonBad

comparisonEquiv def

Uniqueness, up to a trace-preserving star equivalence, of faithful tracial quotients with the same coordinate trace formula. · OAI.KadisonKastler.TraceQuotientMap.comparisonEquiv

ComparisonIndex abbrev

Position, literal label, choice of endpoint, and integer translate. · OAI.QuantitativeVanDerWaerden.CirclePartition.ComparisonIndex

comparisonIso def

The actual strict isomorphism, with the specified comparison as its forward functor. · OAI.Thomason.Construction.DirectedComplex.OrderedAtomBasis.comparisonIso

comparisonRel def

Open-edge adjacency together with the two separate terminal identifications. No virtual identification of common pins is present: those vertices are absent. · OAI.BufferedIsing.comparisonRel

comparisonRelation def

The three contiguous parts of the earlier-departure comparison. · OAI.TwoPointCorrelations.comparisonRelation

Comparisons structure

Data of the original disjoint signed interval comparisons on traversals. The involution and side predicates retain occurrence indices throughout. · OAI.TorsionFreeZeroDivisors.ComparedBlocks.Comparisons

comparisons def

Split every side at the boundaries of both chunks. Empty pieces carry no positions; their chosen endpoint maintains the exact range convention. · OAI.TorsionFreeZeroDivisors.ChunkComparisons.Chunking.comparisons

comparisons def

Every actual maximal run yields a literal integer comparison side. No real-grid or arbitrary interval compatibility is postulated. · OAI.TorsionFreeZeroDivisors.PlanarIntervals.comparisons

comparisonSetoid def

The equivalence closure implements components of the contracted open graph. · OAI.BufferedIsing.comparisonSetoid

comparisonToLocal def

The comparison algebra is mapped by the universal property of its actual cyclotomic quotient, rather than by assuming an arbitrary extension map. · OAI.CirculantHadamard.AlternatingLocalization.comparisonToLocal

comparisonUnit def

The unique unit changing one frame into the other. · OAI.StableCoordinate.FrameLift.comparisonUnit

comparisonX abbrev

The action presentation of X used by the generic native/bar comparison. · OAI.LittleFinitistic.SelectedBarBimodule.comparisonX

compatibility def

The original p-boundary produces compatibility in the actual target vertex. · OAI.ExternalTensor.Section4.ArrowVertex.compatibility

compatibility01 def

Compatibility of the actual original vertex-zero insertion with every 0-to-1 arrow. · OAI.ExternalTensor.Section4.ArrowVertex.compatibility01

compatibility01 def

The homotopy uses exactly the original first and second vertex equivalences. · OAI.LittleFinitistic.CanonicalArrowCompatibility.compatibility01

compatibility01 def

Actual arrow compatibility on the vertex diagram of the full B-valued tensor. · OAI.LittleFinitistic.FullTensorRectification.compatibility01

compatibility12 def

Compatibility of the actual original vertex-one insertion with every 1-to-2 arrow. · OAI.ExternalTensor.Section4.ArrowVertex.compatibility12

compatibility12 def

The homotopy uses exactly the original second and third vertex equivalences. · OAI.LittleFinitistic.CanonicalArrowCompatibility.compatibility12

compatibility12 def

The second arrow family retains its original maps as well. · OAI.LittleFinitistic.FullTensorRectification.compatibility12

Compatible def

Nonempty intersection of the two dyadic bricks. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Prefix.Compatible

Compatible def

Compatibility with all operations of an algebra. · OAI.FiniteCongruence.Compatible

Compatible def

A field-sign map is compatible with each actual edge traversal. · OAI.HyperbolicObstruction.MasterGraph.Path.Compatible

Compatible def

A leaf checks only forced-one indices in its own top block. Forced zeros can be tested against its selected top mask directly. · OAI.LowSpaceSubsetSum.MaskRestriction.Compatible

Compatible def

An assignment obeys all wire equalities. · OAI.OneWayLiveness.Pairing.Compatible

Compatible def

Compatibility is exactly agreement wherever both assignments are defined. · OAI.PartitionConsistency.CohenForcing.Condition.Compatible

Compatible def

Conditions have a common strengthening. · OAI.PartitionConsistency.ForcingQuotient.Compatible

Compatible def

Discrete one-Lipschitz data along the actual tree edges. · OAI.PlanarL1.HeightTree.Compatible

Compatible def

Compatibility of the four signed shifts, expressed independently in each layer. · OAI.Problem335.MomentPairing.Compatible

Compatible def

Exact compatibility of phases through the canonical cyclic quotient. · OAI.SmoothObstruction.PhaseDynamics.Compatible

Compatible def

Alignment with an arbitrary fixed two-sided primitive text. · OAI.Superstring.Hierarchical.Threading.Compatible

Compatible def

Exactly the source compatibility test, before choosing the least u. · OAI.TorsionFreeZeroDivisors.GridAsymptotics.Compatible

compatible def

Derivative compatibility is a closed linear condition on the finite family. · OAI.ForcedComputation.BoundedSpatialJets.compatible

CompatibleBits def

Replaced supported heads are assigned the extremum appropriate to the comparison direction and path parity. Unreplaced heads retain their bit. · OAI.PureTypeSystem.OneChildInterpretation.CompatibleBits

compatibleBlockFrequencyEquiv def

The actual compatible frequency fiber, with its exact kernel-vector parameter. · OAI.MaxCutGames.Inverse.KMSAnalytic.compatibleBlockFrequencyEquiv

compatibleBlockFrequencyEquiv def

The actual compatible frequency fiber, with its exact kernel-vector parameter. · OAI.UniqueGamesTheorem.Inverse.KMSAnalytic.compatibleBlockFrequencyEquiv

CompatibleData structure

Initial data, source, and the three coefficients, represented consistently at every order. · OAI.ForcedComputation.PlaneScalarMild.CompatibleData

CompatibleLocallyCATNegOne def

A metric on the given finite polyhedron, not on a different topology. · OAI.HyperbolicObstruction.CompatibleLocallyCATNegOne

compatibleMatrixFrames def

Choose compatible frames without changing any algebra in the chain. · OAI.KadisonKastler.compatibleMatrixFrames

compatibleMetric abbrev

The compatible metric retains the original topology definitionally. · OAI.PolycyclicRecognition.ProperChainMetric.compatibleMetric

compatibleMetric abbrev

Normalize the metric so that its topology is definitionally the original topology; the existing Lie charts then remain usable. · OAI.PolycyclicRecognition.ProperChainMetric.compatibleMetric

CompatibleOn def

Exactly the usual section compatibility on edges with both endpoints in Ω; coordinates outside Ω are ignored and can be chosen arbitrarily. · OAI.KLInvariance.MomentGraph.Sheaf.CompatibleOn

compatiblePathsEquivSigma def

The compatible quadruples partition uniquely according to their exact pairing word. · OAI.Problem335.Pairings.compatiblePathsEquivSigma

compatibleProjectiveLift def

Correct arbitrary lifts by the retraction so that successive lifts agree. · OAI.PolycyclicRecognition.SequentialRepresentationDiagram.compatibleProjectiveLift

CompatibleSplitting structure

Data of one common coordinate and compatible section vectors. All maps and all smaller spaces below are constructed from these original data. · OAI.MaxCutGames.Inverse.KMSAnalyticCompatibleSplitting.CompatibleSplitting

CompatibleSplitting structure

Data of one common coordinate and compatible section vectors. All maps and all smaller spaces below are constructed from these original data. · OAI.UniqueGamesTheorem.Inverse.KMSAnalyticCompatibleSplitting.CompatibleSplitting

CompatibleSupport def

The union of two positive supports is a chain. · OAI.Thomason.Construction.PosetGeometry.CompatibleSupport

compensatedAdaptiveCoefficients def

Choose any valid completion of the exposed prefix. The precision theorem below proves that the required reduced coefficients do not depend on it. · OAI.Ostmann.compensatedAdaptiveCoefficients

compensatedEquations def

The equation of the moving construction, keeping its compensation product instead of replacing the new giant by a composite pivot. · OAI.Ostmann.PivotDependencyScheme.compensatedEquations

CompensationAbsent def

A bulk coordinate never occurs in any compensation denominator. · OAI.Ostmann.MovingSlotData.CompensationAbsent

CompensationDistinct def

Equality patterns may identify samples at different nodes; only the entries made together in one compensation vector have to be distinct. · OAI.Ostmann.MovingSlotData.CompensationDistinct

compFor def

comp for A play Zplus paired offset+, Zminus paired offset-. Companion columns at each compared sign in opposite sign primal frame. · OAI.HadwigerCounterexample.compFor

compile def

Compile designated references of an AIG, starting only with its input bits. All graph slots and their retained evaluation traces are genuine ancillas. · OAI.ExactQuantumFactoring.AIGCompiler.compile

compile abbrev

A concrete finite local table, using only the three frozen I/O codes and finitely many fixed extra symbols and states. · OAI.WLIdentification.StackModel.compile

compiled def

Every machine with finite control and finitely many finite-alphabet stacks is realized by the multitape Machine, at exactly two local steps per stack step. This includes input preparation and halting semantics. · OAI.ParityWL.Complexity.StackMachine.compiled

compiled def

A complete compiler from raw input bits, with explicit initialization and all serialization work charged. · OAI.ParityWL.Complexity.TreeMachine.compiled

compiledCircuit abbrev

The concrete compressed tableau, now with no semantic seed assumption. · OAI.WLIdentification.Machine.compiledCircuit

compileOp def

The new wire is distinct from every preexisting wire. · OAI.ExactQuantumFactoring.BoolNode.compileOp

compileVec def

The whole output BitVec of a native, already verified, arithmetic bitblaster is now a Boolean network with a linear overhead in its AIG graph size. · OAI.ExactQuantumFactoring.AIGCompiler.compileVec

complement def

The open complement of a closed immersion. · OAI.ActualOpenSupport.complement

complementAmbientTriv def

Extend the locally constructed kernel chart over its open base into the actual global complement total space. · OAI.Kaplansky.ContinuousOrthonormalStageFrame.complementAmbientTriv

complementaryPathCoordinates def

Complement-fiber version, aligned with bidegreeExponentsEquiv. · OAI.Problem335.complementaryPathCoordinates

complementBasis def

Basis obtained from a genuine complementary domain decomposition. · OAI.MaxCutGames.Inverse.KMSAffineRestriction.complementBasis

complementBasis def

Basis obtained from a genuine complementary domain decomposition. · OAI.UniqueGamesTheorem.Inverse.KMSAffineRestriction.complementBasis

complementMatrix def

Candidate missing-basis projection for six bases in dimension six. · OAI.MUB6.Weiner.complementMatrix

complementNormalizer def

The truncated normalizer on complementary divisors. · OAI.Problem337.SelbergOptimal.complementNormalizer

complementParameter def

Values on the omitted domain complement, compressed into range Z. · OAI.MaxCutGames.Appendix.DerivativeInduction.complementParameter

complementParameter def

Values on the omitted domain complement, compressed into range Z. · OAI.UniqueGamesTheorem.Appendix.DerivativeInduction.complementParameter

complementProjection def

Complement of a commutant projection, as a commutant projection. · OAI.KadisonSimilarity.complementProjection

complementRangeEquiv def

A complement to the kernel maps isomorphically onto the range. · OAI.MaxCutGames.Appendix.CoordinateTransport.complementRangeEquiv

complementRangeEquiv def

A complement to the kernel maps isomorphically onto the range. · OAI.UniqueGamesTheorem.Appendix.CoordinateTransport.complementRangeEquiv

complementSplitHomeomorph def

Global topological splitting of the actual bundle. The first factor is the constructed complement vector bundle and the other factor is trivial. · OAI.Kaplansky.ContinuousOrthonormalStageFrame.complementSplitHomeomorph

complementSplitLinearEquiv def

The actual fiber decomposition into the full orthogonal complement and the trivial frame summand. No alignment with the original line summands. · OAI.Kaplansky.ContinuousOrthonormalStageFrame.complementSplitLinearEquiv

ComplementTotal abbrev

The actual complement, with the ambient bundle's subspace topology. · OAI.Kaplansky.ContinuousOrthonormalStageFrame.ComplementTotal

Complete def

Sequential metric completeness, for the intrinsic distance, not the background one. · OAI.HarmonicCounterexample.Complete

Complete def

Every bounded detector row is completely assigned after a finite initial segment. The quantifier over all representatives makes covariance literal. · OAI.PartitionConsistency.TailSelector.TailData.Complete

complete def

Closing the interface uses only an whole-port matching, never a matching of independent child germs. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.PortPatch.complete

complete def

Reconstruct a parity-constrained triple from its retained bit and one free bit. · OAI.MaxCutGames.Soundness.PartnerProjection.complete

complete def

Reconstruct a parity-constrained triple from its retained bit and one free bit. · OAI.UniqueGamesTheorem.Soundness.PartnerProjection.complete

CompleteAt def

At least one labeling satisfies a fraction at least 1 - error. · OAI.MaxCutGames.Foundations.Target.CompleteAt

CompleteAt def

At least one labeling satisfies a fraction at least 1 - error. · OAI.UniqueGamesTheorem.Foundations.Target.CompleteAt

CompleteBound def

A uniform bound on all square matrix amplifications, with the C-star matrix norm on the domain and the Hilbert operator norm on the codomain. · OAI.KadisonSimilarity.CompleteBound

completeCommutatorNorm def

Completely bounded norm of the inner derivation restricted to P. The extended nonnegative real value permits this norm to be infinite before an estimate has been proved. Matrix norms are the concrete Hilbert norms. · OAI.KadisonSimilarity.completeCommutatorNorm

CompleteCopies def

All orthogonal multiplicity copies of one irreducible type. The completeness condition quantifies over genuine intertwiners, i.e. it is exactly the isotypic space condition. · OAI.RowColumn.CompleteCopies

completeCopyEquiv def

Copies are put in bijection without choosing or counting graph embeddings. · OAI.LeanBlast.SecondKahnKalai.completeCopyEquiv

completedAffineChart abbrev

The Cartier chart generated by the degreewise completed cone and the constants of its Proj basic open, inside the coefficient power-series ring. · OAI.ExplicitCone.completedAffineChart

completedAffineExceptionalEquiv def

The exceptional divisor on the completed affine chart is isomorphic to the smooth chart of the projective section ring. · OAI.ExplicitCone.completedAffineExceptionalEquiv

CompleteData abbrev

Complete visible conditioning data: occurrences, public map, projection choices, the entire complement map, and precisely the attainable rows. · OAI.MaxCutGames.Decoder.AdviceLaw.CompleteData

CompleteData abbrev

Complete visible conditioning data: occurrences, public map, projection choices, the entire complement map, and precisely the attainable rows. · OAI.UniqueGamesTheorem.Decoder.AdviceLaw.CompleteData

completedBlowupChartEquiv def

The affine Rees–Proj chart is the algebraic power-series realization of the completed cone. · OAI.ExplicitCone.completedBlowupChartEquiv

completedBlowupMap def

The actual projective Rees model over the actual completed ring. · OAI.ExplicitCone.completedBlowupMap

completedChartBaseMap abbrev

The genuine structure map of the distinguished affine chart of the literal eighteen-section ordinary blowup. · OAI.ExplicitCone.completedChartBaseMap

completedChartInclusion abbrev

The actual open embedding of the distinguished homogeneous Rees chart. · OAI.ExplicitCone.completedChartInclusion

completedChartMap def

The actual completed cone embeds in the formal Cartier chart over the literal z₀z₁z₄ section open. This does NOT assert that the formal chart is the uncompleted affine blowup base change. · OAI.ExplicitCone.completedChartMap

completedChartRestrictionSections abbrev

The exceptional-chart module is the restriction of the double dual of the pullback module to a prime Cartier divisor. · OAI.ExplicitCone.completedChartRestrictionSections

completedCommutationIso def

Commutation with any endofunctor extends uniquely across idempotent completion. · OAI.LittleFinitistic.QuadraticEncoding.completedCommutationIso

completedConeMap def

The coordinate map induced by p on the completed chart Bᵢ ⊗[S] A. · OAI.SourcePullbackChart.completedConeMap

completedDerivedEvaluation def

The manuscript's additive extension of evaluation to all of Kar(Q), with values in the actual bounded finite-dimensional derived category. · OAI.LittleFinitistic.QuadraticEncoding.completedDerivedEvaluation

completedDerivedShiftIso def

This is the formal shift used by the native odd-double construction: shift the underlying object and its idempotent. No shift coherence is assumed. · OAI.LittleFinitistic.QuadraticEncoding.completedDerivedShiftIso

completedEntries def

Sum of the lengths of all completed right dictionaries. · OAI.LowSpaceSubsetSum.completedEntries

completedEval def

The concrete spatial representation on the completed Hilbert tensor product. · OAI.NuclearUltrapower.CuntzAveraging.completedEval

completedExceptionalChartMap abbrev

The literal prime Cartier divisor in the distinguished actual blowup chart, embedded into the GLOBAL proper model. · OAI.ExplicitCone.completedExceptionalChartMap

completedExtensionFactors def

The extension restricts to the original functor under the canonical embedding. · OAI.LittleFinitistic.QuadraticEncoding.completedExtensionFactors

completedExtensionImageIso def

An explicitly split evaluated idempotent identifies the value of the completion. · OAI.LittleFinitistic.QuadraticEncoding.completedExtensionImageIso

completedExtensionLatticeEquiv def

The lattice is isomorphic to the original group, not a quotient by its center. · OAI.PolycyclicRecognition.NilpotentLatticeCoordinates.completedExtensionLatticeEquiv

completedExtensionSplitting def

The completed extension splits if one lift has no fixed infinitesimal vector in the nilpotent kernel. · OAI.PolycyclicRecognition.NilpotentLatticeCoordinates.completedExtensionSplitting

completedFirstAccept def

Complete an exhausted bounded call with an independent eligible backup. This is the ideal marginal in the explicit one-call coupling. · OAI.LowSpaceSubsetSum.completedFirstAccept

completedFractionChart abbrev

The ordinary fraction chart for the original eighteen completed sections. This is the actual affine algebra R[s_i/s_00], not a formal tensor product. · OAI.ExplicitCone.completedFractionChart

completedFractionChartAwayEquiv def

Away from its exceptional equation, every actual fraction chart is the literal completed-cone principal open. These opens cover the puncture. · OAI.ExplicitCone.completedFractionChartAwayEquiv

completedFractionChartEquiv def

The algebraic power-series realization coincides with the literal fraction chart of the blowup of the eighteen-section ideal. · OAI.ExplicitCone.completedFractionChartEquiv

completedGlobalRestrictionIso def

Compatibility with the actual global model; this closes the local/global identity left separate in the preceding positive-rank chart calculation. · OAI.ExplicitCone.completedGlobalRestrictionIso

completedGlobalRestrictionSections abbrev

Sections of the restriction of the GLOBALLY defined extension E. · OAI.ExplicitCone.completedGlobalRestrictionSections

CompletedHeckeFiniteOrder def

An order-one completed function has this deliberately coarse quadratic exponential bound. The regular-point equality respects Lean's zero-valued Gamma at its poles. · OAI.CubicFirstMoment.CompletedHeckeFiniteOrder

completedLeft def

Restore the designated coordinate on every left table read. · OAI.MaxCutGames.Foundations.Games.KernelSampling.completedLeft

completedLeft def

Restore the designated coordinate on every left table read. · OAI.UniqueGamesTheorem.Foundations.Games.KernelSampling.completedLeft

completedMachine def

A fixed finite program over the same Boolean tapes as the raw machine. · OAI.MaxCutGames.Foundations.Complexity.MachineCanonicalOutput.completedMachine

completedMachine def

A fixed finite program over the same Boolean tapes as the raw machine. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineCanonicalOutput.completedMachine

completedModelE abbrev

The literal double-dual extension for an arbitrary module on the completed cone, on the genuine proper Rees modification constructed above. This is not an assumed object with prescribed numerical invariants. · OAI.ExplicitCone.completedModelE

completedOddDoubleImageIso def

Evaluation of the formal odd double used in Section 3. · OAI.LittleFinitistic.QuadraticEncoding.completedOddDoubleImageIso

completedPolynomial def

A polynomial time allowance for the whole prefixRun followed by actual reversal and cleanup. The fixed push bound belongs to the concrete driver. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Finish.completedPolynomial

completedPolynomial def

A polynomial time allowance for the whole prefixRun followed by actual reversal and cleanup. The fixed push bound belongs to the concrete driver. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Finish.completedPolynomial

completedPunctureOpen def

The open punctured spectrum is defined by the eighteen literal source sections; its complement is proved to be precisely the closed point. · OAI.ExplicitCone.completedPunctureOpen

completeDraw def

The actual Draw obtained from the complete-data fiber coordinates. · OAI.MaxCutGames.Decoder.AdviceLaw.completeDraw

completeDraw def

The actual Draw obtained from the complete-data fiber coordinates. · OAI.UniqueGamesTheorem.Decoder.AdviceLaw.completeDraw

completedRight def

Restore the designated coordinate on every right table read. · OAI.MaxCutGames.Foundations.Games.KernelSampling.completedRight

completedRight def

Restore the designated coordinate on every right table read. · OAI.UniqueGamesTheorem.Foundations.Games.KernelSampling.completedRight

completedSection def

The original degree-one section, as an element of the degreewise completion used in MainClaim. · OAI.ExplicitCone.completedSection

completedSectionIdeal def

The ideal generated by the eighteen degree-one sections of the polarization in the completed cone. · OAI.ExplicitCone.completedSectionIdeal

completedSectionKernel def

Complete the conditional kernel by any fixed point on exceptional fibers. · OAI.Problem358.NormalizedSections.completedSectionKernel

completedSegmentEval def

Literal finite segment evaluation, with each coordinate extended to the completion by its proved continuous functional. · OAI.BoundedTreePotentials.Segments.completedSegmentEval

completedShiftedSummandImageIso def

In particular, every formal shift of U evaluates to the same shift of eY. · OAI.LittleFinitistic.QuadraticEncoding.completedShiftedSummandImageIso

completedSphereEquiv def

Extend the actual sphere isometry and its inverse through the sphere completions. · OAI.Tingley.completedSphereEquiv

completedSummandImageIso def

The full completed derived evaluation takes U to the actual finite-dimensional eY stalk. · OAI.LittleFinitistic.QuadraticEncoding.completedSummandImageIso

completedTensorPermutation def

The corresponding permutation on the completed Hilbert tensor power. · OAI.KadisonSimilarity.completedTensorPermutation

CompletedTensorWord abbrev

Completed finite Hilbert tensor powers. · OAI.KadisonSimilarity.CompletedTensorWord

completedTensorWord def

An elementary word in the completed tensor power. · OAI.KadisonSimilarity.completedTensorWord

completedTime def

For fixed raw plans, polynomial raw time gives polynomial cleanup time. · OAI.MaxCutGames.Foundations.Complexity.FinalCNFCleanup.completedTime

completedTime def

For fixed raw plans, polynomial raw time gives polynomial cleanup time. · OAI.UniqueGamesTheorem.Foundations.Complexity.FinalCNFCleanup.completedTime

completedUltraSphereEquiv def

The actual coordinatewise sphere equivalence followed by ambient completion. · OAI.Tingley.completedUltraSphereEquiv

CompletedXiGap def

Uniform separation of completed-zeta zeros on the discriminant scale. · OAI.DukePrimeDegree.CompletedXiGap

completedZeroQuotient def

The zero-section quotient remains exactly D_i after completion. · OAI.SourcePullbackChart.completedZeroQuotient

completeFiberEquiv def

Exact equivalence for a fiber of the complete observation, not just the marginal row observation. This is the input for finite conditioning. · OAI.MaxCutGames.Decoder.AdviceLaw.completeFiberEquiv

completeFiberEquiv def

Exact equivalence for a fiber of the complete observation, not just the marginal row observation. This is the input for finite conditioning. · OAI.UniqueGamesTheorem.Decoder.AdviceLaw.completeFiberEquiv

CompleteLog def

Concrete flat complete-data condition sufficient for retrospective lookup: root present, every required child present, and every non-dummy row a complete prime factorization. Equal labels from different parents are NOT collapsed. · OAI.ExactQuantumFactoring.PhysicalTree.CompleteLog

CompletelyAdditive def

Preservation of arbitrary orthogonal projection sums. The index universe need only contain the set of projections of the algebra. · OAI.BoundedHochschild.KadisonRingrose.CompletelyAdditive

CompletelyBoundedBy def

A complete bound, expressed on all finite rectangular matrix levels. · OAI.KadisonKastler.CompletelyBoundedBy

CompletelyBoundedEstimate def

A uniform bound on all entrywise matrix amplifications of a linear map. · OAI.BoundedHochschild.KadisonRingrose.CompletelyBoundedEstimate

CompletelyIsometric def

Complete isometry in the same rectangular convention. · OAI.KadisonKastler.CompletelyIsometric

completelyPositive_of_embedding def

Positivity at all matrix levels is reflected by a faithful embedding. · OAI.KadisonKastler.completelyPositive_of_embedding

completelyPositive_of_pointwise_weak_limit def

A pointwise weak limit, once bundled linearly, is completely positive. · OAI.KadisonKastler.completelyPositive_of_pointwise_weak_limit

completelyPositiveAmplification def

A matrix amplification of a completely positive map is completely positive. · OAI.KadisonKastler.completelyPositiveAmplification

completelyPositiveComp def

Composition of completely positive linear maps. · OAI.KadisonKastler.completelyPositiveComp

completePrefix def

This helper consumes a proved prefix trace. The final runtime theorem supplies that trace from the concrete header and complete row traversal. · OAI.MaxCutGames.Explicit.MachineSingleOrbitFinish.completePrefix

completePrefix def

Finish the concrete program, including its one-step finalizer entry bridge. The runtime bound uses all actual intermediate tapes, with no output-size or private-space assumption. · OAI.MaxCutGames.Reduction.AddressMachineFinish.completePrefix

completePrefix def

This helper consumes a proved prefix trace. The final runtime theorem supplies that trace from the concrete header and complete row traversal. · OAI.UniqueGamesTheorem.Explicit.MachineSingleOrbitFinish.completePrefix

completePrefix def

Finish the concrete program, including its one-step finalizer entry bridge. The runtime bound uses all actual intermediate tapes, with no output-size or private-space assumption. · OAI.UniqueGamesTheorem.Reduction.AddressMachineFinish.completePrefix

completion def

A completion of the bottom row, with its existence proved above. · OAI.CubicFirstMoment.CubicThetaBottomRow.completion

completionBasis def

A completion, regarded as a basis of the resulting subspace. · OAI.MaxCutGames.Inverse.KMSBasisComparisonPseudorandom.completionBasis

completionBasis def

A completion, regarded as a basis of the resulting subspace. · OAI.UniqueGamesTheorem.Inverse.KMSBasisComparisonPseudorandom.completionBasis

completionCandidates def

Rejected candidates remain as none, allowing later integer sums to ignore them without adding a second enumeration or an ordering oracle. · OAI.ContingencyTables.DensePrograms.completionCandidates

completionCoeff def

Source (completion), with its actual downward integer rounding. · OAI.ExactQuantumFactoring.Exactness.completionCoeff

completionDecision def

A proposal is either a hold or a target state. The sampled completion is discarded after its literal translated-nonnegativity test. · OAI.ContingencyTables.CompletionTranslation.completionDecision

completionEquiv def

Literal identification of the graded coefficient completion and the completion at the positive-degree ideal; no standard grading is assumed. · OAI.SectionCompletion.completionEquiv

completionExpansion def

The expansion map attached to an isomorphism with the maximal-ideal completion. · OAI.CartierSections.completionExpansion

CompletionFibre def

Full row/column views with one prescribed small profile. · OAI.ContingencyTables.PhysicalCompletionFibres.CompletionFibre

completionGraph def

Auxiliary graph construction for a family of invariant partial orbit maps. The support is precisely a common support for the three displayed set names. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.completionGraph

completionIdealCongr def

Equal ideals have canonically isomorphic completions. · OAI.CartierSections.completionIdealCongr

completionOfBasis def

Recover a restricted completion from a basis having the prescribed prefix. · OAI.MaxCutGames.Inverse.KMSBasisComparisonPseudorandom.completionOfBasis

completionOfBasis def

Recover a restricted completion from a basis having the prescribed prefix. · OAI.UniqueGamesTheorem.Inverse.KMSBasisComparisonPseudorandom.completionOfBasis

completionPaddedDraw def

One common reserved word works at every state, despite the different residual margins and quotient lengths. · OAI.ContingencyTables.SmallGraphProfiles.completionPaddedDraw

completionParts def

Index zero denotes the distinguished dependent row or column. · OAI.ContingencyTables.DensePrograms.completionParts

completionRelation def

A ground-indexed family of completed partial maps, with an actual ordered pair as input. Local witnesses are regularized, as in existential forcing. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.completionRelation

completionSpanEquiv def

The unique basis change carrying the first completion to the second. · OAI.MaxCutGames.Inverse.KMSBasisComparisonPseudorandom.completionSpanEquiv

completionSpanEquiv def

The unique basis change carrying the first completion to the second. · OAI.UniqueGamesTheorem.Inverse.KMSBasisComparisonPseudorandom.completionSpanEquiv

completionSpecIso def

Exact affine-cone base-change comparison. · OAI.SourceConeMorphism.completionSpecIso

completionStep def

Rounded completion of all 2m+1 passes, before extending vertically. · OAI.SLEExactGauge.Serpentine.completionStep

completionUnitary def

Extension of a linear isometry equivalence to metric completions. · OAI.NuclearUltrapower.completionUnitary

Complex structure

A finite abstract simplicial complex with every labelled vertex present. · OAI.CoboundaryExpanders.Complex

Complex structure

A finite abstract simplicial complex, including its empty face. Every labelled vertex is a vertex of the complex, so unused labels are disallowed. · OAI.CoboundaryExpanders.Complex

Complex abbrev

The actual category of bounded cochain complexes, with chain maps. · OAI.LittleFinitistic.BoundedHomotopy.Complex

