Glossary: B

Definitions in this project whose names start with B. 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

B abbrev

The literal global ring in the alternating products. · OAI.CirculantHadamard.CyclotomicRings.B

B def

Twice dbar conjugated by exp(t|z|²/2). · OAI.ElasticityPlanar.B

B abbrev

The six path-endpoint components of the manuscript's directed encoding. · OAI.LittleFinitistic.QuadraticEncoding.B

B def

The CH perturbation operator with norm at most 1/10. · OAI.SeparableQuotient.ActualSpace.B

B def

The spatial exponent in the manuscript, with its literal positive root. · OAI.SmoothObstruction.SmoothScales.B

b def

Exactly (1-q)^|U| times the alternating sum over the coordinates in U, with independent Bernoulli expectation on its complement. · OAI.TalagrandDiscreteConvexity.b

b def

The radial coefficient of the modified curve. · OAI.ThompsonNonamenability.Analytic.b

B0 def

Literal matrix in eq:limiting-matrix: row (ell,b), column (j,K), entry K^ell*j^b. Both index types are the finite source constructions. · OAI.Nagata.Workers.W12.B0

back def

The inverse is defined on every simplex, including the empty-suffix case. · OAI.Thomason.Construction.CosieveSimplex.back

back def

The inverse is defined on every simplex, including the empty-prefix case. · OAI.Thomason.Construction.SieveSimplex.back

back def

The inverse on the zero-labelled piece uses the clamped simplex in A. · OAI.Thomason.Construction.SieveSimplex.Cut.back

back def

Outgoing darts at an internal vertex: the preceding edge backwards and following edge forwards, with no arbitrary choices. · OAI.TorsionFreeZeroDivisors.ChainSubdivision.back

backCocone def

The clamping inverses form a cocone on the same full simplex fibres. · OAI.Thomason.Construction.CosieveSimplex.backCocone

backCocone def

The clamping inverses form a cocone on the same full simplex fibres. · OAI.Thomason.Construction.SieveSimplex.backCocone

BackgroundAssignment abbrev

The program's prescribed background allocation gives one such function. · OAI.LowSpaceSubsetSum.RepresentationSetup.BackgroundAssignment

backgroundChild def

Combine each adjacent pair of leaves into its parent child. · OAI.LowSpaceSubsetSum.backgroundChild

backgroundIndexEquiv def

Successive blocks in the sorted outside domain, with the prescribed eight sizes. The library equivalence uses prefix sums of these sizes. · OAI.LowSpaceSubsetSum.backgroundIndexEquiv

backgroundPairColor def

The eight linear labels are the two parents and four leaves of the table. · OAI.LowSpaceSubsetSum.backgroundPairColor

backstep def

A negative coordinate step, needed only to list all finitely many neighbors. · OAI.CriticalZ3.backstep

backward def

A terminal-value Green solution with coefficients at most one when q≥1. · OAI.SmoothObstruction.FiniteShears.backward

backward def

The inverse determinant-one substitution. · OAI.StableCoordinate.Linearization.backward

backwardFiniteSequence def

A finite backwards recursion, extended constantly beyond its terminal index. · OAI.Problem337.backwardFiniteSequence

backwardMatrix def

One backward step, with t = q + n and d = t - M. · OAI.DefocusingNLS.backwardMatrix

backwardProduct def

The ordered product F₀ ⋯ Fₖ₋₁. · OAI.DefocusingNLS.backwardProduct

backwardStep def

The exact backward recurrence, from n = 7 down to n = 0. · OAI.DefocusingNLS.SeparatorArithmetic.backwardStep

bad def

The chain elements which do not lie above the fixed element. · OAI.FiniteChain.bad

bad def

Nongenericity includes the universal lift quantifier in IsGeneric. The field's simultaneous bound includes rank zero as well. · OAI.MaxCutGames.Gadget.FreshGenericity.bad

bad def

Positive integers up to N divisible by a nontrivial integer cube. · OAI.Problem337.CubeFreeDensity.bad

bad def

Nongenericity includes the universal lift quantifier in IsGeneric. The field's simultaneous bound includes rank zero as well. · OAI.UniqueGamesTheorem.Gadget.FreshGenericity.bad

badCatalogCost def

All lengths, all actual slot counts, and all finite equality codes. · OAI.TwoPointCorrelations.badCatalogCost

badCoprimeDivisorMultiples def

The actual exceptional moduli: multiples of a bad divisor coprime to the fixed supply. The unit divisor is excluded, as in the Fourier collision API. · OAI.Problem337.badCoprimeDivisorMultiples

badCycles def

All occurring encoded cycles for a given random outcome. · OAI.Release075.badCycles

badE def

k pair exceptions using disjoint sample positions (ordered pairs). · OAI.HadwigerCounterexample.badE

badEdges def

The exact set minimized by the finite repair argument. · OAI.TorsionFreeZeroDivisors.GirthSwitch.badEdges

badFrequency def

The zero character is excluded from the bad-event count. · OAI.MaxCutGames.Foundations.PCP.Expanders.badFrequency

badFrequency def

The zero character is excluded from the bad-event count. · OAI.UniqueGamesTheorem.Foundations.PCP.Expanders.badFrequency

badGcdThreshold def

Hit threshold used when the exposed gcd makes the reduced modulus small. · OAI.Problem337.RandomProducts.badGcdThreshold

Badge structure

Probe an actual finite-margin numerical question. · OAI.ExactDerandomization.Fret.Badge

badHalfEvent def

A measurable half of the noncrossing event, with the eligibility count read from the conditioning vector. Taking right=false or right=true gives the two orientations used in the contradiction. · OAI.FiniteFactor.CouplingParameters.badHalfEvent

badLeafMass def

The total weight of leaves with at least one failing incident detector. · OAI.ReflexiveFixedPoints.badLeafMass

badListEvent def

The event that every tuple in this particular list is bad. · OAI.MetricEntropyDuality.DirectionSampling.badListEvent

BadModulus def

A global, total predicate independent of any later ambient supply range. The strict energy inequality is encoded without division or typeclass data. · OAI.Problem337.CompositeSupply.BadModulus

badMultiples def

Integers containing a global bad conductor above the logarithmic cutoff. This finite set depends only on the ambient range, not on any chosen modulus. · OAI.Problem337.CompositeSupply.badMultiples

badPiece def

Bad cone roots are exactly the unrestricted size-one part. · OAI.CommonPrices.badPiece

badPiece def

Bad cone roots are exactly the unrestricted size-one part. · OAI.SignedDisk.CommonPrices.badPiece

badPrimeLogCells def

badPrimes is the set removed by the balance or test-error threshold. · OAI.Ostmann.badPrimeLogCells

badProduct def

The finite integer excluded by the auxiliary-prime search. · OAI.ExactQuantumFactoring.Primality.badProduct

badReciprocalCost def

The finite cost left after one nondegenerate prime relation is imposed. · OAI.TwoPointCorrelations.CrudeWordCode.badReciprocalCost

BadRow def

A row is bad when it contains a dense admissible restriction. · OAI.MaxCutGames.Fourier.Deletion.BadRow

BadRow def

A row is bad when it contains a dense admissible restriction. · OAI.UniqueGamesTheorem.Fourier.Deletion.BadRow

badS def

k distinct samples in S (unordered). · OAI.HadwigerCounterexample.badS

badSamples def

The actual union over forms and disjoint witness lists. · OAI.MetricEntropyDuality.DirectionSampling.badSamples

BadScheduledMatching def

This predicate uses the actual leaf coordinates of each bulk prime. · OAI.Ostmann.BadScheduledMatching

badTail def

A finite piece of a global exceptional-modulus predicate above 2^J. · OAI.Problem337.SparseBadDivisorTail.badTail

bagDistance def

Exact finite minimum distance to a nonempty bag in the original weighted metric. · OAI.BoundedTreewidthL1.bagDistance

baker def

The untwisted coordinate baker: move a leading bit from x to y. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Generators.baker

Balanced structure

normal form after covariance updates · OAI.GeneralMahler.ProjField.Balanced

balancedBlockCounts def

The implementation assigns the remainder in the fixed block order. · OAI.LowSpaceSubsetSum.balancedBlockCounts

BalancedColorRestriction abbrev

A restriction in a mixer must have size s; the outside restriction is free. · OAI.LowSpaceSubsetSum.BalancedColorRestriction

BalancedIncidenceCycle structure

The exact combinatorial data required by the quartet projection. · OAI.Ostmann.BalancedIncidenceCycle

balancedLayer def

true selects derivative layers and false selects multiplication layers. · OAI.Problem335.LowerParameters.balancedLayer

balancedMixerJointEquiv def

Conditioning the global sign subset on mixer balance leaves the permutation uniform and independent of the balanced sign subset in the original coordinates. · OAI.LowSpaceSubsetSum.balancedMixerJointEquiv

balancedMixerRestrictionEquiv def

The full conditional sign law is the product of the three uniform halves and the unrestricted outside signs. · OAI.LowSpaceSubsetSum.balancedMixerRestrictionEquiv

BalancedNormalized structure

Literal normalized solution data, produced by the actual centered section maps. · OAI.AffineBernstein.BalancedNormalized

BalancedProfile def

Counts differ by at most one across all labeled blocks. · OAI.LowSpaceSubsetSum.BalancedProfile

BalancedStar structure

The star about a sink bag, with its vertex partition obtained from a tree decomposition. · OAI.ParityWL.BalancedStar

balancedUpdate def

Balanced changes of one-point marginals preserve the exact vertical bound once genuine nestedness has been established. This avoids an unjustified finite-boundary assumption on chart regions. · OAI.PlanarL1.Columns.Model.SetSystem.balancedUpdate

balancedWord def

The actual balanced schedule for the rank-measure parameters of degree n. · OAI.Problem335.LowerParameters.balancedWord

ball def

Closed metric balls. · OAI.KServer.Pilot.ball

ball def

Extended gauge of the unit ball in V. · OAI.PettyProjection.RelativeGauge.ball

ball def

Finite closed combinatorial ball. · OAI.TorsionFreeZeroDivisors.BallCount.ball

ballColumns def

All representatives of the intrinsic closed ball, projected to columns. Using representatives includes columns identified by zero-length paths. · OAI.PlanarL1.Columns.Model.Point.ballColumns

ballIndicator def

The real indicator of a finite-radius ball in the actual partition graph. · OAI.MetricEntropyDuality.CompressionLevels.ballIndicator

ballLineRegion def

The ball restriction in transverse-line and axial-displacement coordinates. · OAI.BoltzmannNonuniqueness.GeometricSlice.ballLineRegion

ballMass def

The manuscript's posterior measure of a closed metric ball, as a finite counting sum. We do not restrict to distinct hidden locations. · OAI.KServer.ParkedCoefficients.ballMass

ballMean def

The equally weighted mean, bundled with its unit-ball bound. · OAI.ThompsonNonamenability.ballMean

ballPredicate def

The union of equal-radius Hamming balls. · OAI.Paper320.ballPredicate

BallSupported def

Support in a fixed closed Euclidean ball. · OAI.ElasticityNegativeOrder.BallSupported

BallTubeCovers def

Covers by balls with mass exponent n and covering exponent n-1. · OAI.DukePrimeDegree.BallTubeCovers

ballVolume def

Riemannian measure of the intrinsic open ball, retaining infinite values. · OAI.HarmonicCounterexample.ballVolume

BallVolumeComparison def

The ball comparison in Sturm (2006), Theorem 2.3, equation (2.5), written without divisions. For the application the model is the radial volume of the constant negative-curvature comparison space. · OAI.RegularPointCoordinates.BallVolumeComparison

band def

Radial bands in the positive-radius factor of polar coordinates. · OAI.HenonLaneEmden.CapGeometry.band

band def

Canonically choose one finite allowed port band for each bounded rank shape. This producer does not take any affine-bundle hypothesis. · OAI.IntegralCharacterVarieties.TwoFlagBand.RankShape.band

band def

The literal closed height bands. Adjacent bands share only their endpoint, where every clipping proposal retains the old field. · OAI.PlanarL1.Tiling.band

BandColor inductive

Distinct occurrences survive even if two original graded spaces are isomorphic, zero, or come from the same repeatedly incident facet. · OAI.IntegralCharacterVarieties.TwoFlagBand.BandColor

bandColor def

Old row and column grades attach to their own ordered old occurrence; only common-graded cells are new facets. Equal old facet names are allowed. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.PortAssembly.bandColor

bandCount def

Number of dyadic bands needed to cover [a,B]. · OAI.Problem337.DyadicBands.bandCount

bandCutChosenJ def

The transport along the new seam with an identity child. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.bandCutChosenJ

bandCutCircleBases def

Boundary gauges determined by the old boundary order. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.bandCutCircleBases

bandCutDiagram def

The topological inverse handle cut, with complete boundary cycles. The identity seam is an intermediate awaiting proper lower-gallery replacement. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.bandCutDiagram

bandCutEarlier def

The two ordered products on either side of the selected old circle. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.bandCutEarlier

bandCutExtraPositions def

There is exactly one additional boundary, at the selected parent facet. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.bandCutExtraPositions

bandCutFacet def

The two sides of the inverse band belong to the SAME original parent surface. All child occurrences, including repeated old facets, remain literal. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.PortAssembly.bandCutFacet

bandCutFrameValues def

Endpoint frames realizing both new splits as refinements of the old flags. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.bandCutFrameValues

bandCutHandleIndex def

Retain the old handle slots after deleting the last handle on the chosen parent. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.bandCutHandleIndex

bandCutIdentityColumns def

The coordinate equivalence for the singleton child of the identity seam. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.bandCutIdentityColumns

bandCutLong def

The oriented old boundary traverses three pieces. Only the middle parent piece at q is replaced by the child side of the inverse band. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.PortAssembly.bandCutLong

bandCutP def

The old positive parent-side matrix, in the simultaneous child basis. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.bandCutP

bandCutRetainedProduct def

The ordered product of the retained handle commutators. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.bandCutRetainedProduct

BandCutSeam abbrev

Three old subintervals and one transverse identity connection. This is the intermediate inverse handle cut, before the two-flag gallery replaces the identity connection by proper lower strips. No fresh facet is introduced. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.PortAssembly.BandCutSeam

bandCutSideEquiv def

Exhaustive side occurrence bijection. The new short boundary is additional, rather than a chosen list that might omit fresh-parent germs. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.PortAssembly.bandCutSideEquiv

bandCutSideValues def

Transports on the cut sides, retaining the old child transports. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.bandCutSideValues

bandCutSolutionAt def

Lift a solution over the coefficient algebra at a chosen parent occurrence. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.bandCutSolutionAt

bandCutSplitChildIndex def

After and branch fiber indices in the full-block split. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.bandCutSplitChildIndex

bandCutSplitChildIndexLast def

After and branch fiber indices in the full-block split. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.bandCutSplitChildIndexLast

bandCutTriplePositions def

Divide each old boundary index into its old side and its phase. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.bandCutTriplePositions

bandD def

exact scale · OAI.GeneralMahler.SCal.Grid.bandD

bandDataOnlineCandidate def

Keep the optimizer opaque while transporting the finite band data. · OAI.EditApproximation.bandDataOnlineCandidate

BandDescent structure

The source descent lives in the BROAD CURRENT band J, whereas the old comparison field and witnesses live in the OLD chart I, possibly in different J-components. The attaining condition refers to J, not to the larger band. · OAI.PlanarL1.Columns.Model.Field.BandDescent

bandFromActionsWithWork def

Every filter callback is the literal binary band predicate. · OAI.EditApproximation.BinaryFraction.bandFromActionsWithWork

bandGlue def

Apply a band intertwiner to every scalar high-coordinate of a colored strip orbit map. This is the concrete tensor product of the two maps. · OAI.Saxl.bandGlue

bandPathEquiv def

The exact path order, starting at the bottom row singleton and ending at the rightmost column singleton. · OAI.Saxl.bandPathEquiv

BandTest def

Exactly the three numeric inequalities of prop:band-test, in one orientation. K=2M-1; height=column lengths and width=row lengths. · OAI.ArithmeticTensorSquares.Capacity.BandTest

BandVertex abbrev

The combined family is finite: lengths may vary with the intersection shape, but there are finitely many shapes and each band is finite. · OAI.IntegralCharacterVarieties.TwoFlagBand.RankShape.BandVertex

bandWord def

EXACT coefficients of the path-ordered alternating band: the two endpoint singletons carry letter 0, and row/column pairs have coefficients J. Here r=m-1. · OAI.Saxl.Path.bandWord

Bank structure

Parameters for one uniform open code guarantee. When doing recursion proofs the index i can range over arbitrary environments including call bounds. · OAI.PowerSaving.RAM.Bank

Bar def

odd primes up to and excluding limit · OAI.PowerSaving.Selection.Bar

barBasis def

The bar image of an actual standard basis vector; inverse units give expression independence before any polynomial coefficients are extracted. · OAI.KLInvariance.Hecke.barBasis

barBasis def

The bar image of an actual standard basis vector; inverse units give expression independence before any polynomial coefficients are extracted. · OAI.KLInvariance.HeckeQ.barBasis

barBicomplex def

The bicomplex for the left-module bar resolution of a nonnegative chain complex. Outer degree is the bar length; inner degree is the input complex degree. The two differentials commute by the proved naturality of the absolute bar … · OAI.DerivedBar.barBicomplex

barComplex def

The explicit two-sided X,D,N bar complex, with no D-flatness assumption. · OAI.DerivedBar.BimoduleTensor.barComplex

barComputesDerivedTensor def

The explicit projective bar resolution computes actual derived tensor on modules, via mathlib's left derived functor to the homotopy category. Extension to bounded-above complexes and repeated totalization remains separate. · OAI.DerivedBar.BimoduleTensor.barComputesDerivedTensor

barDifferential def

The two-sided bar differential for arbitrary bimodule coefficients X. · OAI.DerivedBar.BimoduleTensor.barDifferential

barFunctor def

The explicit free bar resolution is functorial, not just an objectwise choice. · OAI.DerivedBar.barFunctor

barHomologyIso def

Actual left-derived tensor homology (Tor), computed by the explicit two-sided bar. · OAI.DerivedBar.BimoduleTensor.barHomologyIso

barInputAugmentation def

The original bar augmentation, followed by the fixed finite bimodule total. · OAI.LittleFinitistic.BimoduleTotal.barInputAugmentation

barPower def

Every ordinary iterate on the selected first-block stalk is the original bar iterate. · OAI.LittleFinitistic.SelectedOrdinaryOrbit.barPower

barQuotientIso def

The detector's tensor of the actual bar resolution is the bar worker's balanced complex, after restricting only its exterior scalars. · OAI.LittleFinitistic.PDBarQuotient.barQuotientIso

barRaise def

Unit insertion with the exact bicomplex source and target objects. · OAI.DerivedBar.barRaise

