Glossary: S

Definitions in this project whose names start with S. 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

S def: The manuscript's S(ℓ) = sech² ℓ. · OAI.BoltzmannNonuniqueness.JetAxialWeight.S
S def: Spatial form, specified by inverse width · OAI.FourierLLogL.Dyadic.Thermal.S
S def: The coefficient shift S and scalar mixer J in the actual regular Hilbert space. · OAI.KervaireMixing.S
S def: Actual finite-dimensional modules over the fixed selecting algebra. · OAI.LittleFinitistic.FiniteSelectionFamily.S
S abbrev: Exactly the thick closure of Cone(s), Cone(s'). · OAI.LittleFinitistic.SourceVerdier.S
S abbrev: The least thick subcategory containing these two cones. · OAI.LittleFinitistic.TwoConeVerdier.ConePair.S
S def: The excess F s - 2 * s, used for s < L, is extended by zero to negative arguments. · OAI.SoftChannel204.S
s def: Pullback of the source element s_B under x = ab, y = dc. · OAI.StableCoordinate.Bundle.s
Safe def: Every writing subinstruction preserves the bottom track. · OAI.FiniteAlgebraicEnvelopes.MachineRealization.TapeSafety.Safe
SafeSampleClaim def: The source safety guarantee, including the literal constants, all listed groups, global independent comparison set, and the harmless unlisted zero case. · OAI.MatroidProphet.SafeSampleClaim
SafeTaggedTuple def: Enough vacant slots for independent one-step displacements of every tag. · OAI.TraceReconstruction.SafeTaggedTuple
safetyFactor def: A concrete fixed safety factor depending only on the triangle. · OAI.TriangularBilliards.Triangle.safetyFactor
Same def: Eventual equality forced by one Cohen condition. Every comparison is of actual names at their chain diagrams. · OAI.PartitionConsistency.ProductForcing.ChainCode.Same
sameAddress def: Exact local address comparison. A loop identifies the two endpoint tables; the joint edge table is always disjoint from every endpoint table. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Relations.sameAddress
sameAddress def: Exact local address comparison. A loop identifies the two endpoint tables; the joint edge table is always disjoint from every endpoint table. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Relations.sameAddress
SameBase def: Equality of the three quadric base coordinates. · OAI.StableCoordinate.FrameLift.SameBase
sameColorGapTuples def: Cyclic starting points and the first two positive gaps in the source count. · OAI.Paper170.sameColorGapTuples
sameColorResidues def: Residues with the same color after a cyclic shift by k. · OAI.Paper170.sameColorResidues
sameColorResiduesParity def: One parity class of the same-color residues. · OAI.Paper170.sameColorResiduesParity
SameCoordinateFlag def: Equality of parallel identified flags in a fixed split basis: not only preservation of dimensions, but equality of each submodule and each induced coordinate projection on the named grade. · OAI.IntegralCharacterVarieties.SameCoordinateFlag
SameFramedFlag def: Equality includes the quotient maps, not just the underlying flag. · OAI.IntegralCharacterVarieties.SameFramedFlag
sameHighProjection def: Local same-high projection in the ordinary pair-letter coordinate space. · OAI.Saxl.sameHighProjection
SameLengthNoExactOverlaps def: The equivalent fixed-length convention for exact overlaps. · OAI.EntropyRateDimension.Extensions.SameLengthNoExactOverlaps
SamePrefix def: Full permutations describing the same ordered observation prefix. · OAI.MatroidProphet.SamePrefix
samePrimitiveRealCharacter def: Equality on natural arguments compares characters even when their moduli occur with different dependent types. · OAI.Ostmann.samePrimitiveRealCharacter
sameRangeEquiv def: The actual linear change of parameters between two injective presentations. · OAI.MaxCutGames.Appendix.RestrictionTransport.sameRangeEquiv
sameRangeEquiv def: The actual linear change of parameters between two injective presentations. · OAI.UniqueGamesTheorem.Appendix.RestrictionTransport.sameRangeEquiv