Complex abbrev

The actual category of bounded complexes of finitely generated projectives. · OAI.LittleFinitistic.ProjectiveComplexes.Complex

complex def

The unaugmented absolute bar complex, degree n = A ⊗ A^⊗n ⊗ N. · OAI.DerivedBar.complex

complex def

Actual tensor complex, with both B-actions and the actual extended arrow maps. · OAI.LittleFinitistic.OrdinarySimulation.FullTensorTerm.complex

complex def

Actual off-diagonal complex with both actions of the product algebra retained. · OAI.LittleFinitistic.OrdinarySimulation.OffDiagonal.complex

complex def

The manuscript's actual Q_X = O[0] ⊕ Q_T, with both D-actions retained. · OAI.LittleFinitistic.OrdinarySimulation.XResolution.complex

complex def

The actual chain complex of the successive projective covers. · OAI.LittleFinitistic.PDMinimal.complex

complex abbrev

The one fixed genuine D-bimodule resolution Q_X used by the source. · OAI.LittleFinitistic.SelectedOrdinaryResolution.complex

complex abbrev

Evaluate the one fixed selecting bimodule, retaining its full left action. · OAI.LittleFinitistic.SelectedTensor.complex

complex def

Tensor a vector space with every term and differential of the given right-module complex. · OAI.LittleFinitistic.TensorComplex.complex

complex abbrev

The path complex has the exact signed differential of the right tensor. · OAI.Thomason.Construction.DirectedComplex.RightPath.complex

complex def

The concrete directed complex of the pasted pair, before tensoring. · OAI.Thomason.Construction.PastingGlobeBasis.complex

complex def

The actual augmented directed complex associated to a strict category. · OAI.Thomason.Construction.StrictOmegaCategory.Abelianization.complex

complexAlpha def

The dual exponent is a supremum of exact exponent-two rectangular shapes. · OAI.MatrixMultiplication.Arithmetic.complexAlpha

complexBasis def

A given real basis of the conjugation-fixed space is also a complex basis of the original complex space. · OAI.HarmonicCounterexample.RealForm.complexBasis

complexColumnEquiv def

Actual exponent columns and their complex-image columns are equivalent. · OAI.Nagata.W29.complexColumnEquiv

complexCoordinates def

The same coordinate map on the ordinary Pi space used by the Stokes integrals. · OAI.MahlerStokes.complexCoordinates

complexCoreReal def

Restrict scalars while retaining the original complex transition maps as functions, and hence the original bundle atlas and topology. · OAI.Kaplansky.complexCoreReal

complexDerivativeSign def

Sign of a real derivative between complex vector spaces of equal dimension. A complex identification is used to take the determinant; complexDerivativeSign_independent proves that choice immaterial. This is a local linear sign, not an … · OAI.Kaplansky.complexDerivativeSign

complexDiagonal def

The real derivative of multiplication by a unit complex tuple has determinant one. · OAI.EilenbergGanea.complexDiagonal

complexDiagonalHomotopy def

A path through complex units supplies a genuine nonzero-preserving homotopy to the identity; no determinant-to-degree assertion is assumed. · OAI.Kaplansky.complexDiagonalHomotopy

complexDiagonalMap def

Complex diagonal coordinate change with every multiplier invertible. · OAI.Kaplansky.complexDiagonalMap

complexEuclideanRealification def

Realification with the Euclidean norm on both groups of real coordinates. · OAI.SymmetricMahler.complexEuclideanRealification

complexFockBasis def

Complex version of the normalized monomial basis. · OAI.Problem335.complexFockBasis

complexFockMonomial def

Complex normalized monomials are obtained by coefficient extension. · OAI.Problem335.complexFockMonomial

complexGaussian def

Independent centered real and imaginary Gaussian coordinates, each of variance v. The usual standard complex Gaussian takes v = 1/2. · OAI.DefocusingNLS.complexGaussian

complexHessian def

Mixed Hessian in a finite family of directions; the row is the conjugate index. · OAI.Mahler.complexHessian

ComplexHilbert structure

A bundled complex Hilbert space, without any separability assumption. · OAI.NuclearUltrapower.ComplexHilbert

complexificationConjugation def

Conjugation on the complexification as a real Lie automorphism. · OAI.PolycyclicRecognition.complexificationConjugation

complexificationLieHom def

The canonical real Lie homomorphism into the complexification. · OAI.PolycyclicRecognition.complexificationLieHom

complexifiedAdjointMatrix def

The complexified adjoint matrices, restricted to the original real algebra. · OAI.PolycyclicRecognition.complexifiedAdjointMatrix

complexifiedInclusion def

Complexification of the inclusion of a real Lie subalgebra. · OAI.PolycyclicRecognition.complexifiedInclusion

complexifiedSubalgebra def

The actual complex Lie subalgebra generated by the real subalgebra. · OAI.PolycyclicRecognition.complexifiedSubalgebra

complexify def

Extension of a real endomorphism on the actual fixed real form. · OAI.HarmonicCounterexample.RealForm.complexify

complexify def

Embed a real-valued function into complex-valued functions. · OAI.KervaireSpectral.complexify

complexifyLinear def

Complexification respects the REAL vector-space operations. · OAI.HarmonicCounterexample.RealForm.complexifyLinear

complexifyRational def

A continuous rational-linear map commuting with multiplication by i is complex-linear. Real-linearity follows from continuity. · OAI.KadisonKastler.complexifyRational

complexInner def

Polarization of τ(x*x), with the standard conjugate-linear first convention. · OAI.TraceConeClassification.ExtendedTrace.complexInner

complexIso def

The natural comparison gives an actual chain isomorphism on every bimodule complex. · OAI.LittleFinitistic.BalancedVertexFunctor.complexIso

complexIso def

An actual chain comparison with the vertex of the full B-module tensor complex. · OAI.LittleFinitistic.BimoduleTensorVertexBridge.complexIso

complexIso def

Every actual vertex column is its concrete corner-tensor complex. · OAI.LittleFinitistic.CornerColumnComplex.complexIso

complexIso def

Applying an additive module functor commutes with the actual n ↦ -n zero extension. · OAI.LittleFinitistic.IntegerExtensionFunctor.complexIso

complexIso def

The entire actual tensor complex identifies with P on degrees a,a+1,... . · OAI.LittleFinitistic.OrdinarySimulation.FullTensorTerm.complexIso

complexLieEnvelope def

The complex scalar envelope of an actual real Lie algebra, with explicit real and imaginary witnesses. · OAI.HarmonicCounterexample.Control.complexLieEnvelope

complexLieEnvelope def

The complex scalar envelope of an actual real Lie algebra, with explicit real and imaginary witnesses. · OAI.HarmonicCounterexample.Control.VectorRealDescent.complexLieEnvelope

complexLineCoordinateUnit def

The unit comparing two genuine complex coordinates on a line. · OAI.Kaplansky.complexLineCoordinateUnit

complexMap def

An actual chain map between unaugmented free bar resolutions. · OAI.DerivedBar.complexMap

complexMap def

A compatible family of column maps gives an actual map of total complexes. · OAI.LittleFinitistic.FiniteColumnComplex.complexMap

complexMap def

An actual map of off-diagonal complexes retaining both product-ring actions. · OAI.LittleFinitistic.OrdinarySimulation.OffDiagonal.complexMap

complexMap def

The first-factor tensor map is a chain map. · OAI.LittleFinitistic.TensorComplex.complexMap

complexMat def

Complexification of a real matrix, with the actual complex-linear action. · OAI.ArnoldCounterexample.CriticalBound.complexMat

complexMatrix def

Complexification of the actual real coordinate matrix. · OAI.HarmonicCounterexample.Berger.ComplexStructure.complexMatrix

ComplexNormedLieAlgebra class

The analytic and Lie structures share their additive and scalar data. · OAI.PolycyclicRecognition.ComplexNormedLieAlgebra

complexOfCommuteI def

Upgrade a real-linear map commuting with the complex structure. · OAI.Kaplansky.complexOfCommuteI

complexPlane def

The coordinate isometry preserves Euclidean, not max-product, norms. · OAI.CenteredDiskEndpoint.complexPlane

complexPlane def

The coordinate isometry preserves Euclidean, not max-product, norms. · OAI.SignedDisk.CenteredDiskEndpoint.complexPlane

complexPoint def

Coordinatewise standard inclusion ℤ² → ℂ². · OAI.Nagata.W29.complexPoint

ComplexProperty def

Actual bounded complexes of finitely generated projective right/left modules, according to the chosen scalar ring R. · OAI.LittleFinitistic.FiniteProjectiveHomotopy.ComplexProperty

complexShortAverage def

A complex short average with the same endpoint convention. · OAI.JointDickman.PublishedInputs.complexShortAverage

complexShortCombination def

The coefficients in the manuscript's real multiplicative interpolation are complex, even for the principal-character term. · OAI.JointDickman.PublishedInputs.complexShortCombination

ComplexShortIntervalInput def

The quantitative complex short-average input from Matomäki, Radziwiłł, and Tao, An averaged form of Chowla's conjecture, Algebra & Number Theory 9 (2015), Theorem A.1, DOI 10.2140/ant.2015.9.2167. · OAI.JointDickman.PublishedInputs.ComplexShortIntervalInput

ComplexSpace abbrev

Complex Euclidean space of complex dimension n. · OAI.KahlerUniformization.ComplexSpace

complexSphereCoordinates def

Interleaved complex coordinates with the boundary/ambient degree cast shown in the type. · OAI.MahlerStokes.complexSphereCoordinates

ComplexStructure structure

Exactly the two standing equations for an orthogonal complex structure. Both orientation classes are permitted. · OAI.HarmonicCounterexample.Berger.ComplexStructure

complexStructure def

The canonical almost complex structure on the underlying real tangent bundle: multiplication by i in holomorphic tangent coordinates. · OAI.KahlerUniformization.complexStructure

complexStructure def

Multiplication by i as a real continuous linear map. · OAI.Mahler.complexStructure

complexStructure def

Multiplication by i, viewed as a real continuous linear map. · OAI.MahlerStokes.complexStructure

complexValue def

A single complex-valued extension of the spectral construction in every finite matrix size. Abelian linearity is not asserted by this definition. · OAI.Kaplansky.CuntzDimensionState.complexValue

complexValueLinearMap def

Complex linearity for the original spectral value composed with a commutative representation. Agreement is pointwise, ensuring overlap consistency. · OAI.Kaplansky.CuntzDimensionState.complexValueLinearMap

ComplexWeightFactorModel structure

The concrete group part of the complex-weight Cartan model. The ambient overlap is retained, and all lifting conclusions refer to the actual map. · OAI.PolycyclicRecognition.ComplexWeightFactorModel

complexWeightMap def

The same adjoint diagonal on the entire complexification. · OAI.PolycyclicRecognition.ComplexTriangularAdjoint.complexWeightMap

ComplexZeroEnumeration structure

A fixed enumeration with multiplicities used by the published density and explicit-formula statements. Bounded ordinate ranges are finite. · OAI.Ostmann.ComplexZeroEnumeration

Component def

Components are a quotient of the ground set of actual vertices. · OAI.PartitionConsistency.LocalDiagramSystem.Vertex.Component

Component abbrev

The metric component is precisely the separation quotient of the intrinsic pseudometric on representatives. No additional edges are inserted. · OAI.PlanarL1.Columns.Model.Component

component def

The initial homogeneous series of a given weight. · OAI.CartierSections.FormalMonomial.component

component def

Entire weak component, computed by at most the known number of vertices many finite expansions. Unlike running the machine, this always terminates. · OAI.FiniteAlgebraicEnvelopes.Compiler.component

component def

The supported complex's restriction and extension have its actual components. · OAI.LittleFinitistic.OrdinarySimulation.SupportedExtension.component

component def

Tensor the actual right vertex, then identify it with the vertex of the full left B-module tensor, preserving the original balanced tensor classes. · OAI.LittleFinitistic.TensorArrowComparison.component

component def

The component formed by retaining precisely the actual frequencies in P. · OAI.MaxCutGames.Inverse.KMSAnalytic.component

component def

Projection onto one degree in the selected variables. · OAI.Problem335.Bidegree.component

component def

A well-defined linear projection to an integral weight in the quotient. · OAI.StableCoordinate.HomogeneousQuotientWeight.component

component def

Every homogeneous degree-shift component of a derivation is itself a derivation. · OAI.StableCoordinate.LeadingComponent.component

component def

The component formed by retaining precisely the actual frequencies in P. · OAI.UniqueGamesTheorem.Inverse.KMSAnalytic.component

componentAbove def

The parent component under inclusion of vertex sets. · OAI.PlanarL1.ColumnOrder.componentAbove

componentClusterValues def

Finite real cluster values approached from the domain at a component. · OAI.CircleDomainRigidity.componentClusterValues

componentCoefficient def

The integral coefficient of one specified path component. · OAI.Kaplansky.componentCoefficient

componentColoringEquiv def

The exact bijection underlying the factor 2^(number of FK clusters). · OAI.BufferedIsing.componentColoringEquiv

componentColumns def

The vertices of an induced-graph component, as a set in the original vertex type. Components here are Mathlib's actual reachability quotients. · OAI.PlanarL1.ColumnOrder.componentColumns

componentContact def

A contact triple on a finite union, with its ordered component labels. · OAI.Problem356.componentContact

componentCoordinates def

Literal normalized homogeneous coordinates on a selected line. · OAI.Nagata.Workers.W24.componentCoordinates

componentDarts def

Actual endpoint set of the root crossing component. · OAI.FKCRT.Chords.componentDarts

componentEquiv def

If g stabilizes the component, transport is an equivalence to the symbolic component at the original chosen root of the moved system. · OAI.PartitionConsistency.LocalDiagramSystem.ReindexIso.componentEquiv

ComponentInterface structure

Environments carry arbitrary typed raw images. Extension collections are nonempty, and retain old parameters. Product tests range over all listed extensions; specialization can select the normal argument's tested payload. · OAI.PureTypeSystem.ComponentInterface

componentInterface def

The actual observation interpretation, with arbitrary typed raw images and all matching payload extensions. No component interface law is assumed. · OAI.PureTypeSystem.ObservationValues.componentInterface

componentInterface def

One-child components need no relational exclusion: both fresh extremes are tested, and polarity selects one of those tests at specialization. · OAI.PureTypeSystem.OneChildInterpretation.componentInterface

componentLinear def

The shift-j component of a linear map, on an arbitrary ℤ-graded algebra. · OAI.ComplexCancellation.GradedLND.componentLinear

componentLinear def

The part of a linear map that raises homogeneous degree by exactly m. · OAI.StableCoordinate.LeadingComponent.componentLinear

componentMap def

The degreewise map between the actual finite sums of columns. · OAI.LittleFinitistic.FiniteColumnComplex.componentMap

componentPhiNet def

Euler's prime-power formula, implemented by n-bit arithmetic on the computed gcd component. Divisibility by p is a verified-call precondition. · OAI.ExactQuantumFactoring.BitArithmetic.componentPhiNet

ComponentPoint abbrev

Representatives in one component, before taking the identification quotient. The selected vertex determines a component, not an additional edge relation or a different family of fibers. · OAI.PartitionConsistency.LocalDiagramSystem.System.ComponentPoint

componentProductEquiv def

All but one coordinate can be chosen freely in a fixed product fiber. · OAI.Ostmann.componentProductEquiv

componentRepresentationCost def

Literal cost M_{n+1} of the start masses on the component of height n+1. · OAI.BoundedTreePotentials.Segments.componentRepresentationCost

components def

The actual (nonempty) connected components of an open real line layer. · OAI.HenonLaneEmden.LayerPressure.components

components def

Connected components of the actual polygon identifications on darts. · OAI.HyperbolicObstruction.PictureSwitch.PictureMap.components

components def

Connected components of a bipartite rotation system, including edge components consisting of a single edge. Isolated vertices are not encoded. · OAI.KaplanskyCounterexample.Planar.Ribbon.components

components def

The central compact set and its four transported copies. · OAI.Problem356.BranchSeparation.components

components abbrev

Components of the graph generated by two permutations; cycles of their product are contained in these components. · OAI.Release075.PermCycles.components

ComponentSample abbrev

Offsets are fresh for each narrow component. The keep coin, by contrast, is shared by all the components of one surviving group. · OAI.PlanarL1.Columns.Model.SecondaryGeometry.ComponentSample

componentSetoid def

The equivalence relation collapsing exactly the complementary components. · OAI.CircleDomainRigidity.componentSetoid

componentsFunctor def

Path components form a functor on unbased spaces. · OAI.Thomason.Construction.PointedHomotopyGroups.componentsFunctor

componentStabilizer def

Setwise stabilizer of the component of v. Common is already proved to be exactly connectedness under finite contraction zigzags. · OAI.PartitionConsistency.LocalDiagramSystem.Vertex.componentStabilizer

ComponentSumInput def

Component-to-sum finite-codimensional input, exactly at arbitrary unit centers. · OAI.DiamondDistortion.PathApplications.ComponentSumInput

componentTable def

This is a bounded table chosen just once for the component and its chain; in particular it has no filter argument. · OAI.PartitionConsistency.ProductForcing.ChainCode.componentTable

componentTwist def

Separate coordinate twists on the components of a disjoint union of cubes. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.componentTwist

componentVertex def

Original graph-component vertices are carried to the actual restricted rotation system, using representatives only to define the quotient map. · OAI.HyperbolicObstruction.PictureSwitch.PictureMap.componentVertex

componentWeight def

The five component weights are fixed by the counterexample construction. · OAI.Problem356.SmoothBranchMarginal.componentWeight

componentWidth def

A polynomial-width, polynomial-size component modulus circuit. It uses p as an input prime, not an oracle or enumeration of prime factors of m. · OAI.ExactQuantumFactoring.BitArithmetic.componentWidth

compose def

A finite transition coupling. We compose couplings, not stochastic kernels, so zero-mass states require no arbitrary probability distribution. · OAI.BoundedTreewidthL1.ExpandedParticleKernel.compose

compose def

Compose after identifying the two intermediate partitions. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Diagram.compose

compose def

Composition is obtained by the literal common refinement of the middle bricks. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Table.compose

compose def

Composition with range contained in the initial projection. · OAI.Generator.Tracial.PartialIsometry.compose

compose def

Direct composition with an actual preceding simple-bipartite producer. · OAI.MaxCutGames.Explicit.MachineProductRuntime.compose

compose def

General sequential composition with an explicit symbol of the intermediate alphabet to totalize the bridge's unused empty-register expression. Its time polynomial uses the transition-derived output-length bound of the first machine. · OAI.MaxCutGames.Foundations.Complexity.MachineSequential.compose

compose def

Direct composition with an actual preceding simple-bipartite producer. · OAI.UniqueGamesTheorem.Explicit.MachineProductRuntime.compose

compose def

General sequential composition with an explicit symbol of the intermediate alphabet to totalize the bridge's unused empty-register expression. Its time polynomial uses the transition-derived output-length bound of the first machine. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineSequential.compose

compose def

Composition of one-variable open terms, preserving the common ambient context. No logical equivalence is used as a data equality. · PureTypeSystem.Labelled.Expr.compose

composeMachinePhases def

Two verified phases of one transition system compose with added polynomial budgets, both measured in the original input length. This theorem preserves the actual sequence of transition steps, not merely the sizes of configurations. · OAI.MaxCutGames.Foundations.Complexity.composeMachinePhases

composeMachinePhases def

Two verified phases of one transition system compose with added polynomial budgets, both measured in the original input length. This theorem preserves the actual sequence of transition steps, not merely the sizes of configurations. · OAI.UniqueGamesTheorem.Foundations.Complexity.composeMachinePhases

composePartialIso def

Compose two partial isometries when the final support of the second lies in the initial support of the first. · OAI.BoundedHochschild.KadisonRingrose.composePartialIso

composite def

Composition in a fixed finite mapping level. · OAI.Thomason.Construction.StrictOmegaCategory.GrayMapping.composite

Composition abbrev

A weak composition indexed by a finite variable set. · OAI.Problem335.BidegreeAverages.Composition

composition def

Compose arrows, then quotient only the length-two path space by the specified relations. · OAI.LittleFinitistic.QuadraticEncoding.composition

compositionAntidiagEquiv def

The function representation of a weak composition and the finite finsuppAntidiag representation used by occupation moments agree coordinatewise. · OAI.Problem335.BidegreeAverages.compositionAntidiagEquiv

compositionCandidates def

Pascal recursion gives a finite cover of all weak compositions. · OAI.GammaFPP.CrossingBridge.compositionCandidates

compositionExpansionMean def

Average over the finite uniform weak-composition space. · OAI.Problem335.compositionExpansionMean

CompositionExpression inductive

A finite, well-typed expression using generators and strict composition. · OAI.Thomason.Construction.StrictOmegaCategory.CompositionExpression

compositionGlobe def

The universal composite, with its common finite bound made explicit. · OAI.Thomason.Construction.StrictOmegaCategory.GrayMapping.compositionGlobe

compositionMap def

The universal composite is itself represented by a strict globe. · OAI.Thomason.Construction.StrictOmegaCategory.PastingGlobe.compositionMap

compositionPolynomial def

Two reversal transfers each consume intermediateLength + 1 transitions. · OAI.MaxCutGames.Foundations.Complexity.MachineComposition.compositionPolynomial

compositionPolynomial def

Two reversal transfers each consume intermediateLength + 1 transitions. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineComposition.compositionPolynomial

compoundPoisson def

The law of the sum of iid marks with an independent Poisson count. · OAI.DilutedSpinGlass.compoundPoisson

compoundRootLaw def

The actual independent Poisson container of iid root labels. · OAI.DilutedSpinGlass.compoundRootLaw

compress def

Every ambient compression is an element of the corner. · OAI.KadisonSimilarity.CStarCorner.compress

compress def

Compress by restricting to the second domain factor and taking the second codomain coordinate. · OAI.MaxCutGames.Appendix.CompressionCount.compress

compress def

Restriction in the domain and quotient in the codomain, as actual linear maps. This is the Fourier-frequency map of an affine restriction. · OAI.MaxCutGames.Appendix.Level.LinearRank.compress

compress def

The compressed dual frequency Q_A Y|_B. · OAI.MaxCutGames.Appendix.LinearIdentities.compress

compress def

Compress by restricting to the second domain factor and taking the second codomain coordinate. · OAI.UniqueGamesTheorem.Appendix.CompressionCount.compress

compress def

Restriction in the domain and quotient in the codomain, as actual linear maps. This is the Fourier-frequency map of an affine restriction. · OAI.UniqueGamesTheorem.Appendix.Level.LinearRank.compress

compress def

The compressed dual frequency Q_A Y|_B. · OAI.UniqueGamesTheorem.Appendix.LinearIdentities.compress

compressBlock def

The chosen block compression: restrict to B, then forget A. · OAI.MaxCutGames.Inverse.KMSAnalyticHybridCoordinates.compressBlock

compressBlock def

The chosen block compression: restrict to B, then forget A. · OAI.UniqueGamesTheorem.Inverse.KMSAnalyticHybridCoordinates.compressBlock

compressedFiberEquiv def

An exact compressed-fiber parametrization, valid over any ring. · OAI.MaxCutGames.Inverse.KMSAnalyticHybridCoordinates.compressedFiberEquiv

compressedFiberEquiv def

An exact compressed-fiber parametrization, valid over any ring. · OAI.UniqueGamesTheorem.Inverse.KMSAnalyticHybridCoordinates.compressedFiberEquiv

compressedRelativeUnitary def

The moved compression as an element of the specified relative corner. · OAI.WeakPureInfiniteness.compressedRelativeUnitary

compressFrequency def

The actual quotient-and-restriction map on Fourier indices in (A.1). · OAI.MaxCutGames.Appendix.Restriction.compressFrequency

compressFrequency def

The actual quotient-and-restriction map on Fourier indices in (A.1). · OAI.UniqueGamesTheorem.Appendix.Restriction.compressFrequency

Compression def

Ordinary finite compression of the actual dg A-module P on [a,a+l]. Take A=B tensor Bᵐᵒᵖ for a genuine dg B-bimodule. · OAI.LittleFinitistic.OrdinarySimulation.Compression

compression def

Group repetitions at every start/end pair. Triangle inequalities make this operation contractive, rather than an unjustified equality of costs. · OAI.DiamondDistortion.StartModel.compression

compression def

The actual compression word in the Steinberg kernel argument, not a matrix lift assumed to exist. · OAI.FiniteAlgebraicEnvelopes.Steinberg.compression

compression def

Horizontal compression accounts for the number of parking columns. · OAI.ForcedComputation.Recorder.Planar.compression

compression def

Restrict a dual map to ker X and then quotient by range X. · OAI.MaxCutGames.Appendix.FullCompression.compression

compression def

Restrict a dual map to ker X and then quotient by range X. · OAI.UniqueGamesTheorem.Appendix.FullCompression.compression

CompressionCode abbrev

The actual finite data retained by the compression construction. · OAI.MetricEntropyDuality.CompressionCode

compressionDecode def

Decode the common index, pivot labels, and rounded grouped weights into an actual real-valued function on the original domain. · OAI.MetricEntropyDuality.compressionDecode

compressionEquiv def

Reindex the actual maps by their actual compressed frequency. · OAI.MaxCutGames.Appendix.AntecedentCount.compressionEquiv

compressionEquiv def

Reindex the actual maps by their actual compressed frequency. · OAI.UniqueGamesTheorem.Appendix.AntecedentCount.compressionEquiv

compressionImage def

Enumerate the actual image of the finite code space under a fixed decoder. · OAI.MetricEntropyDuality.compressionImage

compressionList def

The actual finite family, fixed before choosing a coefficient vector. · OAI.MetricEntropyDuality.compressionList

compressionPositive def

Two-sided compression of a positive functional. · OAI.BoundedHochschild.KadisonRingrose.compressionPositive

compressionRadius def

A fixed positive radius satisfying the logarithmic compression condition. · OAI.MetricEntropyDuality.compressionRadius

compressionTransferPath def

Move the first compression to the second by rotating the cancelling pair. · OAI.WeakPureInfiniteness.compressionTransferPath

compressionTrial def

The actual compression trial scans every residue; it never uses the solution. · OAI.LowSpaceSubsetSum.CompressionSetup.compressionTrial

compressMap def

Compression to a central corner, which is an algebra homomorphism. · OAI.BoundedHochschild.CentralCorner.compressMap

compulsory def

Compulsory candidates that can actually affect the ramp at a test point. · OAI.KServer.ShortRoster.compulsory

computable def

Actual polynomial-time composition, with the intermediate length bound proved from the first machine's execution rather than postulated. · OAI.ContinuumCoulomb.TM2Composition.computable

computableInPolyTime def

Actual finite-alphabet polynomial-time sparse binary to dense unary renaming. Sparse name magnitudes are never expanded in unary. · OAI.MaxCutGames.BinaryRenameMachine.computableInPolyTime

computableInPolyTime def

A displayed finite machine and linear polynomial in the ordinary input bit length, proved by the transitions above rather than assumed by a caller. · OAI.MaxCutGames.BinaryTokenMachine.computableInPolyTime

computableInPolyTime def

The certificate contains the displayed total machine and its linear trace; it assumes no execution, parser-cost, or malformed-input premise. · OAI.MaxCutGames.BinaryTotalInputMachine.computableInPolyTime

computableInPolyTime def

The polynomial certificate supplies the explicit machine above. There is no assumed parser, verifier, or runtime hypothesis. · OAI.MaxCutGames.BinaryValidatorMachine.computableInPolyTime

computableInPolyTime def

A proved polynomial-time codec conversion, without an assumed time bound, an output-size certificate, or an unbounded integer in finite machine state. · OAI.MaxCutGames.Explicit.MachineSingleOrbitRuntime.computableInPolyTime

computableInPolyTime def

The input subtype records the mathematical involution property, while the same finite transition program is used for every instance of the fixed alphabet. · OAI.MaxCutGames.Explicit.MachineSubdivisionRuntime.computableInPolyTime

computableInPolyTime def

A genuine finite Boolean-stack machine for evaluating any fixed natural polynomial on zero-delimited unary input, with a polynomial transition bound. · OAI.MaxCutGames.Foundations.Complexity.CookLevin.PolynomialMachine.computableInPolyTime

computableInPolyTime def

A concrete finite machine computes the complete CNF serialization from the exact unary graph-table serialization, with a proved polynomial bound. · OAI.MaxCutGames.Foundations.Complexity.FinalCNFMachine.Program.computableInPolyTime

computableInPolyTime def

A polynomial raw run becomes a full canonical-output computation certificate. · OAI.MaxCutGames.Foundations.Complexity.MachineCanonicalOutput.computableInPolyTime

computableInPolyTime def

The actual canonical-output finite machine computes precisely the shared degree-d to degree-2d stage map in polynomial time. · OAI.MaxCutGames.Foundations.Complexity.MachineLazyTableRuntime.computableInPolyTime

computableInPolyTime def

A genuine finite-machine certificate for unary log2(n)+1; its linear polynomial is measured in the actual encoded input length. · OAI.MaxCutGames.Foundations.Complexity.MachineLogCounter.computableInPolyTime

computableInPolyTime def

A concrete machine reaches the canonical haltList encoding of the existing executable preprocessing map in polynomial time. · OAI.MaxCutGames.Foundations.Complexity.MachinePaddedOverlayRuntime.computableInPolyTime

computableInPolyTime def

The literal canonical output configuration computes the shared regular preprocessing map, with the actual complete raw execution as its witness. · OAI.MaxCutGames.Foundations.Complexity.MachineRegularTableRuntime.computableInPolyTime

computableInPolyTime def

One actual finite machine computes the complete logarithmic iteration. · OAI.MaxCutGames.Foundations.Complexity.MachineTableIteration.computableInPolyTime

computableInPolyTime def

A genuine polynomial-time certificate for the actual finite powering constructor, with the existing concrete codecs on both sides. · OAI.MaxCutGames.Foundations.Complexity.PoweringMachineRuntime.computableInPolyTime

computableInPolyTime def

Both stages supply complete executions, cleanup, and polynomial bounds; the sequential composition includes the physical interstage bit transfer. · OAI.MaxCutGames.Foundations.PCP.PreprocessingFinishRuntime.computableInPolyTime

computableInPolyTime def

Actual finite-machine polynomial-time computability, with identity encodings. · OAI.MaxCutGames.Reduction.MachineReverse.computableInPolyTime

computableInPolyTime def

A finite-machine polynomial-time certificate for reversed fixed substitution. · OAI.MaxCutGames.Reduction.MachineSubstitution.computableInPolyTime

computableInPolyTime def

Forward transducer output, using the checked actual reversal composition. · OAI.MaxCutGames.Reduction.MachineTransducer.computableInPolyTime

computableInPolyTime def

A genuine finite-machine certificate with identity encodings and bound X+1. · OAI.MaxCutGames.Reduction.MachineTransfer.computableInPolyTime

computableInPolyTime def

Actual finite-alphabet polynomial-time sparse binary to dense unary renaming. Sparse name magnitudes are never expanded in unary. · OAI.UniqueGamesTheorem.BinaryRenameMachine.computableInPolyTime

computableInPolyTime def

A displayed finite machine and linear polynomial in the ordinary input bit length, proved by the transitions above rather than assumed by a caller. · OAI.UniqueGamesTheorem.BinaryTokenMachine.computableInPolyTime

computableInPolyTime def

The certificate contains the displayed total machine and its linear trace; it assumes no execution, parser-cost, or malformed-input premise. · OAI.UniqueGamesTheorem.BinaryTotalInputMachine.computableInPolyTime

computableInPolyTime def

The polynomial certificate supplies the explicit machine above. There is no assumed parser, verifier, or runtime hypothesis. · OAI.UniqueGamesTheorem.BinaryValidatorMachine.computableInPolyTime

computableInPolyTime def

Its linear polynomial is measured in the actual input bit length. · OAI.UniqueGamesTheorem.Explicit.MachineFixedCopies.computableInPolyTime

computableInPolyTime def

A proved polynomial-time codec conversion, without an assumed time bound, an output-size certificate, or an unbounded integer in finite machine state. · OAI.UniqueGamesTheorem.Explicit.MachineSingleOrbitRuntime.computableInPolyTime

computableInPolyTime def

The input subtype records the mathematical involution property, while the same finite transition program is used for every instance of the fixed alphabet. · OAI.UniqueGamesTheorem.Explicit.MachineSubdivisionRuntime.computableInPolyTime

computableInPolyTime def

A genuine finite Boolean-stack machine for evaluating any fixed natural polynomial on zero-delimited unary input, with a polynomial transition bound. · OAI.UniqueGamesTheorem.Foundations.Complexity.CookLevin.PolynomialMachine.computableInPolyTime

computableInPolyTime def

A concrete finite machine computes the complete CNF serialization from the exact unary graph-table serialization, with a proved polynomial bound. · OAI.UniqueGamesTheorem.Foundations.Complexity.FinalCNFMachine.Program.computableInPolyTime

computableInPolyTime def

A polynomial raw run becomes a full canonical-output computation certificate. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineCanonicalOutput.computableInPolyTime

computableInPolyTime def

The actual canonical-output finite machine computes precisely the shared degree-d to degree-2d stage map in polynomial time. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineLazyTableRuntime.computableInPolyTime

computableInPolyTime def

A genuine finite-machine certificate for unary log2(n)+1; its linear polynomial is measured in the actual encoded input length. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineLogCounter.computableInPolyTime

computableInPolyTime def

A concrete machine reaches the canonical haltList encoding of the existing executable preprocessing map in polynomial time. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachinePaddedOverlayRuntime.computableInPolyTime

computableInPolyTime def

The literal canonical output configuration computes the shared regular preprocessing map, with the actual complete raw execution as its witness. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRegularTableRuntime.computableInPolyTime

computableInPolyTime def

One actual finite machine computes the complete logarithmic iteration. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineTableIteration.computableInPolyTime

computableInPolyTime def

A genuine polynomial-time certificate for the actual finite powering constructor, with the existing concrete codecs on both sides. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringMachineRuntime.computableInPolyTime

computableInPolyTime def

Both stages supply complete executions, cleanup, and polynomial bounds; the sequential composition includes the physical interstage bit transfer. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingFinishRuntime.computableInPolyTime

computableInPolyTime def

A total actual polynomial-time machine, including on malformed input words. The record retains the literal composed machine, without a cast of its machine field when simplifying the two reversals in the semantic output. · OAI.UniqueGamesTheorem.MachineTranslationTable.computableInPolyTime

computableInPolyTime def

Actual finite-machine polynomial-time computability, with identity encodings. · OAI.UniqueGamesTheorem.Reduction.MachineReverse.computableInPolyTime

computableInPolyTime def

A finite-machine polynomial-time certificate for reversed fixed substitution. · OAI.UniqueGamesTheorem.Reduction.MachineSubstitution.computableInPolyTime

computableInPolyTime def

Forward transducer output, using the checked actual reversal composition. · OAI.UniqueGamesTheorem.Reduction.MachineTransducer.computableInPolyTime

computableInPolyTime def

A genuine finite-machine certificate with identity encodings and bound X+1. · OAI.UniqueGamesTheorem.Reduction.MachineTransfer.computableInPolyTime

computableInPolyTimeOfEnumeration def

An explicit finite enumeration supplies the fixed cleanup list. · OAI.MaxCutGames.Foundations.Complexity.MachineCanonicalOutput.computableInPolyTimeOfEnumeration

computableInPolyTimeOfEnumeration def

An explicit finite enumeration supplies the fixed cleanup list. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineCanonicalOutput.computableInPolyTimeOfEnumeration

computation def

The actual sequential machine handles every bitstring, including malformed and noncanonical encodings, and outputs the fixed dense formula serialization. · OAI.MaxCutGames.BinaryInputReduction.computation

computation def

A fixed actual program works for every game and every alphabet size. · OAI.MaxCutGames.Explicit.MachineUniformRun.computation

computation def

The actual sequential machine handles every bitstring, including malformed and noncanonical encodings, and outputs the fixed dense formula serialization. · OAI.UniqueGamesTheorem.BinaryInputReduction.computation

computation def

A fixed actual program works for every game and every alphabet size. · OAI.UniqueGamesTheorem.Explicit.MachineUniformRun.computation

computationalCount def

A full computational-basis count. · OAI.QAC.computationalCount

computed def

The output list itself determines the forest. · OAI.ContingencyTables.ComputedLargeForest.computed

computedFamilyStorageCapacity def

Retained dictionary, all copied coarse inputs, all coarse-array capacities, and all local workspace allocations. Summing all allocations is a conservative bound on simultaneous storage. · OAI.EditApproximation.computedFamilyStorageCapacity

computedFullKernelWork def

Full execution counter: the exact fallback branch or the charged randomized execution, including its concrete local kernel costs. · OAI.EditApproximation.computedFullKernelWork

computedLeafCounts def

The exact two-level integer allocations in the implementation. · OAI.LowSpaceSubsetSum.computedLeafCounts

computedQueryAnswerWithWork def

Binary output conversion, with the explicit equal-string guard. · OAI.EditApproximation.computedQueryAnswerWithWork

computedQueryMemoRaw def

Coarse words here are the stored outputs of the separately charged coarse initializer. The adaptive mixed memo consumes them on demand. · OAI.EditApproximation.computedQueryMemoRaw

computedQueryMemoRawHeap def

Coarse words are borrowed from the separately allocated initializer. Every local buffer, guarded source read and final normalization is annotated. · OAI.EditApproximation.computedQueryMemoRawHeap

computedQueryMemoResult def

The executed mixed binary memo at the final refinement root. · OAI.EditApproximation.computedQueryMemoResult

computedQueryProgramDegree def

Degree of the complete adaptive program when physical reads have degree K. · OAI.EditApproximation.computedQueryProgramDegree

computedQueryReadDegree def

Degree contributed by guards, physical means, and scalar-source reads. · OAI.EditApproximation.computedQueryReadDegree