BarrierFamily structure

Connected barriers around a final component. The inside/outside fields are what permit nested cuts to add one more disjoint barrier at each round. · OAI.BoundedTreewidthL1.Annulus.BarrierFamily

barrierNorm def

Original energy and original band norm of one real coordinate. · OAI.CircleDomainRigidity.FiniteTransfer.CountableRealCoordinates.barrierNorm

barSuffix def

Prefix one original categorical X bar to a linear suffix map. · OAI.DerivedBar.ParityBridge.barSuffix

barTermMap def

The map on the explicit field-tensor bar terms. · OAI.DerivedBar.BimoduleTensor.barTermMap

barTotal def

The ordinary direct-sum total bar complex, with standard totalization signs. The total augmentation quasi-isomorphism must still be proved. · OAI.DerivedBar.barTotal

barTotalAdditiveEquiv def

The total augmentation is a homotopy equivalence after forgetting the A-action. The inverse and homotopy are explicit unit insertions, not A-linear splittings. · OAI.DerivedBar.barTotalAdditiveEquiv

barTotalAugmentationF def

The augmentation is the action in bar length zero and zero elsewhere. · OAI.DerivedBar.barTotalAugmentationF

barTotalAugmentationNat def

The augmentation is natural for all chain maps, not only quasi-isomorphisms. · OAI.DerivedBar.barTotalAugmentationNat

barTotalCochain def

The source's bounded-above cochain model is obtained by the literal n ↦ -n extension. · OAI.DerivedBar.barTotalCochain

barTotalDH def

Horizontal contribution with the declared total object as codomain. · OAI.DerivedBar.barTotalDH

barTotalDV def

Vertical contribution with the declared total object as codomain. · OAI.DerivedBar.barTotalDV

barTotalFunctor def

The total bar resolution is a functor on nonnegative chain complexes. · OAI.DerivedBar.barTotalFunctor

barTotalH def

Natural unit insertion in the total bar, increasing bar length by one. · OAI.DerivedBar.barTotalH

barTotalHomotopy def

The explicit additive contraction, with its degree condition. · OAI.DerivedBar.barTotalHomotopy

barTotalSection def

Unit insertion is an actual additive chain section of the total augmentation. · OAI.DerivedBar.barTotalSection

barTotalSectionF def

Unit insertion in bar length zero gives the additive section. · OAI.DerivedBar.barTotalSectionF

barUnit def

Unit insertion on an individual bar term, viewed as an additive map. · OAI.DerivedBar.barUnit

barycentric def

The barycentric straight-line cone to the newly appended final vertex. · OAI.Thomason.Construction.ConePrism.barycentric

barycentricCellComplexOfMono def

Every realized monomorphism is obtained by attaching actual barycentric n-simplices along their boundaries at stage n. · OAI.Thomason.Construction.RealizationPrism.barycentricCellComplexOfMono

barycentricEdge def

The image of the standard interval inside a barycentric simplex along any ordinal edge. · OAI.Thomason.Construction.barycentricEdge

barycentricPathHomotopy def

Linear interpolation between two barycentric paths fixes their common endpoints. This is the path-level triangle compatibility used in realization. · OAI.Thomason.Construction.barycentricPathHomotopy

Base inductive

The actual finite-scan instruction semantics of the finite compiler. The immutable bases are not part of what an instruction may update. · OAI.FiniteAlgebraicEnvelopes.Compiler.Base

Base abbrev

The three-variable polynomial subalgebra appearing in the rigidity proof. · OAI.StableCoordinate.DeterminantRing.Base

base def

D₀ is precisely one edge. · OAI.ForestSpace.Diamond.base

base def

The old space embeds as a closed subspace of the adjunction. · OAI.GroupRingTopology.Attachment.base

base def

The fixed diagonal leg; initially and during every pulse it is round. · OAI.HarmonicCounterexample.AngularStream.base

base def

The annulus between a single oppositely oriented relator disk and its outside word has a literal genus-zero ribbon witness. · OAI.KervaireRibbon.OuterRibbon.base

base def

Canonical rank-dim U map. · OAI.MaxCutGames.Appendix.CompressionCount.base

base def

The height-zero input, shift and uniform noise alphabet. · OAI.MaxCutGames.Gadget.Stage.base

base def

One larger than the proved bound on every canonical word. · OAI.MaxCutGames.Reduction.CanonicalAddress.base

base def

The product of the first primeCount u primes. · OAI.Problem337.CompositeSupply.base

base def

Number of candidate ordinals consumed before a given level. · OAI.ReflexiveFixedPoints.GlobalCandidateOrdinals.base

base def

Canonical rank-dim U map. · OAI.UniqueGamesTheorem.Appendix.CompressionCount.base

base def

The height-zero input, shift and uniform noise alphabet. · OAI.UniqueGamesTheorem.Gadget.Stage.base

base def

One larger than the proved bound on every canonical word. · OAI.UniqueGamesTheorem.Reduction.CanonicalAddress.base

baseAlgebraVectors def

The algebraic vectors coming from the specified star subalgebra. · OAI.KadisonSimilarity.TracialCornerGeometry.baseAlgebraVectors

baseBar abbrev

The literal two-sided bar over the one fixed square-zero algebra. · OAI.LittleFinitistic.SelectedPDDetection.baseBar

baseCoefficient def

Background diagonal coefficient, independent of all control choices. · OAI.HarmonicCounterexample.LinearODE.HistoryGeometry.baseCoefficient

baseCompression def

The actual compressed left action on arbitrary ambient algebra elements. · OAI.KadisonSimilarity.TracialCornerGeometry.baseCompression

baseCutoff def

Fixed smooth compact cutoff, equal to one on [-1,1]. · OAI.SharpLiebThirring.SobolevProof.baseCutoff

Based def

Fix the selected base corner of every boundary. These are coordinate gauges, not extra equations on an old local system. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.CornerBases.Based

basedCornerEquiv def

Full-ring bijective change of every corner frame in the solution. No vertex, quotient identification or surface equation is lost. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.basedCornerEquiv

basedCornerSolution def

Arbitrary independent internal corner gauges act on the full solution, not just on boundary holonomies or its regular seam equations. The original scalar punctures and handles are unchanged. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.basedCornerSolution

basedDynkinFlowMap def

A rescaled Cartan leg with its moving initial coordinate retained. · OAI.PolycyclicRecognition.basedDynkinFlowMap

basedGroupMap def

The corresponding map on the actual based homotopy quotient. · OAI.Thomason.Construction.HomotopyMaps.basedGroupMap

basedHomeomorph def

Re-basing changes the genuine local-geodesic model by a covering isomorphism. · OAI.HyperbolicObstruction.LocalGeodesic.basedHomeomorph

basedLoopHomotopy def

A homotopy fixed at the basepoint gives homotopic postcomposed loops. · OAI.Thomason.Construction.HomotopyMaps.basedLoopHomotopy

basedLoopMap def

Postcomposition with an explicitly preserved basepoint. · OAI.Thomason.Construction.HomotopyMaps.basedLoopMap

BasedPermutation structure

A finite boundary enumeration change together with an invertible word substitution. Product preservation is NOT the only stored certificate. · OAI.IntegralCharacterVarieties.SurfaceSurgery.Hurwitz.BasedPermutation

BasedRay def

Based geodesic rays, parametrized at integer times. · OAI.CannonRelease.BasedRay

basedReorderSolution def

Full sewn-diagram solution after arbitrary reordering of boundary circles. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.basedReorderSolution

basedTuplePosition def

The tuple positions with the identity adjoined at time zero. · OAI.PolycyclicRecognition.basedTuplePosition

baseEmbed def

Literal insertion of D slots into square-zero-extension slots. · OAI.DerivedBar.ParityBridge.baseEmbed

baseExpectation def

In the ultrapower, the expectation is a linear projection onto the range of the constant embedding. · OAI.KadisonKastler.TraceQuotientMap.baseExpectation

baseExpectationCLM def

The projection onto the constant copy is norm continuous. · OAI.KadisonKastler.TraceQuotientMap.baseExpectationCLM

baseExtension def

Extend a prescribed initial family to unused index counts without adding assumptions. · OAI.Paper320.baseExtension

baseGeometry def

Restrict the trace and left action to the closed base algebra; the Hilbert space is the closure of its actual ambient algebra vectors. · OAI.KadisonSimilarity.TracialCornerGeometry.baseGeometry

baseGood def

Goodness is deliberately independent of observation parameters. · OAI.PureTypeSystem.ObservationValues.baseGood

baseGraph def

Constraint-graph semantics, with each vertex-port pair a distinct dart. · OAI.MaxCutGames.Foundations.PCP.PortTables.baseGraph

baseGraph def

The base port predicates packaged as a constraint graph. · OAI.MaxCutGames.Foundations.PCP.PoweringTest.baseGraph

baseGraph def

Constraint-graph semantics, with each vertex-port pair a distinct dart. · OAI.UniqueGamesTheorem.Foundations.PCP.PortTables.baseGraph

baseGraph def

The base port predicates packaged as a constraint graph. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringTest.baseGraph

baseHaarLetter def

The actual normalized corner Haar vector in the centered base space. · OAI.KadisonSimilarity.TracialCornerGeometry.baseHaarLetter

baseInclusion def

The canonical isometry from the completed base into the ambient space. · OAI.KadisonSimilarity.TracialCornerGeometry.baseInclusion

baseIndex abbrev

The original representable simplex, indexed by its identity map. · OAI.Thomason.Construction.PrismBarycentric.baseIndex

baseline def

The face baseline β=C/2-∑b(h)/6, after D=d+c and s=p+t. · OAI.SidorenkoCounterexample.baseline

baseMap def

The source's quadric-to-determinant-ring map. · OAI.StableCoordinate.BundleCharts.baseMap

baseMap def

The base variables have order a,d,u. · OAI.StableCoordinate.DeterminantRing.baseMap

baseMap def

The source's degree-zero quadric algebra maps into its graded presentation. · OAI.StableCoordinate.LaurentPresentation.baseMap

baseMap abbrev

The source's injective map B → C, with the canonical obstruction-ring API. · OAI.StableCoordinate.Obstruction.baseMap

baseMatrix def

Unnormalized cluster columns d_k h_k. · OAI.HyperedgeComparison.baseMatrix

baseMatrixWordAction def

The genuine endpoint action on the tracial Hilbert space of the other factor. · OAI.KadisonSimilarity.TracialCornerGeometry.baseMatrixWordAction

baseOperator def

Compression of an ambient bounded operator to the completed base. · OAI.KadisonSimilarity.TracialCornerGeometry.baseOperator

BasepointInvariant def

Bijectivity is unchanged along a path of source basepoints in this degree. · OAI.Thomason.Construction.HomotopyMaps.BasepointInvariant

basePolynomialMap def

The map of polynomial rings underlying x=ab, y=dc, z=db, u=u. · OAI.StableCoordinate.BundleCharts.basePolynomialMap

baseRead def

The source base Read clause. For nonliteral expected types this extra coordinate is unused; no additional semantic assertion is made about it. · OAI.PureTypeSystem.ObservationValues.PreResult.baseRead

baseRepresentation def

The restriction to the star subalgebra is a genuine unital star representation on the completed base space. · OAI.KadisonSimilarity.TracialCornerGeometry.baseRepresentation

baseRule def

0 and 1 are constants; 2 compares two labels; 3 and 4 compare a label and a helper value. · OAI.ParityWL.Complexity.baseRule

baseSkeleton def

The machine may alter tags but never these ordered bases or outer flags. · OAI.FiniteAlgebraicEnvelopes.Compiler.Validator.baseSkeleton

baseSlots def

Both address-base phases use the same clean arithmetic work tapes. The right side adds the retained left block; the left side adds a seeded zero. · OAI.MaxCutGames.Foundations.Hastad.SourceRuntimeModel.baseSlots

baseSlots def

Both address-base phases use the same clean arithmetic work tapes. The right side adds the retained left block; the left side adds a seeded zero. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceRuntimeModel.baseSlots

baseSpace def

The actual closed base subspace, with no independent density input. · OAI.KadisonSimilarity.TracialCornerGeometry.baseSpace

baseStageWeights def

A failed or premature code is replaced by the point mass at the first index. · OAI.Tingley.baseStageWeights

baseSymmetryLetter def

A trace-zero base element is an actual centered base Hilbert vector. · OAI.KadisonSimilarity.TracialCornerGeometry.baseSymmetryLetter

baseSymmetryRepeatedInclusion def

The actual base symmetry determines all normalized repeated-letter inclusions required by the word multiplicities. · OAI.KadisonSimilarity.TracialCornerGeometry.baseSymmetryRepeatedInclusion

baseSymmetryUnitary def

The represented symmetry is an actual unitary bounded operator. · OAI.KadisonSimilarity.TracialCornerGeometry.baseSymmetryUnitary

baseTapes def

Canonical frame before the odometer digits are filled. · OAI.MaxCutGames.Explicit.MachineProductHeaders.baseTapes

baseTapes def

Canonical frame before the odometer digits are filled. · OAI.UniqueGamesTheorem.Explicit.MachineProductHeaders.baseTapes

baseToDegeneration def

The canonical algebra homomorphism from the quadric ring into G. · OAI.StableCoordinate.Obstruction.baseToDegeneration

baseToDegenerationPre def

Polynomial map implementing the quadric coordinates in G. · OAI.StableCoordinate.Obstruction.baseToDegenerationPre

baseTorIso def

The canonical base-change Tor is computed by the actual selected bar, independently of which compatible field-scalar instance presents that bar. · OAI.LittleFinitistic.SelectedPDConclusion.baseTorIso

baseToZero def

Blowdown's exact degree-zero ring map on this Rees chart. · OAI.SourcePullbackChart.baseToZero

baseTracialMatrixTower def

The finite matrix tower constructed from its actual tracial embeddings. · OAI.KadisonSimilarity.TracialCornerGeometry.baseTracialMatrixTower

baseTranslations def

The finite affine construction realizes the local interfaces with no assumed graph isomorphism or winning strategy. · OAI.WLIdentification.AffineBlocks.baseTranslations

baseValue def

The same spectral construction on the original coefficient algebra. · OAI.Kaplansky.CuntzDimensionState.baseValue

baseValue def

Registers zero and one start the selected left and right query blocks; register two is the single dummy address. · OAI.MaxCutGames.Foundations.Hastad.SourceAddressDescriptors.baseValue

baseValue def

Registers zero and one start the selected left and right query blocks; register two is the single dummy address. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceAddressDescriptors.baseValue

baseVector def

The dense algebra-vector map into its Hilbert completion. · OAI.KadisonSimilarity.TracialCornerGeometry.baseVector

baseWeight def

The word summand without coincidence corrections. · OAI.Problem335.IsolatedWeight.baseWeight

baseWeight def

Conditional turn weights on side B before setting cancelling turns to zero. · OAI.TorsionFreeZeroDivisors.WordEstimate.baseWeight

baseWidth def

An explicit uniform base interval, so06's finiteness perturbation requires no choice of a finite list of representation realizations. · OAI.CoordinateSweeps.baseWidth

baseWordInclusion def

The base Hilbert space embeds as the scalar first endpoint in degree zero. · OAI.KadisonSimilarity.baseWordInclusion

basic def

Derivations extend to the ring of sections of each principal open. · OAI.AffineDerivation.basic

basicEquiv def

The canonical comparison on each principal open. · OAI.AffineExterior.basicEquiv

BasicFormula inductive

A basis of classical first-order syntax, with last-variable binders. All formulas in the target language translate to this basis. · OAI.PartitionConsistency.ForcingQuotient.BasicFormula

basicMap def

Exterior comparison on sections of a principal open. · OAI.AffineExterior.basicMap

BasicScales structure

A convenient finite list of stronger scale inequalities. Existence is proved below from the actual parameter definitions. · OAI.QuantitativeVanDerWaerden.Parameters.BasicScales

Basis abbrev

A canonical representative of a basis element of two n-globes pasted along a k-face. false denotes the left globe. · OAI.Thomason.Construction.PastingGlobeBasis.Basis

Basis inductive

The direct-sum basis of the right tensor with the oriented interval. · OAI.Thomason.Construction.RightCylinderBasis.Basis

basis def

The fixed simultaneous change of basis used in the determinant degeneration. · OAI.MatrixMultiplication.AuxiliarySeparation.DeterminantBasis.basis

basis def

The corresponding actual basis of the Lie algebra. · OAI.PolycyclicRecognition.WordBasisSelections.basis

basis def

Barycentric coordinates for the exact, nondegenerate table. · OAI.TriangularBilliards.Triangle.basis

basisAtom def

The concrete atom attached to any normalized basis element. · OAI.Thomason.Construction.SimplexComplex.basisAtom

basisBracket def

The same coordinate bracket in the normed space of continuous bilinear maps. · OAI.PolycyclicRecognition.basisBracket

basisBracketLinear def

Coordinate transport of the bracket, initially as a bilinear map. · OAI.PolycyclicRecognition.basisBracketLinear

basisCoefficient def

Actual scalar expression of every basis column, retaining the degree-zero tip. · OAI.Nagata.W22.basisCoefficient

basisCoord def

The three standard coordinate vectors of the quadratic block. · OAI.MaxCutGames.Quadratic.basisCoord

basisCoord def

The three standard coordinate vectors of the quadratic block. · OAI.UniqueGamesTheorem.Quadratic.basisCoord

basisCoordinates def

Ordinary complex coordinates of an orthonormal basis. · OAI.MUB6.Weiner.basisCoordinates

basisExponentialProduct def

The ordered basis-exponential endpoint map. · OAI.PolycyclicRecognition.basisExponentialProduct

basisFromCoefficients def

The basis is constructed from the proved equivalence, not supplied as data. · OAI.Nagata.Workers.W10.basisFromCoefficients

basisFromSupportedOrbits def

The actual centered vectors themselves form a basis. No rescaling of the vectors is needed; normalization occurs only in coefficient extraction. · OAI.Nagata.Workers.W10.basisFromSupportedOrbits

basisHeight def

A common height on the normalized bases of every degree. · OAI.Thomason.Construction.SimplexComplex.basisHeight

basisHessian def

Full ambient Hessian in a constant orthonormal basis. · OAI.AffineBernstein.basisHessian

basisJetMatrix def

The matrix of a linear jet map is its values on a genuine domain basis. · OAI.Nagata.Workers.W11.basisJetMatrix

basisLift def

Lift a normalized basis element to the corresponding actual strict cell. · OAI.Thomason.Construction.SimplexAtomsInput.basisLift

basisNumber def

The physical wire basis is explicitly identified with the natural residue labels; no implicit change in bit order is hidden in the overlap theorem. · OAI.ExactQuantumFactoring.basisNumber

basisPolynomialChange def

Basis coordinates send a coordinate vector to the corresponding original vector. · OAI.Nagata.W01.basisPolynomialChange

basisPostcompose def

Postcomposition with an increasing tuple embeds every normalized basis. · OAI.Thomason.Construction.SimplexComplex.basisPostcompose