ComputedQueryRegisters structure

The remaining scalar premises of the concrete table dispatcher. · OAI.EditApproximation.ComputedQueryRegisters

computedQueryTemporaryCells def

Cumulative concrete local allocations, address workspace, suspended reads, dictionary paths and evaluator frames, over the actual cache trace. · OAI.EditApproximation.computedQueryTemporaryCells

computedTensorComplex def

The computed BC-balanced tensor complex in degrees -l,...,0 with both B-actions. · OAI.LittleFinitistic.OrdinarySimulation.computedTensorComplex

computedTensorIso def

Full bimodule chain comparison, including all zero terms and the genuine integer shift sign. · OAI.LittleFinitistic.OrdinarySimulation.computedTensorIso

ComputesAt def

Finite tape realization at a bound. This is a lemma parameter, not an assumption of either main theorem; the partial-recursive adapter must prove it. · OAI.FiniteAlgebraicEnvelopes.Compiler.Word.ComputesAt

concat def

Concatenation of the *ordered* coordinate sequences. · OAI.CoordinateSweeps.Grid.concat

concatenate def

Concatenation is an actual continuous operation. · OAI.Thomason.Construction.PathSpace.concatenate

concatenate def

The finite-range flatMap presentation used for rows in the manuscript. · OAI.TreeEdit.BinaryLower.BlockExtraction.concatenate

concatenateRows def

Concatenation of a finite ordered array of fixed-width blocks. · OAI.TreeEdit.BinaryLower.BlockExtraction.concatenateRows

concatMatrix def

The concatenated coefficient matrix. · OAI.RationalHitting.Rectangular.concatMatrix

concreteCenter def

The concrete centre, retaining its von Neumann structure. · OAI.KadisonKastler.concreteCenter

concreteDerivation def

The bounded derivation on the original represented algebra. · OAI.KadisonKastler.concreteDerivation

concreteDiagonalSequence def

The diagonal bounded sequence, represented on the concrete Hilbert matrix space. · OAI.KadisonKastler.concreteDiagonalSequence

concreteDiagonalSequenceUnitary def

The diagonal sequence remains an actual unitary before passing to a quotient. · OAI.KadisonKastler.concreteDiagonalSequenceUnitary

concreteDiagonalUnitary def

The concrete diagonal amplification of an inner unitary. · OAI.KadisonKastler.concreteDiagonalUnitary

concreteFactorTrace def

The trace is a complex-valued faithful tracial state with the continuity required by amplification. · OAI.KadisonKastler.concreteFactorTrace

concreteFiniteFactorModel def

The finite-factor presentation has no separate trace-normality or source-trace existence premise. · OAI.KadisonSimilarity.concreteFiniteFactorModel

concreteGNSIsometry def

The native GNS space embeds isometrically into the concrete trace space. · OAI.KadisonSimilarity.concreteGNSIsometry

concreteGroupAlgebra abbrev

The target algebra actually satisfies mathlib's concrete von Neumann algebra definition. · OAI.UltrapowerBridge.concreteGroupAlgebra

concreteHalfTraceProjectionData def

A concrete choice of the half projection used to make the centered reflection in the finite-factor argument. · OAI.KadisonSimilarity.concreteHalfTraceProjectionData

ConcreteLevelPair structure

A literal witness of the source level invariant. halfOrder is half its positive even moment order. countFactor is the prescribed upper multiplier C^(2*B*i+O_B(1)), to be instantiated uniformly in the actual construction. Finite … · OAI.TraceReconstruction.ConcreteLevelPair

concreteMatrixEntries def

Recover the matrix of original-space operators from the concrete amplified algebra. · OAI.KadisonKastler.concreteMatrixEntries

concreteMatrixUnits def

The same matrix units regarded inside their concrete image. · OAI.KadisonKastler.concreteMatrixUnits

concreteNestedTracialMatrices def

The matrix presentation required by the commutator argument comes from the actual increasing matrix subalgebras and their weak closure. · OAI.KadisonSimilarity.concreteNestedTracialMatrices

concretePlethysmDualEquiv def

Canonical duality on a nested symmetric power. · OAI.Problem346.concretePlethysmDualEquiv

concretePredualEquiv def

The given concrete algebra is the conjugate dual of an actual Banach space. · OAI.KadisonSimilarity.concretePredualEquiv

concretePreimage def

Ambient concrete inverse image, contained in S. · OAI.Generator.concretePreimage

concreteRademacher def

The successive dyadic signs, based at the identity projection. · OAI.KadisonSimilarity.concreteRademacher

concreteRealTracialSequence def

A concrete diffuse tracial von Neumann algebra supplies the full real sequence without an independent orthonormal-sequence existence premise. · OAI.KadisonSimilarity.concreteRealTracialSequence

concreteScalarMatrixHom def

The concrete constant outer scalar matrix copy. · OAI.KadisonKastler.concreteScalarMatrixHom

concreteTraceConjugationAction def

The multiplication law follows on the dense bounded algebra vectors. · OAI.KadisonKastler.concreteTraceConjugationAction

concreteTraceConjugationUnitary def

Conjugation by a unitary, acting on the canonical trace Hilbert space. · OAI.KadisonKastler.concreteTraceConjugationUnitary

concreteTraceEmbedding def

The algebraic inclusion into its actual trace Hilbert completion. · OAI.KadisonKastler.concreteTraceEmbedding

ConcreteTraceHilbert abbrev

The completion of the algebra in its canonical trace inner product. · OAI.KadisonKastler.ConcreteTraceHilbert

concreteTraceLeftAction def

Left multiplication in the canonical trace representation. · OAI.KadisonKastler.concreteTraceLeftAction

concreteTraceLeftUnitary def

Left multiplication by a unitary is a unitary on the completion. · OAI.KadisonKastler.concreteTraceLeftUnitary

concreteTraceLinear def

The vector trace bundled only as the linear map needed by the half-projection construction. · OAI.KadisonSimilarity.concreteTraceLinear

concreteTraceRightAction def

Right multiplication in the canonical trace representation. · OAI.KadisonKastler.concreteTraceRightAction

concreteTraceRightUnitary def

Right multiplication by a unitary is a unitary on the completion. · OAI.KadisonKastler.concreteTraceRightUnitary

condensingSequence def

The iteration begins at C and takes actual closed-convex images. · OAI.Tingley.condensingSequence

condensingStep def

The literal closed-convex image operator used in the condensing argument. · OAI.Tingley.condensingStep

Condition structure

Countable partial functions I × κ → 2, represented by their graphs. Countability of the graph is equivalent to countability of the domain for these single-valued graphs. Conditions are ordered by reverse inclusion. · OAI.PartitionConsistency.CohenForcing.Condition

Condition structure

Henkin conditions are genuine consistent first-order contexts. No satisfiability or model-existence assertion occurs in their definition. · OAI.PartitionConsistency.Completeness.Condition

Condition abbrev

The actual product of the Cohen and ground diagram posets. · OAI.PartitionConsistency.ProductForcing.Condition

condition def

The highest Hessian tensor is symmetric in its derivatives, has zero trace, and satisfies the differentiated Bianchi equation. · OAI.PositiveEinsteinFour.HighestHessian.condition

ConditionAction structure

The condition part of a symmetric system. A homomorphism into permutations of the actual set of conditions is enough to lift the action to set names. Preservation of the forcing order is used separately for the union constructor. · OAI.PartitionConsistency.SetForcingNames.ConditionAction

conditional def

The actual conditional distribution on the true prefix fiber. · OAI.KServer.HierarchyCounts.conditional

conditional_main def

The conditional main theorem in each positive finite dimension n+1. · OAI.Thomason.Construction.FiniteReflectionInput.conditional_main

conditional_main def

The conditional main model and Quillen-equivalence conclusion in omega dimension. · OAI.Thomason.Construction.NormalizedSimplexIdentities.conditional_main

conditionalAgreement def

Actual independent response-law success, with the literal projection test. · OAI.MaxCutGames.Decoder.PrivateStrategy.conditionalAgreement

conditionalAgreement def

Actual independent response-law success, with the literal projection test. · OAI.UniqueGamesTheorem.Decoder.PrivateStrategy.conditionalAgreement

conditionalAverage def

Average of the remaining random bijection, conditioned on the entire paths, not only endpoints. The reference affects only unitary identifications. · OAI.CoordinateSweeps.Grid.Holes.conditionalAverage

ConditionalData abbrev

The finite named-field part of a conditional test. · OAI.HyperbolicObstruction.IncidenceTemplate.ConditionalData

conditionalDensity def

The density is chosen only after its concrete geometric existence has been proved. · OAI.Problem358.ConvexTriangularSampler.conditionalDensity

conditionalEndpointProbability def

Conditional probability of an endpoint specification, from the original independent line sample space; the full trajectories of the holes are retained. · OAI.CoordinateSweeps.Grid.Holes.conditionalEndpointProbability

conditionalExpectation def

Conditional expectation under the actual Gibbs law given all other spins. · OAI.SKGap.conditionalExpectation

conditionalFairSplitMass def

Conditional on a site hit, the two choices are independent fair bits; an absent site gives two absent splits. · OAI.JointDickman.conditionalFairSplitMass

ConditionalFamily abbrev

Conditional scalar sampling functions indexed by the length of the existing prefix. · OAI.Problem358.TriangularRecursion.ConditionalFamily

ConditionalFieldOrbitLaw def

Conditional jointly on the spectrum and field, the pressure has its Haar orbit law. This formulation allows arbitrary dependence across dimensions and between the two data sets. · OAI.InvariantIsing.ConditionalFieldOrbitLaw

conditionalKernel def

Finite disintegration, including genuinely zero-probability input states. · OAI.KServer.Rounding.conditionalKernel

ConditionalMain def

Faithful conditional main target; R=2^r. The harmless strengthening z_*≤1/2 makes the probability-law range explicit. · OAI.CoordinateSweeps.ConditionalMain

conditionalMass def

Independent private decoding success on a transferred witness. · OAI.MaxCutGames.Decoder.ConstantSelection.conditionalMass

conditionalMass def

Independent private decoding success on a transferred witness. · OAI.UniqueGamesTheorem.Decoder.ConstantSelection.conditionalMass

conditionalMean def

Exact finite conditional expectation over k future fair bits. Histories are chronological from the right, as in the energy theorems. · OAI.BoundedTreePotentials.Dyadic.conditionalMean

conditionalPairMeasure def

Keep the transverse coordinate and independently resample the leaf coordinate using its conditional law. · OAI.DukePrimeDegree.conditionalPairMeasure

conditionalPrimitiveLaw def

The literal conditional-list oracle used by the rank coefficients. · OAI.TraceReconstruction.conditionalPrimitiveLaw

conditionalProb def

Conditional probability of σ_I=a, with σ_J=b and σ_F=τ prescribed. All sums are finite and all fields are finite real numbers. · OAI.BufferedIsing.conditionalProb

conditionalQuantile def

Actual conditional inverse-CDF sampling on a valid convex prefix. · OAI.Problem358.ConvexTriangularSampler.conditionalQuantile

conditionalScheduledMean def

The scheduled conditional mean on a literal full path, evaluated after e transitions. Before the first branching this is the usual conditional expectation; no independently sampled prefix occurs in this definition. · OAI.DilutedSpinGlass.ReducedTopology.conditionalScheduledMean

conditionalScheduledProduct def

Proper-child mean product at an actual evaluation prefix, expressed without a choice of terminal reserve. · OAI.DilutedSpinGlass.ReducedTopology.conditionalScheduledProduct

conditionalSourceLaw def

Normalize the actual source measure on each observation fiber. · OAI.MatroidProphet.Secretary.conditionalSourceLaw

ConditionalStar def

A conditional star: branches agree exactly on the named fields. The finite law placed on it below need not be uniform. · OAI.HyperbolicObstruction.ConditionalStar

ConditionalTest structure

A literal conditional test of Section 4, with k+1 branches. · OAI.HyperbolicObstruction.IncidenceTemplate.ConditionalTest

conditionalTripleWeight def

The source specifies all seven nonzero triple masses divided by the reveal rate. Outside the observation set the triple is deterministically zero. · OAI.MatroidProphet.Secretary.conditionalTripleWeight

conditioned def

The exact finite space conditioned on girth and prior prescriptions. · OAI.TorsionFreeZeroDivisors.ConditionedMatchings.Matching.conditioned

conditionedChangedEquiv def

The zero perturbation is never a changed perturbation, so conditioning does not remove any element from the change event. · OAI.MaxCutGames.Quadratic.conditionedChangedEquiv

conditionedChangedEquiv def

The zero perturbation is never a changed perturbation, so conditioning does not remove any element from the change event. · OAI.UniqueGamesTheorem.Quadratic.conditionedChangedEquiv

conditionedEquation def

Conditioned storage uses only satisfying-assignment coordinates, so identical restricted canonical inputs are the same variable, even for distinct raw queries. · OAI.MaxCutGames.Foundations.Hastad.FoldedEquation.conditionedEquation

conditionedEquation def

Conditioned storage uses only satisfying-assignment coordinates, so identical restricted canonical inputs are the same variable, even for distinct raw queries. · OAI.UniqueGamesTheorem.Foundations.Hastad.FoldedEquation.conditionedEquation

conditionedFiberEquiv def

Surjectivity makes all output fibers equally large inside an invariant event, without assuming that this event has positive probability. · OAI.MaxCutGames.Gadget.Quotient.conditionedFiberEquiv

conditionedFiberEquiv def

Surjectivity makes all output fibers equally large inside an invariant event, without assuming that this event has positive probability. · OAI.UniqueGamesTheorem.Gadget.Quotient.conditionedFiberEquiv

ConditionedOracle inductive

Storage distinguishes an actually empty legal domain from a folded table over its legal assignments. The empty case needs no nonexistent fold point. · OAI.MaxCutGames.Foundations.Hastad.ConditionedOracle

ConditionedOracle inductive

Storage distinguishes an actually empty legal domain from a folded table over its legal assignments. The empty case needs no nonexistent fold point. · OAI.UniqueGamesTheorem.Foundations.Hastad.ConditionedOracle

conditionedProductEquiv def

An invariant event factors as an independent logical output and one fiber. · OAI.MaxCutGames.Gadget.Quotient.conditionedProductEquiv

conditionedProductEquiv def

An invariant event factors as an independent logical output and one fiber. · OAI.UniqueGamesTheorem.Gadget.Quotient.conditionedProductEquiv

conditionedShiftEquiv def

Translating by a logical shift bijects output fibers even after conditioning on an arbitrary shift-invariant event. · OAI.MaxCutGames.Gadget.Quotient.conditionedShiftEquiv

conditionedShiftEquiv def

Translating by a logical shift bijects output fibers even after conditioning on an arbitrary shift-invariant event. · OAI.UniqueGamesTheorem.Gadget.Quotient.conditionedShiftEquiv

conditionedSigmaEquiv def

Splitting an invariant event into its output fibers. · OAI.MaxCutGames.Gadget.Quotient.conditionedSigmaEquiv

conditionedSigmaEquiv def

Splitting an invariant event into its output fibers. · OAI.UniqueGamesTheorem.Gadget.Quotient.conditionedSigmaEquiv

Conditions structure

All probability hypotheses are the literal finite matching conditions. No expected count or entropy estimate is a field of this certificate. · OAI.TorsionFreeZeroDivisors.TwoSideStage.Conditions

conditionSelector def

This candidate does not yet mention a generic filter. · OAI.PartitionConsistency.TailSelector.TailData.conditionSelector

conditionWeights def

The actual weights obtained by restricting to an event and normalizing. · OAI.MaxCutGames.Foundations.Repetition.conditionWeights

conditionWeights def

The actual weights obtained by restricting to an event and normalizing. · OAI.UniqueGamesTheorem.Foundations.Repetition.conditionWeights

condP def

Conditional distribution on an event; returns the old law when probability is zero. · OAI.FourierLLogL.Law.condP

conductorKernelCandidates def

The eight possible quotients of a conductor by its odd kernel part, with the four possible sign/two factors. · OAI.Ostmann.conductorKernelCandidates

Cone structure

Shared interface of the instructions at a pointer. alpha is the unit ultimately offering the visit (test tag offered by another unit). Knot is a pure capped append on designated paths. · OAI.ExactDerandomization.Chime.Cone

cone def

The solid pointed cone generated by (1,K). · OAI.GeneralMahler.Body.cone

cone def

The bounded compatible families form a cone on the original diagram. · OAI.Thomason.Construction.StrictOmegaCategory.LimitConstruction.cone

coneAdjointAutomorphism def

The induced adjoint Lie automorphism on the cone quotient. · OAI.PolycyclicRecognition.CartanWordModel.coneAdjointAutomorphism

coneBasis def

Append the new final vertex to a basis simplex. · OAI.Thomason.Construction.SimplexComplex.coneBasis

coneCartanSubmodule def

The same mixed ideal, viewed as a submodule for the actual Cartan action. · OAI.PolycyclicRecognition.CartanWordModel.coneCartanSubmodule

coneCell def

Cone a cell by applying the positive prism to its explicit interval table. · OAI.Thomason.Construction.SimplexComplex.coneCell

coneChainMap def

The concrete positive prism from the old simplex to the new vertex. · OAI.Thomason.Construction.SimplexComplex.coneChainMap

ConeData structure

The metric description of nonnegative radial dilations in a Euclidean cone, with the angular cosine eliminated by polarization. No linear or associative addition is included. · OAI.HyperbolicObstruction.ConeData

coneData def

The required radial metric law, with no asserted CAT(0) axiom. The CAT(0) property is a separate construction below. · OAI.HyperbolicObstruction.TangentCone.coneData

ConeDensity structure

An integrable aggregate of vectors in a fixed acute cone. It covers the positive pieces by taking their sum; no direction is imposed on errors. · OAI.LocalCancellation.ConeDensity

ConeDensity structure

An integrable aggregate of vectors in a fixed acute cone. It covers the positive pieces by taking their sum; no direction is imposed on errors. · OAI.SignedDisk.LocalCancellation.ConeDensity

coneDifferential def

Transport the genuine cone differential through this coordinate equivalence. · OAI.LittleFinitistic.FiniteColumnComplex.coneDifferential

coneEnergy def

Squared local polar budget. Supremums in ENNReal avoid any assumption of an optimizing antichain or endpoint. · OAI.BoundedTreePotentials.Segments.coneEnergy

coneEquiv def

Reindex the actual cone of a total map into the actual cones of its column maps. · OAI.LittleFinitistic.FiniteColumnComplex.coneEquiv

coneEvaluation def

The section evaluation, in the coordinate ring of the source's actual fiber product of affine charts. · OAI.SourcePullbackChart.coneEvaluation

ConeFamily def

Finite incomparable tests wholly contained in a specified descendant cone. The component index, prefix, root and endpoint conventions are unchanged. · OAI.BoundedTreePotentials.Segments.ConeFamily

coneForm def

The real cone form in equation free:cone-form. · OAI.DefocusingNLS.coneForm

ConeGroupMaps structure

Exponential and group maps for the one fixed mixed-word ideal. · OAI.PolycyclicRecognition.CartanWordModel.ConeGroupMaps

ConeGroupMeasures structure

Compatible Haar data; no profile, metric bound or offset is included. · OAI.PolycyclicRecognition.CartanWordModel.ConeGroupMeasures

ConeIdeal structure

An ideal in the source's bounded cone: nonempty and downward closed, contained in the cone, and closed under its finite nonempty joins. Closure under binary joins suffices, since singleton joins add no new condition. · OAI.PartitionConsistency.DiagramOrder.ConeIdeal

coneInclusion def

Include the old vertices before a new final vertex. · OAI.Thomason.Construction.SimplexComplex.coneInclusion

coneIndex abbrev

Append the greatest vertex to an original nerve simplex. · OAI.Thomason.Construction.ConePrism.coneIndex

coneInterchange def

The canonical additive comparison for one cone. · OAI.LittleFinitistic.RectificationTotal.coneInterchange

coneIso def

An isomorphism of the two input columns gives an isomorphism of actual cones. · OAI.ExternalTensor.Section4.SourceFiltration.coneIso

coneIso def

The triangular cone model is isomorphic to the native cone, not an assumed cone replacement. · OAI.LittleFinitistic.FiniteColumnComplex.coneIso

ConeLowerSemicontinuous def

Lower semicontinuity is required only on the nonnegative cone. · OAI.Kaplansky.PositiveDimensionFunctional.ConeLowerSemicontinuous

coneMap def

A strictly commuting square induces the ordinary diagonal map of cones. · OAI.LittleFinitistic.RectificationTotal.coneMap

coneMap def

The actual map of cone simplices extending an original simplex morphism. · OAI.Thomason.Construction.ConePrism.coneMap

coneMorphism def

The source's cone-evaluation morphism before completion, constructed by honest gluing on the literal W₀ = X ×[P²] V₀. · OAI.SourcePullbackChart.coneMorphism

ConePair structure

Two specified cone triangles, with the first maps the denominators to invert. · OAI.LittleFinitistic.TwoConeVerdier.ConePair

ConePoint abbrev

Nonzero homogeneous coordinate vectors on the actual equation G=0. · OAI.Nagata.Workers.W24.ConePoint

conePolarTemplate def

The unit-ball slice of the negative polar of the closed weight cone. · OAI.PolycyclicRecognition.conePolarTemplate

coneProbeFirst def

The endpoint carrying both the ideal probe and its quotient section. · OAI.PolycyclicRecognition.coneProbeFirst

coneProbeSecond def

The neighboring endpoint carrying the same ideal probe. · OAI.PolycyclicRecognition.coneProbeSecond

coneProjection def

The canonical Lie algebra projection onto the mixed-word quotient. · OAI.PolycyclicRecognition.CartanWordModel.coneProjection

cones def

The cone choices are made once, in K, before any evaluation module is chosen. · OAI.LittleFinitistic.SourceVerdier.cones

coneTriangular def

The cone of a componentwise total map is triangular with the actual column cones on its diagonal. · OAI.LittleFinitistic.FiniteColumnComplex.coneTriangular

coneVertex def

The constant map to the new final vertex. · OAI.Thomason.Construction.SimplexComplex.coneVertex

coneWordIdeal def

The ideal generated by the letters outside the allowed cone of weights. · OAI.PolycyclicRecognition.CartanWordModel.coneWordIdeal

Config structure

`Config k q` stores either a current state among `q` states or `none` for halting, plus one Boolean list for each of `k` stacks. · OAI.CurrentKS.Config

Configuration structure

All hard-wall, unoriented 0/2-degree configurations with the two specified ports occupied. This is finite graph data, not a probabilistic input. · OAI.CriticalHoneycomb.Transfer.FiniteConfigurations.Configuration

Configuration structure

Fixed ordered bases, a head gap, and finite tags are the finite search data of the finite compiler. Workspace quota insertion is NOT an internal step. · OAI.FiniteAlgebraicEnvelopes.Compiler.Configuration

Configuration abbrev

This declaration is an exact comparator-selected statement. Read the Lean statement below for its mathematical content. · OAI.KServer.Configuration

Configuration abbrev

Labeled configurations; no injectivity condition, so repetitions are allowed. · OAI.KServer.Configuration

configurationChartCoordinates₂ def

Explicit affine coordinates in a simultaneous product chart. · OAI.Nagata.ProjectiveGeometry.configurationChartCoordinates₂

configurationEquiv def

This is a bijection on actual machine configurations, not just on states. · OAI.MaxCutGames.Foundations.Complexity.MachineStateEquiv.configurationEquiv

configurationEquiv def

This is a bijection on actual machine configurations, not just on states. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineStateEquiv.configurationEquiv

configurationIndexEquiv def

Explicit enumeration of the k² original marks followed by n extra marks. · OAI.Nagata.W06.LineArrangement.configurationIndexEquiv

configurationLength def

All stacks are counted, even when their symbol types differ. · OAI.MaxCutGames.Foundations.Hastad.SourceRuntimeSpace.configurationLength

configurationLength def

All stacks are counted, even when their symbol types differ. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceRuntimeSpace.configurationLength

configurationOfNonzero def

This packages derived, not assumed, local degree conditions. · OAI.CriticalHoneycomb.Transfer.CenterPartition.configurationOfNonzero

configurationOfSides def

The exact local 0/2 rule, rather than an assumed path partition, gives an admissible finite configuration on the actual honeycomb graph. · OAI.CriticalHoneycomb.Transfer.FiniteConfigurations.configurationOfSides

confinementSet def

The common compact set obtained from the actual derivative estimates. · OAI.SLEExactGauge.HyperbolicGuideCompactness.confinementSet

conflicts def

Reservation is deterministic and persists even after a random retirement. The previous prefix is also excluded, giving the stipulated one-step cooldown. · OAI.KServer.Chronological.conflicts

conformalMass def

For a conformal identification, w is the continuous weight |J'|^d. · OAI.SLEExactGauge.Multipoint.conformalMass

cong def

Collapse precisely the pairs whose first coordinate is zero. This quotient is the zero product (K \ {0}) × P ⊔ {0} from the algebra construction. · OAI.FiniteAlgebraicEnvelopes.ZeroProduct.cong

congr def

Transport the actual tuple and its actual adjoints through unitary coordinates. The row and column identities are preserved. · OAI.Kaplansky.NormalizedHermitianTuple.congr

congr def

Actual balanced tensor congruence under module equivalences in both factors. · OAI.LittleFinitistic.BalancedTensor.congr

congr def

Actual balanced tensor congruence retaining the outside left action. · OAI.LittleFinitistic.LeftTensor.congr

congr def

An actual right-linear, left-equivariant equivalence retains both actions. · OAI.LittleFinitistic.RetainedBimodule.congr

congr def

Replacing either potential outside the domain preserves the complete integrated Hessian data, including its dense core. · OAI.RegularPointCoordinates.LocalHessianPairing.congr

congrLabel def

Change the displayed port labeling by a literal pointwise equality. · OAI.TorsionFreeZeroDivisors.RealizationTransport.congrLabel

congrLeft def

A genuine bimodule equivalence induces an equivalence of the actual balanced tensors. · OAI.DerivedBar.BimoduleTensor.congrLeft

congrLeft def

Actual equivalences of the right factor on the left preserve the outside bimodule. · OAI.LittleFinitistic.TwoSidedTensor.congrLeft

Congruence def

The full ordered set of congruences, ordered by inclusion of relations. · OAI.FiniteCongruence.Congruence

congruenceOf def

Pullback of equality on G/K, hence an actual equivalence relation on G/H. · OAI.FiniteCongruence.congruenceOf

CongruenceTest def

The source congruence test, represented without an assumed polynomial identity or an assumed primality oracle. · OAI.ExactQuantumFactoring.Primality.CongruenceTest

CongruentAE def

Equality of regions up to ambient flat isometry and a null set. · OAI.CubicTorus.CongruentAE

conjugate def

g P g⁻¹, with membership expressed without existential witnesses. · OAI.HarmonicArtin.ParabolicIntersections.conjugate

conjugate def

Conjugation by a fixed path, on unquotiented loop spaces. · OAI.Thomason.Construction.PathSpace.conjugate

conjugate def

Change coordinates in both the retract and its ambient space. · OAI.Thomason.Construction.TopologicalDeformation.Data.conjugate

conjugate def

Positive conjugation inside the same (possibly nonunital) C*-algebra. · OAI.TraceConeClassification.Positive.conjugate

conjugate def

Conjugating by a second genuine unitary flow preserves all structural properties. For billiards the second flow is angular rotation. · OAI.TriangularBilliards.Analysis.HilbertFlow.conjugate

conjugateAddHom def

Additive notation for conjugation in the Steinberg group. · OAI.FiniteAlgebraicEnvelopes.Steinberg.conjugateAddHom

conjugateBlockPrefix def

A completed alternating prefix: each pair is a conjugated block and a following kernel letter. · OAI.KadisonKastler.conjugateBlockPrefix

conjugateCartanWeight def

Complex conjugation sends a weight to its conjugate after conjugating its Cartan argument. · OAI.PolycyclicRecognition.conjugateCartanWeight

conjugateCharacter def

Conjugation of the values of a Dirichlet character. · OAI.TwoPointCorrelations.conjugateCharacter

conjugateConstantPath def

The image of the constant loop under conjugation is joined to the constant loop. This path supplies the needed basepoint comparison. · OAI.Thomason.Construction.PathSpace.conjugateConstantPath

conjugatedBilinear def

Transport an arbitrary bilinear multiplication by a linear equivalence. · OAI.KadisonKastler.conjugatedBilinear

conjugatedConstantEmbedding def

The prescribed subfactor is embedded as constants, conjugated inside A^ω, then included in B^ω. Thus membership in the prescribed ultrapower is retained. · OAI.KadisonSimilarity.conjugatedConstantEmbedding

conjugatedDiagonal def

The finite family appearing in a word of length r. · OAI.KadisonKastler.conjugatedDiagonal

conjugatedMatrixUnits def

Map the chosen matrix units under the inner star automorphism. · OAI.KadisonKastler.conjugatedMatrixUnits

conjugatedMatrixUnitsSequence def

Conjugated matrix units are chosen separately at every coordinate. · OAI.KadisonKastler.conjugatedMatrixUnitsSequence

conjugatedProjectionWord def

The ith letter is conjugated by the power remaining to its right. · OAI.KadisonKastler.conjugatedProjectionWord

conjugatedWeight def

Constant coordinates on the input and output may differ (Taylor on the input, reverse Taylor on the output in the application). · OAI.RationalHitting.LaurentReduction.conjugatedWeight

conjugateEquiv def

Conjugation gives an actual homotopy equivalence of loop spaces. · OAI.Thomason.Construction.PathSpace.conjugateEquiv

conjugateFunctional def

Conjugating the test-vector slot makes it rational-linear. · OAI.KadisonKastler.ScalarOperatorKernel.CoherentCore.conjugateFunctional

conjugateIndexAction def

The action on the two smooth pieces of the zero-extended conjugate field minus the explicit endpoint bump. · OAI.PositiveEinsteinFour.Jacobi.conjugateIndexAction

conjugateLinearCharacter def

Complex conjugation of a one-dimensional character. · OAI.DukePrimeDegree.conjugateLinearCharacter

conjugateMatrixUnits def

Actual full matrix units in the conjugate right algebra. · OAI.KadisonKastler.conjugateMatrixUnits

conjugateOperatorMap def

Conjugating a complex-linear map twice restores complex linearity. · OAI.KadisonKastler.conjugateOperatorMap

conjugateOrderIso def

Conjugation as an order isomorphism of the subgroup lattice. · OAI.HarmonicArtin.ParabolicIntersections.conjugateOrderIso

conjugateRightAlgebra def

The actual image J R J, as a concrete von Neumann algebra. · OAI.KadisonKastler.conjugateRightAlgebra

Conjugates def

This expresses v M v* = N in the original Hilbert space. · OAI.KadisonKastler.Conjugates

conjugation def

The standard conjugation, constructed from the actual vector trace. · OAI.KadisonKastler.StandardTracialPosition.conjugation

conjugationDifferentialHom def

The adjoint action as a monoid homomorphism into continuous linear endomorphisms of the actual tangent Lie algebra. · OAI.PolycyclicRecognition.conjugationDifferentialHom

conjugationFamily def

The orbit family of an operator under an actual unitary representation. · OAI.KadisonKastler.ScalarInvariantMean.conjugationFamily

conjugationHom def

The root-inverting endomorphism fixes the coefficient algebra. · OAI.CirculantHadamard.RamifiedConjugation.conjugationHom

conjugationLift def

Faithful ambient conjugation forces the action laws. · OAI.PolycyclicRecognition.conjugationLift

conjugationLiftOfNormal def

A normal image supplies the pointwise automorphisms by transport from the actual subgroup. In particular, no action is part of this premise. · OAI.PolycyclicRecognition.conjugationLiftOfNormal

conjugationOperator def

Conjugating a linear operator by a conjugate-linear isometry gives a complex-linear operator. · OAI.KadisonKastler.conjugationOperator

conjugationOrbitMap def

The conjugation orbit map exists because its value is unchanged by the actual lattice point stabilizer. · OAI.DukePrimeDegree.conjugationOrbitMap

conjugationRealStarAlgEquiv def

An involutive antiunitary acts by a real star-algebra automorphism. · OAI.KadisonKastler.conjugationRealStarAlgEquiv

conjugationUnitary def

Conjugating a unitary by an involutive conjugation is unitary. · OAI.KadisonKastler.conjugationUnitary

conjugationUnitaryHom def

The right-action operation is a homomorphism on the unitary group. · OAI.KadisonKastler.conjugationUnitaryHom

Connected def

Connectedness of the nonempty one-skeleton. · OAI.CoboundaryExpanders.Complex.Connected

Connected def

Connectedness of the one-skeleton, including nonemptiness. · OAI.CoboundaryExpanders.Complex.Connected

Connected def

Finite paths in the undirected marked graph, including paths of length zero. · OAI.FiniteCongruence.Connected

Connected def

`G.Connected T` says every nonempty proper set of vertices has at least one edge of `T` crossing to its complement. · OAI.StrongThinTree.MultiGraph.Connected

Connected def

Used-support connectivity in a form that exactly expresses closure under closed walks. It ignores incidental isolated vertices and records a root. · OAI.Superstring.Hierarchical.Loop.Connected

connected def

The equivalence relation whose classes are the pairs. · OAI.OneWayLiveness.Pairing.connected

connectedDecision def

Operational component equality has a finite decision procedure. Its only primitives are the finite table, tagged cells, and bounded finite search. · OAI.FiniteAlgebraicEnvelopes.Compiler.Scan.connectedDecision

connectedMatchingNumber def

The supremum of the cardinalities of pairwise touching matchings. For finite vertex types, this supremum is attained. · OAI.HadwigerMatching.connectedMatchingNumber

connecting def

The ideal action constructs the connecting morphism and exact sequences of coherent sheaves on the Cartier divisor. · OAI.ActualCartier.connecting

connectionCoeff def

Cartesian Christoffel coefficients written in their invariant six-term form. · OAI.HarmonicCounterexample.Berger.connectionCoeff

connectionIndexValue def

Exactly seeds and three comparisons per private buffer, with two distinct copies of A's full-product identity self-connection. · OAI.WLIdentification.ProductCircuit.connectionIndexValue

connectionMatrix def

The endomorphism v ↦ Γ(u,v), after raising the first-kind connection. This nine-low-rank decomposition permits exact finite-dimensional traces. · OAI.HarmonicCounterexample.Berger.connectionMatrix

conormalBaseChangeIso def

The conormal identity i*J ≅ f*(z*I) for the two kernel ideals. · OAI.SourcePullbackChart.conormalBaseChangeIso

conormalFunctor def

Tensor by the conormal line, on coherent sheaves of the divisor. · OAI.ActualCartier.conormalFunctor

conormalOverlapRatio def

The linear transition, in the common-open ring. · OAI.SourceSymmetry.conormalOverlapRatio

conormalTensor def

The conormal twist, as a coherent sheaf ON the Cartier divisor. · OAI.ActualCartier.conormalTensor

cons def

Binding a raw actual term does not normalize or evaluate its syntax. · OAI.PureTypeSystem.ObservationValues.Environment.cons

Consecutive def

Neighbors in the cyclic order, including the wrap-around edge. · OAI.PlanarL1.Columns.Consecutive

consecutive def

Extract two consecutive jets, currying the higher jet as a derivative. · OAI.ForcedComputation.BoundedSpatialJets.consecutive

consecutiveBlocks def

Consecutive blocks, with their lengths prescribed explicitly. · OAI.Problem335.consecutiveBlocks

Conservative def

Rank conservation on every ordered port (including side branches). · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.Decoration.Conservative

Conservative structure

Conservation of a submonoid means more than an injective homomorphism: right factors of old comparisons, entire old intersections, and absence of old common right multiples are preserved. · OAI.PartitionConsistency.DiagramOrder.Conservative

Consistent def

Ordinary syntactic consistency: there is no finite natural-deduction proof of falsity from the theory, with any finite supply of free variables. · OAI.PartitionConsistency.Consistent

Consistent def

Consecutive edge endpoint agreement. A marked exterior break can be encoded by one extra formal item with distinct prescribed before/after endpoints. · OAI.TorsionFreeZeroDivisors.BoundarySurgery.Consistent

constant def

The actual constant copy of the original factor. · OAI.KadisonKastler.TraceQuotientMap.constant

constant def

A constant has a single order-zero term. · OAI.Problem335.FactorialExpansion.constant

constant def

Constant bounded sequences. · OAI.Tingley.UltraSequence.constant

constantAlgebraUnitaryFamily def

A family of algebra unitaries determines a family of constant unitary fields. · OAI.KadisonKastler.constantAlgebraUnitaryFamily

constantCopies def

The unnormalized constant-coordinate inclusion. · OAI.KadisonSimilarity.constantCopies

constantEmbedding def

Constant sequences embed the original space isometrically and linearly. · OAI.Tingley.constantEmbedding

constantFiberIso def

The fibre of a constant label over its value is the entire original category. · OAI.Thomason.Construction.StrictOmegaCategory.constantFiberIso

constantFieldRepresentation def

A constant operator representation in the actual field quotient. · OAI.KadisonKastler.constantFieldRepresentation

constantHilbertInclusion def

The isometric constant-vector inclusion in the completed Hilbert ultrapower. · OAI.KadisonSimilarity.constantHilbertInclusion

constantInterpolationWeight def

The constant-coefficient functional expressed in the Lagrange basis. · OAI.MatrixMultiplication.AuxiliarySeparation.constantInterpolationWeight

constantL2 def

Constant observables as actual L2 vectors. · OAI.DukePrimeDegree.constantL2

constantLabel def

The unshifted constant-slope label used at a fixed minimum level. · OAI.Problem326.Affine.constantLabel

constantLabel def

The constant strict label, including every positive-dimensional cell. · OAI.Thomason.Construction.StrictOmegaCategory.constantLabel

constantLabels def

The fixed labels at either external vertex. · OAI.Problem335.Pairings.constantLabels

constantPath def

A constant quantile has deficit 1 - max q r. · OAI.InvariantIsing.constantPath

constantPointedMatrixProduct def

A constant two-dimensional product with its actual matrix-state moments. · OAI.KadisonKastler.constantPointedMatrixProduct

constantPreHilbert def

The ultralimit seminorm of a constant family is its original norm. · OAI.KadisonSimilarity.constantPreHilbert

constants def

Constants on every open come from the given structure morphism. · OAI.ActualCotangent.constants

constants def

The field of constants of a derivation on a field. · OAI.ComplexCancellation.DifferentialSlice.constants

constants def

Constant Laurent polynomials as a base-algebra map. · OAI.StableCoordinate.DeterminantLiftWeights.constants

constantTracialGNSInclusion def

The trace-preserving constant inclusion gives the actual GNS inclusion. · OAI.KadisonSimilarity.constantTracialGNSInclusion

constituentFinalCountRate def

This single constant bounds both the prime-test count and the quadratic pairwise-coprimality count of the literal final histories. · OAI.Ostmann.constituentFinalCountRate

ConstituentHProductLower def

The endpoint is required only where the full inherited coefficient is live. · OAI.Ostmann.ConstituentHProductLower

constituentOriginalMatchedPair def

The pair is integrated before the external-pivot sum. The second H prior is kept explicitly, with the inverse of the matching in the diagonal. · OAI.Ostmann.constituentOriginalMatchedPair

constituentPivot def

Choose the first prime constituent solely to name the pivot column. All constituents remain present in the grouped phase and prior. · OAI.Ostmann.constituentPivot

constituentTransferWeight def

Pairwise prime support is retained inside each grouped atom as well as between atoms. The reconstructed pivot remains a single integer parameter. · OAI.Ostmann.constituentTransferWeight

constPowMod def

Static modulus and exponent; exponent wiring is reversed explicitly. · OAI.ExactQuantumFactoring.BitArithmetic.constPowMod

constrainedFieldValue def

The prescribed-magnetization field functional from (mag:constrained-functional). · OAI.InvariantIsing.constrainedFieldValue

constrainedHeightFieldValue def

The centered constrained field value on the strict height cone. · OAI.InvariantIsing.constrainedHeightFieldValue

constrainedHistory def

The exact signed constrained history sum for an anchored old tree. · OAI.DilutedSpinGlass.PrescribedTree.constrainedHistory

Constraint structure

A `Constraint vertices alphabet` selects a source vertex, a target vertex, and the permutation table that tells which target label matches each source label. · OAI.UniqueGamesTheorem.Foundations.Target.Constraint

constraint def

The finite section constraint whose kernel is the actual section module. · OAI.KLInvariance.MomentGraph.Sheaf.constraint

constraintEquiv def

The coordinates outside the prescribed right endpoints are free. · OAI.TreeEdit.BinaryLower.RandomMatching.constraintEquiv

constraintGraph def

Keep the original tests and add always-accepting tests on the overlay ports. · OAI.MaxCutGames.Foundations.PCP.Overlay.constraintGraph

constraintGraph def

Keep the original tests and add always-accepting tests on the overlay ports. · OAI.UniqueGamesTheorem.Foundations.PCP.Overlay.constraintGraph

constraintPolytope def

The degree and odd-cut constraint system. · OAI.MatchingEntropy.LooplessGraph.constraintPolytope

constraintWords def

`constraintWords` emits the source index, target index, and every forward permutation image as natural numbers. · OAI.UniqueGamesTheorem.Foundations.Complexity.constraintWords

constructedBoundaryAssembly def

The published interface is now supplied by the constructed analytic index. · OAI.CoarseAssembly.constructedBoundaryAssembly

constructedChainModelComparison def

The simplex comparison with the global chain model follows from normalized simplex identities. · OAI.Thomason.Construction.NormalizedSimplexIdentities.constructedChainModelComparison

constructedExtensionLocalizer def

The extension functor on the actual realization localizer uses no model-category input. · OAI.Thomason.Construction.constructedExtensionLocalizer

constructedFiniteFactorError def

Displacement from the initial embedding and the proved finite-factor estimate. · OAI.KadisonKastler.constructedFiniteFactorError

constructedFiniteReflection def

Positive finite reflection, obtained from the quotient universal property. · OAI.Thomason.Construction.constructedFiniteReflection

constructedInitialFaceIso def

The explicit initial ordinal face is an isomorphism onto the full subcategory on its vertices, proved directly from positive chain support. · OAI.Thomason.Construction.SimplexComplex.constructedInitialFaceIso

constructedModularConjugationData def

Modular conjugation data from the Tomita commutant identity. · OAI.KadisonKastler.constructedModularConjugationData

constructedModularSpectralResolution def

Spectral modular data for a modular group normalizing the algebra. · OAI.KadisonKastler.constructedModularSpectralResolution

constructedRightGrayInput def

The right interval adjunction with its natural ordered simplex comparison. · OAI.Thomason.Construction.NormalizedSimplexIdentities.constructedRightGrayInput

constructedRightSimplexComparison def

The explicit right chain simplex cylinder identifies with the global extended interval tensor, naturally in every simplex operator. · OAI.Thomason.Construction.NormalizedSimplexIdentities.constructedRightSimplexComparison

constructedSet def

The fixed subset defined by finite-support displacements and open-strip conditions. · OAI.DoublingHilbert.constructedSet

ConstructionStage structure

One finite-coordinate stage of the construction. · OAI.ExactModuli.KW.ConstructionStage

contactCharge def

Each letter contact is charged to an endpoint other than the class maximum. This uses actual side capacities, not a forest assumption on the contact graph. · OAI.HyperbolicObstruction.Coincidence.contactCharge

ContactGridEdge structure

The canonical key of a lattice edge consists of its direction and its lower/left endpoint. The final Boolean records the input orientation. · OAI.ContinuumCoulomb.ContactGridEdge

contactGridPoint def

The original square-lattice vertices are spaced seventeen units apart. · OAI.ContinuumCoulomb.contactGridPoint

ContactStripSite structure

Coordinates of an internal site in an oriented length-17 lattice edge. Off-axis sites stay farther from both original endpoints. · OAI.ContinuumCoulomb.ContactStripSite

containing def

A head containing any given finite list of vertices. · OAI.IndependentProducts.FiniteHead.containing

containingCopyCount def

Copies of I containing the fixed observed set U. · OAI.LeanBlast.SecondKahnKalai.containingCopyCount

containingTripleEquiv def

Repackaging as the finite set of three-element supersets of {c}. · OAI.PixtonChow.containingTripleEquiv

containingWord def

The actual symbol word containing the origin, not an independently assigned label. Its starting time is minus the finite phase representative. · OAI.SmoothObstruction.PhaseDynamics.containingWord

containmentAvoidanceEquiv def

Containment and exact intersection become ordinary avoidance in the containment quotient. This is a literal equivalence of finite sample spaces. · OAI.SidorenkoCounterexample.containmentAvoidanceEquiv

containmentProb def

Probability that a uniform actual copy contains an observed finite set. · OAI.LeanBlast.SecondKahnKalai.containmentProb

containmentProbability def

Probability that G(n,p) contains at least one non-induced copy. · OAI.LeanBlast.SecondKahnKalai.containmentProbability

Contains def

The actual chart domain, with the connected-component requirement made explicit. Different copies of a quotient point satisfy the same condition. · OAI.PlanarL1.Columns.Model.OrdinaryData.Contains

Contains def

Prior prescriptions are undirected matching edges, counted once. · OAI.TorsionFreeZeroDivisors.ConditionedMatchings.Matching.Contains

ContainsSegment def

A path contains a translate of the fixed segment. In a tree the endpoint condition also puts the entire translated segment in the path. · OAI.HarmonicArtin.ParabolicIntersections.TreeAction.ContainsSegment

Content def

The actual vertices of a fixed base layer, independent of its coordinate presentation. This is not the larger set of all recorded prefixes. · OAI.Superstring.Hierarchical.Layer.Content

content def

An actual compact content obtained by infima of positive dimension values. · OAI.Kaplansky.PositiveDimensionFunctional.content

Context inductive

A single literal equation step in arbitrary word context. · OAI.FiniteAlgebraicEnvelopes.Compiler.Path.Context

ContextConsistent def

Consistency of an open context, with its displayed finite variable supply. · OAI.PartitionConsistency.Completeness.ContextConsistent

contextEntry def

Initial traversal of the outer context, including recurrence closure. · OAI.TwoWayComplementation.Diagram.contextEntry

contextEquation def

The private query law depends only on the two visible contexts. In particular, the hidden sampled slots are not inputs to pi. · OAI.MaxCutGames.Foundations.Hastad.SourceOccurrences.contextEquation

contextEquation def

The private query law depends only on the two visible contexts. In particular, the hidden sampled slots are not inputs to pi. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceOccurrences.contextEquation

contextEquiv def

The exact small-coordinate bijection required by physical context transport. The suffix enumeration introduces no extra assignment. · OAI.ContingencyTables.SmallContextCoordinates.contextEquiv

ContextEquivalent structure

Context conversion as exact equality of all judgments on unchanged syntax. Both contexts are required valid. This auxiliary relation carries no semantic or normalization assumption. · OAI.PureTypeSystem.ContextEquivalent

contextExit def

Final traversal of the outer context, including recurrence closure. · OAI.TwoWayComplementation.Diagram.contextExit

contextInTime def

The exact trace packaged for composition with neighboring cloning stages. · OAI.MaxCutGames.Explicit.MachineClone100Context.contextInTime

contextInTime def

The exact trace packaged for composition with neighboring cloning stages. · OAI.UniqueGamesTheorem.Explicit.MachineClone100Context.contextInTime

contextNameProfile def

The equality matrix can be supplied by a separate field-comparison procedure after an explicit reindexing of the 3*u positions. · OAI.MaxCutGames.Foundations.Hastad.SourceLocalSignature.contextNameProfile

contextNameProfile def

The equality matrix can be supplied by a separate field-comparison procedure after an explicit reindexing of the 3*u positions. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceLocalSignature.contextNameProfile

ContextTower structure

Finite prefixes of a common context, using fixed-level names when viewed outside a prefix. Every sortable fixed-level expression is declared arbitrarily late. Bound variables remain de Bruijn indices and are never declared here. · OAI.PureTypeSystem.ContextTower

Contextual def

Precisely the bounded contextual tests used at source bootstrap:273–297. · OAI.FiniteAlgebraicEnvelopes.Bootstrap.Static.Contextual

Continuation def

A continuation is tested after every legal extension; callbacks are actual open terms with one proof input, not arbitrary functions on inhabitation. · OAI.PureTypeSystem.Labelled.Continuation

continuation def

Product of the turn weights from a starting letter through a tail. · OAI.TorsionFreeZeroDivisors.WordEstimate.continuation

continuationCost def

Expected continuation from a specified (possibly null) history. · OAI.KServer.continuationCost

continuationRoute def

The actual compact set traced by the vertical piece, the circular piece, and the remainder of the guide. · OAI.SLEExactGauge.GuideSplice.continuationRoute

Continues inductive

Reaches another frame at the same level, with any children finished. (For use proving Executes via a schedule of loops.) · OAI.Fifths.Flow.Continues

continuityCore def

An explicit countable intersection of open sets of continuity points. · OAI.Generator.continuityCore

continuousAd def

The adjoint operator, with its finite-dimensional continuous structure. · OAI.PolycyclicRecognition.continuousAd

continuousAdditiveToComplexLinear def

A continuous additive map that commutes with i is complex linear. · OAI.KadisonSimilarity.continuousAdditiveToComplexLinear

continuousDetRows def

The determinant, as a continuous map multilinear in its rows. · OAI.AffineBernstein.continuousDetRows

continuousDimension def

Transport the existing regularized dimension through a real star algebra homomorphism. On negative functions the auxiliary value is zero and is unused. · OAI.Kaplansky.CuntzDimensionState.continuousDimension

continuousDimensionCutoff def

Continuous scalar cutoff, used only for uniform approximation of functions. · OAI.Kaplansky.continuousDimensionCutoff

ContinuousDrawing structure

Actual injective continuous edge arcs, with unrestricted routing. · OAI.Paper170.ContinuousDrawing

continuousFiber def

The literal continuous fibre, with its subspace topology. · OAI.Thomason.Construction.continuousFiber

continuousFiberBaseChangeMap def

A genuinely commutative square induces the literal map on its point fibres, retaining the total-space point and the specified base point. · OAI.Thomason.Construction.continuousFiberBaseChangeMap

continuousFiberHomotopy def

The full parameterized homotopy restricts to the fibre when its base coordinate is constant throughout. · OAI.Thomason.Construction.continuousFiberHomotopy

continuousFiberHomotopyEquiv def

A homotopy equivalence with both homotopies over the base induces an actual homotopy equivalence on every continuous fibre. · OAI.Thomason.Construction.continuousFiberHomotopyEquiv

continuousFiberIso def

An isomorphism over the base restricts to an isomorphism of fibres. · OAI.Thomason.Construction.continuousFiberIso

continuousFiberMap def

Any exact map over the base restricts to each literal fibre. · OAI.Thomason.Construction.continuousFiberMap

continuousGoodMass def

Gibbs mass of realized initial states still farther than 1 / 4 from equilibrium. · OAI.CriticalSK.continuousGoodMass

ContinuousHeatKernelInput def

Existence of a continuous kernel representing the Cheeger heat semigroup. · OAI.RegularPointCoordinates.ContinuousHeatKernelInput

ContinuousHermitianMetric def

Continuity of a Hermitian metric, expressed by its values on local continuous sections of an ordinary complex vector bundle. · OAI.Kaplansky.ContinuousHermitianMetric

continuousHomotopyFiber def

The homotopy fibre consists of a point and a path from its image to b. · OAI.Thomason.Construction.continuousHomotopyFiber

continuousHomotopyFiberContraction def

Interpolation contracts each path to the canonical one, fixing both endpoints. This uses an explicit diagonal-preserving interpolation, not a homotopy-fibre preservation assumption. · OAI.Thomason.Construction.continuousHomotopyFiberContraction

continuousHomotopyFiberEquiv def

Over a continuously interpolated space the homotopy fibre projects by an actual homotopy equivalence to the original source. · OAI.Thomason.Construction.continuousHomotopyFiberEquiv

continuousHomotopyFiberSection def

A continuous interpolation provides a canonical path into the fibre. · OAI.Thomason.Construction.continuousHomotopyFiberSection

continuousLinearEquivPuncturedHomeomorph def

Nonzero vectors transport under an actual continuous linear equivalence. · OAI.Kaplansky.continuousLinearEquivPuncturedHomeomorph

continuousLinearMapOfUnitaryImages def

A linear map preserving unitaries is automatically bounded. · OAI.KadisonKastler.continuousLinearMapOfUnitaryImages

continuousLinearTransfer def

The transferred algebra map is bounded as soon as its unitary images are uniformly close to the original unitaries. · OAI.KadisonKastler.continuousLinearTransfer

ContinuousNormalizedStep structure

The normalized maps and their true adjoints vary continuously over the specified ordinary topological bundles. · OAI.Kaplansky.ContinuousNormalizedStep

continuousOrbitSubmodule def

Vectors with continuous orbit form a linear subspace. · OAI.KadisonSimilarity.continuousOrbitSubmodule

continuousPathConstant def

The constant-path section of initial evaluation. · OAI.Thomason.Construction.continuousPathConstant

continuousPathContraction def

Shrinking the time coordinate contracts each path to its initial point. · OAI.Thomason.Construction.continuousPathContraction

continuousPathFamilyEquiv def

Exponential adjunction with the compact interval, in the order used by the realization cylinder. No local compactness of the parameter space is needed. · OAI.Thomason.Construction.continuousPathFamilyEquiv

continuousPathMap def

Postcomposition on actual compact-open path spaces. · OAI.Thomason.Construction.continuousPathMap

continuousPathSpace def

The ordinary compact-open space of all paths, with unrestricted ends. · OAI.Thomason.Construction.continuousPathSpace

continuousPathStart def

Evaluation at the initial endpoint of an unrestricted continuous path. · OAI.Thomason.Construction.continuousPathStart

continuousPathStartEquiv def

Initial evaluation is an actual homotopy equivalence, naturally in the space. · OAI.Thomason.Construction.continuousPathStartEquiv

continuousPointFiberComparison def

The canonical map from the literal continuous point fibre into the homotopy fibre appends its constant path. · OAI.Thomason.Construction.continuousPointFiberComparison

continuousProductFiberEquiv def

Interpolation in the base makes each literal product fibre a homotopy-equivalent subspace of the entire product. · OAI.Thomason.Construction.continuousProductFiberEquiv

continuousProductFiberHomeomorph def

The fibre of the ordinary second projection is canonically its first factor. This identification retains the literal subspace topology. · OAI.Thomason.Construction.continuousProductFiberHomeomorph

continuousRealInclusion def

Real continuous functions viewed inside the complex function algebra. · OAI.Kaplansky.continuousRealInclusion

ContinuousStageMap structure

A linear map between the actual bundles, continuous on total spaces. · OAI.Kaplansky.ContinuousStageMap

continuousWedge def

The shuffle product bundled with its actual joint-vector continuity. · OAI.Mahler.continuousWedge

continuousWedge def

Continuity is proved from the actual determinant polynomial. · OAI.SmoothObstruction.ExteriorStability.continuousWedge

continuousWedgeCLM def

Exterior multiplication is an actual continuous bilinear map. · OAI.Mahler.continuousWedgeCLM

continuousWedgePower def

The recursive shuffle power as a continuous form. · OAI.Mahler.continuousWedgePower

continuousWedgePowerVariation def

The exact finite product-rule variation of a bundled exterior power. · OAI.Mahler.continuousWedgePowerVariation

contra def

Inverse transpose in Euclidean metric. · OAI.GeneralMahler.contra

contraCallback def

The other slot of odd evaluation, exposed as an actual raw callback. · PureTypeSystem.Labelled.DataWrapper.contraCallback

Contract def

Contract assumed on external calls inside the open body proof only. · OAI.PowerSaving.RAM.Contract

contract def

Contract two adjacent frontier strands with the ordinary transpose cap. · OAI.CriticalHoneycomb.Transfer.PlanarNormalization.contract

contract def

Contraction on x, computed in full table coordinates; at the small table only its restriction to C_i is initially meaningful. · OAI.HadwigerCounterexample.contract

contract def

Fix the first vector of a symmetric form, retaining its multiset coefficients. · OAI.MetricEntropyDuality.SymmetricForm.contract

contract def

Contraction of an actual bounded ideal, with no new choice of an ideal. · OAI.PartitionConsistency.DiagramOrder.ConeIdeal.contract

contract def

A connected induced set is contracted by choosing its literal spanning submatching and replacing the rotations by the contour permutation. · OAI.TorsionFreeZeroDivisors.PlanarMap.Model.contract

contractAt def

The explicit contraction vertex. The right quotient is specified only to avoid a hidden choice; left cancellation makes it unique. · OAI.PartitionConsistency.LocalDiagramSystem.Vertex.contractAt

contractBoundedProjective def

Bounded-above acyclic complexes of projectives admit an actual contraction. · OAI.LittleFinitistic.RectificationAcyclic.contractBoundedProjective

contractedPinnedWeight def

The actual contracted marginal retains arbitrary fields and all effects of the common pins. It is distinct from the zero-field comparison weight. · OAI.BufferedIsing.contractedPinnedWeight

contractedVertexInclusion def

A chosen vertex as a map from the terminal standard simplex. · OAI.Thomason.Construction.contractedVertexInclusion

contractedZeroWeight def

Zero-field Ising weight on the vertex-deleted graph, with I,J separately contracted by restricting each to constant spin assignments. · OAI.BufferedIsing.contractedZeroWeight

contractibleProductProjection def

Projection from a product with an actually contractible base, keeping the projection itself as the forward map. · OAI.Kaplansky.contractibleProductProjection

contractibleRelativeBoundaryIso def

In degrees at least two, relative homology in a contractible ambient space is the preceding homology of the subspace, via the actual boundary. · OAI.Kaplansky.contractibleRelativeBoundaryIso

contractibleSubspaceLocalizationOneIso def

The contractible-subspace localization theorem also holds in degree one; the low-degree proof uses H₀ injectivity, not a false H₀ vanishing claim. · OAI.Kaplansky.contractibleSubspaceLocalizationOneIso

Contraction structure

An augmented chain contraction on the actual ordered cliques. · OAI.SimplicialChains.Contraction

contraction def

Unit insertion; this is k-linear, generally not A-linear. · OAI.DerivedBar.contraction

contraction def

An augmented integral chain contraction of the actual103-vertex source cell-order complex, not an assumed homology property of the seed. · OAI.EilenbergGanea.SeedChainCertificate.contraction

contraction def

Literal face homotopy. Its domain is the actual closed square prism. · OAI.HyperbolicObstruction.BasedSquareContraction.contraction

contraction def

All finite matrix certificate equations are used to obtain a genuine augmented contraction of every actual clique of the graph. · OAI.SimplicialChains.TwoDimensionalIndices.contraction

contraction def

The actual continuous homotopy from identity to the greatest vertex. · OAI.Thomason.Construction.ConePrism.contraction

contractionSymmetryUnitary def

A centered four-by-four symmetry containing an arbitrary contraction. The blocks use the existing selfadjoint off-diagonal contraction lemma. · OAI.KadisonSimilarity.contractionSymmetryUnitary

contractLinearMap def

The actual contraction map used as a partition label. · OAI.MetricEntropyDuality.SymmetricForm.contractLinearMap

Control abbrev

A common, countably coded state space for all unary-initialized programs. · OAI.FiniteAlgebraicEnvelopes.MachineRealization.ParameterCode.Control