BasisProbability structure

A separate wrapper retains both probability topologies without overriding the existing weak topology. · OAI.DukePrimeDegree.BasisProbability

basisProductLog def

Its logarithm in a global exponential bijection. · OAI.PolycyclicRecognition.basisProductLog

basisProjectiveMap def

A genuine linear coordinate change induces an injective actual projective map. · OAI.Nagata.W01.basisProjectiveMap

basisResidue def

The fixed little-endian physical representation of a modular residue. · OAI.ExactQuantumFactoring.basisResidue

basisScaleTransition def

Coordinates of the same Lie-algebra point in the second scaled basis. · OAI.PolycyclicRecognition.basisScaleTransition

basisSupport def

The support exponent contributed by one bracket-word basis. · OAI.PolycyclicRecognition.SupportDatum.basisSupport

BasisSystem structure

The purely set-theoretic assumptions supplied by the prototype assembly. · OAI.InfiniteMatroidCounterexample.BasisSystem

basisToAbelianization def

Read the unit's actual cell tables on each basis. · OAI.Thomason.Construction.PastingGlobeBasis.basisToAbelianization

basisVector def

Computational basis vectors and the normalized minus vector. · OAI.QAC.basisVector

basisVolumeContinuous def

Continuous form for a fixed basis covector determinant. · OAI.Mahler.basisVolumeContinuous

basisWeights def

The decreasing ternary exponents associated to a simplex basis. · OAI.Thomason.Construction.SimplexComplex.basisWeights

bAt def

Here a h is the sum over the diagonal coordinates of the dyadic squared chord errors. Its square root replaces the optional convolution upper bound in the manuscript's example of the choice of sequence. · OAI.SmoothObstruction.GoodStationary.bAt

batch def

The simultaneous main mode is now genuinely defined for every old state and every finite sample. Its family contains exactly the available kept cells. · OAI.PlanarL1.Columns.Model.MainFamily.batch

batch def

Literal maximum within an old B identifier. Neither disjoint domains nor an assumed winning branch replaces the actual overlapping maximum. · OAI.PlanarL1.Columns.Model.SecondaryGeometry.batch

Bazaar structure

Tables already fixed on population. bag M is total for convenience but hypotheses on positive mass only apply on the indicated chart. · OAI.HadwigerCounterexample.Bazaar

bCoefficient def

The negative coefficient of the side input B. · OAI.UniformKServer.SideStability.bCoefficient

BCoordinates def

Equivalence of coordinates when B is moved past n physical sites, with the A auxiliary fixed at the front. · OAI.CriticalHoneycomb.Transfer.PlanarNormalization.BCoordinates

BData inductive

System B has precisely the ordinary and ordered chart alternatives. · OAI.PlanarL1.Columns.Model.BData

beamCoefficient def

The same passive rotation is performed on all mode indices. · OAI.EntropyPhotonNumber.beamCoefficient

Beat structure

Local output paths, excluding terminal removals (unnecessary for transform). · OAI.FourierLLogL.Dyadic.Procedure.Thread.Beat

Before abbrev

Iterates before and through stop. · OAI.Fifths.Before

before def

The ancestral interval [a,g), ending immediately before g along its path. · OAI.ForestSpace.before

beforeBlock def

The actual root just before the entire descendant block switches. · OAI.ReflexiveFixedPoints.TreeEvaluation.beforeBlock

beforeFinish def

The actual work tapes after all vertex blocks, before final headers. · OAI.MaxCutGames.Foundations.Complexity.PoweringMachineRuntime.beforeFinish

beforeFinish def

The actual work tapes after all vertex blocks, before final headers. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringMachineRuntime.beforeFinish

beforeFinishBudget def

Initialization and the actual outer loop's proved budgets. · OAI.MaxCutGames.Foundations.Complexity.PoweringMachineRuntime.beforeFinishBudget

beforeFinishBudget def

Initialization and the actual outer loop's proved budgets. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringMachineRuntime.beforeFinishBudget

beforeFinishExecution def

Actual execution from the literal single-input configuration through every vertex. No graph-dependent instruction or supplied body-execution premise. · OAI.MaxCutGames.Foundations.Complexity.PoweringMachineRuntime.beforeFinishExecution

beforeFinishExecution def

Actual execution from the literal single-input configuration through every vertex. No graph-dependent instruction or supplied body-execution premise. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringMachineRuntime.beforeFinishExecution

beforeGrade def

Source-order grades, not an arbitrary enumeration. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.LocalRanks.beforeGrade

behind def

Lift an operator through a spectator wire. · OAI.CriticalHoneycomb.Transfer.PlanarNormalization.behind

bellMediatorHamiltonian def

The actual full matrix in the complete Bell-coordinate spin space. · OAI.ContinuumCoulomb.bellMediatorHamiltonian

bellMemberPauli def

Actual Pauli action on either member of a mediator pair in Bell coordinates. · OAI.ContinuumCoulomb.bellMemberPauli

below def

Cubes lying entirely in the nonpositive halfspace. · OAI.CubicalChains.Reflection.below

below def

Called branching. · OAI.FourierLLogL.Dyadic.Node.below

below def

A stronger principal quotient retains every literal representation witness. In particular its three ground record entries do not change. · OAI.PartitionConsistency.ProductForcing.LocalWitness.below

BenesCoins abbrev

Two genuine independent butterfly coin fields; the first routes outward on the output side, the second outward on the input side. · OAI.CubeShuffle.BenesCoins

bergBottomCoefficient def

The bottom coefficient determined by an exact row extension. · OAI.WeakPureInfiniteness.bergBottomCoefficient

bergCornerIterHom def

Iteration as an actual nonunital star homomorphism. · OAI.WeakPureInfiniteness.bergCornerIterHom

bergCornerLift def

Adjoin the complementary identity to a nonunital corner map. · OAI.WeakPureInfiniteness.bergCornerLift

bergerMetric def

A concrete smooth metric from the manuscript's logarithmic Berger data. The local constancy alternative encodes switching J only inside round gaps. · OAI.HarmonicCounterexample.Cartesian.bergerMetric

bergerSlowGenerator def

The source's first-order slow generator, including its precise Riccati denominator. The angular infinitesimal is r B Id+D. · OAI.HarmonicCounterexample.LinearODE.bergerSlowGenerator

bergFiniteRotation def

The sum of rotations in orthogonal corners is a single unitary. · OAI.WeakPureInfiniteness.bergFiniteRotation

bergFiniteTower def

A finite tower is assembled with its entire terminal transfer coefficient. · OAI.WeakPureInfiniteness.bergFiniteTower

bergFiniteTowerUnitary def

The bundled unitary represented by a finite tower. · OAI.WeakPureInfiniteness.bergFiniteTowerUnitary

bergRectangle def

One rectangle in the product of the two orthogonal layer towers. · OAI.WeakPureInfiniteness.bergRectangle

bergRotationCore def

Polynomial form of a rotation; c and s will be cosine and sine. · OAI.WeakPureInfiniteness.bergRotationCore

bergRowsCommute def

The tensor-factor commutation relations, including adjoints. · OAI.WeakPureInfiniteness.bergRowsCommute

bergShiftAngles def

Angles shifted by one layer, with a zero at the new bottom. · OAI.WeakPureInfiniteness.bergShiftAngles

bergSkew def

The skew-adjoint generator of the corner rotation. · OAI.WeakPureInfiniteness.bergSkew

bergSpareCorner def

Two unused ranges give a square-zero partial isometry. · OAI.WeakPureInfiniteness.bergSpareCorner

bergSupport def

The sum of the orthogonal initial and final projections. · OAI.WeakPureInfiniteness.bergSupport

bergTent def

A tent of height π / 2, centered at L, with endpoints 0 and 2L. · OAI.WeakPureInfiniteness.bergTent

bergTowerLayer def

One orthogonal difference of successive range projections. · OAI.WeakPureInfiniteness.bergTowerLayer

bergTowerTail def

The range projection of the nth iterate of the corner map. · OAI.WeakPureInfiniteness.bergTowerTail

bernoulli def

Bernoulli weights, with true denoting that the record survives. · OAI.KServer.RosterExpectation.bernoulli

bernoulli def

true is the singleton branch. · OAI.MaxCutGames.Clean.bernoulli

bernoulli def

true is the singleton branch. · OAI.UniqueGamesTheorem.Clean.bernoulli

bernoulliAverage def

Expectation of f evaluated at the number of successes in n independent Bernoulli trials. The recursion itself specifies the product distribution. · OAI.MaxCutGames.Soundness.RepetitionUpper.bernoulliAverage

bernoulliAverage def

Expectation of f evaluated at the number of successes in n independent Bernoulli trials. The recursion itself specifies the product distribution. · OAI.UniqueGamesTheorem.Soundness.RepetitionUpper.bernoulliAverage

bernoulliBitMass def

One Bernoulli coordinate, written as a mass on Bool. · OAI.JointDickman.bernoulliBitMass

bernoulliMass def

Independent Bernoulli coordinates, including every absent coordinate. · OAI.TalagrandDiscreteConvexity.bernoulliMass

bernoulliMass def

The mass of a subset under independent Bernoulli coordinates. · OAI.TalagrandDiscreteConvexity.bernoulliMass

bernoulliProduct def

Product of the independent Bernoulli trials in the source table model. · OAI.HyperbolicObstruction.FiniteLaw.bernoulliProduct

bernoulliProductMass def

Independent site hits, independent also between coordinates. · OAI.JointDickman.bernoulliProductMass

bernoulliSiteMass def

Independent Bernoulli mass of the original site. · OAI.JointDickman.bernoulliSiteMass

bernoulliSubsetMass def

The independent-selection mass of a subset of P. All sums of these masses below are restricted to P.powerset. · OAI.JointDickman.bernoulliSubsetMass

bernoulliWeight def

Product weight on the actual finite space of all bit assignments. · OAI.QuantitativeVanDerWaerden.FiniteProbability.bernoulliWeight

beside def

parallel sum · OAI.PowerSaving.Slant.Stroke.beside

bestCoset def

Minimization is on pattern objects with their fixed wellordering, not on an arbitrary ordering of the cosets. · OAI.PartitionConsistency.TailSelector.TailData.bestCoset

bestResponse def

Copy a satisfying Bob triple; otherwise flip its first bit. · OAI.MaxCutGames.Reduction.Incidence.bestResponse

bestResponse def

Copy a satisfying Bob triple; otherwise flip its first bit. · OAI.UniqueGamesTheorem.Reduction.Incidence.bestResponse

beta def

Literal true and held beta parameters. Their coefficient is chosen only once; these functions do not depend on the horizon or posterior law. · OAI.KServer.CoarseParameters.beta

beta def

The actual number of injective linear maps, with no asymptotic convention. · OAI.MaxCutGames.Inverse.KMSInjectionCount.beta

beta def

Fourier coefficient of the residue indicator, probability normalization. · OAI.OptimalMaxCut.Unweighted.beta

beta def

The cutoff supported inside the half-radius ball. · OAI.ThompsonNonamenability.Analytic.beta

beta def

Clipped β parameters are unclipped on active vertices. · OAI.UniformKServer.ParameterGaps.beta

beta def

The actual number of injective linear maps, with no asymptotic convention. · OAI.UniqueGamesTheorem.Inverse.KMSInjectionCount.beta

betaLinear def

The deformation beta, represented as a real-linear one-form. · OAI.Mahler.betaLinear

BEval def

One common budget bounds both the number of functional instructions and every intermediate serialized value. The compiler uses a universal quadratic power, independent of the finite program's syntax. · OAI.ParityWL.Complexity.BEval

bias def

Unique bias at an interior mean; arbitrary value outside interior. · OAI.GeneralMahler.bias

biases def

Bias tagged to subgroup characters χ_gx, side coordinate and the fixed child. · OAI.FourierLLogL.Dyadic.Thermal.biases

bidegreeAntidiagEquiv def

Reindex a bidegree exponent by its two finite occupation antidiagonals. · OAI.Problem335.bidegreeAntidiagEquiv

bidegreeComplexFockBasis def

The complex factorial-normalized monomial basis at fixed bidegree. · OAI.Problem335.bidegreeComplexFockBasis

bidegreeComponent def

Linear projection to a fixed bidegree. · OAI.Problem335.bidegreeComponent

bidegreeExponentsEquiv def

Split a bihomogeneous monomial into its two independent exponent vectors. · OAI.Problem335.bidegreeExponentsEquiv

bidegreeFinsuppExponentsEquiv def

A fixed bidegree fiber is the product of two ordinary degree fibers. · OAI.Problem335.bidegreeFinsuppExponentsEquiv

bidegreeFockBasis def

The real factorial-normalized monomial basis at fixed bidegree. · OAI.Problem335.bidegreeFockBasis

BidegreeIndex abbrev

Exponent vectors at a fixed derivative/multiplication bidegree. · OAI.Problem335.BidegreeIndex

bidegreeMonomialBasis def

Ordinary monomial basis of a fixed bidegree subspace. · OAI.Problem335.bidegreeMonomialBasis

bidegreeOperatorFamily def

Project the defining polynomial to bidegree (k,m) before substitution. · OAI.Problem335.bidegreeOperatorFamily

bidegreeRank def

The rank measure from the manuscript, with an ambient codomain. · OAI.Problem335.bidegreeRank

BidegreeSource abbrev

The canonical source exponent subtype used by the bidegree monomial basis. · OAI.Problem335.ShiftedOccupation.BidegreeSource

bidegreeSubmodule def

The polynomial subspace of fixed bidegree. · OAI.Problem335.bidegreeSubmodule

bidegreeWeight def

Degree in derivative variables and in multiplication variables, respectively. · OAI.Problem335.bidegreeWeight

bids def

fixed companion Z restrict frequent pair to this list of small Φ values · OAI.HadwigerCounterexample.Bazaar.bids

BieriDualityInput def

Bieri, Theorem 9.23, in the integral case needed here. · OAI.PolycyclicRecognition.BieriDualityInput

BieriPolycyclicDualityInput def

The precise polycyclic specialization of Bieri's duality theorems. · OAI.PolycyclicRecognition.BieriPolycyclicDualityInput

big def

Label abbreviations (at fixed j) · OAI.PowerSaving.Cluster.big

biHarmonic def

Actual bihomogeneous harmonic polynomials. · OAI.HarmonicCounterexample.ComplexAngular.biHarmonic

BiHolderEstimate def

The two global estimates have the very same exponent and constant. · OAI.WeakMTWTransport.BiHolderEstimate

biIndexEquiv def

The actual monomials of bidegree (p,q) are independently chosen left and right multisets. This is an equivalence, not an assumed multiplicity. · OAI.HarmonicCounterexample.ComplexAngular.biIndexEquiv

bijection def

A rectangular permutation diagram; arbitrary equivalences also allow renumbering finite port sets without changing the underlying diagram. · OAI.OneWayLiveness.Pairing.bijection

BijectionSystem structure

A finite-slot bijective extension system. The extra slot is genuine: a k-tuple is extended before any one of its slots is discarded. · OAI.WLIdentification.BijectionSystem

bijectionSystem def

Combining independently selected owner bijections gives one announced vertex bijection before the next vertex is selected. · OAI.WLIdentification.AffineBlocks.OwnedTranslations.bijectionSystem

BilateralGauge structure

A finite two-sided gauge of the ACTUAL derivative, based at time zero. This carries no splitting, threshold, switch, Jacobian or measure conclusion. · OAI.SmoothObstruction.ManifoldCoordinates.BilateralGauge

bilateralShiftEquiv def

The shift sends the basis vector at n to the basis vector at n+1. · OAI.CoarseAssembly.Analytic.bilateralShiftEquiv

BilinAut def

Linear automorphisms preserving a specified bilinear form. · OAI.SidorenkoCounterexample.BilinAut

bilinear def

Complex-bilinear pairing defining the quadric. · OAI.ArnoldCounterexample.Quadric.bilinear

bilinear def

Pointwise application of a bounded bilinear map to two compatible jets. · OAI.ForcedComputation.BoundedSpatialJets.bilinear

bilinearCLM def

Dependence of the multiplication operator on its coefficient jet is linear and bounded. · OAI.ForcedComputation.BoundedSpatialJets.bilinearCLM

bilinearDynkinPolynomial def

The finite Dynkin expression evaluated with a variable bilinear law. · OAI.PolycyclicRecognition.bilinearDynkinPolynomial

bilinearProduct def

The row-column extension of a bilinear product to rectangular matrices. · OAI.KadisonKastler.bilinearProduct

bilinearToAlternating def

Alternating bilinear maps expressed in the standard exterior universal-property format. · OAI.ActualExterior.bilinearToAlternating

Bill structure

One basic work unit below pays for O(1) primitive RAM operations; includes loop and stack overhead. peak tracks produced/needed integer quantities beyond inputs (total allocation for stack and array bases is covered separately by work). · OAI.PowerSaving.RAM.Bill

bill def

Noisy bill: cross entropy with fixed comparator law per position, entropy of letter vector conditional on noisy sum, plus error cost. · OAI.FourierLLogL.Compress.bill

billiardFlow def

The actual specular flow on complete regular chains. Singular initial states are assigned their initial state; the theorem must prove these form a null set. No nonemptiness of the regular set is postulated. · OAI.TriangularBilliards.billiardFlow

bimap def

The actual balanced tensor map in both variables. · OAI.LittleFinitistic.BalancedTensor.bimap

bimap def

The actual balanced bimap retains the outside left action. · OAI.LittleFinitistic.LeftTensor.bimap

bimodule def

One explicit bimodule complex chosen before any evaluation module. · OAI.LittleFinitistic.SelectedRectification.bimodule

bimoduleAction def

Both outside actions on a balanced tensor, combined without forgetting either factor. · OAI.LittleFinitistic.BalancedTensor.bimoduleAction

bimoduleAugmentationMap def

The actual augmentation of the genuine finite bimodule resolution. · OAI.LittleFinitistic.OrdinarySimulation.TensorResolution.bimoduleAugmentationMap

BimoduleCoefficients def

A genuine tensor-algebra module with both scalar restrictions recorded together. · OAI.LittleFinitistic.BimoduleCoefficients

bimoduleComplex def

The actual finite resolution retains the left B-action as a genuine tensor-algebra complex. · OAI.LittleFinitistic.OrdinarySimulation.TensorResolution.bimoduleComplex

bimoduleEquiv def

Degreewise comparison as a genuine B-bimodule equivalence. · OAI.LittleFinitistic.OrdinarySimulation.VertexTensor.bimoduleEquiv

BimoduleMean structure

A contractive, unital, star-preserving mean with both module identities. The map ι includes fields independent of the averaged variable. · OAI.KadisonKastler.BimoduleMean

bimoduleMean def

The scalar construction supplies the precise bimodule mean used by the already proved boundary correction. · OAI.KadisonKastler.ScalarInvariantMean.bimoduleMean

BimoduleO def

O from the genuine dg B-bimodule, as an ordinary finite sum of its terms. · OAI.LittleFinitistic.OrdinarySimulation.BimoduleO

bimoduleTensorFunctor def

Actual ordinary tensor with a fixed bimodule, preserving both outside actions. · OAI.LittleFinitistic.bimoduleTensorFunctor

BimoduleX abbrev

The actual ordinary Section 5 bimodule X over D=B×(B tensor C). · OAI.LittleFinitistic.OrdinarySimulation.BimoduleX

bin def

Literal nearest-bin rule floor(jQ/d+1/2), expressed entirely with integers. · OAI.ExactQuantumFactoring.OrderTrial.bin

binary def

Splitting p at an explicitly given subprojection q. · OAI.Generator.Tracial.CornerPartition.binary

binary3SAT def

The ordinary binary 3SAT language, including rejection of malformed words. · OAI.OptimalMaxCut.binary3SAT

binaryAnchorBelow def

Product of all lower sibling-group factors. Each group still averages its own single common global draw, not two independent global draws. · OAI.TraceReconstruction.binaryAnchorBelow

binaryAnchorKernel def

Product of the first-child unsigned anchor at every forced nonleaf, evaluated using the literal complete-tree orientation assignment. · OAI.TraceReconstruction.binaryAnchorKernel

binaryBlock def

Block diagonal operators, on the actual Hilbert sum. · OAI.CoarseAssembly.Analytic.binaryBlock

binaryCellEquiv def

Any fixed bijective labeling of the 2^B cells by B bits suffices: the decoded equalities are represented by their truth tables. · OAI.TwoPointCorrelations.binaryCellEquiv

BinaryChartNeighbors abbrev

Coordinate matrices adjacent to a fixed graph matrix. · OAI.MaxCutGames.Inverse.MatrixChart.BinaryChartNeighbors

BinaryChartNeighbors abbrev

Coordinate matrices adjacent to a fixed graph matrix. · OAI.UniqueGamesTheorem.Inverse.MatrixChart.BinaryChartNeighbors

BinaryChoices abbrev

Exactly one independent fair bit for every stage-edge. · OAI.CoordinateSweeps.BinaryChoices

BinaryCoefficient abbrev

Binary vectors of length m, the actual coefficient alphabet. · OAI.MaxCutGames.Soundness.RepeatedGameBounds.BinaryCoefficient

BinaryCoefficient abbrev

Binary vectors of length m, the actual coefficient alphabet. · OAI.UniqueGamesTheorem.Soundness.RepeatedGameBounds.BinaryCoefficient

BinaryCorrectedElliott def

Theorem 1.2 (thm:elliott): the factors are ordinary multiplicative functions; either original factor satisfies the growing-height condition. · OAI.TwoPointCorrelations.BinaryCorrectedElliott

binaryDegree def

Binary paths really sample critical, variance-one even offspring. · OAI.FKCRT.TreeCoding.binaryDegree

binaryDescendantEnergy def

Exact dynamic sum of squared group anchors over the allowed finite labels below a fixed forced split. Distinct groups retain their own metadata. · OAI.TraceReconstruction.binaryDescendantEnergy

BinaryEdge def

Edge addresses in a complete binary tree, independent of all actual numeric labels. False and true are its two children. · OAI.TraceReconstruction.OccupiedParityTree.BinaryEdge

binaryEntropy def

Binary entropy measured in bits. · OAI.LowSpaceSubsetSum.binaryEntropy

binaryExponent def

The base-two exponent needed by the terminal binary range. · OAI.Problem337.TerminalDepth.binaryExponent

binaryFactorsEquivGraphNeighbors def

The complete nonzero-factor sampling equivalence, ending in actual Grassmann subspaces in the chart. · OAI.MaxCutGames.Inverse.MatrixChart.binaryFactorsEquivGraphNeighbors

binaryFactorsEquivGraphNeighbors def

The complete nonzero-factor sampling equivalence, ending in actual Grassmann subspaces in the chart. · OAI.UniqueGamesTheorem.Inverse.MatrixChart.binaryFactorsEquivGraphNeighbors

BinaryField abbrev

The degree-d binary Galois field. Cardinality statements below require d > 0; the degenerate exponent zero is deliberately excluded. · OAI.MaxCutGames.Quadratic.BinaryField

BinaryField abbrev

The degree-d binary Galois field. Cardinality statements below require d > 0; the degenerate exponent zero is deliberately excluded. · OAI.UniqueGamesTheorem.Quadratic.BinaryField

BinaryGapReduction structure

Every parameter, coordinate identification, program, and polynomial is fixed before the input quantifier. The identity input encoding measures runtime in the original raw bit length, including sparse binary variable names. · OAI.MaxCutGames.Explicit.MachineOutputContract.BinaryGapReduction

BinaryGapReduction structure

Every parameter, coordinate identification, program, and polynomial is fixed before the input quantifier. The identity input encoding measures runtime in the original raw bit length, including sparse binary variable names. · OAI.UniqueGamesTheorem.Explicit.MachineOutputContract.BinaryGapReduction

BinaryGraphChartNeighbors abbrev

The actual adjacent subspaces that lie in the graph chart. · OAI.MaxCutGames.Inverse.MatrixChart.BinaryGraphChartNeighbors

BinaryGraphChartNeighbors abbrev

The actual adjacent subspaces that lie in the graph chart. · OAI.UniqueGamesTheorem.Inverse.MatrixChart.BinaryGraphChartNeighbors

binaryIn def

Arguments individually binary. · OAI.Fifths.binaryIn

BinaryInput structure

`BinaryInput` lists `p` distinct ordered endpoint pairs and assigns a natural-number multiplicity to each pair instead of enumerating every parallel edge copy. · OAI.AlgorithmicThinTrees.BinaryInput

binaryInputs def

The two actual little-endian input words of the bitblaster. · OAI.ExactQuantumFactoring.BitArithmetic.binaryInputs

binaryLabel def

A binary vertex label extends to a strict walking-arrow label. · OAI.Thomason.Construction.SimplexComplex.binaryLabel

binaryLabelsEquiv def

Exact coordinates of the recursively defined forced label tuple. · OAI.TraceReconstruction.OccupiedParityTree.binaryLabelsEquiv

binaryLaw def

The source's B_(2^d), as the pushforward of the independent fair switches. · OAI.CoordinateSweeps.binaryLaw

binaryLaw def

The common binary probability law used for every permuted character. · OAI.MatrixMultiplication.AuxiliarySeparation.binaryLaw

binaryLayer def

The parallel-edge switch layer in one specified bit direction. · OAI.CoordinateSweeps.binaryLayer

BinaryLowerBound def

The binary main endpoint, for every sufficiently large cap (natural logarithms). · OAI.TreeEdit.BinaryLowerBound

binaryLr def

The coefficient of the ratio d / c along 2. Division by c means multiplication by the inverse of the supplied unit. · OAI.CirculantHadamard.binaryLr

binaryLw def

The coefficient of the ratio g / c along 1 + i. · OAI.CirculantHadamard.binaryLw

binaryMomentBound def

The successive bounds are E, 2 E², 8 E⁴, and so on. · OAI.Ostmann.binaryMomentBound

binaryPacket def

The literal serializer receives precisely the finite data of this source. · OAI.ContinuumCoulomb.SquareLatticeHeisenberg.binaryPacket

binaryProductEquiv def

The explicit binary vector coordinates of the final alphabet. The coordinate order is (base bit, repetition coordinate). · OAI.MaxCutGames.Explicit.BipartiteGame.binaryProductEquiv

binaryProductEquiv def

The explicit binary vector coordinates of the final alphabet. The coordinate order is (base bit, repetition coordinate). · OAI.UniqueGamesTheorem.Explicit.BipartiteGame.binaryProductEquiv

binaryRankOneEquiv def

Bijection justifying uniform rank-one sampling by independent uniform nonzero factors. The inverse uses choice only to select the proved unique pair. · OAI.MaxCutGames.Inverse.MatrixChart.binaryRankOneEquiv

binaryRankOneEquiv def

Bijection justifying uniform rank-one sampling by independent uniform nonzero factors. The inverse uses choice only to select the proved unique pair. · OAI.UniqueGamesTheorem.Inverse.MatrixChart.binaryRankOneEquiv

BinaryRankOneFactors abbrev

Ordered nonzero factors, with functional coordinate first. · OAI.MaxCutGames.Inverse.MatrixChart.BinaryRankOneFactors

BinaryRankOneFactors abbrev

Ordered nonzero factors, with functional coordinate first. · OAI.UniqueGamesTheorem.Inverse.MatrixChart.BinaryRankOneFactors

BinaryRankOneMaps abbrev

The actual linear maps whose range has dimension one. · OAI.MaxCutGames.Inverse.MatrixChart.BinaryRankOneMaps

BinaryRankOneMaps abbrev

The actual linear maps whose range has dimension one. · OAI.UniqueGamesTheorem.Inverse.MatrixChart.BinaryRankOneMaps

BinaryRelation structure

A regular, equality-respecting binary forcing relation. This auxiliary structure is used to construct an actual bounded set graph by collection; it is not an assumed function or an assumed semantic model. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.BinaryRelation

binarySetDistortion def

Distortion of a finite collection of equal-length binary words. · OAI.TreeEdit.BinaryLower.binarySetDistortion

binarySiblingAnchor def

One correlated sibling-group anchor, using the actual child metadata and their parent's one shared global orientation collection. · OAI.TraceReconstruction.binarySiblingAnchor

binarySign def

The binary character is multiplicative across addition of its exponents. · OAI.MaxCutGames.Appendix.Restriction.binarySign

binarySign def

The unique nontrivial complex-valued character of the binary field. · OAI.MaxCutGames.Fourier.MatrixCharacters.binarySign

binarySign def

The binary character is multiplicative across addition of its exponents. · OAI.UniqueGamesTheorem.Appendix.Restriction.binarySign

binarySign def

The unique nontrivial complex-valued character of the binary field. · OAI.UniqueGamesTheorem.Fourier.MatrixCharacters.binarySign

binarySpectralCorrection def

One binary spectral correction. · OAI.KadisonSimilarity.binarySpectralCorrection

binarySpectralRadius def

The prescribed geometric error radius. · OAI.KadisonSimilarity.binarySpectralRadius

binarySpectralResidual def

Repeated binary rounding, performed inside the given operator algebra. · OAI.KadisonSimilarity.binarySpectralResidual

BinarySplitPlan def

Parameters for an ordered binary tree, with adaptive descendant data. · OAI.Ostmann.BinarySplitPlan

binaryStep def

A fair binary step of length l. · OAI.WeakPinned.Grid.binaryStep

binarySweep def

The coordinate layers are applied in increasing coordinate order. · OAI.BinaryCoordinateSweeps.binarySweep

binarySweep def

One binary coordinate sweep, with all individual switches specified. · OAI.CoordinateSweeps.binarySweep

BinarySystem structure

Binary measurable assignments with integrable pair differences; individual sets may have infinite measure, exactly as for the paper's label half-lines. · OAI.PlanarL1.FiniteL1.BinarySystem

binaryTail def

An explicit computable constant-width binary tail family. Choosing the generator order by its lexicographic order implements the source convention. The generous width still obeys D₀>M=2, with no machine-size restriction. · OAI.FiniteAlgebraicEnvelopes.Bootstrap.binaryTail

binaryTreeExponent def

Exact exponent of the full binary forced skeleton, in the same bottom-up order as binaryTreeBudget. · OAI.TraceReconstruction.binaryTreeExponent

binaryTreeSum def

Exact bottom-up summation of any finite binary anchor model. The state contains its actual index path, so group-dependent kernels are allowed. · OAI.TraceReconstruction.binaryTreeSum

binaryValue def

Little-endian evaluation; leading zeroes at the high end are allowed. · OAI.ExactQuantumFactoring.BitStackProgram.binaryValue

BinaryWindowLabels def

A concrete binary selection of labels in the complete index tree. Unlike a free abstract shape this retains every actual column label. · OAI.TraceReconstruction.BinaryWindowLabels

BinaryWitness def

A finite nonempty subset of binary words of one common length. · OAI.EditDistortion.BinaryWitness

binaryWord def

Little-endian encoding of a finite Boolean word. · OAI.EditApproximation.binaryWord

binaryWord def

Canonical most-significant-first binary representation. · OAI.WLIdentification.binaryWord

binCheckExpr def

Input order Q,B,k,d,j; coprimality is already guaranteed by the convergent recurrence and does not require a second gcd computation. · OAI.ExactQuantumFactoring.OrderTrial.binCheckExpr

binCount def

The number of factors in a finite bin, with multiplicity. · OAI.JointDickman.binCount

bind def

A finite mixture, to represent the root and the independent kernels exactly. · OAI.DilutedSpinGlass.FiniteLaw.bind

bind def

The output retains the preceding draw, making its dependence explicit. · OAI.LowSpaceSubsetSum.FiniteLaw.bind

bind def

Finite conditional sampling; both the law and all output masses are rational. · OAI.OptimalMaxCut.LongCode.Law.bind

binErrorRat def

Exact rational form of the nearest-bin error. · OAI.ExactQuantumFactoring.OrderTrial.binErrorRat

binExponent def

The dyadic exponent of a positive weight. Its value at a nonpositive weight is irrelevant to the bin partition and is fixed to zero. · OAI.TalagrandDiscreteConvexity.binExponent

binGood def

Finite-bin event with crossing trials and two whole attachment edge sets. · OAI.GammaFPP.Grid.binGood

binLabel def

A bin label, viewed as an arithmetic function (so its value at zero is zero). · OAI.JointDickman.binLabel

binomialBits def

The implementation computes the bit length of a positive binomial count. · OAI.LowSpaceSubsetSum.binomialBits

binomialIteration def

Categorical binomial iteration with a common shift offset. · OAI.ExternalTensor.Binomial.binomialIteration

binomialLengthDifference def

Forward length differences of the actual binomial probability polynomial, before imposing the conditional rank total. · OAI.TraceReconstruction.binomialLengthDifference

binomialObject def

The manuscript's repeated finite biproduct, including the multiplicity index. · OAI.ExternalTensor.Binomial.binomialObject

binomialWeight def

The symmetric binomial probability weights, presented as a finite real function. · OAI.LeanBlast.GotsmanLinial.binomialWeight

binomRat def

The binomial polynomial, evaluated in exact rational arithmetic. · OAI.HarmonicCounterexample.Certificate.binomRat

binWeight def

Bin majorant for the one-sided reciprocal distance. · OAI.Problem337.RationalPhaseSpacing.binWeight

binWords def

Exact logarithmic bins, excluding zero weights just as in the source. · OAI.TorsionFreeZeroDivisors.WordEstimate.binWords

bipartitionFlip def

Flipping one bipartition converts opposite matching assignments into constant colorings. The construction includes the unmarked-value convention. · OAI.CubeShuffle.CycleColoring.bipartitionFlip

biprodAction def

Direct sum of two actions of the same algebra. · OAI.LittleFinitistic.FormalSummandAction.biprodAction

biprodIso def

The actual coordinate complex is the native binary biproduct. · OAI.LittleFinitistic.PairComplex.biprodIso

biproductBiprod def

Distribute a finite biproduct of binary biproducts and retain the two branch labels. · OAI.ExternalTensor.Binomial.biproductBiprod

Bit def

The Boolean assignment actually contained in a product condition. The adjoined top condition supplies no assignment. · OAI.PartitionConsistency.ProductForcing.Bit

bitCoefficient def

Depends on the fixed repetition count alone. · OAI.MaxCutGames.Foundations.Hastad.SourceBounds.bitCoefficient

bitCoefficient def

Depends on the fixed repetition count alone. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceBounds.bitCoefficient

bitCommands def

The finite relation table is encoded one actual Boolean branch at a time. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Emitter.bitCommands

bitCommands def

The finite relation table is encoded one actual Boolean branch at a time. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Emitter.bitCommands

bitCountFormula def

The two padded long-code tables and one global dummy bit. · OAI.MaxCutGames.Foundations.Hastad.SourceBounds.bitCountFormula

bitCountFormula def

The two padded long-code tables and one global dummy bit. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceBounds.bitCountFormula

bitFlip def

The elementary swap in a single cube coordinate. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.bitFlip

BitGood def

The guard implements infinity at zero; D's totalized value at zero is unused. · OAI.GammaFPP.BK.BitGood

bitLaw def

The actual terminal-bit law used at parameter p. · OAI.Problem310.ConcreteLabels.bitLaw

bits def

Full explicit vertex and arc serialization, including isolated vertices. · OAI.DirectedFeedback.Digraph.bits

bits def

Each output bit uses one testBit operation and one list record. · OAI.LowSpaceSubsetSum.CountedWeights.bits

bits def

Canonical delimited binary vertex count and scale, followed by the full adjacency table in lexicographic order. · OAI.OptimalMaxCut.ScaledGraph.bits

bitsCoordinatesEquiv def

The coordinate map is a bijection all the way to the actual bit-row type. · OAI.MaxCutGames.Clean.AnswerBridge.bitsCoordinatesEquiv

bitsCoordinatesEquiv def

The coordinate map is a bijection all the way to the actual bit-row type. · OAI.UniqueGamesTheorem.Clean.AnswerBridge.bitsCoordinatesEquiv

bitSign def

The real character of a single bit. · OAI.QAC.bitSign

bitSources def

Residue coordinates on which a given Walsh character depends. · OAI.TwoPointCorrelations.bitSources

bitsValue def

Little-endian computational wires, with the same bit ordering as the manuscript and input/output interface. · OAI.ExactQuantumFactoring.bitsValue

bitsValue def

`bitsValue` converts Lean's canonical Boolean digit list back to a natural number. · OAI.UniqueGamesTheorem.BinaryEncoding.bitsValue

bitWeight def

A single Bernoulli coordinate gives weight p to true. · OAI.QuantitativeVanDerWaerden.FiniteProbability.bitWeight

bitWeight def

The mass of a Bernoulli bit. · OAI.TalagrandDiscreteConvexity.bitWeight

bivariateNestedEquiv def

The two concrete polynomial coordinate rings are isomorphic. · OAI.Nagata.W18.bivariateNestedEquiv

bivariateToNested def

Coordinate 0 is the base variable U; coordinate 1 is the fiber variable W. · OAI.Nagata.W18.bivariateToNested

blankNumbers def

Parsing natural-number fields is unchanged by trailing tape blanks. · OAI.ContingencyTables.Algorithms.blankNumbers

blend def

cosine series Taylor argument · OAI.FourierLLogL.Dyadic.Thermal.blend

Block def

Finite processed broad-component/narrow-band blocks. In particular no infinite family of unused tail bands is randomized. · OAI.PlanarL1.Columns.Model.RetainedSelection.Block