Control structure

The encoded vertex of firstH already stores both square ports. The two graph-return registers and final-H register are overwritten by the corresponding callbacks. Every field is bounded independently of table size. · OAI.MaxCutGames.Foundations.PCP.ExpanderRowControl.Control

Control structure

The encoded vertex of firstH already stores both square ports. The two graph-return registers and final-H register are overwritten by the corresponding callbacks. Every field is bounded independently of table size. · OAI.UniqueGamesTheorem.Foundations.PCP.ExpanderRowControl.Control

controlCoordinates def

Seven output coordinates and twenty-eight complementary parameters. · OAI.BoltzmannNonuniqueness.FiveGainMap.controlCoordinates

Controlled def

Uniform finite orbit control, not compactness of the infinite free orbit. · OAI.GroupRingTopology.Milnor.Controlled

controlledAssembly def

The constructed whole-union Roe boundary descends through every relation of this controlled analytic cycle model. · OAI.CoarseAssembly.Rips.controlledAssembly

controlledBoundaryClass def

An actual class of the reduced locally compact Roe algebra. · OAI.CoarseAssembly.DiscreteRoe.controlledBoundaryClass

controlledCoarseAssembly def

An actual compatible assembly map on the controlled component model. · OAI.CoarseAssembly.Rips.controlledCoarseAssembly

controlledCoboundary def

The homogeneous coboundary, as an actual map on controlled cochains. · OAI.PolycyclicRecognition.controlledCoboundary

controlledCochainComplex def

Locally finite scalar homogeneous cochains, with the standard alternating differential. · OAI.PolycyclicRecognition.controlledCochainComplex

controlledCochainHomotopy def

The controlled-support estimate promotes the simplicial prism to an actual homotopy between the two cochain maps. · OAI.PolycyclicRecognition.controlledCochainHomotopy

controlledCochainHomotopyEquiv def

The coarse inverse produces inverse controlled cochain maps up to homotopy. · OAI.PolycyclicRecognition.controlledCochainHomotopyEquiv

controlledCochainMap def

Coarse pullback commutes with the actual homogeneous differential. · OAI.PolycyclicRecognition.controlledCochainMap

controlledCochainPullback def

Pullback is well-defined because controlled tuples map to controlled tuples and the map on tuples has finite fibers. · OAI.PolycyclicRecognition.controlledCochainPullback

controlledCochains def

Homogeneous scalar cochains, locally finite on every controlled set. · OAI.PolycyclicRecognition.controlledCochains

controlledCohomologyIso def

Coarse-equivalent groups have isomorphic controlled scalar cohomology. · OAI.PolycyclicRecognition.controlledCohomologyIso

ControlledContraction structure

The quantitative contraction data used by the interpolation algorithm. It is constructed from coordinates below, rather than assumed to supply a quasi-isometry representative. · OAI.PolycyclicRecognition.ControlledContraction

controlledCoordinate def

Ordinary analytic K-homology of each compact component. Injectivity or surjectivity of the resulting coordinate family is not asserted here. · OAI.CoarseAssembly.Rips.controlledCoordinate

controlledFace def

Face pullback preserves locally finite support. · OAI.PolycyclicRecognition.controlledFace

ControlledFiberMaps def

Uniform controlled scalar maps preserving the normalized constants. · OAI.CoarseAssembly.ControlledFiberMaps

controlledGroupCohomologyIso def

Controlled scalar homogeneous cohomology is group-ring cohomology. · OAI.PolycyclicRecognition.controlledGroupCohomologyIso

controlledGroupRingValue def

The group-ring value of a scalar homogeneous cochain. · OAI.PolycyclicRecognition.controlledGroupRingValue

controlledHadamardAt abbrev

One of the literal fixed alphabet gates; the extra control is the first wire and is never modified. · OAI.ExactQuantumFactoring.controlledHadamardAt

controlledPrism def

The negative alternating prism, with the sign convention of Mathlib's chain homotopy, lowers cochain degree by one. · OAI.PolycyclicRecognition.controlledPrism

controlledPrismTerm def

Each prism summand is defined on the controlled cochains themselves. · OAI.PolycyclicRecognition.controlledPrismTerm

controlledStandardCochainIso def

The controlled complex is the actual standard-resolution cochain complex. · OAI.PolycyclicRecognition.controlledStandardCochainIso

controlStateEquiv def

Expose the existing arithmetic head register to the odometer and final cleanup. This is a static finite-state equivalence, with no machine steps. · OAI.MaxCutGames.Foundations.Hastad.SourceGeneratorModel.controlStateEquiv

controlStateEquiv def

Expose the existing arithmetic head register to the odometer and final cleanup. This is a static finite-state equivalence, with no machine steps. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceGeneratorModel.controlStateEquiv

ControlsTwoPaths def

The path property proved for the measured regions. This records geometric input paths and the single fixed lift controlling the whole right cell, without choosing points or tile frames. · OAI.PolycyclicRecognition.MeasuredProbeRegions.ControlsTwoPaths

ConullConjugate def

Measurable conjugacy on invariant conull measurable subsets, not a continuous conjugacy and not merely a factor map. · OAI.SmoothObstruction.ConullConjugate

ConullIsomorphic def

Measure-space isomorphism after removing measurable null sets. · OAI.SmoothObstruction.ConullIsomorphic

conullProjectionFiber abbrev

The genuine old-unit range at every base point, including zero fibers on the null complement of the original representations. · OAI.KadisonSimilarity.conullProjectionFiber

convergentPair def

Reduced nonnegative numerator/positive denominator representation; the zero case is essential because the total rational inverse sends zero to zero. · OAI.ExactQuantumFactoring.OrderTrial.convergentPair

conversion def

The affine conversion constants, without manufacturing a point source. · OAI.PlanarL1.Columns.Model.OrdinaryData.conversion

convertedTransitions def

Recode moves and remove only impossible outward marker transitions. · OAI.TwoWayComplementation.CompanionBridge.convertedTransitions

convexCombination def

An actual finite pointwise convex combination, for arbitrary finite index universes. · OAI.ReflexiveFixedPoints.convexCombination

convexDifferenceSpace def

Every vector is one nonnegative multiple of a difference of two points. This representation retains the points needed for the distance pairing. · OAI.RegularPointCoordinates.convexDifferenceSpace

ConvexFamily def

Convexity of an actual map family is measured in its ambient function vector space. The subtype of points of K need not be a vector space. · OAI.ReflexiveFixedPoints.WeakSelfMap.ConvexFamily

convexMix def

A binary convex combination, with membership proved in the ambient space. · OAI.ReflexiveFixedPoints.convexMix

convolution def

Convolution as an element of the same finite-order jet space. · OAI.ForcedComputation.BoundedSpatialJets.convolution

convolution def

The tensor C(a,b) of Section 5, in monomial coefficient coordinates. · OAI.MatrixMultiplication.AuxiliarySeparation.convolution

convolution def

Convolution is the algebra homomorphism extending the left regular representation. · OAI.TorsionFreeBass.Regular.convolution

convolutionCLM def

The bounded linear heat/convolution operator; its norm is at most the kernel's L1 mass. · OAI.ForcedComputation.BoundedSpatialJets.convolutionCLM

convolutionFamily def

Pointwise convolution of the finite family of bounded derivatives. · OAI.ForcedComputation.BoundedSpatialJets.convolutionFamily

convolutionOneInputEquiv def

Relabel the single input coordinate of C(1,b) by Unit. · OAI.MatrixMultiplication.AuxiliarySeparation.convolutionOneInputEquiv

convolutionOneOutputEquiv def

Relabel the output of C(1,b) by its actual dimension b. · OAI.MatrixMultiplication.AuxiliarySeparation.convolutionOneOutputEquiv

convolutionOperator def

The convolution operator in the fixed normed operator space. · OAI.ForcedComputation.BoundedSpatialJets.convolutionOperator

convolutionOutput def

Every pair of input monomials has a valid output coordinate. · OAI.MatrixMultiplication.AuxiliarySeparation.convolutionOutput

convolutionPower def

The law of independent repetitions, with the empty product at the identity. · OAI.BinaryCoordinateSweeps.convolutionPower

convolutionProfile def

The polynomial profile P(a,b) normalized by the actual mean exponent. · OAI.MatrixMultiplication.AuxiliarySeparation.Character.convolutionProfile

convolutionRightOneOutputEquiv def

Relabel the output of C(a,1) by its actual dimension a. · OAI.MatrixMultiplication.AuxiliarySeparation.convolutionRightOneOutputEquiv

convolutionWindow def

A finite subset of ℤ, exactly the two simultaneous coefficient cutoffs. · OAI.FiniteFactor.convolutionWindow

ConvProgram def

Convolution in the same model extended by pointwise field multiplication for two inputs. We even require the bound on the supplied root from the proof of the corollary. · OAI.PowerSaving.ConvProgram

Coord abbrev

Coordinates on the root-including, countably branching height-h tree. · OAI.DiamondDistortion.RecursiveModel.Coord

coord def

Actual coordinate derivative, as a derivation of the smooth function algebra. · OAI.ElasticityAugmented.coord

coord def

Inverse of the source Euler compactification, in the normalized radius. · OAI.SepticProfile.SonicShooting.coord

coord abbrev

The affine coordinate x in the exact source rational function field. · OAI.SourceSymmetry.coord

coordAnnulus def

Open-ball puncture; the inner sphere has zero volume. · OAI.MahlerStokes.coordAnnulus

coordBall def

An actual Euclidean ball in coordinates; the ambient Pi norm is irrelevant. · OAI.MahlerStokes.coordBall

coordDeriv def

The coordinate directional derivative. · OAI.HarmonicCounterexample.coordDeriv

Coordinate abbrev

The finite coordinate algebra, retaining all surface relations, all regular seam identifications and all vertex refinements/complements. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.Coordinate

coordinate def

The actual CRT coordinate, without a chosen-decomposition hypothesis. · OAI.CirculantHadamard.PrimeCharacterEvaluations.coordinate

coordinate def

The kernel coordinate of an element relative to the chosen section. · OAI.PolycyclicRecognition.GroupExtensionSection.coordinate

coordinate def

Lift an explicit univariate expansion to one selected coordinate. · OAI.Problem335.FactorialExpansion.coordinate

coordinate def

The selected coordinate in the class containing a given position. · OAI.Problem348.Construction.HostSampling.coordinate

coordinate def

Fixed row-major block coordinate. · OAI.RationalHitting.Generator.coordinate

coordinate def

The literal increasing coordinate on a contiguous block of integer edge indices (the manuscript's half-integer centers shifted by 1/2). · OAI.TorsionFreeZeroDivisors.IntervalSelfLinks.coordinate

coordinateAction def

The fixing subgroup, with precisely the three remaining scalars. · OAI.KummerLines.coordinateAction

coordinateAlgebra def

The ramified coordinate part of the literal cover at (x,y)=(0,1). · OAI.KummerNode.coordinateAlgebra

coordinateAlgebra def

The two ramified coordinates form an actual polynomial plane in the original field. No independent auxiliary Kummer field is substituted. · OAI.KummerShifted.coordinateAlgebra

coordinateAlgebra def

The two ramified coordinates form an actual polynomial plane in the original field. No independent auxiliary Kummer field is substituted. · OAI.KummerShiftedDiagonal.coordinateAlgebra

coordinateAlgebra def

The two ramified coordinates form an actual polynomial plane in the original field. No independent auxiliary Kummer field is substituted. · OAI.KummerTriple.coordinateAlgebra

coordinateAlgebra def

The two ramified coordinates form an actual polynomial plane in the original field. No independent auxiliary Kummer field is substituted. · OAI.KummerTripleDiagonal.coordinateAlgebra

coordinateAlgEquiv def

The scheme of all right inverses, on each explicit splitting chart, is literally affine space over its base. This is a coefficient-algebra isomorphism, not merely a point-level or generic-fiber parametrization. · OAI.IntegralCharacterVarieties.ExactSequenceChart.MatrixSplitData.coordinateAlgEquiv

coordinateAllowance def

A linear allowance depending on the Gamma shape and rate. · OAI.GammaFPP.coordinateAllowance

coordinateAverage def

A concrete signed coordinate Type I expression. · OAI.SeparableQuotient.Norming.coordinateAverage

coordinateBasis def

Standard coordinate basis as a tuple of vectors. · OAI.EilenbergGanea.coordinateBasis

coordinateBasis def

Ordered coordinate basis for dx_0 wedge ... wedge dx_n. · OAI.MahlerStokes.coordinateBasis

coordinateBilinearUnit def

A single elementary coordinate bilinear map. · OAI.PolycyclicRecognition.coordinateBilinearUnit

coordinateBlockSpan def

The binary span of the three coordinate block spaces. · OAI.MaxCutGames.Quadratic.coordinateBlockSpan

coordinateBlockSpan def

The binary span of the three coordinate block spaces. · OAI.UniqueGamesTheorem.Quadratic.coordinateBlockSpan

coordinateCell def

Sixteen cells of width t/8 cover the coordinate projection of a radius-t ball. · OAI.DoublingHilbert.coordinateCell

coordinateChangeFirst def

First derivative of the composed map, linear in the supplied jet. · OAI.ClosedSurfaceR4.JetPolynomial.coordinateChangeFirst

coordinateChangeSecond def

Ordered second derivative, with both the Hessian and acceleration terms. · OAI.ClosedSurfaceR4.JetPolynomial.coordinateChangeSecond

coordinateChartPoint def

The chart sample uses the actual factor exponentials and includes the deterministic enlarged suffix. · OAI.PolycyclicRecognition.coordinateChartPoint

coordinateClamp def

Coordinatewise projection onto a closed finite-dimensional box. · OAI.Problem326.coordinateClamp

coordinateComplement def

The complemented Laplacian as one uniformly bounded operator sequence. · OAI.NuclearUltrapower.coordinateComplement

coordinateCornerEntry def

Extract the original-algebra entry of an operator in the represented matrix algebra. Membership is part of the returned subtype. · OAI.KadisonKastler.coordinateCornerEntry

coordinateCornerOperator def

Place one operator in a diagonal coordinate of a Hilbert array. · OAI.KadisonSimilarity.coordinateCornerOperator

coordinateDirection def

Coordinate directions are included alongside all complex differences. · OAI.Problem326.BarrierConstruction.coordinateDirection

coordinateDot def

Real dot product as a continuous linear functional on coordinate space. · OAI.MahlerStokes.coordinateDot

coordinateEquiv def

The nested coordinate sums contain exactly the retained coordinates and one doubled coordinate for each physical large cell. · OAI.ContingencyTables.LargePairEnumeration.coordinateEquiv

coordinateEquiv def

The chart coordinates of a finite dyadic partition. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Partition.coordinateEquiv

coordinateEquiv def

Relabel coordinates using e; no coordinate is removed or adjoined. · OAI.MetricEntropyDuality.coordinateEquiv

coordinateEquiv def

Literal free coordinates on every degree of the actual right cylinder. · OAI.Thomason.Construction.PastingGlobeCylinderBasis.coordinateEquiv

coordinateEvent def

Each coordinate may restrict its bit independently. · OAI.QuantitativeVanDerWaerden.FiniteProbability.coordinateEvent

coordinateExponent def

Embedding one integral exponent into its own gap coordinate. · OAI.TraceReconstruction.coordinateExponent

coordinateFamily def

All tuples of independently allowed scalar words. Dependent coordinate types permit first words and truncated second words to have different types. · OAI.QuantitativeVanDerWaerden.coordinateFamily

coordinateFinite def

Coordinate evaluation on finite arrays, bounded by the singleton test. · OAI.DiamondDistortion.SegmentSystem.coordinateFinite

coordinateFloor def

Source coordinate floors. Integer times include both directions. · OAI.FiniteFactor.coordinateFloor

coordinateFrame def

A physical orthogonal frame whose first vector is the inward normal. · OAI.Elasticity.coordinateFrame

coordinateFrame def

The pointwise coordinate change associated to a coefficient frame. · OAI.RegularPointCoordinates.coordinateFrame

coordinateFullLinearized def

The actual metric derivative plus the actual polynomial derivative, written in the coordinates used by the finite solver. · OAI.ClosedSurfaceR4.JetPolynomial.Perturbation.coordinateFullLinearized

coordinateFunctional def

Coordinate linear functional on the Euclidean space. · OAI.ProjectionCounterexample.coordinateFunctional

coordinateGame def

A single tested coordinate under an arbitrary full-tuple question law. · OAI.MaxCutGames.Foundations.Games.Game.coordinateGame

coordinateGame def

A single tested coordinate under an arbitrary full-tuple question law. · OAI.UniqueGamesTheorem.Foundations.Games.Game.coordinateGame

coordinateGauss def

Gaussian curvature times the coordinate metric determinant. · OAI.ClosedSurfaceR4.RealModes.coordinateGauss

coordinateGrid def

The square lattice in the displayed (possibly nonorthonormal) parameter coordinates. · OAI.RuledApproximation.coordinateGrid

coordinateIndex def

The other ordinary affine node, and the open part of the diagonal. Together with the node (0,1), these will cover the complement of the two actual affine triple points. · OAI.KummerAffineExtra.coordinateIndex

coordinateIntertwinerData def

Compose a scalar intertwiner with a fixed Hilbert coordinate and retain its intertwining identity without unfolding the scalar construction. · OAI.KadisonSimilarity.coordinateIntertwinerData

coordinateIso def

The native cone term in its actual two module coordinates. · OAI.LittleFinitistic.ConeCoordinates.coordinateIso

coordinateJoin def

Concatenate the two coordinate blocks of the chart ambient space. · OAI.MaxCutGames.Inverse.MatrixChart.coordinateJoin

coordinateJoin def

Concatenate the two coordinate blocks of the chart ambient space. · OAI.UniqueGamesTheorem.Inverse.MatrixChart.coordinateJoin

coordinateL2 def

Insert one scalar L² function in a Euclidean coordinate direction. · OAI.PositiveEinsteinFour.Geometry.ScalarEuclideanSobolev.coordinateL2

coordinateLayer def

Independent switches on the edges parallel to one coordinate. · OAI.BinaryCoordinateSweeps.coordinateLayer

coordinateLeft def

Restore the left input coordinate even on completion tables of weight zero. · OAI.MaxCutGames.Foundations.Repetition.CompletedSampling.coordinateLeft

coordinateLeft def

Restore the left input coordinate even on completion tables of weight zero. · OAI.UniqueGamesTheorem.Foundations.Repetition.CompletedSampling.coordinateLeft

coordinateLinearMap def

The weighted Laplacian on coordinate functions, without a change of norm. · OAI.Problem358.CellLaplacian.coordinateLinearMap

coordinateLineProjection def

A concrete basepoint in every projective factor. · OAI.Kaplansky.coordinateLineProjection

coordinateMatrixUnits def

The canonical matrix units, with the identity of B in the selected entry. · OAI.KadisonKastler.coordinateMatrixUnits

coordinateMissEvent def

The finite union event specifying at least q coordinate misses. · OAI.FiniteFactor.coordinateMissEvent

coordinateMulEquiv def

Multiplication by pointwise nonzero coefficients is a linear equivalence. · OAI.LeanBlast.GotsmanLinial.coordinateMulEquiv

coordinateMultiplier def

The actual coordinate multiplier on Euclidean space. · OAI.LeanBlast.GotsmanLinial.coordinateMultiplier

coordinateN def

The same increasing coordinate, transported along the exact cardinality. · OAI.TorsionFreeZeroDivisors.IntervalSelfLinks.coordinateN

coordinateNorm def

The actual Frobenius norm in these twelve real coordinates. · OAI.PositiveEinsteinFour.Bianchi.coordinateNorm

coordinateNorm def

The Euclidean norm of an ordinary finite coordinate tuple. · OAI.SubsphereRegression.coordinateNorm

coordinateNormal def

Normal covector of a linear boundary-coordinate map, represented in the fixed Euclidean coordinates by Riesz duality. · OAI.Elasticity.coordinateNormal

CoordinateOutcome abbrev

(question occurrences, binary map matrix, noise index, binary functional). The matrix is flattened and has (2*k+1)*dimR bits. · OAI.MaxCutGames.Reduction.Explicit.CoordinateOutcome

CoordinateOutcome abbrev

(question occurrences, binary map matrix, noise index, binary functional). The matrix is flattened and has (2*k+1)*dimR bits. · OAI.UniqueGamesTheorem.Reduction.Explicit.CoordinateOutcome

coordinatePairEquiv def

Split a family of pairs into a pair of families. · OAI.TwoPointCorrelations.coordinatePairEquiv

coordinatePartial def

One fixed coordinate derivative of a scalar function. · OAI.ClosedSurfaceR4.JetPolynomial.coordinatePartial

coordinatePartial def

Partial differentiation in a specified member of a polynomial coordinate system, expressed intrinsically on the target algebra. · OAI.StableCoordinate.CoordinateDerivation.coordinatePartial

coordinatePerm def

Relabeling the tensor sites. · OAI.GAD.coordinatePerm

coordinatePerm def

Define the coordinate action on the full graph, including both the ordinal input and its Boolean output. · OAI.PartitionConsistency.CohenForcing.Condition.coordinatePerm

coordinatePermuteCLM def

Reindexing by a permutation, as a bounded complex linear operator. · OAI.Generator.coordinatePermuteCLM

coordinatePolar def

Polar body for the coordinate pairing. · OAI.SymmetricMahler.coordinatePolar

CoordinatePolynomial def

Each coordinate-line restriction is a polynomial of degree at most D. · OAI.PolycyclicRecognition.CoordinatePolynomial

CoordinatePredual def

The norm-closed coordinate span, not the full infinite-tree dual. · OAI.DiamondDistortion.SegmentSystem.CoordinatePredual

coordinatePrefix def

The coordinate prefix attached to a split finite filtration. · OAI.IntegralCharacterVarieties.coordinatePrefix

coordinateProduct def

Restrictions of ambient coordinate quadratics. Their span contains the constants on the unit sphere and all degree-two spherical harmonics. · OAI.PettyProjection.Spherical.coordinateProduct

coordinateProjection def

The actual coordinate functional through the Euclidean-space equivalence. · OAI.AffineBernstein.coordinateProjection

coordinateProjection def

Projection to a scalar coordinate, on the actual product-index regular Hilbert space. · OAI.KervaireMatrix.coordinateProjection

coordinateProjection def

Coordinate projection; equivariance is asserted only for preserved predicates. · OAI.Saxl.coordinateProjection

coordinateProjectionWeights def

Integral zero-sum weights detecting the k-th coordinate projection. · OAI.DukePrimeDegree.coordinateProjectionWeights

coordinatePullback def

Pull back a finite expression through a fixed coordinate change. The low-jet substitution is supplied by the finite-dimensional chain rule. · OAI.ClosedSurfaceR4.JetPolynomial.Expression.coordinatePullback

coordinatePullback def

Pull back both the scalar jet expressions and their tensor basis. · OAI.ClosedSurfaceR4.JetPolynomial.TensorExpression.coordinatePullback

coordinatePunctureIntersectionHomotopyEquiv def

Every nonempty coordinate-puncture intersection retracts to the actual projective product with one dimension removed in each punctured factor. · OAI.Kaplansky.ComplexGrassmann.coordinatePunctureIntersectionHomotopyEquiv

coordinateQuadratic def

The actual Euclidean quadratic model associated to bounded harmonic coordinates. Its gradient is used to compare distance gradients. · OAI.RegularPointCoordinates.coordinateQuadratic

coordinateQuarterRadius def

The quarter scales used in the constructive surjectivity argument. · OAI.RegularPointCoordinates.coordinateQuarterRadius

coordinateQuotientSection def

The section selected in linear quotient coordinates is an actual continuous section on the quotient group. · OAI.PolycyclicRecognition.coordinateQuotientSection

coordinateRealLinearized def

Polynomial first variation in the product coordinates of the metric solver. · OAI.ClosedSurfaceR4.JetPolynomial.Perturbation.coordinateRealLinearized

coordinateRight def

Restore the right input coordinate even on completion tables of weight zero. · OAI.MaxCutGames.Foundations.Repetition.CompletedSampling.coordinateRight

coordinateRight def

Restore the right input coordinate even on completion tables of weight zero. · OAI.UniqueGamesTheorem.Foundations.Repetition.CompletedSampling.coordinateRight

coordinateRoot def

Literal y-chart over the other affine triple point (1,1). · OAI.KummerShifted.coordinateRoot

coordinateRoot def

Literal y-chart over the other affine triple point (1,1). · OAI.KummerShiftedDiagonal.coordinateRoot

coordinateRoot def

Coordinates at the exceptional divisor over the actual triple point (0,0): u is the original root of x, v is root(y)/root(x). · OAI.KummerTriple.coordinateRoot

coordinateRoot def

Coordinates at the exceptional divisor over the actual triple point (0,0): u is the original root of x, v is root(x−y)/root(x). · OAI.KummerTripleDiagonal.coordinateRoot

coordinateRotate def

Integer iterates of the source rotation, in [0,1) representatives. · OAI.FiniteFactor.coordinateRotate

Coordinates structure

Coordinates of a map whose compression is (z,0). · OAI.MaxCutGames.Inverse.KMSAnalyticHybridCoordinates.Coordinates

Coordinates structure

Coordinates of a map whose compression is (z,0). · OAI.UniqueGamesTheorem.Inverse.KMSAnalyticHybridCoordinates.Coordinates

coordinates def

First the positively rotated square, then the negatively rotated square, and finally the original coordinate square. · OAI.GrahamSpherical.coordinates

coordinates def

Coordinates in a fixed basis of actual harmonic polynomials. · OAI.HarmonicCounterexample.Angular.coordinates

coordinates def

The coordinate equivalence obtained from the actual matrix units. · OAI.KadisonKastler.FullMatrixCorner.coordinates

coordinates def

The actual invariant-ring basis 1, α, with no finiteness assumption. · OAI.KLInvariance.ReflectionFrobenius.coordinates

coordinates def

Extract the fifty independent entries of a derivative-symmetric tensor. · OAI.PositiveEinsteinFour.HighestHessian.coordinates

coordinates def

The actual selected orthonormal coordinates, not a formal minor proxy. · OAI.SmoothObstruction.EndpointFrame.coordinates

coordinates def

Combine the zero-simplex coordinates using the corresponding poset elements. · OAI.Thomason.Construction.PosetGeometry.coordinates

coordinates def

Continuous coordinates on the realization of a finite simplicial cube. · OAI.Thomason.Construction.RealizedCube.coordinates

coordinates def

The two actual realized cylinder coordinates. · OAI.Thomason.Construction.RealizedCylinder.coordinates

coordinatesDataEquiv def

The coordinate record has precisely three independent entries. · OAI.MaxCutGames.Inverse.KMSAnalyticHybridEnergy.coordinatesDataEquiv

coordinatesDataEquiv def

The coordinate record has precisely three independent entries. · OAI.UniqueGamesTheorem.Inverse.KMSAnalyticHybridEnergy.coordinatesDataEquiv

coordinateSet def

The image of a measurable carrier under a measurable coordinate map with measurable inverse. Unlike an arbitrary image, this is visibly measurable. · OAI.SmoothObstruction.ChartDomination.coordinateSet

coordinatesHomeomorph def

The actual finite-poset realization is homeomorphic to its barycentric image. · OAI.Thomason.Construction.PosetGeometry.coordinatesHomeomorph

coordinateSignMatrix def

The actual cube coordinate-sign matrix Z_i. · OAI.LeanBlast.GotsmanLinial.coordinateSignMatrix

coordinateSimplex def

The coordinate standard simplex with nonnegative coordinates and sum at most r. · OAI.ProjectionCounterexample.coordinateSimplex

coordinateSimplexProduct def

The coordinate product, with the first p and last q coordinates separated. · OAI.ProjectionCounterexample.coordinateSimplexProduct

coordinatesL1 def

Coordinate restriction induces an actual L1 linear isometry. · OAI.ExactModuli.KW.coordinatesL1

coordinateSphereDensity def

Outward cofactor density in the fixed ordered coordinate basis. The argument x is the unit normal; r*x is the evaluation point of the form. · OAI.MahlerStokes.coordinateSphereDensity

coordinatesToFin def

The terminal ambient-space identification is a genuine linear homeomorphism, not a cardinality-only relabelling. · OAI.MetricEntropyDuality.coordinatesToFin

coordinateSubspace def

The norm-closed coordinate span inside J(T)*. No identification with the full dual is made for a general forest. · OAI.ForestSpace.coordinateSubspace

coordinateSwap def

Swap the two assignments outside S, retaining them on S. · OAI.MetricEntropyDuality.FiniteProductCounting.coordinateSwap

coordinateSwap def

Exchange the complementary coordinates between two complete assignments. · OAI.QuantitativeVanDerWaerden.FiniteProbability.coordinateSwap

coordinateTail def

Closed finite-codimensional coordinate tail, with no topological assumption on an arbitrary algebraic complement. · OAI.BoundedTreePotentials.coordinateTail

coordinateTensorEmbedding def

The coordinate-field tensor square embeds in the rational function field on eighteen variables. · OAI.EuclideanRamsey.NineField.coordinateTensorEmbedding

CoordinateTest structure

One compact-chart seminorm test for the usual smooth topology of metrics. · OAI.YauCounterexamples.CoordinateTest

coordinateUpdateLaw def

A single-coordinate update, as a linear operation on the remaining law. · OAI.FiniteFactor.coordinateUpdateLaw

coordinateValue def

Coordinate value of the bounded infinite path functional. · OAI.SeparableQuotient.Norming.InfinitePath.coordinateValue