Block abbrev

Blocks: none is the common vector, some 0 is the fixed first row, and some i.succ is the independent vector in row i. · OAI.Problem346.SecondTransfer.Block

Block inductive

Structured bit-stack code. Loops are ordinary operational loops, not unbounded-cost primitives. Their iteration count is charged in Exec. · OAI.Superstring.BitCode.Block

Block def

The inner occurrence is strictly inside the two single-letter mismatches. · OAI.Superstring.Counts.Block

block structure

entries in frame of A, eigenvalues x_i · OAI.GeneralMahler.block

block def

A genuine model in every even dimension, not a nonemptiness assumption. · OAI.HarmonicCounterexample.Berger.ComplexStructure.block

block def

The actual dyadic block, not a rectangular surrogate without the cutoff. · OAI.Problem337.VaughanDyadic.block

block def

A self-delimiting binary block: unary length, terminator, then the bits. · OAI.UniformSparsestCut.block

blockAllowance def

Scalar capacity; computing it does not scan either completed table. · OAI.LowSpaceSubsetSum.PreparedIncidenceCells.blockAllowance

BlockAnnihilates def

Literal annihilation of every vector in the block attached to v. · OAI.MaxCutGames.Quadratic.BlockAnnihilates

BlockAnnihilates def

Literal annihilation of every vector in the block attached to v. · OAI.UniqueGamesTheorem.Quadratic.BlockAnnihilates

blockAt def

The normal-space matrix at one pair of mesh nodes. · OAI.PositiveEinsteinFour.Jacobi.blockAt

blockAtom def

Normalize a local block to radius R. A zero block stays zero. · OAI.DiamondDistortion.SegmentSystem.blockAtom

blockBoundaryEdge def

The boundary edge joining a vertex of the block to a fixed exterior vertex. · OAI.Problem313.blockBoundaryEdge

blockCharacter def

The binary character with separate horizontal and vertical coefficients. · OAI.MaxCutGames.Quadratic.blockCharacter

blockCharacter def

The binary character with separate horizontal and vertical coefficients. · OAI.UniqueGamesTheorem.Quadratic.blockCharacter

blockCharacterLinear def

The actual trace character as a binary linear functional. · OAI.MaxCutGames.Quadratic.blockCharacterLinear

blockCharacterLinear def

The actual trace character as a binary linear functional. · OAI.UniqueGamesTheorem.Quadratic.blockCharacterLinear

blockCoefficient def

Curvature coefficients with the final two slots in covector order. · OAI.PositiveEinsteinFour.CurvatureContraction.blockCoefficient

blockCommands def

One old dart emits every local event, then all six slots, false orientation before true. These loops are fixed by q and form part of the finite template. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.EmitRows.blockCommands

blockCommands def

One old dart emits every local event, then all six slots, false orientation before true. These loops are fixed by q and form part of the finite template. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.EmitRows.blockCommands

blockCommandsIn def

Preserve additional finite parser/driver registers through a projection. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.EmitRows.blockCommandsIn

blockCommandsIn def

Preserve additional finite parser/driver registers through a projection. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.EmitRows.blockCommandsIn

BlockCoordinate abbrev

The source coordinates, retaining the block and the position in that block. · OAI.FiniteFactor.CouplingParameters.BlockCoordinate

blockCoordinateAlgebra def

Coordinate functions constant on each root block, viewed over ℚ. · OAI.DukePrimeDegree.blockCoordinateAlgebra

blockCoordinateMap def

Extract the source coordinates after the proved coefficient-image equivalence. · OAI.Nagata.Workers.W10.blockCoordinateMap

blockCoordinates def

The two-form contraction for a tensor with no derivative slots. · OAI.PositiveEinsteinFour.Geometry.blockCoordinates

blockCoordinatesLinear def

The fixed two-form contraction is linear in the four-tensor. · OAI.PositiveEinsteinFour.Geometry.blockCoordinatesLinear

blockCorePolynomial def

The inner port loop can grow the old prefix by at most q complete rows. · OAI.MaxCutGames.Foundations.Complexity.MachineRegularExecutionBounds.blockCorePolynomial

blockCorePolynomial def

The inner port loop can grow the old prefix by at most q complete rows. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRegularExecutionBounds.blockCorePolynomial

blockCount def

Number of portions, including an inherited nonempty raw block. · OAI.UniformKServer.EpochShadow.blockCount

blockCouplingWeight def

The actual finite probability weights supplied by the Hilbert--Schmidt blocks of an operator in a binomial orthogonal grading. · OAI.LeanBlast.GotsmanLinial.blockCouplingWeight

blockCovariance def

Ambient commutation and the actual inclusion identities supply the whole cross-multiplicity covariance of every compressed word block. · OAI.KadisonSimilarity.WordLevelMatrixActions.blockCovariance

blockDegreeBound def

A separate degree bound for each point-coordinate block. · OAI.Nagata.W27.blockDegreeBound

blockDegreeEquiv def

Reindex the finite block-length diagonal by the existing tensor shapes. The shift by r occurs in the degree, not in a hidden tensor-power replacement. · OAI.DerivedBar.Multibar.Indexed.blockDegreeEquiv

blockDehomogenize def

Restrict every point block to [1 : z₀ : z₁]. · OAI.Nagata.W16.blockDehomogenize

BlockDetects def

A block detects a support when the restricted Fourier degree is positive and no larger than the specified real cutoff. · OAI.Problem313.BlockDetects

blockDiagonal def

The diagonal matrix with entries a and b. · OAI.KadisonKastler.blockDiagonal

blockDifferential def

Transport of the actual count differential to signed tensor blocks. · OAI.DerivedBar.Multibar.Indexed.blockDifferential

blockDualHilbertEquiv def

Hilbert coefficients of every finite-height linear-cost dual functional, with its genuine continuous topology. Its norm is p_n, not the Hilbert norm. · OAI.BoundedTreePotentials.blockDualHilbertEquiv

blockEdgeOrbit def

The edge orbit under permutations fixing the complement of B pointwise. · OAI.Problem313.blockEdgeOrbit

blockedOccurrence def

The blocked-occurrence map. · OAI.Superstring.Counts.blockedOccurrence

BlockEliminationChain inductive

An elimination schedule records only zeros that must be preserved; nonzero pivots are supplied locally by convergence to the identity. · OAI.DukePrimeDegree.BlockEliminationChain

blockEnergy def

Full block energy when the low-low perturbation block vanishes. · OAI.ContinuumCoulomb.Perturbation.blockEnergy

blockEntropy def

Shannon entropy of the finite joined partition, with natural logs. · OAI.DukePrimeDegree.blockEntropy

blockEntropy def

The entropy of the length-n join of the partition with labels in Fin q. · OAI.SmoothObstruction.blockEntropy

blockEval def

The ordinary Kronecker evaluation, with the two vector indices curried in reverse order. This is only a coordinate convention. · OAI.RationalHitting.Rectangular.blockEval

blockFlip def

The unitary that swaps the two coordinates. · OAI.KadisonKastler.blockFlip

blockFrequency def

Adjoin a new identity coordinate and an arbitrary off-diagonal vector. · OAI.MaxCutGames.Inverse.KMSAnalytic.blockFrequency

blockFrequency def

Adjoin a new identity coordinate and an arbitrary off-diagonal vector. · OAI.UniqueGamesTheorem.Inverse.KMSAnalytic.blockFrequency

blockFunctional def

The functional with Hilbert coefficients f, acting on the L_n block. · OAI.BoundedTreePotentials.blockFunctional

blockGraph def

Disjoint red cliques: the first coordinate records the block. · OAI.CycleClique.Construction.blockGraph

BlockIdempotent def

The block idempotents in a ring. · OAI.Donovan.BlockIdempotent

blockIndex def

Fixed coordinate identification for a rank sum. The choice is retained uniformly throughout the finite incidence presentation. · OAI.IntegralCharacterVarieties.MatrixExpression.blockIndex

blockIndex def

The index of a stored row within the consecutive block of vertex v. · OAI.MaxCutGames.Foundations.PCP.PoweringTableLayout.blockIndex

blockIndex def

Position z in full block b. · OAI.QuantitativeVanDerWaerden.BlockCoverage.blockIndex

blockIndex def

The index of a stored row within the consecutive block of vertex v. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringTableLayout.blockIndex

blockIndexEquiv def

Actual equivalence used to flatten the recursively nested finite diagonals. · OAI.DerivedBar.RecursiveTotal.blockIndexEquiv

blockInTime def

Actual execution of all rows belonging to one old dart. The finite relation register is supplied by the checked row parser; only code equations are assumed about the enclosing program. The trace restores the five operand tapes. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.EmitRows.blockInTime

blockInTime def

Actual execution of all rows belonging to one old dart. The finite relation register is supplied by the checked row parser; only code equations are assumed about the enclosing program. The trace restores the five operand tapes. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.EmitRows.blockInTime

blockLengths def

Full blocks followed by one possible shorter final block. · OAI.Problem335.blockLengths

blockLetters def

A positive-power corner block, after absorbing its first reflection. · OAI.KadisonSimilarity.DihedralCornerWords.blockLetters

BlockLowerIndex abbrev

A lower product gate chooses a block, its two endpoints, and all internal indices. · OAI.Problem335.BlockLowerIndex

blockMatrixFlatten def

Flatten square blocks using the standard product enumeration. · OAI.KadisonSimilarity.blockMatrixFlatten

blockMatrixGroup def

The determinant-one matrices commuting with every block-scalar diagonal. This is the usual block-diagonal matrix subgroup. · OAI.DukePrimeDegree.blockMatrixGroup

blockMatrixProductEquiv def

First flatten block matrices to a product index. · OAI.KadisonSimilarity.blockMatrixProductEquiv

BlockMiddleIndex abbrev

A middle sum gate computes one entry of one block product. · OAI.Problem335.BlockMiddleIndex

blockNoiseCoordinates def

Coordinates in which the nonlinear output difference is just the second coordinate, for every fixed input x. · OAI.MaxCutGames.Quadratic.blockNoiseCoordinates

blockNoiseCoordinates def

Coordinates in which the nonlinear output difference is just the second coordinate, for every fixed input x. · OAI.UniqueGamesTheorem.Quadratic.blockNoiseCoordinates

blockNonbacktrackingMatrix def

The manuscript's concrete block matrix H[d,d'] = 1[d≠d'] A[d']. · OAI.TwoPointCorrelations.blockNonbacktrackingMatrix

blockObservable def

The block's nonlinear observable. · OAI.MaxCutGames.Quadratic.blockObservable

blockObservable def

The block's nonlinear observable. · OAI.UniqueGamesTheorem.Quadratic.blockObservable

blockOffDiagonal def

The off-diagonal matrix with entries a and b. · OAI.KadisonKastler.blockOffDiagonal

blockOffset def

Number of D entries strictly before block j, the ordinary totalization offset. · OAI.DerivedBar.Multibar.blockOffset

blockOffset def

Starting index of the block belonging to b. · OAI.QuantitativeVanDerWaerden.blockOffset

blockOrder def

Affine degree in one point-coordinate block. · OAI.Nagata.W27.blockOrder

BlockOrientation abbrev

An orientation identifies the common binary alphabet with a line's block. · OAI.MaxCutGames.Quadratic.BlockOrientation

BlockOrientation abbrev

An orientation identifies the common binary alphabet with a line's block. · OAI.UniqueGamesTheorem.Quadratic.BlockOrientation

blockOrientationEquiv def

Composing with one fixed orientation identifies all orientations with automorphisms of the common binary alphabet. · OAI.MaxCutGames.Quadratic.blockOrientationEquiv

blockOrientationEquiv def

Composing with one fixed orientation identifies all orientations with automorphisms of the common binary alphabet. · OAI.UniqueGamesTheorem.Quadratic.blockOrientationEquiv

blockOverlap def

Unnormalized projected overlap; division by N is applied after contraction. · OAI.InvariantIsing.blockOverlap

blockParametrization def

The parametrization (a,w) ↦ (a,Q(a)+w) is a binary-linear bijection. It is not asserted to be field-linear: squaring along a field line is only binary-linear. · OAI.MaxCutGames.Quadratic.blockParametrization

blockParametrization def

The parametrization (a,w) ↦ (a,Q(a)+w) is a binary-linear bijection. It is not asserted to be field-linear: squaring along a field line is only binary-linear. · OAI.UniqueGamesTheorem.Quadratic.blockParametrization

BlockPencil structure

The actual finite blocks after common-kernel compression and fixed diagonal congruence. · OAI.Paper256.BlockPencil

blockPermutationSubgroup def

The subgroup of permutations supported on a given vertex block. · OAI.Problem313.blockPermutationSubgroup

blockPolygon def

Lexicographic subdivision of a cyclic polygon by routed blocks. · OAI.IntegralCharacterVarieties.FiniteCycles.blockPolygon

BlockPresentation structure

Pick one block and one leaf index for the two edges in the spanning tree. · OAI.Release075.BlockPresentation

blockProbability def

Probability of an event for a uniformly chosen fixed-size subset. · OAI.Problem313.blockProbability

blockProductCoupling def

Actual augmented law on the block-indexed coordinates. · OAI.FiniteFactor.CouplingParameters.blockProductCoupling

blockQuadratic def

The plus part of the genuine quadratic integrand before the scalar derivatives of κ and the variance correction are added. · OAI.PositiveEinsteinFour.Bianchi.blockQuadratic

blockQuadratic def

The row of the four-index operator is (i,j) and the column (k,l). · OAI.PositiveEinsteinFour.ScalarHessian.blockQuadratic

blockRelation def

A directed graph partitioned into two kinds of vertices. · OAI.TwoWayComplementation.blockRelation

blockRestriction def

Restriction to one block, linear in the logical character z. · OAI.MaxCutGames.Quadratic.blockRestriction

blockRestriction def

Restriction to one block, linear in the logical character z. · OAI.UniqueGamesTheorem.Quadratic.blockRestriction

Blocks def

The r+1 block lengths, with no empty-list ambiguity at r=0. · OAI.DerivedBar.Multibar.Blocks

blocks def

Fin-four indexing agrees with the coefficient arithmetic used by the halves. · OAI.CirculantHadamard.BinaryBlocks.blocks

blocks def

Literal placement blocks, with no normalized trace convention. · OAI.CoordinateSweeps.BlockCycles.blocks

blocks def

Recursive total diagonals record exactly r+1 block lengths; the final input degree is the last block length. · OAI.DerivedBar.RecursiveTotal.blocks

blocks def

The executed omega-one many saturation rounds form a literal block tower inside the very same final monoid. · OAI.PartitionConsistency.MonoidConstruction.Saturation.blocks

blocks def

Consecutive segments with the original global position parities. · OAI.Saxl.Band.blocks

blocks def

A block sequence approximating unit vectors in the reduced closed subspace, in the selected pure or mixed family. · OAI.SeparableQuotient.ActualSpace.BlockApprox.blocks

blocks def

A row obtained by concatenating equal-width blocks in increasing index order. · OAI.TreeEdit.BinaryLower.TaggedExtraction.blocks

blockScalarAlgebra def

The real algebra of diagonal matrices constant on each root block. · OAI.DukePrimeDegree.blockScalarAlgebra

blockScalarAlgHom def

The diagonal representation of functions on the coordinate blocks. · OAI.DukePrimeDegree.blockScalarAlgHom

blockScalarNormalizer def

The concrete normalizer of the real block-scalar matrix algebra. · OAI.DukePrimeDegree.blockScalarNormalizer

BlockSegmentSeparation def

Concrete separation of segments in a selected pair of width-P blocks. The words here are indexed functions solely to avoid casts in total length. · OAI.TreeEdit.BinaryLower.Code.BlockSegmentSeparation

blocksEmbedding def

The D-1 full blocks after first alignment are disjoint and remain in [0,D*m), including the case of a nonzero advance. · OAI.SmoothObstruction.RecognitionWindow.blocksEmbedding

blockSensitivity def

Maximum block sensitivity over all Boolean inputs. · OAI.Paper320.blockSensitivity

blockSensitivityAt def

Maximum size of a family of disjoint nonempty sensitive blocks at an input. The empty family is allowed and contributes zero. Invalid families contribute zero. · OAI.Paper320.blockSensitivityAt

BlockSequence structure

Admissible block sequences with real coefficients. Pure mode requires one color, mixed mode both orders. · OAI.SeparableQuotient.ActualSpace.BlockSequence

blockShift def

Move one unit of polynomial degree from block u to block v. The other coordinate blocks are spectators. · OAI.Problem346.blockShift

blockSize def

One adjacent orientation pair for each recorded port word. · OAI.MaxCutGames.Foundations.PCP.PoweringTableLayout.blockSize

blockSize def

One adjacent orientation pair for each recorded port word. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringTableLayout.blockSize

blockSlotEquiv def

Each block is enumerated in increasing color order. · OAI.PixtonChow.blockSlotEquiv

blocksPrefix def

Bits emitted by the first k blocks. · OAI.MaxCutGames.Foundations.PCP.PreprocessingRegularLoopWords.blocksPrefix

blocksPrefix def

Bits emitted by the first k blocks. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingRegularLoopWords.blocksPrefix

blockSqrtTest def

The exact polynomial lower bound, not the non-polynomial square root. · OAI.ThreeTorus.blockSqrtTest

blocksSuffix def

Bits remaining after the first k blocks. · OAI.MaxCutGames.Foundations.PCP.PreprocessingRegularLoopWords.blocksSuffix

blocksSuffix def

Bits remaining after the first k blocks. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingRegularLoopWords.blocksSuffix

blockStmt def

Read a fixed block, write its fixed finite transformation, reset the local register, and execute the next finite statement. · OAI.MaxCutGames.Foundations.Complexity.MachineFixedBlockMap.blockStmt

blockStmt def

Read a fixed block, write its fixed finite transformation, reset the local register, and execute the next finite statement. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineFixedBlockMap.blockStmt

blockSuccEquiv def

A label block split at its first interaction occurrence. · OAI.DilutedSpinGlass.blockSuccEquiv

blockSupport def

The sum of the two block support functions, with a common scale. · OAI.ProjectionCounterexample.blockSupport

blockTerm abbrev

The source block tensor, including the exterior D coefficient. · OAI.DerivedBar.Multibar.blockTerm

BlockTower structure

An iterated cellular construction before flattening. All stages are literal submonoids of the same final monoid. · OAI.PartitionConsistency.ConservativeMonoids.BlockTower