CoordinateVarianceControl def

Only a uniform commutator test for the scalar part of each coordinate is needed for the factor property. · OAI.KadisonKastler.CoordinateVarianceControl

coordinateVector def

A coordinate vector, pushed from a chart back into the tangent bundle. · OAI.YauCounterexamples.coordinateVector

coordinateVolumeForm def

Determinant form with the standard coordinate orientation. · OAI.EilenbergGanea.coordinateVolumeForm

coordinateWin def

One coordinate's success may depend on the entire local question tuples. · OAI.MaxCutGames.Foundations.Games.Game.coordinateWin

coordinateWin def

One coordinate's success may depend on the entire local question tuples. · OAI.UniqueGamesTheorem.Foundations.Games.Game.coordinateWin

copiedPhases def

The actual restored phase, allowing repeated copies of a chart. · OAI.ClosedSurfaceR4.FiniteOrderSmoothing.CircularFamilyGeometry.copiedPhases

copiedSlotCharacter def

Characters keep their original slot identity when a slot is copied. · OAI.Ostmann.copiedSlotCharacter

copiedWordLeafOutside def

Every other H coordinate is among the primes held fixed in the conditional leaf expectation, including the currently sampled pivot groups. · OAI.Ostmann.copiedWordLeafOutside

copies def

List of output IDs, preserving different occurrences with equal records. · OAI.UniqueGamesTheorem.Explicit.RationalWeights.copies

copiesCost def

A positive choice of costs below all the source's required finite bounds. The finite product is at most each represented old cost. · OAI.PartitionConsistency.DiagramForcing.OverlapPrescription.copiesCost

copiesFintype def

A concrete finite enumeration exists because an injection is determined by its finite function table. · OAI.MaxCutGames.Gadget.EnlargementConstruction.copiesFintype

copiesFintype def

A concrete finite enumeration exists because an injection is determined by its finite function table. · OAI.UniqueGamesTheorem.Gadget.EnlargementConstruction.copiesFintype

copiesHeight def

The literal finite average used in the merger height construction. · OAI.PartitionConsistency.DiagramForcing.OverlapPrescription.copiesHeight

copiesHeightCost def

Actual heights and costs on the prescribed quotient, satisfying the strict supremum margin. · OAI.PartitionConsistency.DiagramForcing.OverlapPrescription.copiesHeightCost

copiesMap def

The genuine copy quotient embeds under conservative base simulation. · OAI.PartitionConsistency.DiagramForcing.OverlapPrescription.copiesMap

coprimeDivisorProductEquiv def

Multiplication bijects the divisor lists of two coprime integers with the divisor list of their product. · OAI.Problem337.coprimeDivisorProductEquiv

coproductData def

A whole coproduct of relative deformations remains a relative deformation. · OAI.Thomason.Construction.TopologicalDeformation.coproductData

coproductHomotopy def

Homotopies out of the summands assemble through the actual degreewise coproduct, so equality after the homotopy quotient is still detected summandwise. · OAI.DerivedBar.coproductHomotopy

coproductTrack def

Assemble the tracks using preservation of coproducts by the interval cylinder. · OAI.Thomason.Construction.TopologicalDeformation.coproductTrack

copy def

Copy without destruction using a disjoint zeroed backup register. · OAI.OptimalMaxCut.CounterMachine.Command.copy

copy def

Copying is compiled, not a primitive constant-time operation of the tape model. · OAI.ParityWL.Complexity.TreeCode.copy

copy def

Copy an arbitrary binary word, restoring the operand and clearing temporary storage; all registers have fixed indices. · OAI.RationalHitting.StackLanguage.copy

copy def

Reindex an explicit walk along equality of its endpoint words. · OAI.Superstring.Hierarchical.Path.copy

copy def

Copy a complete word, preserving its order and the source, using one private work stack. · OAI.WLIdentification.StackModel.Command.copy

copyConstituentEquiv def

The copied constituent labels are the current atom labels paired with an original index inside that atom. No new prime slots appear. · OAI.Ostmann.copyConstituentEquiv

copyCorner def

A diagonal corner of a bounded operator on two finite Hilbert sums. · OAI.KadisonSimilarity.copyCorner

copyEmbedding def

Zero extension, now bundled as a linear isometry. · OAI.KadisonSimilarity.copyEmbedding

copyEmbeddingMap def

The actual zero extension along an injection of finite coordinate sets. · OAI.KadisonSimilarity.copyEmbeddingMap

copyField def

Consume one complete unary field, including its delimiter, appending its reverse to the output accumulator. All other tapes are unchanged. · OAI.MaxCutGames.Explicit.MachineSingleOrbitProgram.copyField

copyField def

Consume one complete unary field, including its delimiter, appending its reverse to the output accumulator. All other tapes are unchanged. · OAI.UniqueGamesTheorem.Explicit.MachineSingleOrbitProgram.copyField

copyHead def

Copy a nonempty head to another register without consuming it. · OAI.RationalHitting.StackLanguage.copyHead

copyIndices def

Output occurrence indices; each integer count gives that many separate copies. · OAI.MaxCutGames.Reduction.WeightRounding.copyIndices

copyIndices def

Output occurrence indices; each integer count gives that many separate copies. · OAI.UniqueGamesTheorem.Reduction.WeightRounding.copyIndices

copyInsert def

Insertion into one coordinate of a finite Hilbert sum. · OAI.KadisonSimilarity.copyInsert

copyInTime def

A timed witness whose stored step count is the exact two-loop trace length. · OAI.MaxCutGames.Foundations.Complexity.MachineCopy.copyInTime

copyInTime def

A timed witness whose stored step count is the exact two-loop trace length. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineCopy.copyInTime

copyLaw def

The uniform law on distinct copies, rather than on labeled embeddings. · OAI.LeanBlast.SecondKahnKalai.copyLaw

copyMatrix def

Exact normalized copy matrix, including the within-group diagonal correction. · OAI.ContingencyTables.FactorialSignatures.copyMatrix

copyMatrixUnit def

The scalar matrix unit which sends coordinate j to coordinate i. · OAI.KadisonSimilarity.copyMatrixUnit

copyObserver def

Restriction of a linear observer on the quotient to one base-input copy. · OAI.MaxCutGames.Gadget.EnlargementConstruction.copyObserver

copyObserver def

Restriction of a linear observer on the quotient to one base-input copy. · OAI.UniqueGamesTheorem.Gadget.EnlargementConstruction.copyObserver

copyOrbit def

The distinct copies of a finite subset under the given action. · OAI.LeanBlast.SecondKahnKalai.copyOrbit

copyPrefix def

Copy disjoint source wires into a separate target register, by XOR. · OAI.ExactQuantumFactoring.BooleanNetwork.copyPrefix

copyProjection def

Sum of the genuine range projections for the given isometric copies. · OAI.RowColumn.copyProjection

copyRestriction def

Restriction to the coordinates in the image of the injection. · OAI.KadisonSimilarity.copyRestriction

copyScheduleAnchor def

A fresh anchor is retained until its reserved step, copied exactly once, and then retained at every later step. · OAI.Ostmann.copyScheduleAnchor

copyScheduleMap def

Apply a map of original labels without changing any copy/retain choice. · OAI.Ostmann.copyScheduleMap

copyScheduleOrigin def

Copying never conjugates the underlying Dirichlet character. · OAI.Ostmann.copyScheduleOrigin

copySchedulePredecessor def

Collapse only the erased pivot. Both surviving alternatives have their original predecessor, as in the manuscript's H/Y partition. · OAI.Ostmann.copySchedulePredecessor

copyScheduleRole def

The role of an actual copy, including the inactive negative pivot copies. · OAI.Ostmann.copyScheduleRole

CopyScheduleSurvives def

Exactly the surviving labels. Copied labels come from H, retained labels come from Y, and every constituent of the current pivot is removed. · OAI.Ostmann.CopyScheduleSurvives

copyScheduleSurvivorEquiv def

This is a literal equivalence, not a count of an oversized ambient space. · OAI.Ostmann.copyScheduleSurvivorEquiv

CopyScheduleVertex def

A copied label remembers its sign; a retained label remembers its old slot. · OAI.Ostmann.CopyScheduleVertex

copySetoid def

The quotient of the finitely many prescribed copies. Validity, rather than an equivalence closure, provides its setoid. · OAI.PartitionConsistency.DiagramForcing.OverlapPrescription.copySetoid

copyWord def

Prepending a copy preserves the source and clears the temporary stack. · OAI.ParityWL.Complexity.copyWord

core def

Charts retain the positive barycentric coordinate of their chosen vertex. Multiplicative transition functions are locally constant, with the literal triangular relation providing their cocycle identity. · OAI.HyperbolicObstruction.FinitePolyhedron.Voltage.core

core def

The kernel bundle core has the actual rank dim(V)-dim(W), with local transitions constructed from the family and its right inverses. · OAI.Kaplansky.SmoothKernel.SurjectiveFamily.core

core def

Deleting empty, duplicate and properly contained input words. · OAI.Superstring.Input.core

CoreAdmissible def

Admissibility of core-field sets, retaining the exact three/two quotas. · OAI.HyperbolicObstruction.IncidenceTemplate.CoreAdmissible

CoreCell def

Root and dummy cells needed for Boolean propagation. · OAI.Problem348.GuardedPath.Selection.CoreCell

coreChildrenEquiv def

An exact product description of all nonempty genuine words. · OAI.FKCRT.Inventory.coreChildrenEquiv

coreDualEquiv def

The segment estimate extends to the completed dual. · OAI.BoundedTreePotentials.Segments.coreDualEquiv

coreEdge def

The unoriented field of a core adjacency. · OAI.HyperbolicObstruction.MasterGraph.coreEdge

coreEvaluation def

Evaluation on the whole open flat core, as an actual function space. · OAI.HarmonicCounterexample.Angular.coreEvaluation

coreFinishTapes def

Tail, selected destination, return port, scratch and emitted row fields. · OAI.MaxCutGames.Foundations.Complexity.MachineRegularInternalRow.coreFinishTapes

coreFinishTapes def

Tail, selected destination, return port, scratch and emitted row fields. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRegularInternalRow.coreFinishTapes

coreIncrement def

Spine records index the chosen child from the right. Impossible arities and slot labels are totalized by zero and never sampled by the actual law. · OAI.FKCRT.Inventory.coreIncrement

coreInstruction def

The whole internal-row program has only fixed q and p in its code. · OAI.MaxCutGames.Foundations.Complexity.MachineRegularInternalRow.coreInstruction

coreInstruction def

The whole internal-row program has only fixed q and p in its code. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRegularInternalRow.coreInstruction

coreLinearMap def

The Riesz images form a rational-linear map on the core. · OAI.KadisonKastler.ScalarOperatorKernel.CoherentCore.coreLinearMap

coreMachineInTime def

The concrete finite machine executes the actual internal row in the proved affine budget and returns the reusable clean frame. · OAI.MaxCutGames.Foundations.Complexity.MachineRegularInternalRow.coreMachineInTime

coreMachineInTime def

The concrete finite machine executes the actual internal row in the proved affine budget and returns the reusable clean frame. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRegularInternalRow.coreMachineInTime

coreRotorTapes def

The fixed core rotor roles use the agreed reusable 27-tape frame. · OAI.MaxCutGames.Foundations.Complexity.MachineRegularInternalRow.coreRotorTapes

coreRotorTapes def

The fixed core rotor roles use the agreed reusable 27-tape frame. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRegularInternalRow.coreRotorTapes

coreStep def

Core adjacency corresponds to the unique actual core field. · OAI.HyperbolicObstruction.MasterGraph.coreStep

coreTimeBound def

For fixed cloud degree q this is affine in the lengths and unary values on the actual input tapes. All relation/cleanup constants are included. · OAI.MaxCutGames.Foundations.Complexity.MachineRegularInternalRow.coreTimeBound

coreTimeBound def

For fixed cloud degree q this is affine in the lengths and unary values on the actual input tapes. All relation/cleanup constants are included. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRegularInternalRow.coreTimeBound

CoreTree inductive

The genuine finite ordered connected-core tree. Leaves carry exactly the trivial word; every nonleaf has one child for each ordered core dart. · OAI.FKCRT.Inventory.CoreTree

coreVector def

Riesz representation supplies the image of a core vector. · OAI.KadisonKastler.ScalarOperatorKernel.CoherentCore.coreVector

CoreWord def

A connected nonempty typed core, expressed without an arbitrary pairing choice: any matching certificate is the unique literal stack matching. · OAI.FKCRT.Inventory.CoreWord

Corner structure

The local monoid at an idempotent, with its genuine local identity. · OAI.OneWayLiveness.Corner

corner def

The actual arc element in the normalized half corner. · OAI.KadisonSimilarity.DihedralSpectralArc.corner

corner def

The distinguished corner of a full-column operator. · OAI.KadisonSimilarity.FullColumnCoordinates.corner

corner def

The actual Hilbert corner, defined by its two support equations. · OAI.KadisonSimilarity.TracialCornerGeometry.corner

corner def

The restriction is trace preserving for the normalized corner trace. · OAI.KadisonSimilarity.TracialExpectation.corner

corner def

Four cyclic corners, with side labels r,r',r,r'. · OAI.KaplanskyCounterexample.FiniteGeometry.HoleEvents.corner

corner def

The coefficient on the corner immediately following a dart. · OAI.KervaireRibbon.corner

corner def

The actual two-sided corner e_i B e_j as a vector subspace. · OAI.LittleFinitistic.CornerSpaces.corner

corner def

The corner membership equations for an idempotent e. · OAI.TwoWayComplementation.Diagram.corner

CornerBases abbrev

A independent basis at each boundary corner. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.CornerBases

cornerBasesOfBoundary def

Transport the exhaustively indexed boundary bases to the side occurrences, including every repeated facet and every cyclic base position. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.cornerBasesOfBoundary

cornerCancellationPath def

The cancellation uses the rotation from the first range to the second. · OAI.WeakPureInfiniteness.cornerCancellationPath

cornerCoefficient def

Matrix-unit coefficient with its only nonzero entry in the top row. · OAI.WeakPureInfiniteness.cornerCoefficient

cornerColumns def

The endpoint is shared by the whole ordered child list, with its own opposite-oriented boundary corner for each child. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.cornerColumns

cornerCommutator def

The actual rectangular commutator of a corner operator. · OAI.KadisonSimilarity.cornerCommutator

cornerCompression def

Compression of an arbitrary ambient operator. · OAI.KadisonKastler.cornerCompression

cornerCompression def

Extend a unitary in the initial corner of a by the identity elsewhere. · OAI.WeakPureInfiniteness.cornerCompression

cornerEquiv def

The actual vertex of an induced column is its path-corner tensor. · OAI.LittleFinitistic.VertexColumnModels.cornerEquiv

cornerExtension def

Extension by zero on the orthogonal complement. · OAI.KadisonKastler.cornerExtension

cornerFlatten def

Taking a corner inside an absorbing corner gives the same smaller corner. · OAI.TwoWayComplementation.cornerFlatten

cornerGaugeGenerators def

Path transport transforms from its start corner to its finish; seam frames transform at the precise endpoint, not per boundary circle. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.cornerGaugeGenerators

cornerGeometry def

The corner representation is constructed without a new GNS or free-product existence assumption. Only the scalar normalization is chosen. · OAI.KadisonSimilarity.TracialCornerGeometry.cornerGeometry

cornerGraph def

Shared names give a single edge. Vector bit coordinates are not counted as separate horizontal edges in the corner graph. · OAI.WLIdentification.RingLayout.cornerGraph

cornerInclusion def

A corner element of a subalgebra is the same ambient corner element, with exactly the same norm. · OAI.KadisonSimilarity.TracialCornerGeometry.cornerInclusion

cornerInsert def

Literal corner insertion, keeping the root skeleton letters in order. · OAI.FKCRT.Inventory.cornerInsert

cornerMap def

Cutting down a nonunital coefficient map by a commuting idempotent. · OAI.FiniteAlgebraicEnvelopes.Steinberg.cornerMap

cornerMap def

Right multiplication by the original path on the actual vertex corner. · OAI.LittleFinitistic.CornerColumnPaths.cornerMap

cornerMap def

A multiplicative map restricts to a unital map between matching idempotent corners. · OAI.TwoWayComplementation.cornerMap

cornerMatrixEntry def

Matrix entries of an arbitrary algebra element, as actual elements of the unital corner algebra. · OAI.KadisonSimilarity.TracialCornerGeometry.cornerMatrixEntry

CornerPartition structure

A finite projection partition of a corner, allowing zero pieces. · OAI.Generator.Tracial.CornerPartition

cornerRepresentation def

The identity of ePe is e, acting as the identity on eL²(P)r. · OAI.KadisonSimilarity.TracialCornerGeometry.cornerRepresentation

cornerRestriction def

Compression of the ambient operator to the actual domain and output corners. The domain may have a smaller right support. · OAI.KadisonSimilarity.TracialCornerGeometry.cornerRestriction

Corners structure

Frame-independent incidence data before imposing the allowed vertex tables. Facets and germs stay separate. Corner maps cannot accidentally identify two occurrences merely because they sit on the same facet. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.Corners

corners def

Explicit equivalence data between two orthogonal supported corners. · OAI.FiniteAlgebraicEnvelopes.Steinberg.SupportedPair.corners

corners def

The global corner involution is constructed from the source vertex tables. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexAssembly.corners

cornerScalar def

The scalar conditional expectation in pMp, expressed in ambient coordinates. · OAI.Generator.Tracial.cornerScalar

cornerSign def

Orientation sign of the vertex correspondence. · OAI.ArtinCAT0.cornerSign

cornerSpectralPartition def

The finite corner partition constructed from the shifted positive part. · OAI.Generator.Tracial.cornerSpectralPartition

cornerStabilizationFirstPath def

Add an adjoint-cancelling pair to the two spare branches. · OAI.WeakPureInfiniteness.cornerStabilizationFirstPath

cornerStabilizationSecondPath def

Contract the folded first two branches against the third branch. · OAI.WeakPureInfiniteness.cornerStabilizationSecondPath

cornerStarAlgHom def

A unital star homomorphism restricts to a unital homomorphism between the actual corners of corresponding projections. · OAI.KadisonSimilarity.cornerStarAlgHom

cornerSubgroup def

Subgroup generated by root coefficients in an idempotent corner. · OAI.FiniteAlgebraicEnvelopes.Steinberg.cornerSubgroup

cornerSuccessorPermutation def

Extend a finite permutation by fixing the added last coordinate. · OAI.KadisonKastler.cornerSuccessorPermutation

cornerSum def

Evaluate once at every oriented corner. For cyclic face values this is three times the ordinary integral face chain; no unproved chain-complex property is used. · OAI.Release075.TriangleSphere.cornerSum

cornerTensorEquiv def

The concrete corner tensor is the fixed submodule of the induced column. · OAI.LittleFinitistic.CornerTensor.cornerTensorEquiv

cornerToIdeal def

Embed an actual corner into the principal left ideal of its source vertex. · OAI.LittleFinitistic.CornerColumnComplex.cornerToIdeal

cornerToRightColumn def

Forget only the left support of a two-sided corner. · OAI.KadisonSimilarity.TracialCornerGeometry.cornerToRightColumn

CornerUnitary structure

An actual operator with identical initial and final support p. · OAI.Generator.Tracial.CornerUnitary

cornerUnitary def

The unitary equal to v on E and to the identity on Eᗮ. · OAI.KadisonKastler.cornerUnitary

cornerUnitaryExtension def

A unitary in a represented corner is a unitary on the whole space after adding the complementary identity. · OAI.KadisonKastler.cornerUnitaryExtension

cornerVertexBases def

Every local vertex gets a coherent basis system without any new compatibility hypothesis. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.cornerVertexBases

cornerWeight def

Count one marked corner of every interior face. Exterior corners contribute zero. This uses independent cell coefficients, not the word image. · OAI.HyperbolicObstruction.WordPictures.WordPicture.Indexing.cornerWeight

CornulierTesseraModelInput def

The solvable-source specialization of CT Lemma 3.A.1. The intermediate Lie groups are retained only through the locally compact data needed to compare their metrics. · OAI.PolycyclicRecognition.CornulierTesseraModelInput

corr def

The literal finite average of the real inner product. · OAI.ThompsonNonamenability.corr

Correct def

Exact operational contract: all scratch counters are returned to zero, original arguments and input are preserved, and a fixed explicit clock holds. · OAI.OptimalMaxCut.CounterMachine.Expr.Correct

Corrected def

The persistent selected-coset version of the assignment predicate. · OAI.PartitionConsistency.ProductForcing.ChainCode.Corrected

correctedAlgEquiv def

The genuine algebra equivalence resulting from a correction of the transported product. Its underlying map is exactly F ∘ L⁻¹. · OAI.KadisonKastler.correctedAlgEquiv

correctedCubeDensity def

Add a constant after removing low Walsh modes, then renormalize. · OAI.TwoPointCorrelations.correctedCubeDensity

correctedDispersionPolynomial def

The corrected row polynomial. · OAI.CubicFirstMoment.correctedDispersionPolynomial

correctedFamily def

The integer carrier contributes only a deck translation. Removing it makes the limit the literal old twist lift, including at derivative order 0. · OAI.ThreeTorus.PassageGeometryFamily.correctedFamily

correctedFinite def

The finite interpolation coefficients with the finite correction added. · OAI.AtomicTriangular.Construction.correctedFinite

correctedHom def

Finite-dimensional bounded-defect correction, packaged as a genuine homomorphism. · OAI.PolycyclicRecognition.correctedHom

correctedMap def

Adjoining the unchanged complementary identity to a nonunital map. · OAI.FiniteAlgebraicEnvelopes.correctedMap

correctedPerturbLog def

The manuscript's random convex correction, still before any root expectation is taken. · OAI.DilutedSpinGlass.KernelTower.correctedPerturbLog

correctedPrimitiveTensor def

The corrected tensor is obtained by actual pointwise inverse. · OAI.ClosedSurfaceR4.FiniteOrderSmoothing.SmoothingAtlas.correctedPrimitiveTensor

correctedStarAlgEquiv def

The star-algebra equivalence furnished by a star-preserving linear correction of the transported multiplication. · OAI.KadisonKastler.correctedStarAlgEquiv

CorrectEncoding def

Complete nondecreasing prime factorization, multiplicities included, padded by zeros, as in Section 6. There is no primality oracle in the circuit. · OAI.ExactQuantumFactoring.CorrectEncoding

correction def

All coordinate labels cancel when a table is compared with its double along a coordinate unused by its source prefixes. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Table.correction

correction def

The coincidence correction factors at the normal V positions. · OAI.Problem335.IsolatedWeight.correction

correction def

The alternating factor for a coordinate inside the differentiated set. · OAI.TalagrandDiscreteConvexity.correction

correctionData def

The literal radial, angular, and rotation inputs to the correction formula. · OAI.ThompsonNonamenability.Analytic.correctionData

correctionMap def

The actual quadratic fixed-point map from the stationary correction argument, written for arbitrary Banach spaces. I is the modified inverse, T = I+G₀, and N is the projected tensor-divergence bilinear map. · OAI.StationaryNS.correctionMap

correctionOrder def

Construction starts at order ten, as required by the scale estimates. · OAI.ClosedSurfaceR4.FiniteOrderSmoothing.correctionOrder

correctionRadius def

Radius of the contraction ball, r₀ = 2 F₀ / η. · OAI.StationaryNS.correctionRadius

correctionScale def

The fast-decaying scale used in successive smoothing steps. · OAI.ClosedSurfaceR4.ExactCorrection.correctionScale

correctProbability def

Unnormalized Born sum on correct outputs; success is exactly one, not positivity, postselection, or a limiting high probability. · OAI.ExactQuantumFactoring.Circuit.correctProbability

correlated def

Correlated coordinates keep both input images full while prescribing the sum. · OAI.WLIdentification.BufferAlgebra.correlated

correlationShift def

A single multiplicative difference of a complex sequence. · OAI.Problem337.correlationShift

CorrelationSide inductive

Which of the two squarefree divisors contains a prime of their union. · OAI.Ostmann.CorrelationSide

correlationSum def

An ordinary sum, with weight one on each positive integer at most N. · OAI.TwoPointCorrelations.correlationSum

coset_action_correspondence def

The full congruence lattice of the coset algebra is order-isomorphic to the subgroup interval. · OAI.FiniteCongruence.coset_action_correspondence

cosetAlgebra def

One unary operation for each element of G; no omitted operations. · OAI.FiniteCongruence.cosetAlgebra

cosetCocycle def

The transversal cocycle, with source coordinate i and target g • i. · OAI.NuclearUltrapower.cosetCocycle

cosetConjugate def

The formal inverse to the graph-group splitting, on its free generators. · OAI.EilenbergGanea.cosetConjugate

cosetMap def

A homomorphism induces a map on left cosets of a subgroup, without a normality hypothesis on that subgroup. · OAI.EilenbergGanea.cosetMap

cosetPoints def

Full finite affine cosets; no subset or segment convention is used. · OAI.KaplanskyCounterexample.FiniteGeometry.cosetPoints

cosetRep def

A normalized transversal; the identity coset is represented by the identity. · OAI.NuclearUltrapower.cosetRep

cosimplicial def

The actual oriental cosimplicial object. · OAI.Thomason.Construction.NormalizedSimplexIdentities.cosimplicial

cosine def

The actual cosine integral, with its constant fixed geometrically. · OAI.PettyProjection.Spherical.cosine

cosineConstant def

The true normalized cosine kernel's constant, not an assumed multiplier. · OAI.PettyProjection.Spherical.cosineConstant

cosineSeminorm def

The actual cosine-transform norm furnished by a spanning measure. · OAI.PettyProjection.cosineSeminorm

cost def

The exact falling-factorial potential C(H), with (m)_0=1. · OAI.CoordinateSweeps.Grid.Holes.cost

cost def

Literal infimum over all decompositions z=h+Jμ. · OAI.DiamondDistortion.PathModel.cost

cost def

Syntactic duplication cost; independent of the numerical constants. · OAI.ExactQuantumFactoring.NatExpr.cost

cost def

The cost of a finite real-valued transport matrix. · OAI.Problem358.FiniteCoupling.cost

cost def

Each distinct generator is charged once at the original density. · OAI.TalagrandDiscreteConvexity.cost

cost def

Cost at the specified density, counting each generator once. · OAI.TalagrandDiscreteConvexity.cost

cost def

The source component cost with the actual occurrence matching and its actual self-links, not a supplied abstract component partition. · OAI.TorsionFreeZeroDivisors.ActualStageLabels.cost

cost def

Source vertex-assignment cost, separating the finitely many marked vertices. · OAI.TorsionFreeZeroDivisors.ChainRealization.cost

cost def

The exact vertex/table/prescription factor, before the retained turn cost. · OAI.TorsionFreeZeroDivisors.StageCost.cost

cost def

Squared-distance cost with factor one half. · OAI.WeakMTWTransport.cost

CostAdmissible def

Admissibility for the finite coding-measure family, with the prescribed residual-image null constraint but without any tuple exclusions. · OAI.CircleDomainRigidity.CostAdmissible

costCoefficientCoordinates def

The three quaternion coefficients of the total action on (7-K)L. · OAI.PositiveEinsteinFour.Bianchi.costCoefficientCoordinates

CostDomain def

The domain consists of all literal arrays of finite upward start cost. · OAI.BoundedTreePotentials.Segments.CostDomain

costParameters def

Only this fixed frame is needed for the invariant real-vector cost. · OAI.PositiveEinsteinFour.Bianchi.costParameters

costs def

Costs of trails in the finite candidate family. · OAI.GammaFPP.FiniteLocal.costs

costValue def

A clipped distance-to-set ramp, expressed as the minimum of all live center costs and the empty-roster cost one. · OAI.UniformKServer.ShortRoster.costValue

cotangentBasis def

The smooth normalization has cotangent basis du,dv. · OAI.KummerAffineZero.cotangentBasis

cotangentBasis def

The smooth normalization has cotangent basis du,dv. · OAI.KummerShifted.cotangentBasis

cotangentBasis def

The smooth normalization has cotangent basis du,dv. · OAI.KummerShiftedDiagonal.cotangentBasis

cotangentBasis def

The smooth normalization has cotangent basis du,dv. · OAI.KummerTriple.cotangentBasis

cotangentBasis def

The smooth normalization has cotangent basis du,dv. · OAI.KummerTripleDiagonal.cotangentBasis

cotangentDivergence def

Coordinate divergence in the position and momentum basis. · OAI.PositiveEinsteinFour.Geometry.cotangentDivergence

cotreeColor def

Colors are ACTUAL components of the complementary dual graph after isolating one specified face; the face itself receives its isolated color. · OAI.HyperbolicObstruction.PictureSwitch.PictureMap.cotreeColor

coulombForm def

Electronic form with kinetic coefficient 1/2; nuclear repulsion omitted. · OAI.ContinuumCoulomb.coulombForm

coulombLineKernel def

A nonsingular Coulomb line section, with the other two squared coordinates (and optional regularization) collected in A. · OAI.ContinuumCoulomb.coulombLineKernel

coulombPairEnergy def

The pair interaction energy of an independently sampled pair. · OAI.Problem356.coulombPairEnergy

CouniversalStreetWeakEquivalence def

Every cobase change of a strict map is a Street weak equivalence. · OAI.Thomason.Construction.NormalizedSimplexIdentities.CouniversalStreetWeakEquivalence

Count structure

The same data describe a product projection and its mismatch count. Normalization is exactly the sum of the squared moduli being one. · OAI.QAC.Count

count def

Common denominator, inverse branch probability, number of tables, and the ordinary draw's numerator. · OAI.ContingencyTables.ResidualWeightPrograms.count

count def

Occupancies of single alphabet symbols in an ordinary tensor word. · OAI.CoordinateSweeps.OrdinaryTensor.count

count def

false records a D slot, true an X slot. The leading D is separate. · OAI.DerivedBar.Multibar.count

count def

Number of marked boundaries before n. · OAI.FKCRT.CutOutput.count

count def

All ranks queried by the allocation are actual conditional hidden counts. The online map may depend on the partition tape and request requestPrefix; it never changes how the old hidden quantity is filtered. · OAI.KServer.FiniteExperiment.count

count abbrev

The original summand with exactly r square-zero factors in the actual bar complex. · OAI.LittleFinitistic.SelectedCountCohomology.count

count def

The stage random variable is a literal sum of feasible injective graph realizations, restricted to the full-word projection and to immersed labels. · OAI.TorsionFreeZeroDivisors.StageExpectation.count

CountableAdditivityInput def

The exact countable-additivity specialization used in this application. This is a published input on a specified family of spaces. · OAI.CoarseAssembly.Analytic.CountableAdditivityInput

countableAmplify def

The actual coordinatewise countable amplification of an operator. · OAI.KadisonSimilarity.countableAmplify

CountableCopies abbrev

The countable Hilbert direct sum of a fixed Hilbert space. · OAI.KadisonSimilarity.CountableCopies

CountableInteriorHits def

Countably many open hit tests encode dense interior trace times. · OAI.SLEExactGauge.StoppedBoundaryRestart.CountableInteriorHits

CountableProjectionCopies structure

Actual equivalent projection copies and their strong sum. · OAI.KadisonKastler.CountableProjectionCopies

countableRepresentation def

The countable standard amplification of a represented algebra. · OAI.KadisonSimilarity.countableRepresentation

countableSelector def

With countability and regularity in hand the defining existence proof is provided by distinguished_bound, rather than assumed of the chosen patterns. · OAI.PartitionConsistency.TailSelector.TailData.countableSelector

countAfter def

Double internal ranks and the singleton top, with no singleton bottom term because upward arrows never end at the bottom. · OAI.KLInvariance.SignedDihedral.countAfter

countCap def

A proved polynomial workspace guard. The cap is part of the implemented loop; the mathematical bound proves it never fires on an actual input. · OAI.Superstring.Computable.countCap

countedAdvance def

The same winning-row advance, with one counted row call and a counted root path replacement. Eight units per visited node cover field access, path tests, rebuilding and a cached comparison; key evaluation is separate. · OAI.LowSpaceSubsetSum.RowMerge.countedAdvance

countedArray def

The filter counter records predicate calls and output-array writes. The cost of the predicate itself is handled separately. · OAI.LowSpaceSubsetSum.RepresentationLeaf.countedArray

countedArrays def

Counted generation, prefix filtering, and sorting of the two stored leaf arrays. No row is materialized by this setup. · OAI.LowSpaceSubsetSum.RepresentationPairData.countedArrays

countedCertificate def

The generic repeat machine executes only the actual successive graph words. Its polynomial bound uses the proved linear growth of their semantic sizes. · OAI.MaxCutGames.Foundations.Complexity.MachineTableIteration.countedCertificate

countedCertificate def

The generic repeat machine executes only the actual successive graph words. Its polynomial bound uses the proved linear growth of their semantic sizes. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineTableIteration.countedCertificate

countedCopy def

A full snapshot copy, charged by the size of one node record. · OAI.LowSpaceSubsetSum.Tournament.countedCopy

countedLeaf def

A raw leaf is shared by the plus and minus names on both axes. · OAI.Problem348.Construction.countedLeaf

countedMean def

Quenched mean at fixed physical and perturbation counts, under one common countable type law. Marks retain their own finite alphabets. · OAI.DilutedSpinGlass.SizeCoupling.countedMean

countedMotifLaw def

Initial observations retain the central count with its original binomial law. · OAI.TraceReconstruction.countedMotifLaw

countedNext def

The three reader branches use no traversal of the backing array. The nonempty branch performs one indexed load and one cyclic cursor update. · OAI.LowSpaceSubsetSum.SliceCursor.countedNext

countedReplace def

Rebuild one path, counting touched nodes. · OAI.LowSpaceSubsetSum.Tournament.countedReplace

countedSnapshot def

Deep-copy the mutable reader tree while retaining the immutable array parameters. The extra record allowance in snapshotCells covers the root. · OAI.LowSpaceSubsetSum.RepresentationPairData.countedSnapshot

countEquiv def

The switch-count restriction of the word encoding. · OAI.Problem335.WordSwitches.countEquiv

Counterexample def

`Counterexample r` says that every vertex fails `GoodVertex`: its exact second out-neighborhood is strictly smaller than its first. This pointwise strict inequality is the negation normal form used by the minimal-counterexample argument. · OAI.SeymourSecondNeighborhood.Counterexample

counterOverhead def

Exact concrete-counter overhead for a complete positive-radix nesting. · OAI.MaxCutGames.Foundations.Complexity.MachineTupleOdometer.counterOverhead

counterOverhead def

Exact concrete-counter overhead for a complete positive-radix nesting. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineTupleOdometer.counterOverhead

counterTapes def

An arbitrary caller frame with one tape holding the unary counter and its unread suffix. Only this tape depends on the counter value. · OAI.MaxCutGames.Foundations.Complexity.MachineUnaryCounter.counterTapes

counterTapes def

An arbitrary caller frame with one tape holding the unary counter and its unread suffix. Only this tape depends on the counter value. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineUnaryCounter.counterTapes

countingDivisorSplitSet def

Append the two denominators indexed by a divisor to a fixed unchanged set. · OAI.Problem337.countingDivisorSplitSet

countingGoodSplitDivisors def

A split is good when neither new denominator collides with an unchanged term. · OAI.Problem337.countingGoodSplitDivisors

countingMass def

Finite posterior counting mass; repeated hidden locations contribute their full weight and do not require an injective posterior support. · OAI.KServer.Rounding.countingMass

CountingRegular def

These are exactly the regularity conditions used in the band count; the preimage conditions hold for every preimage of the value. · OAI.TotientAsymptotic.CountingRegular

CountingStatement def

Paper 200-02, Theorem main. The input length includes the rational accuracy parameters, and infeasible inputs always produce exactly zero. · OAI.ContingencyTables.Algorithms.CountingStatement

countObject abbrev

The original r-count object, with exactly the original scalar and inflated actions. · OAI.DerivedBar.ParityBridge.countObject

countObjectComparison def

The proved genuine shift comparison, with its target named for downstream use. · OAI.DerivedBar.ParityBridge.countObjectComparison

countPrefix def

An exact antiderivative of the harmonic multiplicity kernel for s=8. · OAI.HarmonicCounterexample.Certificate.countPrefix

countRank def

Source integer ranks: rank a (with zero-based index) is P(N ≤ a). The probability law is joint, without independent-child assumptions. · OAI.KServer.RankOrder.countRank

countRanks def

The rank inputs of the recursion can be instantiated directly from the actual finite hidden counts, not from a postulated martingale interface. · OAI.KServer.AdaptiveMinimization.countRanks

CountRowData abbrev

Shared unary allowance and binary denominator, current exponent data, and the enumeration of all states. · OAI.ContingencyTables.DensePrograms.CountRowData

counts def

All arithmetic hypotheses of the earlier curvature identity are now derived from actual edge/face/vertex permutations and face-color bijections. · OAI.Release075.TriangleRibbon.counts

counts def

A complete invariant of the simultaneous permutation orbit of finitely many words: the counts of their joint symbols. · OAI.RowColumn.WordOrbits.counts

countSatisfied def

`countSatisfied labeling constraints` recursively counts how many entries in the constraint list evaluate to `true` under one labeling. · OAI.UniqueGamesTheorem.Foundations.Target.countSatisfied

CountsLE def

All counters allocated in the position trie are bounded. · OAI.EditApproximation.PositionCounts.CountsLE

countsLetter def

Unsimplified positive/negative counters avoid assuming a computation model on G. · OAI.FiniteAlgebraicEnvelopes.GroupInput.countsLetter

countSplitting def

Chain-level count splitting of the actual two-sided absolute bar complex. This is the first chain isomorphism in the proof of manuscript prop:bar-splitting. Identification of these summands with derived tensor iterates is a separate … · OAI.DerivedBar.SquareZero.countSplitting

CountsPositive def

Every decision to divide is preceded by the literal positive-count check used in the source algorithm. · OAI.ContingencyTables.CompanionRationalUpdates.CountsPositive

countTensorBarInclusion def

The original count subcomplex included into the actual balanced tensor bar. · OAI.DerivedBar.ParityBridge.countTensorBarInclusion

coupledQuadratic def

Both block contributions, the differentiated scalar multipliers, and the derivative term of the variance correction. · OAI.PositiveEinsteinFour.Bianchi.coupledQuadratic

CoupledState structure

The two literal cut/chart systems together with every B chart's recorded increment history. The radius bound is an invariant, not a separation promise. · OAI.PlanarL1.Columns.Model.CoupledState

coupling def

Forgetting labels gives the one-coordinate pair law. · OAI.FiniteFactor.CouplingParameters.coupling

coupling def

The monotone coupling parametrized by the common unit-interval coordinate. · OAI.Problem358.IntervalMidpoint.coupling

couplingCoefficient def

The coefficient c in the paper's equation (15). · OAI.PositiveEinsteinFour.couplingCoefficient

couplingDecode def

Lookup needs only the seed and the observed label set, not its weights or the future suffix order. · OAI.MatroidProphet.Secretary.couplingDecode

couplingKernel def

Transition obtained from a prescribed adjacent-pair marginal. A zero-mass row is filled with the next marginal; the row is never visited. · OAI.BoundedTreewidthL1.MarkovProduct.couplingKernel

couplingKernel def

The matrix kernel of Σ I_outer,s (I_inner,s + Σ_s). · OAI.PositiveEinsteinFour.HighestHessian.couplingKernel

couplingLength def

The cutoff is fixed before arrivals using only the first part of the seed. · OAI.MatroidProphet.Secretary.couplingLength

CouplingParameters structure

One block of the actual coupling, with two distinct width labels. The boolean records whether this is the tagged block. · OAI.FiniteFactor.CouplingParameters

couplingPotential def

The scalar potential used in the one-sided coupling argument. · OAI.LeanBlast.GotsmanLinial.couplingPotential

CouplingSample abbrev

The augmented countable product retains each coordinate's label and pair. · OAI.FiniteFactor.CouplingParameters.CouplingSample

CouplingSeed abbrev

The full source seed together with every conditional draw, all predrawn. · OAI.MatroidProphet.Secretary.CouplingSeed

couplingSeedLaw def

This initial seed law depends on the matroid but not on any weights. · OAI.MatroidProphet.Secretary.couplingSeedLaw

covariance def

The covariance operator applied to a vector. · OAI.ClosedSurfaceR4.CovarianceCorrector.covariance

covarianceCLM def

The covariance as a quadratic expression in a continuous periodic map. · OAI.ClosedSurfaceR4.CovarianceCorrector.covarianceCLM

covarianceEnergy def

Squared conditional covariance density. · OAI.DilutedSpinGlass.FiniteLaw.covarianceEnergy

covarianceLinearMap def

The covariance defines a linear operator on the orthogonal complement. · OAI.ClosedSurfaceR4.CovarianceCorrector.covarianceLinearMap

CovariantData structure

Intermediate data: covariance plus only zero-argument algebraic relations. This is constructed below from a finite presented group, not postulated for it. · OAI.LittleFinitistic.FinitePresentation.CovariantData

covariantDerivative def

The induced Levi-Civita derivative on covariant coordinate tensors. The leading slot records the direction of differentiation. · OAI.PositiveEinsteinFour.Geometry.covariantDerivative

covariantSlotAction def

Apply a matrix in one covariant slot. · OAI.PositiveEinsteinFour.Geometry.covariantSlotAction

covectorDet def

Determinant of the same Gram tensor. · OAI.CircleDomainRigidity.FiniteTransfer.covectorDet

covectorTrace def

Trace of the Gram tensor formed from a finite list of planar covectors. · OAI.CircleDomainRigidity.FiniteTransfer.covectorTrace

covectorVolume def

The alternating product of a family of real-linear complex covectors. · OAI.Mahler.covectorVolume

covectorWedges def

Each unordered pair is counted exactly once. · OAI.CircleDomainRigidity.FiniteTransfer.covectorWedges

Covenant structure

Specific contract per version, Good row assumed.Pulse.EI at any m scoreboard. If inactive potential, no TC or T services. End serves zeros. · OAI.ExactDerandomization.Umbra.Elect.Covenant

Cover abbrev

The manuscript's projective cover, realized as an actual direct summand. · OAI.LittleFinitistic.DirectedBound.DirectedSystem.Cover

Cover structure

Projective covers with their actual maps and radical-kernel property. · OAI.LittleFinitistic.PDMinimal.Cover

Covered def

A dart lying in one of the actual maximal marked chains. · OAI.TorsionFreeZeroDivisors.ChainCoverage.Covered

covered def

Particles incident to a particle-sharing edge or a fixed hole. · OAI.CoordinateSweeps.Coverage.covered

covered def

All letters have support contained in comparison law's. · OAI.FourierLLogL.Compress.covered

covered def

The union after exactly n selections, including any padding selections. · OAI.MetricEntropyDuality.GreedyPivots.covered

covered def

Portion of sigma occupied by the smaller network. · OAI.PowerSaving.Binary.covered

CoveredBy def

The event that some full path is covered by a given set. · OAI.LeanBlast.SecondKahnKalai.ProbabilityTree.CoveredBy

coveringNormalVelocity def

Vertical velocity lifting a prescribed covering-frame normal coordinate. · OAI.Nagata.Workers.W17.coveringNormalVelocity

coveringPow def

Bounded priority search for the least covering power of two. The bounds are compile-time constants, not exponentiation gates with variable exponents. · OAI.ExactQuantumFactoring.OrderTrial.Expressions.coveringPow

coverMap abbrev

Evaluation restricted to the projective summand. · OAI.LittleFinitistic.DirectedBound.DirectedSystem.coverMap

coverRadii def

Positive radii admitting a finite diameter cover. · OAI.Tingley.coverRadii

Covers def

Every covered set contains a generator. · OAI.TalagrandDiscreteConvexity.Covers

Covers def

A cover supplies a contained generator for every target set. · OAI.TalagrandDiscreteConvexity.Covers

CoversPoints def

The direction partition covers every nonzero element of each point group. · OAI.KaplanskyCounterexample.GluedGroup.CoversPoints

CoverState structure

The actual retained tree, its spread budget, and recovery to the fixed original tree. Only positive-capacity levels are stored. · OAI.LeanBlast.SecondKahnKalai.ProbabilityTree.CoverState

coverTuple def

The fixed lifts followed by the q moving lifts in the common exponential chart. · OAI.Nagata.W06.TorusConfiguration.coverTuple

coverX def

Cover as a tensor space; used to count tiny-cover span. · OAI.HadwigerCounterexample.coverX

cpM def

returns derivative of f(x)*y with f derivatives supplied by w · OAI.GeneralMahler.Jet.cpM

cpMatrixVectorFunctional def

Applying a completely positive map entrywise before a vector quadratic form yields a positive functional on the original matrix algebra. · OAI.BoundedHochschild.KadisonRingrose.cpMatrixVectorFunctional

Crack abbrev

Enumerate the increments by a pool in which each piece of a telescope, variant of the walk, and multiplicand port have one reserved index. · OAI.ExactDerandomization.Rain.Crack

crack def

Consume frame prefix of k, pushing first component reversed onto t. · OAI.Fifths.SM.crack

cRat def

continued fraction · OAI.GeneralMahler.SCal.cRat

create def

Insert an adjacent cup at a frontier position. States are zero on words too short for the requested position. · OAI.CriticalHoneycomb.Transfer.PlanarNormalization.create

creed def

selected pure excludes second tags · OAI.HadwigerCounterexample.creed

Critical def

The two critical configurations, checked only inside the current length ball. · OAI.FiniteAlgebraicEnvelopes.Bootstrap.Critical

criticalNumber def

The minimum critical number, including degenerate functions (ENat allows infinity). · OAI.ArnoldCounterexample.criticalNumber

criticalProbability def

The infimum of parameters with positive probability of an infinite cluster. · OAI.CriticalPercolation.BondGraph.criticalProbability

criticalValues def

Critical values of a map between two-dimensional coordinate spaces. · OAI.ClosedSurfaceR4.PublishedInputs.criticalValues

cropped def

Crops of Type II expressions retain the original paths and coding. · OAI.SeparableQuotient.Norming.TypeII.cropped

CropStable def

Stability of a norming set under the specified coordinate crop. · OAI.SeparableQuotient.NormConstruction.NormingSet.CropStable

cross def

Across the final facet, pass to the neighboring cube and toggle its endpoint. · OAI.SharpLiebThirring.CubeFlags.Flag.cross

crossBilinear def

The symmetric mixed metric coefficient as a bounded bilinear map of the two differentials. · OAI.ClosedSurfaceR4.ExactCorrection.crossBilinear

CrossBound def

The full cross-mass property; sets may overlap. · OAI.CliqueFreeIndependence.WeightedGraph.CrossBound

crosscapParameterLinear def

Reorder the linearized direction variables into the standard crosscap target coordinates. · OAI.ClosedSurfaceR4.FiniteOrderSmoothing.crosscapParameterLinear

crosscapTargetShear def

A quadratic target shear with identity differential at the origin. · OAI.ClosedSurfaceR4.FiniteOrderSmoothing.crosscapTargetShear

crossCloud def

Cross the outer graph edge and swap the two inner ports. · OAI.MaxCutGames.Foundations.PCP.ZigzagGraphs.crossCloud

crossCloud def

Cross the outer graph edge and swap the two inner ports. · OAI.UniqueGamesTheorem.Foundations.PCP.ZigzagGraphs.crossCloud

crossE def

cross entropy (only meaningful for comparison positive on support). · OAI.FourierLLogL.Law.crossE

Crosses def

Exactly one endpoint in S; edge identities are kept, including parallels. · OAI.MatchingEntropy.LooplessGraph.Crosses

CrossFrequencyUnits def

Cross-branch frequency tests are retained explicitly at every node. · OAI.Ostmann.MovingSlotData.CrossFrequencyUnits

crossingAntiderivative def

The polynomial whose finite difference evaluates the crossing sum. · OAI.Paper170.crossingAntiderivative

CrossingGerms def

Crossing anchor sequences identified precisely when they agree eventually. · OAI.BoundedTreewidthL1.CrossingGerms

crossingGraph def

Edges whose midpoint fibers contain a compact continuum joining the two boundaries. · OAI.EuclideanFiveColor.crossingGraph

crossingLabel def

Conditioning on y uses a bottom label; conditioning on x uses a top label. · OAI.FiniteFactor.CouplingParameters.crossingLabel

crossingPartner def

The interval in which the unconditioned partner crosses an eligible threshold. · OAI.FiniteFactor.CouplingParameters.crossingPartner

crossingPoints def

Spatial intersection points. Different meetings of the same pair remain distinct. · OAI.Paper170.crossingPoints

crossings def

Actual crossing pairs read from the literal vertex decoration, with the right (moving left) label first. Splits, merges and continuations do not add crossings. This retains occurrence labels when the colours label occurrences. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.Decoration.crossings

CrossingSequence def

A reservation interval whose effective start is early and terminal time is late. Validity is explicit; no anchor is created merely by taking a quotient. · OAI.BoundedTreewidthL1.CrossingSequence

crossingTuples def

Increasing four-tuples whose alternating edges occupy the same page. · OAI.Paper170.crossingTuples

crossMap def

Oriented angular momentum about the origin against a fixed velocity. · OAI.TriangularBilliards.crossMap

CrossMultiplicityMatrixCovariance structure

Actual word covariance for an operator between independent multiplicity spaces. The word formulas refer only to the common endpoint and tail factors. · OAI.KadisonSimilarity.CrossMultiplicityMatrixCovariance

crossMultiplicitySlice def

A bounded scalar multiplicity slice with independently typed source and target multiplicities. · OAI.KadisonSimilarity.crossMultiplicitySlice

crossOperator def

The rectangular operator associated with an algebraic Hilbert tensor. · OAI.NuclearUltrapower.crossOperator

crossPairingEquiv def

A cross pairing determines the bijection indexing its contraction. · OAI.KadisonSimilarity.crossPairingEquiv

crossPairingOperator def

The explicit finite sum of the operators indexed by cross pairings. · OAI.KadisonSimilarity.crossPairingOperator

crossPairings def

The finite family of matchings which join source slots to test slots. · OAI.KadisonSimilarity.crossPairings

crt def

The resulting algebra equivalence preserves the actual convolution. · OAI.CirculantHadamard.BinaryBlocks.crt

crtIndex def

The concrete CRT map on the exponent group. · OAI.CirculantHadamard.BinaryBlocks.crtIndex

crtJitterCode def

Evaluate joint finite jitter directly on CRT residue coordinates. · OAI.TwoPointCorrelations.crtJitterCode

crtJointProbability def

The actual joint residue law of an interval under the CRT equivalence. · OAI.TwoPointCorrelations.crtJointProbability

crtRing def

The actual finite Chinese remainder equivalence, with no chosen-map premise. · OAI.CirculantHadamard.PrimeComponents.crtRing

crtScale def

The positive source normalization in original edge units. · OAI.FKCRT.Inventory.crtScale

crtSplitExternalEquiv def

Simultaneously split the integer and unit giant residues. · OAI.Ostmann.crtSplitExternalEquiv

crtSplitUnitEquiv def

The retained frequency block and all other unit residues in the actual modulus. · OAI.Ostmann.crtSplitUnitEquiv

crtUniform def

Uniform independent residue coordinates. · OAI.TwoPointCorrelations.crtUniform

crtUnitEquiv def

CRT on the actual unit residue classes, including prime-square moduli. · OAI.Ostmann.crtUnitEquiv

Crude structure

recursive premises needed for one order data/pulses. · OAI.ExactDerandomization.Chron.Moon.Weather.Crude

crudePaddingWeight def

The actual padding weight u(q)=4^ω(q). · OAI.TwoPointCorrelations.crudePaddingWeight

crudeTraceWeight def

The positive trace factors left after discarding all denominator cutoffs. This includes the full mixed-difference multiplicity. · OAI.TwoPointCorrelations.crudeTraceWeight

CrudeWordCode abbrev

Signs, a partition code, bounded records per slot, and lit designations. · OAI.TwoPointCorrelations.CrudeWordCode

cshift def

Complex translation; no ordinary measurability is asserted or needed. · OAI.PlaneFiveColor.InvariantMean.cshift

cstarAlgebra def

The separated completion of a C*-seminormed star algebra is a C*-algebra. · OAI.NuclearUltrapower.UniversalCompletion.cstarAlgebra

cstarColumn def

Extract a column without changing the number of rows. · OAI.KadisonSimilarity.cstarColumn

cstarMatrixComp def

Flattening blocks of C⋆-matrices is a star-algebra equivalence. · OAI.KadisonKastler.cstarMatrixComp

cstarMatrixFlatten def

Flatten the two genuine C*-matrix levels into one product-indexed matrix algebra, preserving multiplication, star, and complex scalars. · OAI.Kaplansky.cstarMatrixFlatten

cstarMatrixPadding def

Zero padding with the actual C-star matrix norms. · OAI.KadisonSimilarity.cstarMatrixPadding

CStarSimple def

Simplicity in the C-star sense: there are no proper nonzero closed two-sided ideals. Nontriviality is supplied separately where used. · OAI.KadisonKastler.CStarSimple

ctxt def

Context part of a vector of full rows. · OAI.FourierLLogL.Compress.ctxt

Cube abbrev

Binary positions with their actual bit-coordinate labels. · OAI.CoordinateSweeps.Cube

Cube abbrev

The Cantor cube used by the twisted Brin--Thompson construction. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Cube

Cube abbrev

The sign cube {−1,1}^n, with Boolean coordinates encoding the signs. · OAI.LeanBlast.GotsmanLinial.Cube

cube def

The actual unit cube in the coordinate space. · OAI.DiagonalZonotope.cube

cube def

Multiplicities in the cube are explicitly retained through composition. · OAI.PlaneFiveColor.InvariantMean.cube

cube def

The formalization proves one-third Hölder stability of Brenier maps from the uniform measure on a compact convex body with nonempty interior in $\mathbb R^d$, for every $d\ge2$. For targets supported in one fixed nonempty compact set, the … · OAI.Problem358.cube

CubeAdjacent def

Two cube vertices differ by a single coordinate flip. · OAI.LeanBlast.GotsmanLinial.CubeAdjacent

cubeAverage def

Uniform average on the finite Boolean cube. · OAI.TwoPointCorrelations.cubeAverage

cubeBoundary def

The closed-cube vertex boundary used by the finite-scale argument. · OAI.CriticalZ3.cubeBoundary

cubeCoord def

Interpret a Boolean cube coordinate as the corresponding real sign. · OAI.LeanBlast.GotsmanLinial.cubeCoord

cubeCoordinates def

The product of closed unit intervals is exactly the coordinate box. · OAI.Thomason.Construction.SimplexPolytopes.cubeCoordinates

cubeEquiv def

Explicit binary rank/unrank; the inverse extracts digits by division and remainder, as supplied by the executable finFunctionFinEquiv. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Enumeration.cubeEquiv

cubeEquiv def

Fixed binary radix indexing, rather than a chosen Fintype.equivFin. · OAI.OptimalMaxCut.LongCode.InstanceAdapter.cubeEquiv

cubeEquiv def

Explicit binary rank/unrank; the inverse extracts digits by division and remainder, as supplied by the executable finFunctionFinEquiv. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Enumeration.cubeEquiv

cubeEquivSignVectors def

The Boolean encoding represents every sign vector exactly once. · OAI.LeanBlast.GotsmanLinial.cubeEquivSignVectors

cubeHomeomorph def

A finite simplicial cube realizes to the ordinary topological cube, with the original constructed cubical section as its inverse coordinates. · OAI.Thomason.Construction.RealizedCube.cubeHomeomorph

cubePattern def

Indicator of a prescribed pattern on a set of coordinates. · OAI.TwoPointCorrelations.cubePattern

cubeProfileIntegral def

The normalized planar integral; the measure of an Eisenstein fundamental parallelogram is one. · OAI.CubicFirstMoment.cubeProfileIntegral

cubeSet def

The subset represented by a Boolean coordinate vector. · OAI.TalagrandDiscreteConvexity.cubeSet

cubeSimplexRank def

Count the interval coordinates of a cube simplex, pointwise along its ordinal parameter. This is monotone in that parameter. · OAI.Thomason.Construction.cubeSimplexRank

cubeToSimplex def

The actual simplicial map from the cube to the simplex counting its ordered interval coordinates. · OAI.Thomason.Construction.cubeToSimplex

cubeWeight def

Product mass on the Boolean cube, including signed parameters. · OAI.TalagrandDiscreteConvexity.cubeWeight

cubicBesselHeat def

The positive heat integrand for 2*K_(1/3)(2*sqrt x). · OAI.CubicFirstMoment.cubicBesselHeat

cubicBesselKernel def

Fixed Bessel order required by the cubic metaplectic Fourier series. · OAI.CubicFirstMoment.cubicBesselKernel

cubicContraction def

The literal matrix contraction tr(Z_a Σ_j Z_j²)/6. Each inner product contracts the derivative (four-vector) index. · OAI.PositiveEinsteinFour.Bianchi.cubicContraction

cubicCoverMap def

The actual projective map on the nonzero complex covering space. · OAI.Nagata.W07.cubicCoverMap

cubicCurveMap def

The actual projective curve-valued map, without a surjectivity claim. · OAI.Nagata.W21.cubicCurveMap

cubicEvaluationSpan def

Degree-three evaluation space. Repeated indices recover all lower positive degrees on binary points. · OAI.MaxCutGames.Quadratic.cubicEvaluationSpan

cubicEvaluationSpan def

Degree-three evaluation space. Repeated indices recover all lower positive degrees on binary points. · OAI.UniqueGamesTheorem.Quadratic.cubicEvaluationSpan

cubicEvaluationVector def

Three actual evaluation coordinates associated to a basis of D(3,γ). · OAI.Nagata.W07.cubicEvaluationVector

cubicExponentialMap def

The covering map in a genuine local exponential parameter. · OAI.Nagata.W07.cubicExponentialMap

cubicHom abbrev

Naming this morphism keeps kernel checking from repeatedly expanding the iterated tensor-module structures at successor degrees. · OAI.ArExplicit.BarBalanced.cubicHom

CubicInput def

Exact finite-head cubic inequality, with its positive radius denominator. · OAI.DiamondDistortion.RecursiveApplications.CubicInput

cubicMapFromDimensions def

The genuine projective map, with basepoint exclusion obtained from D2 dimensions. · OAI.Nagata.W07.cubicMapFromDimensions

cubicOrbitScale def

The scalar accumulated along nonnegative iterates of the period. · OAI.Nagata.Workers.W25.cubicOrbitScale

cubicPairing def

Coefficient obtained by pairing a polynomial row with a cubic binary evaluation. · OAI.MaxCutGames.Quadratic.cubicPairing

cubicPairing def

Coefficient obtained by pairing a polynomial row with a cubic binary evaluation. · OAI.UniqueGamesTheorem.Quadratic.cubicPairing

cubicPeriod def

The actual nonzero real period as a complex unit. · OAI.Nagata.W07.cubicPeriod

cubicPrimeCoefficient def

The normalized cubic exponential-sum contribution of the nonzero frequencies at a prime modulus. · OAI.Problem337.ThreePrimeLocalFactor.cubicPrimeCoefficient

CubicPrimeSiegelWalfisz def

Exact logarithmic-conductor prime-sum input. The conductor is the product of the coprime squarefree conductors of the two character factors. Constants are uniform over that family. Both conjugate primes are counted. · OAI.CubicFirstMoment.CubicPrimeSiegelWalfisz

CubicSupplementaryPeriodicity def

Cubic reciprocity plus its supplementary laws, specialized to periodicity of the actual cubic symbol on primary denominators. · OAI.CubicFirstMoment.CubicSupplementaryPeriodicity

cubicThetaAngularCoefficient def

The frequency multiplier produced by the derivative of order |ell|, with its signed angular component kept explicit. · OAI.CubicFirstMoment.cubicThetaAngularCoefficient

cubicThetaArithmeticAngularAxis def

Extend the actual derivative on the positive vertical axis by zero. Only positive heights enter its Mellin transform. · OAI.CubicFirstMoment.cubicThetaArithmeticAngularAxis

cubicThetaGaussCuspMatrix def

Lambda cubed cancels the selected frequency denominator. · OAI.CubicFirstMoment.cubicThetaGaussCuspMatrix

cubicThetaKubotaCharacter def

The published multiplier is a genuine multiplicative character. · OAI.CubicFirstMoment.cubicThetaKubotaCharacter

cubicThetaKubotaValue def

The cubic residue value in the original Kubota convention. · OAI.CubicFirstMoment.cubicThetaKubotaValue

CubicThetaPoint def

Hyperbolic three-space in the coordinates of the manuscript. · OAI.CubicFirstMoment.CubicThetaPoint

cubicThetaResidueChar def

The source trace character, without the inverse-different factor. For primary q the removed factor is a unit modulo q. · OAI.CubicFirstMoment.cubicThetaResidueChar

cubicThetaShiftedLatticeCoefficient def

The actual translated-cusp coefficient, with the single global residue scalar removed. This is derived, not postulated as a new theta family. · OAI.CubicFirstMoment.cubicThetaShiftedLatticeCoefficient

cubicThetaSiegelSet def

A single full-group cusp chart with a uniform positive lower height. · OAI.CubicFirstMoment.cubicThetaSiegelSet

cubicThetaTau def

The source uses tau(mu), while our Whittaker coefficient is tau(mu)/|mu|. · OAI.CubicFirstMoment.cubicThetaTau

cubicThetaTwistedGauss def

The normalized Gauss sum at an invertible additive frequency. · OAI.CubicFirstMoment.cubicThetaTwistedGauss

cubicThetaTwistTransform def

Finite numerator Fourier transform in the exact trace normalization. · OAI.CubicFirstMoment.cubicThetaTwistTransform

cubicThetaUnitPhase def

Unit phases in the first three cases of (5.7). Raising a sixth root of unity to the fourth power removes its sign. · OAI.CubicFirstMoment.cubicThetaUnitPhase

cubicThetaWhittaker def

The heat-integral realization of v*K_(1/3)(4*pi*v). · OAI.CubicFirstMoment.cubicThetaWhittaker

cubicTorusMap def

The map on the actual multiplicative torus quotient. · OAI.Nagata.W07.cubicTorusMap

cumulative def

Barycentric mass before the kth vertex. · OAI.Thomason.Construction.PrismBarycentric.cumulative

cumulative def

The manuscript's distribution function, without a normalization convention for non-probability inputs. · OAI.ThreeTorus.cumulative

cumulativeMinimum def

The actual cumulative minimum, with the initial cap equal to one. · OAI.ReflexiveFixedPoints.OrderedTree.cumulativeMinimum

cuntzBlock def

Block addition before passing to Cuntz classes. · OAI.Kaplansky.cuntzBlock

cuntzBlockHom def

The genuine block-diagonal unital star homomorphism. · OAI.Kaplansky.cuntzBlockHom

cuntzCutoff def

The genuine continuous spectral cutoff (a-ε)₊. · OAI.Kaplansky.cuntzCutoff

CuntzDimensionState structure

A normalized dimension state on actual positive finite matrices. This is a semigroup state, not yet a quasitrace or its regularization. · OAI.Kaplansky.CuntzDimensionState

CuntzEquivalent def

Mutual actual rectangular comparison; no order or state is assumed. · OAI.Kaplansky.CuntzEquivalent

CuntzFamily structure

A finite family of isometries with orthogonal ranges summing to one. · OAI.KadisonKastler.CuntzFamily

cuntzFamily def

This construction is independent of the algebra that contains the matrix units, so the same family can be used on both sides of a gap. · OAI.KadisonKastler.FullMatrixCorner.cuntzFamily

CuntzLE def

Rectangular Cuntz comparison: norm closure of actual conjugations. The finite product topology here is the topology of the C*-matrix norm. · OAI.Kaplansky.CuntzLE

CuntzMonoid abbrev

The ordered Cuntz monoid of actual positive finite matrices over A. The order is rectangular norm comparison, not an assumed abstract relation. · OAI.Kaplansky.CuntzMonoid

CuntzPositiveMatrix structure

An actual positive matrix with a finite, explicitly stored size. · OAI.Kaplansky.CuntzPositiveMatrix

CuntzTwoRelations def

The two Cuntz relations, including unit preservation explicitly. · OAI.NuclearUltrapower.CuntzTwoRelations

cuntzUnit def

Class of the n-by-n identity. · OAI.Kaplansky.cuntzUnit

CupHistory inductive

A planar cup-only normal form. b is the number of through strands (0 for the empty boundary, 1 for one specified oriented path). · OAI.CriticalHoneycomb.Transfer.PlanarNormalization.CupHistory

current def

The coproduct of the actual spin-raising current, with direction weights attached to the boundary wires in their planar contour order. · OAI.CriticalHoneycomb.Transfer.FluxWard.current

current def

The compact horizontal current evaluated on a smooth unitary map. · OAI.KervaireIncidenceCurrent.current

curry def

Currying is an actual positive augmented chain map into the path complex. · OAI.Thomason.Construction.DirectedComplex.RightPath.curry

curryLoop def

Curry the first coordinate of a based cube into a based path. · OAI.Thomason.Construction.HomotopyMaps.curryLoop

curryLoopHomotopy def

A cubical homotopy curries continuously and remains fixed on the boundary. · OAI.Thomason.Construction.HomotopyMaps.curryLoopHomotopy

curryMap def

Split the first slot of a jet value, with the fixed normed-space structures. · OAI.ForcedComputation.BoundedSpatialJets.curryMap

currySigned def

Read a signed exponent by its layer and matrix-entry coordinates. · OAI.Problem335.FockLayerProfile.currySigned

curvature def

The curvature operator R(u,v)w = ∇_U ∇_V W - ∇_V ∇_U W - ∇_[U,V] W, using local smooth extensions at x. This is the convention where g(R(u,v)v,u) is the sectional-curvature numerator. · OAI.KahlerUniformization.SmoothKahlerMetric.curvature

curvatureAction def

The matrix-slot curvature operator on the concrete Bianchi space. · OAI.PositiveEinsteinFour.Bianchi.curvatureAction

curvatureBilinear def

The symmetric bilinear version of the actual matrix contraction. · OAI.PositiveEinsteinFour.Bianchi.curvatureBilinear

curvatureBlock def

The factor 1/8 accounts for antisymmetric coordinate pairs and the 1/sqrt(2) normalization of the quaternionic two-form basis. · OAI.PositiveEinsteinFour.Geometry.curvatureBlock

curvatureContraction def

Direct matrix-slot quadratic contraction with diagonal Weyl block v*s. The second sum is the off-diagonal contribution, because the spectral trace vanishes. · OAI.PositiveEinsteinFour.Bianchi.curvatureContraction

curvatureCoordinates def

Coordinate components of <R(e_a,e_b)e_c,e_d>. · OAI.PositiveEinsteinFour.Geometry.curvatureCoordinates