BlockTridiagonal def

Tridiagonality means that blocks at grading distance greater than one vanish. · OAI.LeanBlast.GotsmanLinial.BlockTridiagonal

blockUnit def

Direct-sum transport, with all summand indices retained. · OAI.IntegralCharacterVarieties.MatrixExpression.blockUnit

BlockUpperIndex abbrev

An upper product gate chooses the indices at the boundaries between blocks. · OAI.Problem335.BlockUpperIndex

blockVector def

Vectors constant on the groups of three and four remaining labels. · OAI.DistinctLifts.blockVector

blockWeight def

Weight one on one coordinate block, and zero on all spectator blocks. · OAI.Problem346.PolynomialWeights.blockWeight

blockWeightedGraphEnergy def

The coefficient is precisely the fraction of origins retaining both endpoints. · OAI.JointDickman.blockWeightedGraphEnergy

blockWeights def

Retain a full pair with probability 1-beta, or choose one of three singleton positions with probability beta/3 and its uniform slope. · OAI.MaxCutGames.Decoder.PaddedLaw.blockWeights

blockWeights def

Retain a full pair with probability 1-beta, or choose one of three singleton positions with probability beta/3 and its uniform slope. · OAI.UniqueGamesTheorem.Decoder.PaddedLaw.blockWeights

BlockWitness abbrev

A recorded bracketing word occurrence, and a counted placement inside it. · OAI.Superstring.Counts.BlockWitness

BlockWitness abbrev

Bracketing witnesses over the whole fixed family of layers. · OAI.Superstring.Hierarchical.Family.BlockWitness

BlockWitness abbrev

A recorded complete bracketing word and one of its eligible inner placements. · OAI.Superstring.Hierarchical.Layer.BlockWitness

blowupCoordinate def

Affine blowup coordinates inside the original rational function field. · OAI.SourcePullbackChart.blowupCoordinate

blowupGraph def

The actual simple unweighted finite output graph, not an abstract promise. · OAI.OptimalMaxCut.Unweighted.blowupGraph

blowupOutput def

The scale is p^12, not the output number of edges. · OAI.OptimalMaxCut.Unweighted.blowupOutput

bobTriple def

Bob receives only a variable name: the strategy has no occurrence input. · OAI.MaxCutGames.Reduction.Incidence.bobTriple

bobTriple def

Bob receives only a variable name: the strategy has no occurrence input. · OAI.UniqueGamesTheorem.Reduction.Incidence.bobTriple

bochnerDrift def

The absolutely continuous part of the Gram Laplacian after removing its full Bochner measure. · OAI.RegularPointCoordinates.bochnerDrift

bochnerGramSource def

Full source of a Gram entry with its Bochner measure separated out. · OAI.RegularPointCoordinates.bochnerGramSource

Body inductive

The right suffix tested from J': anchors, C, workspace, D, anchors. · OAI.FiniteAlgebraicEnvelopes.Compiler.Validator.Body

Body structure

Paper K is a real convex body with interior. · OAI.GeneralMahler.Body

Body def

The lower-right two-by-two pattern of the fixed matrix fixedH. · OAI.Problem348.PathPropagation.Body

body def

The whole tape during copying: read and unread anchor prefixes on either side of the reserved workspace, with a filled prefix followed by blanks. · OAI.FiniteAlgebraicEnvelopes.Compiler.Simulator.body

body def

The actual unit nearest-neighbor Lipschitz polytope in its free coordinates. The graph and pinned set are arbitrary here; the finite lattice polygon is a special case. Fixed--fixed inequalities are retained. · OAI.TriangularSurface.FiniteModel.body

bodyDerivativeFamily def

Jointly measurable actual inverse section-density diagonal derivative. · OAI.Problem358.FinitePrefix.bodyDerivativeFamily

bodyExecution def

Lift the actual certified body run while preserving the remaining counter. · OAI.MaxCutGames.Foundations.Complexity.MachineRepeat.bodyExecution

bodyExecution def

Lift the actual certified body run while preserving the remaining counter. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRepeat.bodyExecution

bodyIndex def

The row body's nine tapes retain their numbering; tape 9 is the stream. · OAI.MaxCutGames.Foundations.Complexity.MachineOverlayRows.bodyIndex

bodyIndex def

The row body's nine tapes retain their numbering; tape 9 is the stream. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineOverlayRows.bodyIndex

bodyInTime def

Complete actual .startU → .guard execution, without a phase-run premise. · OAI.MaxCutGames.Explicit.MachineSubdivisionBody.bodyInTime

bodyInTime def

Complete actual .startU → .guard execution, without a phase-run premise. · OAI.UniqueGamesTheorem.Explicit.MachineSubdivisionBody.bodyInTime

BodyNodeOrder def

The necessary parent comparison for an ordered body in each of the six modes. · OAI.Problem348.Construction.BodyNodeOrder

bodyOrder def

Slow spectator, then column positions in α each with an f-bit digit. · OAI.PowerSaving.Binary.bodyOrder

bodyQuantileFamily def

The concrete conditional family, extended by zero beyond the ambient dimension. · OAI.Problem358.FinitePrefix.bodyQuantileFamily

BodyReads abbrev

Every raw tester and every matching payload has its own body Read. This quantification is over all payloads, not only semantic realizers. · OAI.PureTypeSystem.ObservationValues.ProductInstance.BodyReads

bodyReturn def

Transfer the literal body output back to the input, preserving its order. · OAI.MaxCutGames.Foundations.Complexity.MachineRepeat.bodyReturn

bodyReturn def

Transfer the literal body output back to the input, preserving its order. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRepeat.bodyReturn

BodyTraces def

The hypotheses are actual traces of the caller's body in the same program. Both endpoints have the same unary remainder and unread suffix. · OAI.MaxCutGames.Foundations.Complexity.MachineCountedLoop.BodyTraces

BodyTraces def

The hypotheses are actual traces of the caller's body in the same program. Both endpoints have the same unary remainder and unread suffix. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineCountedLoop.BodyTraces

bodyVolume def

Lebesgue volume in the ambient Euclidean space. · OAI.ProjectionCounterexample.bodyVolume

BondGraph structure

An undirected graph with labelled bonds, including loops and parallel bonds. · OAI.CriticalPercolation.BondGraph

booleanFinish def

A short proposal word and an independently drawn cached completion reserve their full combined word length. · OAI.ContingencyTables.IntegerLaw.booleanFinish

BooleanNetwork structure

Typed interfaces for network composition, with explicitly retained workspace. · OAI.ExactQuantumFactoring.BooleanNetwork

BooleanOp inductive

A Boolean node is implemented by at most two of the named fixed gates. The target is excluded from the sources, but repeated sources are permitted. Its action toggles the target, so the semantics are reversible on all states. · OAI.ExactQuantumFactoring.BooleanOp

booleanPartitionMap def

Pull a step function back to a finer partition. · OAI.PolycyclicRecognition.booleanPartitionMap

booleanPartitionRefine def

Forget some of the Boolean coordinates of a realized partition cell. · OAI.PolycyclicRecognition.booleanPartitionRefine

booleanPartitionRetraction def

Choose one refined cell above each old cell to obtain a retraction. · OAI.PolycyclicRecognition.booleanPartitionRetraction

BooleanProgram abbrev

The classical network explicitly retains all values already computed. In a fresh-output DAG each node's target is a previously zero ancilla. · OAI.ExactQuantumFactoring.BooleanProgram

booleanSign def

The sign representation of a Boolean coordinate. · OAI.TwoPointCorrelations.booleanSign

booleanStepMulSemiringAction def

The action is by actual ring automorphisms, given by left translation. · OAI.PolycyclicRecognition.booleanStepMulSemiringAction

boolKindEquiv def

Boolean words use false for the normal pairing and true for the diagonal pairing. · OAI.Problem335.Pairings.boolKindEquiv

BoolNet inductive

Straight-line Boolean computation retaining its whole trace. Width increases by exactly one per node, so there is no truth-table synthesis overhead. · OAI.ExactQuantumFactoring.BoolNet

BoolNode inductive

A bounded-fan-in Boolean node, with no hidden arithmetic operation. · OAI.ExactQuantumFactoring.BoolNode

boolOuterStatement def

Uniform-alphabet presentation of the actual outer code. All arithmetic parameters here are fixed program constants; runtime sizes are scanned. · OAI.MaxCutGames.Foundations.Complexity.MachineExpanderFamily.boolOuterStatement

boolOuterStatement def

Uniform-alphabet presentation of the actual outer code. All arithmetic parameters here are fixed program constants; runtime sizes are scanned. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineExpanderFamily.boolOuterStatement

boolReal def

Real value of a Boolean indicator. · OAI.MaxCutGames.Fourier.MatrixEnergy.boolReal

boolReal def

Real value of a Boolean indicator. · OAI.UniqueGamesTheorem.Fourier.MatrixEnergy.boolReal

boolSign def

A Boolean encoding of a real sign. · OAI.AsymptoticallyMinimalLittlewood.boolSign

boolTypeSign def

The Boolean representation of the source Prouhet signs; true denotes a negative type. No sign convention is changed in passing to literal words. · OAI.TraceReconstruction.boolTypeSign

borelChart def

A chart has no prescribed behaviour outside its source. This Borel extension, used only to state global relations, agrees with it on the source. · OAI.SmoothObstruction.ManifoldCoordinates.borelChart

BorelRules def

Borel probabilities, including counterfactual states. · OAI.SubsphereCurrent.KernelLearner.BorelRules

bottIndexFlip def

Exchange the two branches of the first subexpression slot. · OAI.KLInvariance.TitsSpace.bottIndexFlip

Bottom abbrev

Recursive axis/fiber indexing; no extra workspace is introduced. · OAI.ExactFourier.Triangular.Bottom

bottomAddress def

The actual full-tree leaf address of a bottom edge, including the edge's own numeric label. · OAI.TraceReconstruction.OccupiedParityTree.Edge.bottomAddress

bottomEnergy def

Bottom Rayleigh energy, equal to the bottom of the matching spectrum in a faithful representation. This definition retains arbitrary Hilbert dimension. · OAI.NuclearUltrapower.bottomEnergy

bottomGraph def

Adjoin a vertex below the old vertices, with specified outgoing edges. Only the edge orientations, not the extra comparabilities, enter sections. · OAI.KLInvariance.MomentGraph.bottomGraph

bottomPairValue def

Bottom-pair value with difference d and ratio t, zero at the singular ratio. · OAI.Ostmann.bottomPairValue

bottomRoleWeight def

At the last internal level every selected leaf has to be active. Role zero has weight zero; unsigned and signed roles are respectively χ/2 and -σ/2 in (χ-στ)/2. · OAI.TraceReconstruction.bottomRoleWeight

bottomSeed def

A bottom flag with its cube chosen within one mesh of any given spatial point. · OAI.SharpLiebThirring.CubeFlags.bottomSeed

bottomSides def

The bottom of the prism together with all its vertical sides. · OAI.HyperbolicObstruction.CubicalHomotopyExtension.bottomSides

bottomUnary def

Extending the final unary edge keeps every internal branching depth. In contrast, unary S delays all of them by exactly one. · OAI.DilutedSpinGlass.PrescribedTree.bottomUnary

bottomValue def

Value of a bottom gate before relabelling the gate types. · OAI.Problem335.IndexedDepth5Circuit.bottomValue

bottSimpleFlip def

Reflect the first tensor factor. It is not asserted to be left A-linear. · OAI.KLInvariance.TitsSpace.bottSimpleFlip

bounceRelation def

Returns from an exit port through the surrounding left or right context. · OAI.TwoWayComplementation.Diagram.bounceRelation

bound def

room for any fixed power of the allowed accessor logarithm and for start-up costs. · OAI.Fifths.Cost.bound

bound abbrev

A fixed numerical bound determined solely by the fixed selecting bimodule. · OAI.LittleFinitistic.SelectedOrdinaryDimension.bound

Boundary structure

Connecting morphism and exactness on the Cartier divisor. · OAI.CartierConnecting.Boundary

Boundary structure

An actual finite closed edge boundary for each cell, with indexed sides. The cell type may be infinite (as in the universal cover); every cell boundary is finite. Orientation reversal is constructed, not postulated. · OAI.HyperbolicObstruction.PictureSwitch.PolygonTarget.Boundary

Boundary def

First-exit vertices. · OAI.IndependentProducts.FiniteHead.Boundary

boundary def

The snake connecting morphism descends through a fully faithful exact closed pushforward to the divisor category. · OAI.CartierConnecting.boundary

boundary def

The partial sweep at a stage boundary, including boundaries 0 and b. · OAI.CoordinateSweeps.Grid.boundary

boundary def

An initial segment of the facial boundary, with two edge sides per step. · OAI.KaplanskyCounterexample.Planar.Balanced.boundary

boundary def

Negative and positive boundary, with no cancellation between them. · OAI.Thomason.Construction.PastingGlobeBasis.boundary

boundary def

All coordinate faces of the (n+1)-dimensional simplicial cube. · OAI.Thomason.Construction.RealizedCube.boundary

boundary def

Boundary action of a signed interval isometry. On half-integer centers it agrees with the manuscript's letter-index affine action. · OAI.TorsionFreeZeroDivisors.AffineIntervals.boundary

boundaryArrowIso def

The whole realized boundary arrow agrees with the barycentric pair. · OAI.Thomason.Construction.RealizationPrism.boundaryArrowIso

boundaryArrowIso def

An isomorphism of the complete boundary-inclusion arrows. · OAI.Thomason.Construction.RealizedCube.boundaryArrowIso

boundaryArrowIso def

Transport of lifting problems retains the entire boundary on both sides. · OAI.Thomason.Construction.SimplexPolytopes.boundaryArrowIso

boundaryArrowRetract def

A retract in the actual topological arrow category, preserving both boundaries and ambient cubes. · OAI.Thomason.Construction.RealizedCube.boundaryArrowRetract

boundaryBandExposure def

Universal exposure used by every chronological band step of a whole-port move. Its branch is decided by old occurrence circles, not colors. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.boundaryBandExposure

boundaryChart def

Unique normalized full solution plus all independent based corner matrices. This is one global path-coordinate chart, not the missing cutting map. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.boundaryChart

BoundaryChord structure

Literal chord components with endpoints on any two distinct rectangle ports. · OAI.CriticalHoneycomb.Transfer.RasterGrid.BoundaryChord

boundaryCoeff def

The (possibly zero) head sum carried by the unique segment at a gate. · OAI.DiamondDistortion.Ancestry.boundaryCoeff

boundaryConstant def

Fields independent of the averaged group coordinate. · OAI.KadisonKastler.boundaryConstant

boundaryCoordinates def

The barycentric coordinates of the actual boundary inclusion. · OAI.Thomason.Construction.RealizationPrism.boundaryCoordinates

BoundaryData abbrev

Variational boundary data: the quotient by the closure of compactly supported smooth jets. · OAI.Elasticity.BoundaryData

boundaryExclusionSet def

This is the open set excluded by a single retained period coordinate. · OAI.CircleDomainRigidity.boundaryExclusionSet

boundaryFacePoint def

The literal face point in the full cube boundary. · OAI.Thomason.Construction.RealizedCube.boundaryFacePoint

boundaryFold def

Literal free boundary cancellation, valid even when either canceled side was paired to an interior side or to another part of the outside. It is a genus-zero operation, not an assumed disk replacement. · OAI.HyperbolicObstruction.PictureSwitch.PictureMap.boundaryFold

boundaryForcePair def

The actual L2 force pairing, with no regularity required of the test. · OAI.Elasticity.boundaryForcePair

boundaryForm def

The full shuffle boundary form, using actual exterior differentiation. · OAI.Mahler.boundaryForm

boundaryFormFin def

Actual boundary form, in its standard ordered slots. · OAI.Mahler.boundaryFormFin

boundaryGraph def

Delete precisely the edges with both ends in the excluded vertex set. · OAI.CycleClique.Construction.boundaryGraph

boundaryHeight def

The two boundary arcs; vertices in plus get lam, the other pinned vertices get -lam. Membership outside pins has no effect on body or law. Each pinned vertex receives exactly one prescribed value. · OAI.TriangularSurface.FiniteModel.boundaryHeight

boundaryHomeomorph def

The actual boundary realization is homeomorphic to the zero-coordinate locus of the barycentric simplex. · OAI.Thomason.Construction.RealizationPrism.boundaryHomeomorph

boundaryHomeomorph def

The exact boundary homeomorphism is compatible with the ambient cube. · OAI.Thomason.Construction.RealizedCube.boundaryHomeomorph

boundaryHomeomorph def

The full subdivided boundary realizes to the entire geometric boundary. · OAI.Thomason.Construction.SubdivisionBarycentric.boundaryHomeomorph

boundaryHomotopy def

The prescribed first boundary becomes null-homotopic in the actual total. · OAI.LittleFinitistic.InducedRectification.boundaryHomotopy

boundaryHomotopy def

The first-column boundary vanishes by a specified chain homotopy in the total. For the induced B-arrow columns this is exactly the arrow compatibility homotopy. · OAI.LittleFinitistic.RectificationTotal.boundaryHomotopy

boundaryInclusion def

The literal inclusion of the barycentric boundary subspace. · OAI.Thomason.Construction.RealizationPrism.boundaryInclusion

boundaryIndex def

A literal indexed match between two enumerated boundaries. Repeated edge letters are not quotiented: this map uses the unique side positions. · OAI.HyperbolicObstruction.PictureSwitch.boundaryIndex

boundaryLength def

Numbers of sectors in the two coned boundary polygons. · OAI.EilenbergGanea.boundaryLength

boundaryLinear def

The actual finite-array boundary operator consists of both recurrence residuals, the middle matching equation, and the two terminal projections. · OAI.SmoothObstruction.FiniteGreen.Matching.boundaryLinear

boundaryLocus def

The boundary locus, including the empty boundary in dimension zero. · OAI.Thomason.Construction.RealizationPrism.boundaryLocus

boundaryMap def

Boundary values supplied by sections strictly above a vertex. · OAI.KLInvariance.MomentGraph.Sheaf.boundaryMap

BoundaryMapFamily structure

Data constructed by the ray-limit theorem. The compact set is common to all maps in the family and all ambient basepoints. · OAI.PolycyclicRecognition.BoundaryMapFamily

boundaryMax def

The real maximum requires an actual member of the fixed transport set. · OAI.ThompsonNonamenability.boundaryMax

BoundaryNaturality def

The classical naturality identity for alternating normalized chains. This is the normalized simplex chain-map assertion of Steiner (2007), Section 4. · OAI.Thomason.Construction.SimplexComplex.BoundaryNaturality

boundaryNext def

The next-side permutation of the oriented facet boundaries. All finite cycles, in particular self-incidences and one-side circles, are retained. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.Corners.boundaryNext

boundaryNext def

Follow the first uncancelled side counterclockwise around the arriving corner. The two tests inspect the other two squares meeting that corner. · OAI.PlanarFPP.boundaryNext

boundaryNullhomotopyOfExtension def

Any extension over a standard simplex supplies an explicit simplicial nullhomotopy, by the actual initial-vertex contraction. · OAI.Thomason.Construction.boundaryNullhomotopyOfExtension

boundaryNullhomotopyOfExtension def

Restrict the simultaneous cube contraction through a given extension. This is a homotopy of the whole cubical boundary, with both endpoints exact. · OAI.Thomason.Construction.RealizedCube.boundaryNullhomotopyOfExtension

boundaryObject def

The object at either end of a cell. · OAI.Thomason.Construction.StrictOmegaCategory.boundaryObject

boundaryOutput def

The actual output butterfly in one central block. · OAI.CubeShuffle.boundaryOutput

boundaryPairing def

Contract all path components onto their boundary. The degree assumptions are precisely degree one at the exposed ports and degree zero or two elsewhere. Closed components disappear. · OAI.OneWayLiveness.FiniteWiring.boundaryPairing

boundaryPerm def

Cancellation of shared sides leaves a permutation of the remaining sides; its cycles are directed closed walks with no repeated directed side. · OAI.PlanarFPP.boundaryPerm

boundaryPhase def

All boundary-position phases form a measurable ambient-null set. · OAI.TriangularBilliards.boundaryPhase

boundaryPinch def

Pinch the two consecutive boundary sides off as a genuine bigon. The other side pairs are unchanged, and the remaining boundary bypasses them. · OAI.HyperbolicObstruction.PictureSwitch.PictureMap.boundaryPinch

boundaryPrefix def

The ordered prefix preceding a chosen boundary occurrence. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.boundaryPrefix

boundaryResidualFin def

The residual variation, retaining all terms and the actual exterior spatial derivatives. Its sphere integral has not been asserted to vanish. · OAI.Mahler.boundaryResidualFin

BoundaryRetraction structure

A section and an actual simplicial homotopy to its other composite. · OAI.Thomason.Construction.BoundaryRetraction

boundaryRetraction def

Every map lifting against boundaries admits explicit simplicial retraction data. · OAI.Thomason.Construction.boundaryRetraction

boundaryRing def

The actual restriction of a polynomial to the flat boundary. · OAI.ElasticityBoundaryMVNormal.boundaryRing

boundarySection def

The ordinary boundary section, with values in the actual realized simplicial boundary. · OAI.Thomason.Construction.RealizedCube.boundarySection

BoundarySelection def

Occupied actual midpoint edges, with only the two prescribed exits. · OAI.CriticalHoneycomb.Transfer.FiniteConfigurations.BoundarySelection

BoundarySheaf structure

The outcome of the actual minimal boundary construction. Full upward-kernel freeness, integral duality and KL characters are NOT part of this structure. · OAI.KLInvariance.MomentGraph.BoundarySheaf

boundarySingleResidualFin def

The residual with one explicit d beta factor and no suppressed placements. The intermediate slot type keeps the original recursive order. · OAI.Mahler.boundarySingleResidualFin

BoundarySquared def

The usual differential identity on these concrete normalized chains. · OAI.Thomason.Construction.SimplexComplex.BoundarySquared

boundaryState def

A fresh row call, with its residue and bit register reset. · OAI.MaxCutGames.Foundations.Complexity.MachineExpanderTable.boundaryState

boundaryState def

A fresh row call, with its residue and bit register reset. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineExpanderTable.boundaryState

boundaryTranslate def

Transport combines left translation of the group coordinate and the original action on the value algebra. · OAI.KadisonKastler.boundaryTranslate

boundaryTranslationAction def

The induced translations themselves form a group action. · OAI.KadisonKastler.boundaryTranslationAction

boundaryTransport def

The transport field used before averaging an approximate cocycle. · OAI.KadisonKastler.boundaryTransport

boundaryTree def

A fixed central map in every block, with the genuine input-side fair layers above the boundary. The coins below L are ignored on this side. · OAI.CubeShuffle.boundaryTree

boundaryValue def

Exact old edge matrix viewed in the chosen facet fiber. Outside that facet it is irrelevant and set to 1; no old edge in that facet is replaced. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.boundaryValue

BoundaryVolumeComparison structure

The measures, common window, and estimates produced by the compact boundary shadows. These data are constructed below, not published inputs. · OAI.PolycyclicRecognition.BoundaryVolumeComparison

boundaryWeightIdeal def

The ideal generated by boundary monomials of weight strictly above L. · OAI.CartierSections.boundaryWeightIdeal

BoundaryWire def

The actual four rectangle cuts; internal edges are not constrained. · OAI.CriticalHoneycomb.Transfer.RasterGrid.BoundaryWire

boundaryWord def

Matrix transport reads a positive boundary path from right to left. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.boundaryWord

BoundAtDegree def

The strong-induction motive quantifies over the actual changing domains. Quotients and subspaces preserve the fixed universe levels. · OAI.MaxCutGames.Appendix.A5Induction.BoundAtDegree

BoundAtDegree def

The strong-induction motive quantifies over the actual changing domains. Quotients and subspaces preserve the fixed universe levels. · OAI.UniqueGamesTheorem.Appendix.A5Induction.BoundAtDegree

Bounded structure

Uniformly bounded pointed sections of the actual varying action spaces. · OAI.HyperbolicObstruction.MetricLimit.Bounded

Bounded inductive

A structural draw bound, independent of the stream's values. · OAI.LowSpaceSubsetSum.WordProgram.Bounded

Bounded def

Every monomial appearing in p has integer weight at most d. · OAI.StableCoordinate.WeightedInitial.Bounded

Bounded def

All stack lengths are bounded by a common number. · OAI.Superstring.BitCode.Program.Bounded

Bounded def

The literal finite-dimensional condition on the mapping-family carrier. · OAI.Thomason.Construction.StrictOmegaCategory.GlobeMappingCells.Bounded

bounded def

Bounded representatives, with homotopy classes of chain maps. · OAI.LittleFinitistic.BoundedHomotopy.bounded

bounded def

The caller's capacity guard also bounds the terminal cost on every input. · OAI.LowSpaceSubsetSum.SeparatorTerminal.bounded

bounded def

Bounds are included in the relation when proving uniqueness, so no functional behavior outside the specified sets is required. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.BinaryRelation.bounded

bounded def

The actual subcategory of cells of dimension at most n. · OAI.Thomason.Construction.StrictOmegaCategory.bounded

bounded def

Bounding a literal system adds no realizability or word constraints. · OAI.TorsionFreeZeroDivisors.SampleGraph.PathSystem.bounded

BoundedBimoduleResolution structure

A genuine ordinary bimodule supplied with an actual bounded right-projective resolution. · OAI.LittleFinitistic.BoundedBimoduleResolution

boundedBoundaryTransport def

The exact transport field, bundled as a bounded unitary family. · OAI.KadisonKastler.boundedBoundaryTransport

BoundedC1 abbrev

A bounded function with bounded continuous derivative, in its graph norm. · OAI.ForcedComputation.BoundedC1

boundedCells def

The cells fixed by both n-dimensional boundary maps. · OAI.Thomason.Construction.StrictOmegaCategory.FilteredColimit.boundedCells

boundedCellsIsColimit def

A filtered strict colimit is a colimit on each bounded-cell set. · OAI.Thomason.Construction.StrictOmegaCategory.FilteredColimit.boundedCellsIsColimit

boundedChainEndpoints def

Endpoints of at most n bounded-height steps from a fixed starting point. · OAI.PolycyclicRecognition.boundedChainEndpoints

boundedClasses def

Morita classes represented by blocks with a defect group of order at most M. · OAI.Donovan.boundedClasses

BoundedCode def

Literal finite parameter space of bounded ordered grids; no model of irreducibles or of conditioned complex measures is involved. · OAI.CoordinateSweeps.Grid.BoundedCode

boundedCoefficientPolynomials def

Genuine bounded polynomials, with each coefficient in its specified submodule. · OAI.Nagata.Workers.W10.boundedCoefficientPolynomials

boundedCommutator def

The bounded commutator operator on a normed algebra. · OAI.KadisonKastler.boundedCommutator

boundedComplex def

Strict boundedness of the underlying complex, at both ends. · OAI.LittleFinitistic.BoundedHomotopy.boundedComplex

BoundedConeRelator def

Only locally PROVED fillings are admitted to this auxiliary relator set. It is used for derivation substitution, not to alter either presentation. · OAI.HyperbolicObstruction.ConeSkeleton.BoundedConeRelator

boundedConstant def

A constant bounded family, with the ordinary supremum-norm domain. · OAI.KadisonSimilarity.boundedConstant

boundedDerivativeGraph def

The bounded derivative graph, with its inherited uniform product norm. · OAI.ForcedComputation.boundedDerivativeGraph

boundedDerivedEquivalence def

Bounded complexes of f.g. projectives form a full subcategory of the actual derived category. · OAI.LittleFinitistic.FiniteProjectiveHomotopy.boundedDerivedEquivalence

boundedDerivedIso def

The comparison for exactly the same bounded Q and its chosen interval. · OAI.LittleFinitistic.BimoduleTotalPresentation.boundedDerivedIso

BoundedDiameter def

Weak diameter, measured in the original weighted graph metric. · OAI.BoundedTreewidthL1.PaddedPartitions.BoundedDiameter

BoundedDistance def

A bounded-test distance, with the scale written outside the test. · OAI.DilutedSpinGlass.BoundedDistance

boundedEntropyTest def

Outside the bounded logarithmic window use the zero test. At a zero density, use the negative endpoint, which recovers the Maxwellian contribution. · OAI.BoltzmannNonuniqueness.boundedEntropyTest

boundedEquiv def

Evaluation on an n-globe exactly represents the bounded mapping cells. · OAI.Thomason.Construction.StrictOmegaCategory.GlobeMappingCells.boundedEquiv

boundedFieldOperator def

The bounded L² operator associated to an essentially bounded strongly operator measurable field. · OAI.KadisonSimilarity.boundedFieldOperator

boundedFieldVector def

The L² section obtained by applying an essentially bounded field. · OAI.KadisonSimilarity.boundedFieldVector

boundedFirstHeightTime def

Finite first-height clock, stopped at the prescribed deterministic bound. · OAI.SLEExactGauge.FirstHeightSupport.boundedFirstHeightTime

boundedForestCode def

Fixed-length bit vectors enumerate every forest with at most N vertices. · OAI.TwoPointCorrelations.boundedForestCode

boundedHeat def

The continuous representative constructed from bounded-data smoothing. · OAI.RegularPointCoordinates.NoncollapsedRCD.boundedHeat

BoundedHeightChain inductive

A finite chain whose steps and heights are uniformly bounded. · OAI.PolycyclicRecognition.BoundedHeightChain

boundedIncrement def

The increment of a function of uniformly bounded additive defect is bounded. · OAI.PolycyclicRecognition.boundedIncrement

boundedIndex def

Extend finite indices harmlessly outside the range used by the machine. · OAI.MaxCutGames.Foundations.Hastad.SourceFieldArray.boundedIndex

boundedIndex def

Extend finite indices harmlessly outside the range used by the machine. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceFieldArray.boundedIndex

boundedIndices def

A finite index set that contains every hat meeting a fixed bounded region. · OAI.DyadicHats.boundedIndices

boundedIndices def

A finite index set that contains every hat meeting a fixed bounded region. · OAI.SignedDisk.DyadicHats.boundedIndices

BoundedK abbrev

The exact bounded-projective source category used in Section 3. · OAI.LittleFinitistic.QuadraticEncoding.BoundedK

boundedKernelAverage def

The averaged kernel is globally bounded by reciprocal testing volume. This makes it a legitimate test for a finite signed source, even at singular source poles. · OAI.RegularPointCoordinates.boundedKernelAverage

boundedLabel def

Bounded integer labels are encoded without enlarging the physical cells. · OAI.SmoothObstruction.GridMargin.boundedLabel

boundedLevel def

An executable exponent, computed with at most k multiplications. · OAI.MaxCutGames.Foundations.PCP.PreprocessingLevels.boundedLevel

boundedLevel def

An executable exponent, computed with at most k multiplications. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingLevels.boundedLevel

boundedLists def

An explicit bounded list enumeration, used only for finite internal search. This bound counts cells, not grade, and thus never admits quota insertion. · OAI.FiniteAlgebraicEnvelopes.Compiler.Scan.boundedLists

boundedLists def

All lists of length at most n over a finite alphabet. · OAI.Generator.Tracial.boundedLists

boundedLocalVersion def

Clipping only the global extension leaves a locally bounded coordinate unchanged. · OAI.RegularPointCoordinates.boundedLocalVersion

boundedLpSquare def

The square of a bounded L² function is again an actual L² function. · OAI.RegularPointCoordinates.boundedLpSquare

boundedMap def

Forget continuity (automatic on the discrete index set) and retain boundedness. · OAI.Generator.Tracial.boundedMap

boundedMatrixOperator def

Extension of the bounded finite matrix action to every l2 vector. · OAI.FiniteFactor.boundedMatrixOperator

BoundedMatrixTerm structure

One term of a rectangular star polynomial, with its actual word and its degree bound. Zero coefficients and empty words are allowed. · OAI.Kaplansky.BoundedMatrixTerm

boundedMultiply def

The same first monomial multiplies both components. · OAI.MatrixMultiplication.AuxiliarySeparation.DeterminantFiltration.boundedMultiply

boundedNonpositiveDerivedIso def

Generic bridge for a shifted P as well as Q_X. No augmentation or target module is needed to compare two presentations of the very same bounded complex. · OAI.LittleFinitistic.BimoduleTotalPresentation.boundedNonpositiveDerivedIso

boundedNonpositivePowerIso def

Direct iterate transport for any same-complex bounded model on [-length,0]. · OAI.LittleFinitistic.BimoduleTotalPresentation.boundedNonpositivePowerIso

boundedNonpositiveTotalIso def

Any native bounded model with these endpoints uses exactly the ordinary finite degree list. · OAI.LittleFinitistic.BimoduleTotalPresentation.boundedNonpositiveTotalIso

BoundedOn def

Bounded multipliers supported on A, regardless of the lottery weight of the entry. · OAI.CliqueFreeIndependence.WeightedGraph.BoundedOn

boundedOrbit def

Restrict a bounded continuous function to an orbit, as an actual bounded function on the acting group. · OAI.PolycyclicRecognition.boundedOrbit

BoundedPair abbrev

Affine coefficient pairs of forms of bidegree (e,1). · OAI.MatrixMultiplication.AuxiliarySeparation.DeterminantBasis.BoundedPair

BoundedPair abbrev

Pairs representing forms of bidegree (e, 1). · OAI.MatrixMultiplication.AuxiliarySeparation.DeterminantBounds.BoundedPair

boundedPolynomialBasis def

Bases of coefficient subspaces assemble into a basis of genuine polynomials. · OAI.Nagata.Workers.W10.boundedPolynomialBasis

boundedPolynomialCoordinates def

Read the complete finite tuple of actual coefficients. · OAI.Nagata.Workers.W10.boundedPolynomialCoordinates

boundedPolynomialEquiv def

The direct product description is a proved linear equivalence. · OAI.Nagata.Workers.W10.boundedPolynomialEquiv

boundedPositiveTripleSum def

The same sum after the genuine degree-support cutoff. · OAI.MaxCutGames.Appendix.A13Reduction.boundedPositiveTripleSum

boundedPositiveTripleSum def

The same sum after the genuine degree-support cutoff. · OAI.UniqueGamesTheorem.Appendix.A13Reduction.boundedPositiveTripleSum

BoundedProjective structure

Exactly the bounded-right-projective data needed to compute tensor on every complex. · OAI.LittleFinitistic.BimoduleTotal.BoundedProjective

BoundedQ abbrev

The actual localization of K^b(proj-B) by the thick subcategory generated by the two cones. · OAI.LittleFinitistic.QuadraticEncoding.BoundedQ

boundedQuadraticCost def

A conveniently clipped quadratic cost, bounded and continuous on the full product space. · OAI.Problem358.boundedQuadraticCost

boundedQuadraticDomain def

The evaluation restriction is essential: unrestricted coefficient triples have infinitely many literal integer floor words. · OAI.QuantitativeVanDerWaerden.boundedQuadraticDomain

boundedQuotientEvaluation def

Tensor evaluation on the manuscript's fixed quotient; no generator-vanishing assumption remains. · OAI.LittleFinitistic.QuadraticEncoding.boundedQuotientEvaluation

boundedRowPlan def

Each command of the actual row plan has a fixed bound of twice the radius. · OAI.MaxCutGames.Foundations.Complexity.PoweringMachinePlan.boundedRowPlan

boundedRowPlan def

Each command of the actual row plan has a fixed bound of twice the radius. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringMachinePlan.boundedRowPlan

BoundedSamplingStatement def

Paper 200-01, Theorem main(i): bounded runtime on every random tape and total variation at most 2⁻ᵏ, with freely varying dimensions. · OAI.ContingencyTables.Algorithms.BoundedSamplingStatement

boundedSectors def

The finite search universe consists of actual list-zipper sectors. · OAI.FiniteAlgebraicEnvelopes.Compiler.Scan.boundedSectors

boundedSelfAdjointGroup def

Exponentiation of a bounded self-adjoint generator. · OAI.KadisonKastler.boundedSelfAdjointGroup

BoundedSeq abbrev

Bounded sequences, with their supremum norm. N is discrete. · OAI.NuclearUltrapower.BoundedSeq

boundedSeqToLp def

The two standard bounded-sequence presentations agree coordinatewise. · OAI.NuclearUltrapower.boundedSeqToLp

boundedSequenceConst def

Constant sequences as a unital star homomorphism. · OAI.KadisonKastler.boundedSequenceConst

boundedSequenceMapHom def

Apply a contractive star homomorphism at every coordinate. · OAI.KadisonKastler.boundedSequenceMapHom

boundedSequences def

Operator-norm bounded sequences, as an actual star subalgebra of pointwise sequences. · OAI.UltrapowerBridge.boundedSequences

boundedSequenceStarHom def

Apply a star homomorphism at every coordinate of a bounded sequence. · OAI.KadisonSimilarity.boundedSequenceStarHom

boundedSequenceSubmodule def

The algebraic submodule of uniformly bounded sequences. · OAI.Tingley.boundedSequenceSubmodule

BoundedShape abbrev

Each count is bounded separately. Thus every collection of at most M occurrences is included, with all its retained fan slots. This harmless stronger finite family needs no union over diagram copies or equality guesses. · OAI.HyperbolicObstruction.IncidenceTemplate.Occurrences.BoundedShape

boundedShift def

Multiplication by a fixed first-input monomial between the bounded spaces. · OAI.MatrixMultiplication.AuxiliarySeparation.DeterminantFiltration.boundedShift

boundedStep def

A bounded-state one-epoch transducer. Neither its state bound nor its transition function contains a raw word or a request horizon. · OAI.UniformKServer.EpochControl.boundedStep

boundedSubmodule def

The bounded polynomials form the usual weighted filtration submodule. · OAI.StableCoordinate.WeightedInitial.boundedSubmodule