CurvatureDimension def

Unreduced CD(K,N), written with the standard dynamical optimal-plan formula. · OAI.WeakHessian.CurvatureDimension

curvatureEndomorphism def

Slots are arranged as the endomorphism R(e_i,e_j) on vectors. · OAI.PositiveEinsteinFour.Geometry.curvatureEndomorphism

curvatureEulerDensity def

The curvature polynomial whose integral is the Euler number in the Chern--Gauss--Bonnet theorem. That global theorem is not used here. · OAI.PositiveEinsteinFour.Geometry.curvatureEulerDensity

curvatureForm def

The quadratic form of the compressed matrix-slot action, normalized by |Z|² / 6. · OAI.PositiveEinsteinFour.HessianAlgebra.curvatureForm

curvaturePairing def

Curvature as a bilinear form on two-form coordinates. · OAI.PositiveEinsteinFour.Geometry.curvaturePairing

curvaturePairing def

The symmetric form of the concrete curvature operator. · OAI.PositiveEinsteinFour.HighestHessian.curvaturePairing

curvaturePontryaginDensity def

The first Pontryagin density; its third is the signature density. · OAI.PositiveEinsteinFour.Geometry.curvaturePontryaginDensity

CurveChart abbrev

Coordinates on the curve's actual projective affine chart. · OAI.Nagata.W16.AssociatedBundle.CurveChart

curveChartIdeal def

The genuine principal ideal on every affine plane chart; no radical is taken. · OAI.Nagata.Workers.W10.curveChartIdeal

curveEquation def

An equation realizing the entire cycle, including repeated components. · OAI.Nagata.Workers.W10.curveEquation

curveEquation def

Reconstruct an actual homogeneous equation from the ideal cycle. · OAI.Nagata.Workers.W10.curveEquation

curveForm def

The chosen reconstruction is packaged as a genuine nonzero homogeneous form. · OAI.Nagata.Workers.W10.curveForm

curveMap def

The one-dimensional extension-field line, viewed over the ground field. · OAI.KaplanskyCounterexample.FiniteGeometry.curveMap

curveMultiplicityAtLeast def

Local multiplicity uses the actual affine chart maximal-ideal powers, equivalently the actual regular local-ring order used by the manuscript. · OAI.Nagata.Workers.W10.curveMultiplicityAtLeast

curveNormalParameter def

Actual reparametrization and homogeneous normal-fiber rescaling. · OAI.Nagata.Workers.W28.curveNormalParameter

curveOfEquation def

An actual positive-degree projective equation class gives its zero-divisor cycle. · OAI.Nagata.Workers.W10.curveOfEquation

curveOrdinaryMultiplicity def

Actual ordinary multiplicity of the reconstructed curve equation. · OAI.Nagata.W15.curveOrdinaryMultiplicity

CurvePoint abbrev

Actual points on the projective zero locus of G. · OAI.Nagata.W16.AssociatedBundle.CurvePoint

CurveSecondDerivativeBound def

The literal second-distributional-derivative conclusion on (0,1). · OAI.WeakHessian.CurveSecondDerivativeBound

Cut structure

The zero-labelled vertices of a poset simplex form one initial segment. · OAI.Thomason.Construction.SieveSimplex.Cut

cut def

A compact cutoff acts on every smooth amplitude, without any temperate assumption on the amplitude itself. · OAI.ElasticityCGO.cut

cut def

Coordinate restriction, before the boundedness hypothesis is imposed. · OAI.ForestSpace.TestVector.cut

cut def

cut t groups of R.n fibers from the peeled coordinates · OAI.PowerSaving.RAM.cut

cut def

`G.cut T S` keeps exactly those edge copies in `T` with one endpoint in the vertex set `S` and the other endpoint outside it. · OAI.StrongThinTree.MultiGraph.cut

cut def

Construct the last vertex of the nonempty initial segment. · OAI.Thomason.Construction.SieveSimplex.cut

cut def

The actual cut comparison data, with its carrier exposed once. · OAI.TorsionFreeZeroDivisors.OriginalPattern.cut

cut def

The nonunital positive cut (a - ε)₊, defined using functional calculus. · OAI.TraceConeClassification.Positive.cut

cutApproach def

Tangential approach to the phase cut, from the prescribed side. · OAI.KervaireScalarPhase.cutApproach

cutCapacity def

Each unordered separated pair occurs once, with its endpoint in S first. · OAI.UniformSparsestCut.Graph.cutCapacity

CutColumnCoordinatesAligned def

Column coordinates at two ports are compatible with transport across the cut seam. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.CutColumnCoordinatesAligned

cutEdges def

Oriented edges crossing a two-color cut. · OAI.CoordinateSweeps.Coverage.cutEdges

CutFamily structure

All final metrics, and all adaptive averages of them, use the same finite cut index type. No finite approximation to the label space is made. · OAI.PlanarL1.FiniteL1.CutFamily

cutFraction def

Fraction of a cluster on the positive side, on the exact p^6 scale. · OAI.OptimalMaxCut.Unweighted.cutFraction

cutFromOldFrames def

Connected-strip incidence constructed from a pair of seam frames. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.cutFromOldFrames

cutGeneratorWitness opaque

A simultaneous splitting for arbitrary original generator coordinates, including the circle- gauged coordinates of the surface equation. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.cutGeneratorWitness

cutIndicator def

The indicator of a nonnegative affine cut. · OAI.Problem358.MovingCellTrace.cutIndicator

CutInheritedChildLabels def

The inherited column coordinates keep the original child label. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.CutInheritedChildLabels

CutInheritedColumnValues def

The inherited column coordinates keep the position inside each child. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.CutInheritedColumnValues

CutInheritedFrameCoordinates def

Every inherited port frame agrees with the original frame in the original seam coordinates. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.CutInheritedFrameCoordinates

CutInheritedSeamColumnCoordinates def

Inherited port columns agree with the column coordinates of their cut seam. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.CutInheritedSeamColumnCoordinates

CutInheritedSeamGrades def

Inherited seam columns retain the grades of their original child coordinates. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.CutInheritedSeamGrades

CutInheritedSeamRowCoordinates def

Inherited port rows agree with the row coordinates of their cut seam. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.CutInheritedSeamRowCoordinates

cutInnerLinear def

The inner boundary is the inverse-cut word T⁻¹ J⁻¹ P T. · OAI.IntegralCharacterVarieties.NamedBandGrades.cutInnerLinear

cutLastOldPort def

The old positive port in the refined cut diagram. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.cutLastOldPort

cutMap def

The final finite cut-coordinate formula, valued in the ACTUAL real L1 space requested by the main theorem, rather than an abstract cut cone. · OAI.PlanarL1.FiniteL1.cutMap

cutMass def

The barycentric amount assigned to the lower copy of the cut vertex. · OAI.Thomason.Construction.PrismBarycentric.cutMass

cutMirrorColumnCoordinates def

The child label and the coordinate within that child at a mirror port. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.cutMirrorColumnCoordinates

CutMirrorFrameCoordinates def

Equality of the transported and aligned mirror frames in the original port coordinates. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.CutMirrorFrameCoordinates

CutMirrorLastOutputAttached def

The final mirror output belongs to the final end of the short mirror seam. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.CutMirrorLastOutputAttached

CutMirrorSeamColumnsCoordinates def

Mirror column coordinates agree with the short-seam enumeration. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.CutMirrorSeamColumnsCoordinates

CutMirrorSeamGrade def

The short mirror seam grading is the original child index. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.CutMirrorSeamGrade

CutMirrorSeamRowsCoordinates def

Mirror row coordinates agree with the short-seam enumeration. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.CutMirrorSeamRowsCoordinates

cutOfBits def

A cut is returned as one bit per vertex, in vertex order. · OAI.UniformSparsestCut.cutOfBits

cutoff def

The fixed cutoff is independent of all pulse data. It is zero already on (-infty,5/2] and one on [7/2,infty), as allowed by geometry.tex. · OAI.HarmonicCounterexample.Scale.cutoff

cutoff def

The cutoff remains an actual positive matrix of the same finite size. · OAI.Kaplansky.CuntzPositiveMatrix.cutoff

cutoff def

The common geometric cutoffs used in the random middle-level selection. · OAI.Problem337.GeometricRandomRanges.cutoff

cutoffAverageHeatMass def

Average the kernel first, then integrate the cutoff average against the full mass measure. · OAI.RegularPointCoordinates.cutoffAverageHeatMass

cutoffAverageHeatMassAt def

The actual averaged heat mass, set to zero outside positive times. · OAI.RegularPointCoordinates.cutoffAverageHeatMassAt

cutoffCorrection def

All cutoff product terms, before taking an absolute value or norm. · OAI.ElasticityBoundaryCutoffLayer.cutoffCorrection

cutoffEnergy def

Squared energy retained by a coordinate projection. · OAI.MaxCutGames.Fourier.MatrixEnergy.cutoffEnergy

cutoffEnergy def

The actual bounded Sobolev product representing the cutoff energy. · OAI.RegularPointCoordinates.LocalHarmonicEnergy.cutoffEnergy

cutoffEnergy def

Squared energy retained by a coordinate projection. · OAI.UniqueGamesTheorem.Fourier.MatrixEnergy.cutoffEnergy

cutoffHeatGram def

The actual cutoff heat average of the Gram field, with the manuscript's positive scalar regularization. The weight is constructed from the kernel. · OAI.RegularPointCoordinates.cutoffHeatGram

cutoffHeatGramAt def

An all-time extension used only to state ordinary derivatives at positive times; it agrees with the actual Gram wherever it is used. · OAI.RegularPointCoordinates.cutoffHeatGramAt

cutoffHeatGramEquiv def

Strict regularization constructs the inverse of every actual heat Gram operator in finite dimension, at every testing point and positive time. · OAI.RegularPointCoordinates.cutoffHeatGramEquiv

cutoffHeatMass def

Heat integration of the full mass measure on a finite coordinate span. · OAI.RegularPointCoordinates.cutoffHeatMass

cutoffHeatMassAt def

The actual pointwise heat mass as a function of all real times. · OAI.RegularPointCoordinates.cutoffHeatMassAt

cutoffHeatWeight def

The actual cutoff heat weight, as a bounded continuous function. · OAI.RegularPointCoordinates.cutoffHeatWeight

cutoffLift def

The actual cutoff sends the original pullback cover to the pullback of the modified circle coordinate, over the identity of the base. · OAI.CircleDomainRigidity.FiniteTransfer.CircleLiftCover.cutoffLift

cutoffMass def

Colliding mass in the size-2^H central blocks, read along the actual input routes. · OAI.CubeShuffle.cutoffMass

cutoffMass def

Unnormalized radius-two distance cutoffs in the sum norm. · OAI.PolycyclicRecognition.cutoffMass

cutoffMeasureSource def

The full signed source of the cutoff function. · OAI.RegularPointCoordinates.cutoffMeasureSource

cutoffPackets def

Build literal smooth passage input coefficients from compact smooth cylinder cutoffs and smooth periodic two-torus coefficients. · OAI.ThreeTorus.cutoffPackets

cutoffSource def

The canonical full signed source, not a density-only replacement. · OAI.RegularPointCoordinates.LocalHarmonicEnergy.cutoffSource

cutoffSourceDensity def

The absolutely continuous part introduced by the cutoff product. · OAI.RegularPointCoordinates.cutoffSourceDensity

cutoffSquareFactor def

A continuous cutoff factor, regular at zero for a strictly positive threshold. · OAI.Kaplansky.cutoffSquareFactor

cutoffTargetMass def

Actual target marginal restricted to the event that the density is at least δ. · OAI.LeanBlast.SecondKahnKalai.cutoffTargetMass

cutOldSideValues def

The selected original parent side changes from P to J; children retain their transports over the coefficient ring. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.cutOldSideValues

cutPaddingWeight def

The departure cutoff retains a row only when its total mass is at most K. · OAI.TwoPointCorrelations.cutPaddingWeight

cutPartialIsometryTarget def

Restrict the final projection of a partial isometry. · OAI.KadisonKastler.cutPartialIsometryTarget

CutPartition structure

A finite partition with a recorded set of removed separator vertices. Edges between distinct labels must meet that set. This is only the recursive combinatorial stage; no planar separator is supplied by this definition. · OAI.HyperbolicObstruction.PlanarRecursion.CutPartition

CutPortAttached def

A port of the cut diagram is attached to a specified seam end. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.CutPortAttached

CutPortsAdjacent def

Two ports of the cut diagram meet the same seam. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.CutPortsAdjacent

cutRowWord def

The cut's two layers, with their exact retained row and column permutations. · OAI.Saxl.cutRowWord

Cuts inductive

Finite sequences of actual vertex cuts. This retains the whole matching; only cyclic boundary order changes. Cuts must occur on the same current cycle. · OAI.TorsionFreeZeroDivisors.BoundaryRefinement.Cuts

cuts def

Cut immediately after an omega, omitting the empty final sector. On paths every interior cut is exactly omega-alpha, including headless end sectors. · OAI.FiniteAlgebraicEnvelopes.Compiler.Path.cuts

cuts def

Cut every bad gap and every prescribed marking together with its opposite. · OAI.TorsionFreeZeroDivisors.PlanarRotation.cuts

cutSet def

Either side of a cut inside one cluster. · OAI.OptimalMaxCut.Unweighted.cutSet

CutShapes abbrev

Multiple cuts are indexed simultaneously by a finite product of bounded rank arrays, not by an unbounded fresh family chosen for each old solution. · OAI.IntegralCharacterVarieties.TwoFlagBand.CutShapes

cutShift def

Cumulative shifts select a local family on the corresponding exponent cell. · OAI.Problem326.Affine.cutShift

cutShortSideValues def

New strip sides have identity transport except for the mirror principal short seam, which carries Y = T⁻¹ J⁻¹ P T. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.cutShortSideValues

CutSide abbrev

Oriented side copies cut by a partition. Parallel contacts and self-contacts remain literal side copies; no underlying-simple-graph collapse is made. · OAI.HyperbolicObstruction.PictureSwitch.CutSide

cutSign def

Signs used in the finite cut maximum. · OAI.JointDickman.cutSign

cutSize def

Each unordered distinct edge crossing the cut is counted exactly once. · OAI.OptimalMaxCut.Graph.cutSize

CutSystem structure

A finite circular boundary cut at least once in each boundary component. · OAI.TorsionFreeZeroDivisors.PlanarRuns.CutSystem

cutTable def

Truncate a table at degree n, retaining the chosen boundary twice in that degree. The lower-dimensional entries remain unchanged. · OAI.Thomason.Construction.DirectedComplex.Cell.cutTable

cutTerm def

Signed contribution from changing one cut in the finite intersection. · OAI.Problem358.FiniteIntersectionVariation.cutTerm

cutTime def

A concrete sequence in the open disk approaching the required side of the cut. · OAI.KervaireScalarPhase.cutTime

cutTransfer def

A continuous extension of the manuscript's gδ away from the nonnegative spectrum. The denominator is never zero when δ>0. · OAI.TraceConeClassification.cutTransfer

cwComplex def

The ordinary CW structure follows from the genuine normalized-cube gluing; no combinatorial homology or contractibility assertion enters this definition. · OAI.EilenbergGanea.CubeRealization.cwComplex

cwComplex def

Every ordinary CW structure lifts through any covering, with exactly the actual sheets times the base cells as cell indices. · OAI.EilenbergGanea.CWCover.cwComplex

cwComplexOfT2 def

The ordinary CW structure follows from the genuine normalized-cube gluing; only Hausdorffness and compact characteristic disks are used. · OAI.EilenbergGanea.CubeRealization.cwComplexOfT2

cX def

The actual blowup chart open containing s=t=0 (the intersection of its exceptional divisor with the strict transform of y=0). · OAI.KummerShifted.cX

cX def

The actual blowup chart open containing s=t=0 (the intersection of its exceptional divisor with the strict transform of y=0). · OAI.KummerShiftedDiagonal.cX

cX def

The actual blowup chart open containing s=t=0 (the intersection of its exceptional divisor with the strict transform of y=0). · OAI.KummerTriple.cX

cX def

The actual blowup chart open containing s=t=0 (the intersection of its exceptional divisor with the strict transform of x−y=0). · OAI.KummerTripleDiagonal.cX

cX def

Determinant-ring variables a,d,b,c,u, in this order. · OAI.StableCoordinate.Obstruction.cX

cycleBudget def

Uniform budget for one vertex iteration, including unary addition and cleanup. Its input argument bounds the complete table length. · OAI.MaxCutGames.Foundations.Complexity.MachineCloudPrefix.cycleBudget

cycleBudget def

Uniform budget for one vertex iteration, including unary addition and cleanup. Its input argument bounds the complete table length. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineCloudPrefix.cycleBudget

CycleCandidate def

An over-count of short cycles: parameters, rather than marks, are enumerated. Marks at punctured endpoints are deliberately permitted for the upper bound. · OAI.Release075.CycleCandidate

CycleConstantLabelings abbrev

Labels constant along each cycle are exactly functions on the cycle quotient. This is the counting mechanism behind trace normalization. · OAI.KadisonSimilarity.CycleConstantLabelings

cycleConstantLabelingsEquiv def

A concrete equivalence, rather than an assumed formula for orbit counts. · OAI.KadisonSimilarity.cycleConstantLabelingsEquiv

cycleCoordinates def

Cyclic permutation from the transverse physical normalization to (p+rv,r). · OAI.RuledApproximation.cycleCoordinates

cycleEdge def

Extension by 1 is solely for evaluating the finite prefix; no selected path ever encounters an extended edge. · OAI.IntegralCharacterVarieties.SurfaceSurgery.cycleEdge

cycleEquation def

Choose an actual polynomial representing the product of prime classes with weights. · OAI.Nagata.Workers.W30.cycleEquation

cycleFiber def

Ownership by distinguishable cycle slots; none means unselected. · OAI.CubeShuffle.cycleFiber

cycleFiber def

Ownership by distinguishable cycle slots; none means unselected. · OAI.CubeShuffle.RoutingNetwork.cycleFiber

cycleMap def

Cycles map under a coarsening of the orbit partition. · OAI.KaplanskyCounterexample.Planar.cycleMap

cycleMap def

An additional all-ones case supplies the cyclic seam for positions only. · OAI.WLIdentification.BinaryAddress.cycleMap

CycleMembers abbrev

The vertices in one cycle. · OAI.KadisonSimilarity.CycleMembers

cycleNext def

Cyclic successor, defined also for the empty index type. · OAI.CycleClique.Construction.cycleNext

cyclePairing def

Cyclic consecutive edge incidences. succ is the permutation of edge positions; opening its passage at one position handles an interval path. · OAI.TorsionFreeZeroDivisors.PathIncidences.cyclePairing

cyclePoolIndex def

Disjoint countable pools can be selected from one orthonormal sequence. · OAI.KadisonSimilarity.cyclePoolIndex

Cycles abbrev

Vertex cycles of a rotation, including the cycles of length one. · OAI.KaplanskyCounterexample.Planar.Cycles

CycleSchedule structure

Explicit caller state and tape endpoints for one level of the nested loop. Compatibility with its child schedules is checked by their types. · OAI.MaxCutGames.Foundations.Complexity.MachineTupleOdometer.CycleSchedule

CycleSchedule structure

Explicit caller state and tape endpoints for one level of the nested loop. Compatibility with its child schedules is checked by their types. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineTupleOdometer.CycleSchedule

cyclesEquiv def

Actual boundary cycles are exactly the nonempty boundary words. · OAI.TorsionFreeZeroDivisors.BoundaryRotation.cyclesEquiv

cycleSlotsEquivAssignments def

Ownership and labelled distinct full cycles are equivalent, not merely bounded by one another. This identifies the factorial-moment combinatorics. · OAI.CubeShuffle.cycleSlotsEquivAssignments

cycleSlotsEquivAssignments def

Ownership and labelled distinct full cycles are equivalent, not merely bounded by one another. This identifies the factorial-moment combinatorics. · OAI.CubeShuffle.RoutingNetwork.cycleSlotsEquivAssignments

cyclesOn def

Cycles and boundaries are actual submodules of normalized singular chains. · OAI.EilenbergGanea.CubicalSingular.cyclesOn

cycleSubtypeMap def

The inclusion of the cycles of an invariant subset. · OAI.KaplanskyCounterexample.Planar.cycleSubtypeMap

cycleSumEquiv def

A trivial extra generator contributes an independent free summand to the cycle module, with no cardinality restriction. · OAI.EilenbergGanea.cycleSumEquiv

cycleTensor def

The completed-cycle contribution plus any subset of the current cycle. · OAI.ClosedSurfaceR4.FiniteOrderSmoothing.SmoothPrimitiveFamily.cycleTensor

cycleTrace def

Deterministic semantics of the adaptive cycle validator. The result records all visited inputs and the unique terminal (closing-path input). · OAI.CubeShuffle.cycleTrace

cycleWords def

The six nonempty tape roles that may occur between expansion phases. · OAI.MaxCutGames.Foundations.Complexity.MachineExpanderFamily.cycleWords

cycleWords def

The six nonempty tape roles that may occur between expansion phases. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineExpanderFamily.cycleWords

Cyclic abbrev

Exactly s coefficients with cyclic convolution, the concrete algebra (Z/mZ)[X]/(X^s-1) used in the appendix. · OAI.ExactQuantumFactoring.Primality.Cyclic

cyclic def

The cyclic subrepresentation generated by an actual vector. · OAI.Saxl.cyclic

cyclic def

The possible interval is designated once, not independently at each stage. · OAI.TorsionFreeZeroDivisors.StageRoot.cyclic

CyclicallyReduced def

Including the closing turn, not just linear immersion. · OAI.TorsionFreeZeroDivisors.SystemLeaves.CyclicallyReduced

cyclicBlockRealization def

The actual cyclic representatives at the beginning of a full block, with the original displacement vectors retained. · OAI.QuantitativeVanDerWaerden.CyclicBlockLabels.cyclicBlockRealization

cyclicBundledGNSIsometry def

The cyclic isometry with a bundled source Hilbert space. · OAI.WeakPureInfiniteness.cyclicBundledGNSIsometry

cyclicCharacter def

Precomposing a character with a cyclic permutation gives a character. · OAI.MatrixMultiplication.AuxiliarySeparation.Character.cyclicCharacter

cyclicColumnRoot def

The positive square root of a finite column square. · OAI.KadisonSimilarity.cyclicColumnRoot

CyclicColumnTests def

The approximation needed by the matrix estimate; the input vector may vary. · OAI.KadisonSimilarity.CyclicColumnTests

cyclicCyclotomicEvaluation def

Evaluation of the actual cyclic group ring at the root in its cyclotomic quotient, using the root's proved power relation. · OAI.CirculantHadamard.cyclicCyclotomicEvaluation

CyclicDecompositionData structure

A chosen maximal family, together with its proved density. · OAI.KadisonSimilarity.CyclicDecompositionData

cyclicEquiv def

The actual cyclic group algebra is the polynomial quotient by X ^ n - 1. · OAI.CirculantHadamard.CyclicPolynomial.cyclicEquiv

cyclicGramEquivalence def

The Hilbert-space identification of two dense families with identical Gram matrices, including the prescribed action on each family vector. · OAI.KadisonKastler.cyclicGramEquivalence

cyclicGraph def

The closed cyclic graph of the orbit map σ(a)Ω ↦ ρ(a)ξ. · OAI.KadisonSimilarity.cyclicGraph

cyclicHeckeFactor def

The principal factor is the incomplete Dedekind zeta function; the other factors are the entire continuations of the realized characters. · OAI.DukePrimeDegree.cyclicHeckeFactor

CyclicLowerBound def

One threshold supplies actual avoiding cyclic colorings independently of the eventual number of colors. This is a proposition, not an axiom. · OAI.QuantitativeVanDerWaerden.CyclicLowerBound

CyclicOrbitOrthogonal def

Pairwise orthogonality of the full closed cyclic spaces. · OAI.KadisonSimilarity.CyclicOrbitOrthogonal

CyclicOrder structure

A cyclic order on every nonempty fiber of a finite incidence map. · OAI.OneWayLiveness.CyclicOrder

cyclicPolarColumn def

The regularized polar column stays in the original algebra. · OAI.KadisonSimilarity.cyclicPolarColumn

cyclicPolygon def

A cycle of patch routes is an embedded polygon for the boundary permutation. There is no extra topological closure hypothesis. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.PortPatch.ParentPath.cyclicPolygon

cyclicPower def

n-bit repeated squaring in the coefficient array; no exponential polynomial or algebraic root is part of the computation. · OAI.ExactQuantumFactoring.BitArithmetic.cyclicPower

cyclicProjection def

The quotient of cyclic group rings induced by reduction of indices. · OAI.CirculantHadamard.cyclicProjection

cyclicProjectionAlgHom def

The cyclic quotient also preserves the coefficient algebra structure. · OAI.CirculantHadamard.cyclicProjectionAlgHom

cyclicQuotientMap def

The quotient always surjects onto the expected cyclic exponent quotient. · OAI.Kervaire.cyclicQuotientMap

cyclicRow def

The cyclic group-ring element whose coefficients are the given first row. · OAI.CirculantHadamard.cyclicRow

CyclicSaxlConjecture def

Every Specht type occurs in the specified staircase cyclic module. · OAI.Saxl.CyclicSaxlConjecture

cyclicShiftVector def

The shifted square root is the single input vector for the regularized column. · OAI.KadisonSimilarity.cyclicShiftVector

cyclicSignature def

Literal signatures of the actual cyclic samples. · OAI.QuantitativeVanDerWaerden.SparsePerturbation.cyclicSignature

CyclicSignRow def

Every literal integer coefficient is a sign. · OAI.CirculantHadamard.CyclicSignRow

CyclicStarRepresentation def

A representation has a vector whose algebra orbit is dense. Because the representation is linear in the algebra variable, no additional span is needed. · OAI.KadisonSimilarity.CyclicStarRepresentation

cyclicStart def

Sectors with identical programs start cyclically at the F possible starts. · OAI.ExactFourier.Packing.cyclicStart

CyclicSubgroups abbrev

The finite set of cyclic subgroups, retaining their actual subgroup carrier. No choice of generators enters the coefficients. · OAI.DukePrimeDegree.CyclicSubgroups

cyclicTest def

Literal 8s AKS congruence on s coefficient words, m in the input word. · OAI.ExactQuantumFactoring.BitArithmetic.cyclicTest

cyclicX def

The residue class of X. No exponentially long polynomial is constructed. · OAI.ExactQuantumFactoring.Primality.cyclicX

cyclicZeroSign def

The sign detected by the zero order of the cyclic fixed field. · OAI.DukePrimeDegree.cyclicZeroSign

cyclotomicConjugation def

The actual involutive algebra automorphism inverting the cyclotomic root. · OAI.CirculantHadamard.RamifiedConjugation.cyclotomicConjugation

CyclotomicRing abbrev

The actual cyclotomic quotient used by the ramified construction. · OAI.CirculantHadamard.RamifiedConjugation.CyclotomicRing

cylinder def

The selector cylinder expressing that a prescribed path is followed. · OAI.Problem310.FirstDefault.cylinder

cylinder def

Assign prescribed bits on a named finite set of distinct keys. · OAI.QuantitativeVanDerWaerden.FiniteProbability.cylinder

cylinder def

Small-colimit extension of the explicit cylinders. · OAI.Thomason.Construction.StrictOmegaCategory.GrayChainExtension.cylinder

cylinder def

Product with the ordinary unit interval. · OAI.Thomason.Construction.TopologicalInterval.cylinder

CylinderBound def

Abstract inclusion moments m(S) and their coordinate marginal majorant. This definition does not postulate independence. · OAI.CubeShuffle.CylinderBound

cylinderComposition def

The concrete signed chain prism of the universal composition map. · OAI.Thomason.Construction.StrictOmegaCategory.PastingGlobe.cylinderComposition

cylinderContraction def

Fixed-direction contraction and slice restriction, bundled in norm topology. · OAI.ArnoldCounterexample.Forms.cylinderContraction

cylinderCoordinates def

Coordinates are formed by the actual endpoint and mixed direct sums. · OAI.Thomason.Construction.DirectedComplex.OrderedAtomBasis.cylinderCoordinates

cylinderCosimplicial def

The cosimplicial diagram of the actual strict chain cylinders. · OAI.Thomason.Construction.NormalizedSimplexIdentities.cylinderCosimplicial

cylinderCutoff def

Periodization with a smooth saturation gives a [0,1]-valued cylinder function. · OAI.ThreeTorus.cylinderCutoff

cylinderFirstPeriodize def

Periodization in the REAL coordinate of a compact cylinder cutoff. · OAI.ThreeTorus.cylinderFirstPeriodize

cylinderHomeomorph def

The natural comparison is an actual homeomorphism, with the proved coordinate formulas as its inverse. · OAI.Thomason.Construction.RealizedCylinder.cylinderHomeomorph

cylinderIso def

The same cylinder comparison as an isomorphism of actual topological spaces. · OAI.Thomason.Construction.RealizedCylinder.cylinderIso

cylinderMaxHomotopy def

Contract the simplicial cylinder to its one endpoint. · OAI.Thomason.Construction.cylinderMaxHomotopy

cylinderMinHomotopy def

Contract the simplicial cylinder to its zero endpoint. · OAI.Thomason.Construction.cylinderMinHomotopy

cylinderOrderedAtomBasis def

The concrete right interval tensor on each normalized simplex has an actual ordered atom basis, including all mixed prisms. · OAI.Thomason.Construction.SimplexComplex.cylinderOrderedAtomBasis

cylinderPeriodize def

Periodization only in the circle coordinate. · OAI.ThreeTorus.cylinderPeriodize

cylinderProjection def

The covering map for the real-frequency/slow-circle cylinder. · OAI.ThreeTorus.cylinderProjection

cylinderRealizationMap def

Pull a continuous cylinder back along the actual realized projections. · OAI.Thomason.Construction.cylinderRealizationMap

cylinderRestrictionEquiv def

The adjunction chain giving restriction of cylinders along the reflection unit. Its target is an actual finite-dimensional strict category. · OAI.Thomason.Construction.RightGrayInput.cylinderRestrictionEquiv

cylinders def

Cylinders pulled back along the actual coordinate map. · OAI.SmoothObstruction.CylinderApproximation.cylinders

CylinderSmooth def

Smoothness on this cylinder is smoothness of the periodic covering lift. · OAI.ThreeTorus.CylinderSmooth

cylinderTorusCutoff def

The cutoff on the actual two-torus, not a chart-only surrogate. · OAI.ThreeTorus.cylinderTorusCutoff

cylinderVector def

The basis vectors in the actual tensor coordinates. · OAI.Thomason.Construction.DirectedComplex.OrderedAtomBasis.cylinderVector

cylinderVolume def

Fundamental cylinder with the exact normalized slow-coordinate measure. · OAI.ThreeTorus.cylinderVolume

cylindricalDensity def

Gaussian density in cylindrical coordinates, with the axial coordinate remaining. · OAI.FKCRT.SimpleWalk.cylindricalDensity