BoundedTable def

The entry constraints for the companion paper, including structural zeros. · OAI.ContingencyTables.BoundedTable

boundedTotalCycles def

Actual total cycles supported in columns strictly below p. · OAI.PolycyclicRecognition.boundedTotalCycles

boundedTotalLeadingCycles def

The leading component as a map to the cycles in its row. · OAI.PolycyclicRecognition.boundedTotalLeadingCycles

boundedTotalLeadingHomology def

The leading row's homology class. · OAI.PolycyclicRecognition.boundedTotalLeadingHomology

boundedTotalSecondPageCycles def

Leading total cycles mapped into the cycles of the first-page complex. · OAI.PolycyclicRecognition.boundedTotalSecondPageCycles

boundedTraceProjection def

A total common finite projection. The fallback is never used by a genuine trace of a length-at-most-n input. · OAI.TraceReconstruction.boundedTraceProjection

BoundedTracks def

Uniform bounded spatial support; used only to define the local Sobolev class. · OAI.WeakHessian.BoundedTracks

boundedTuple def

Bounded requests have an explicit finite coordinate representation; the copy coordinate is retained even though the progress rank ignores it. · OAI.EditApproximation.PhysicalTableRequest.boundedTuple

BoundedUnitalHom abbrev

A complex-linear unital algebra homomorphism which is bounded. · OAI.KadisonSimilarity.BoundedUnitalHom

boundedUnitary def

A unitary family is uniformly bounded, independently of Hilbert dimension. · OAI.NuclearUltrapower.HilbertUltraproduct.boundedUnitary

boundedUnitaryFamily def

Any family of unitaries is uniformly bounded, hence defines an actual unitary of the bounded-function C-star algebra. · OAI.KadisonKastler.boundedUnitaryFamily

boundedUnitarySeq def

Every unitary sequence is bounded by one. · OAI.NuclearUltrapower.boundedUnitarySeq

boundedWeakNormalEquiv def

A faithful representation continuous on bounded weak nets identifies the original algebra with its generated von Neumann algebra. · OAI.KadisonKastler.boundedWeakNormalEquiv

boundedWordProducts def

Products with at most the specified number of letters. · OAI.PolycyclicRecognition.boundedWordProducts

boundingFace def

The part of the body on the bounding hyperplane of inequality i. · OAI.ProjectionCounterexample.boundingFace

boundState def

The exact unnormalized extremizing bound state, as an element of the weak form domain. · OAI.SharpLiebThirring.ExtremizerProof.boundState

Bow structure

Common path-table scripts. The prefix of the digit stream attached to an endpoint will be assembled with repeated stationary queries from the initiation. use steps=0 for the plain cap kernel. · OAI.ExactDerandomization.Sub.Bow

box def

A finite rectangular neighbourhood of a cube point. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Prefix.box

box def

The physical reciprocal box for a constructed affine barrier. · OAI.Problem326.BarrierConstruction.AffineBarrierData.box

BoxBasis abbrev

Throughout one invocation a product basis for each physical wire. · OAI.PowerSaving.Cluster.BoxBasis

boxBasis def

The actual scaled word basis, selected before taking its exponential image. · OAI.PolycyclicRecognition.CartanWordModel.boxBasis

boxCost def

The exponent of a word is the sum of the separately supported letters. · OAI.PolycyclicRecognition.CartanWordModel.boxCost

BoxProbeChains structure

The data retained for each of the three probe templates. The common length and constants are independent of the scale and the probe point. · OAI.PolycyclicRecognition.CartanWordModel.BoxProbeChains

BPL def

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.BPL

bracketWordLength def

Number of input letters in a bracket word. · OAI.PolycyclicRecognition.bracketWordLength

bracketWordNodes def

Number of bracket operations in the word. · OAI.PolycyclicRecognition.bracketWordNodes

braidLift def

Universal property, with equations rather than escaped relator words. · OAI.ArtinCAT0.braidLift

BraidRelations def

The length-three and distant commutation relations on m successive letters. The letters are zero-indexed here. · OAI.ArtinCAT0.BraidRelations

braidRelators def

Positive braid presentation on n generators, i.e. the group B_(n+1). · OAI.ArtinCAT0.braidRelators

braidTokenRepresentation def

Token representation. Unlike naive strand deletion, it is defined on every selected-position state, so respects arbitrary signed prefixes and suffixes. · OAI.ArtinCAT0.braidTokenRepresentation

Branch def

A branch records the choices of the two summands during iteration. · OAI.ExternalTensor.Binomial.Branch

branch def

The two graded blocks, padded with zeros in common coordinate spaces. · OAI.MatrixMultiplication.AuxiliarySeparation.DeterminantFiltration.branch

branch def

Branching tests only the finitely many currently scanned stack symbols. · OAI.ParityWL.Complexity.StackMachine.branch

branch def

At node i, region i is the parent branch; a child is indexed by itself. All other regions are empty. · OAI.ParityWL.RootedCuts.branch

branch def

Deterministic Boolean branching on a serialized pair. Only the selected finite subprogram runs; the tag is physically consumed. · OAI.RationalHitting.StackLanguage.Procedure.branch

BranchContact def

A deliberately broad contact relation: any pair of outer branches is allowed. A more precise geometric contact set can imply this relation. · OAI.Problem356.ContactSelection.BranchContact

branchContact def

Symmetric union of the two actual local branch graphs. The parameter remains in the corresponding chart source. · OAI.Problem356.FiveComponentGeometry.branchContact

branchCost def

Cost of the unconditioned entries of one star branch supported in A∪V. · OAI.HyperbolicObstruction.SparsePattern.Pattern.branchCost

branchCoupling def

Each branch is averaged over its three cyclic coordinate orderings. · OAI.Problem356.branchCoupling

branchDensity def

Zero-extended Jacobian density on a single injective good arc. · OAI.ThreeTorus.branchDensity

branchDensityWeight def

The weights of the central and four outer squares. · OAI.Problem356.branchDensityWeight

branchExcess def

Higher-degree factors, after removing the one grid increment per branching vertex. · OAI.DilutedSpinGlass.PrescribedTree.branchExcess

branchExponent def

The explicit manuscript token branch, specialized to the n=16 witness. · OAI.HarmonicCounterexample.Control.branchExponent

branchFamily def

The actual three retained tensors in their original ambient spaces. · OAI.MatrixMultiplication.AuxiliarySeparation.Sector.branchFamily

branchFamily def

The branch Jacobian with an arbitrary compact slow parameter space. · OAI.ThreeTorus.branchFamily

branchGraph def

A graph with all three coordinates explicitly specified. · OAI.Problem356.branchGraph

BranchingTree inductive

Finite occupied trees with at least two continuing odd children at each positive depth. This records exactly the no-singleton branch of the parity argument; the arity and all extra children remain arbitrary. · OAI.TraceReconstruction.BranchingTree

branchIteration def

Apply the splitting independently at each object; no naturality in S is present. · OAI.ExternalTensor.Binomial.branchIteration

branchKernel def

Parametrized old-frequency Jacobian density, extended by zero off its branch. · OAI.ThreeTorus.branchKernel

branchMarginal def

The marginal used by the two-branch construction. · OAI.Problem356.branchMarginal

BranchModel structure

A genuine finite branch-set model of a graph minor. Each branch is nonempty and connected; branches are disjoint and edges are witnessed by actual ambient edges. No topological or minor-exclusion axiom is hidden here. · OAI.PlanarL1.ColumnOrder.BranchModel

branchOperator def

The derivative of the stationary branch: central and opposite components. · OAI.Problem356.LocalHessian.branchOperator

branchPath def

The prescribed-weight exact-pair construction yields actual admissible paths on all infinite binary branches; records include all original metadata. · OAI.SeparableQuotient.ActualSpace.TreeNode.branchPath

branchPins def

All fields belonging to the specified branch are fixed in an extension. · OAI.HyperbolicObstruction.StarModel.branchPins

branchProduct def

The local product of reference factors for k represented children. The empty and unary induced vertices contribute no factor. · OAI.DilutedSpinGlass.PrescribedTree.branchProduct

branchProjectionSq def

The error is measured in the genuine two-copy branch law, in the same coupling as the physical spin products, not in a surrogate law. · OAI.DilutedSpinGlass.PrescribedTree.branchProjectionSq

branchRate def

Literal branchwise Bernoulli rates from sections/secretary.tex. false is the maximum branch and true is the main branch. · OAI.MatroidProphet.Secretary.branchRate

branchRebase def

Rebasing a genuine strict Lipschitz height branch, with no differentiability assumption. The old sign is averaged; its conditional laws are not asserted Lebesgue preserving. · OAI.PlanarL1.Threshold.branchRebase

BranchRegular def

Every vertex with at least three children has its higher-degree reference factors bounded away from zero. Binary vertices need only the grid increment. · OAI.DilutedSpinGlass.PrescribedTree.BranchRegular

branchWordTensor def

A word prescribes one branch tensor at each tensor-product position. · OAI.MatrixMultiplication.AuxiliarySeparation.branchWordTensor

brauerProjection def

Delete coefficients outside the centralizer of D. On the D-fixed group algebra in characteristic p, for a p-subgroup D, this is the Brauer map. We only use it on central idempotents; no multiplicativity on the entire group … · OAI.Donovan.brauerProjection

BravermanDepth22Input def

Braverman's Corollary 2 at depth twenty-two. The constants absorb a choice of logarithm base. Distributions are represented by nonnegative densities relative to the uniform law on the Boolean cube. · OAI.TwoPointCorrelations.BravermanDepth22Input

bridge def

A positive (n+1)-chain and common lower faces form a literal cell. · OAI.Thomason.Construction.DirectedComplex.Cell.bridge

bridgeLaplace def

The height Laplace transform of the strict bridge mass. · OAI.CriticalHoneycomb.bridgeLaplace

BrinKarcherAveraging structure

Generic spherical disintegration, parameterized by the frame measure. · OAI.PositiveEinsteinFour.Geometry.BrinKarcherAveraging

BruhatLE def

The actual strong Bruhat order, not the weak order. · OAI.KLInvariance.BruhatLE

BruhatStep def

An upward edge in the strong Bruhat graph, with left reflection labels. · OAI.KLInvariance.BruhatStep

BrunnMinkowskiRoot def

The dimension-induction predicate, used only with a proved lower-dimensional case. It is not an extra hypothesis on the projection-volume main theorem. · OAI.PettyProjection.BrunnMinkowskiRoot

Bud structure

The finite path-kernel: after q name val = next_raw_name + 1, zero for no edge. Modes 0 = static, 1 = opaque stationary, any other = mobile. Every operation-dependent guard is generated in LF before calling a lower query. · OAI.ExactDerandomization.Swirl.Bud

Budget structure

Budget laws are used only where the unreduced concatenation has bounded length. They are exactly the totality/length/context interface of the source. · OAI.FiniteAlgebraicEnvelopes.Compiler.RawRuns.Budget

budget def

A charging potential for the recursive separator construction. The constant is deliberately inessential: only existence of K is required. · OAI.HyperbolicObstruction.PlanarRecursion.budget

budget def

Extend the nonroot budget by zero at the root, which has no detector. · OAI.ReflexiveFixedPoints.SwitchingTreeData.budget

Budgeted def

Each completed stage releases scratch space and retains its bounded seed. · OAI.LowSpaceSubsetSum.WordSeedMeasured.Budgeted

budgeted def

No terminal allocation occurs before its guard succeeds. · OAI.LowSpaceSubsetSum.StoredTerminalCells.budgeted

budgetPolynomial def

A polynomial representing the exact elementary block-map cost. · OAI.ContingencyTables.LiteralProgram.budgetPolynomial

BudgetProfiles structure

The numerical part of the local mean budgets, without the map or scale. · OAI.ClosedSurfaceR4.PhaseMean.BudgetProfiles

BufferAbsorptionCorner structure

The exact corner data used by the absorption formula. · OAI.WeakPureInfiniteness.BufferAbsorptionCorner

bufferedAbsorptionValue def

The exact target after n buffered absorptions. · OAI.WeakPureInfiniteness.bufferedAbsorptionValue

bufferedIterationUnitary def

The actual unitary recursion used in buffered absorption. · OAI.WeakPureInfiniteness.bufferedIterationUnitary

build def

Unfold actual finite successor arrays. Local progress makes the construction terminate even when earlier edges also descend physically. · OAI.EditApproximation.DependencyExpansion.build

build def

The source's factor-and-recurse producer, with a depth cap. The Finsupp is only a representation of the multiplicities in the returned prime list; primeRecord never computes an unknown prime factorization. · OAI.ExactQuantumFactoring.DataProducer.build

build def

One complete preparation and initialization; a is evaluated once. · OAI.LowSpaceSubsetSum.PreparedReaderContext.build

build def

Actual executable output table. Its validity proof is erased at execution. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Table.build

build def

The final finite gadget, with K embedded as its actual shift space. · OAI.MaxCutGames.Gadget.EnlargementConstruction.build

build def

Actual executable output table. Its validity proof is erased at execution. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Table.build

build def

The final finite gadget, with K embedded as its actual shift space. · OAI.UniqueGamesTheorem.Gadget.EnlargementConstruction.build

buildIntegerCellPartition def

A fully constructed partition of the actual integers in the selected log cells. Separation is a numerical/residue property, not an assertion about the distribution of integers. · OAI.Ostmann.buildIntegerCellPartition

buildMovingGiantTree def

small contains only the fixed regular factors at the top of this history. Bulk values and the two moving giants do not occur in this constructor. · OAI.Ostmann.buildMovingGiantTree

buildMovingSlotData def

This carries the source's compensation product into both child regular lists while retaining the per-leaf distribution of the four samples. · OAI.Ostmann.buildMovingSlotData

buildPrimeCellPartition def

A fully constructed partition of the actual primes in the selected log cells. Separation is a numerical/residue property, not an assertion about the distribution of primes. · OAI.Ostmann.buildPrimeCellPartition

bulkResidueEquiv def

Simultaneous CRT on all bulk slots, transposed into one frequency block and one block for each spectator. · OAI.Ostmann.bulkResidueEquiv

bulkResidueModuli def

The only moduli remaining after the internal line comparison. · OAI.Ostmann.bulkResidueModuli

bumps def

A fixed sequence of positive-radius smooth normalized mollifiers. · OAI.Bernoulli.SobolevDensity.bumps

Bundle structure

Graphs of different finite sizes inhabit one fixed type for iteration. · OAI.MaxCutGames.Foundations.PCP.FiniteGraph.Bundle

Bundle structure

Graphs of different finite sizes inhabit one fixed type for iteration. · OAI.UniqueGamesTheorem.Foundations.PCP.FiniteGraph.Bundle

bundleAdjoint def

The actual fiberwise adjoint, with finite-dimensionality supplied by the original bundle rather than a new hypothesis. · OAI.Kaplansky.bundleAdjoint

bundleAt def

Inward (false) and outward (true) bundles with the central edge at -1. · OAI.InfiniteMatroidCounterexample.bundleAt

bundleBody def

Source Bundle is a fixed data term; no non-dummy proof enters its code. · OAI.PureTypeSystem.Labelled.DoubleCode.bundleBody

BundleChart abbrev

The part of the associated total space above that affine chart. · OAI.Nagata.W16.AssociatedBundle.BundleChart

bundleChartFrame def

The ordinary frame obtained from a linear trivialization. · OAI.Kaplansky.bundleChartFrame

bundleComponent def

Fiber coordinates, used only on the corresponding base chart. · OAI.ClosedSurfaceR4.FiniteOrderSmoothing.SmoothingAtlas.bundleComponent

bundleCurvatureAction def

Curvature acting on the matrix bundle in the Ric=3g normalization. The row M r s is the Weyl block in the same fixed frame. · OAI.PositiveEinsteinFour.TraceHessian.bundleCurvatureAction

bundleCurvatureDerivative def

Differentiate the matrix coefficient in the bundle curvature action, keeping the tensor acted on fixed in a parallel frame. · OAI.PositiveEinsteinFour.Geometry.bundleCurvatureDerivative

bundleFiniteForm def

Bundle an actual finite-dimensional form using its exact normalized basis expansion. Forgetting continuity returns the original form. · OAI.Mahler.bundleFiniteForm

bundleMapCoordinates def

Coordinates of genuine fiber maps in the constructed orthonormal frames. · OAI.Kaplansky.bundleMapCoordinates

bundleMatrixPolynomial def

The intrinsic finite matrix restriction. Its continuity is proved from actual normalized bundle maps and adjoints, including the constant term. · OAI.Kaplansky.bundleMatrixPolynomial

bundleOrthonormalFrame def

The actual orthonormal frame on the chart, extended by zero outside. · OAI.Kaplansky.bundleOrthonormalFrame

bundleParameterIndex def

First all first samples, then all second samples; each block is grouped by bundle before the generator position. · OAI.PolycyclicRecognition.bundleParameterIndex

bundleSectionHomotopy def

The genuine global affine homotopy of two continuous bundle sections. It permits intermediate degenerate zeros. · OAI.Kaplansky.bundleSectionHomotopy

bundleUnitaryTransition def

Transition from the orthonormalization of chart a to that of chart b. · OAI.Kaplansky.bundleUnitaryTransition

BundleVector structure

Representatives of the weight-k associated bundle. · OAI.Nagata.W16.AssociatedBundle.BundleVector

BundleWordValue structure

One monomial restricted to the bottom level, as an actual continuous bundle map into its single output level. · OAI.Kaplansky.BundleWordValue

butterflies def

One butterfly for each of m pairs: charge m scalar multiplications, m additions, and m subtractions, retaining the scaled terms for both output signs. · OAI.ExactFourier.LinearDAG.butterflies

Butterfly def

Two independently routed half-sized butterflies, followed by the final layer of first-coordinate pair switches. Coordinates are named in the reverse of their editing order, as in the manuscript's recursive proof. · OAI.CubeShuffle.Butterfly

butterflyCoinEquiv def

Thus the bit experiment and independent recursive butterfly experiment are exactly the same finite probability space, up to a bijection. · OAI.CubeShuffle.butterflyCoinEquiv

bX def

Quadric variables x,y,z,u, in this order. · OAI.StableCoordinate.Obstruction.bX

bypass def

Fix the punctured point and bypass it in its original cycle. Restricting this permutation to the complement literally erases that indexed corner. · OAI.HyperbolicObstruction.PictureSwitch.bypass

bypass def

Delete an occurrence by bypassing it in its boundary cycle. · OAI.TorsionFreeZeroDivisors.DeleteCycle.bypass