sameRootScheduledPairEmbedding def: Two descendant histories in a coefficient square share their root. Their inclusion in the full pair of scheduled histories is injective. · OAI.Ostmann.sameRootScheduledPairEmbedding
sameSiteTwoSplitMass def: The actual same-site two-split law, integrating out the Bernoulli hit. · OAI.JointDickman.sameSiteTwoSplitMass
sameSiteTwoSplitProductMass def: Product law of the actual pairs of fair splits. · OAI.JointDickman.sameSiteTwoSplitProductMass
SameText def: Full-text translation equivalence. Finite prefixes alone do not identify groups. · OAI.Superstring.Hierarchical.Threading.SameText
sameTwoVal def: Equality of exact dyadic valuations by bounded divisibility tests. The parameter is the word width, not an input-dependent search cap. · OAI.ExactQuantumFactoring.BitArithmetic.sameTwoVal
Sample def: Samples at separate sibling vertices are independent; repeated values are allowed. The full unary parts of each prescribed tree are retained. · OAI.DilutedSpinGlass.PrescribedTree.Sample
Sample abbrev: A sampling outcome specifies precisely the pairs selected. · OAI.TalagrandDiscreteConvexity.Covering.Sample
sample def: Sampling along the straight line through the output point. · OAI.LipschitzHilbert.sample
sample def: Successive conditional sampling, using the last input coordinate at each step. · OAI.Problem358.TriangularRecursion.sample
sampleComplexity def: Natural fixed sample budget; infimum of the empty set is infinity. · OAI.TraceReconstruction.sampleComplexity
sampleCount def: Number of integers x<Q congruent to t modulo d, when t<d≤Q. · OAI.ExactQuantumFactoring.OrderTrial.sampleCount
sampledAgreement def: The actual experiment, before identifying its visible-map marginal with the clean-coordinate law. The two private answer draws are already averaged inside observedAgreement. · OAI.MaxCutGames.Decoder.ActualAdviceSampling.sampledAgreement
sampledAgreement def: The actual experiment, before identifying its visible-map marginal with the clean-coordinate law. The two private answer draws are already averaged inside observedAgreement. · OAI.UniqueGamesTheorem.Decoder.ActualAdviceSampling.sampledAgreement
sampledBlockProducts def: Products of complete sampled blocks, extended by the identity. · OAI.PolycyclicRecognition.sampledBlockProducts
sampledBlockWord def: The actual difference word of one block of the global sample vector. · OAI.PolycyclicRecognition.sampledBlockWord
sampledDirectionalVisitPath def: The root curve uses the normalized interval amplitude. · OAI.PolycyclicRecognition.CartanWordModel.sampledDirectionalVisitPath
sampleDensity def: Density relative to uniform measure of a finite uniformly sampled map. · OAI.TwoPointCorrelations.sampleDensity
sampleDepth def: Anchor and dummy positions use their independent axis seed. · OAI.Problem348.Construction.HostSampling.sampleDepth
sampledFamilies def: Sample occurrences keep their indices, including repeated subsets. · OAI.LowSpaceSubsetSum.SeparatorCheck.sampledFamilies
sampledFamily def: Duplicate unions are identified in the family of sampled generators. · OAI.TalagrandDiscreteConvexity.Covering.sampledFamily
sampledFourierMean def: The actual Boolean product Fourier average, extended by zero at modulus zero only to permit a single function on all natural moduli. · OAI.Problem337.RandomProducts.sampledFourierMean
sampledKernel def: Before freezing the slow coordinate, the exact rescaled kernel density. · OAI.ThreeTorus.sampledKernel
sampledMean def: Total version of the normalized sampled character average. · OAI.Problem337.RandomProducts.sampledMean
SampledPivotHistory structure: Actual integer products together with their exact unit residues. · OAI.Ostmann.SampledPivotHistory
sampledProbeProfile def: This is a continuous map on the whole Cartan chart, before restricting to the compact set chosen for all three paths. · OAI.PolycyclicRecognition.CartanFactorModel.sampledProbeProfile
sampledProbeSuffix def: The original sample is multiplied by the actual probe prefix. · OAI.PolycyclicRecognition.CartanFactorModel.sampledProbeSuffix
sampledRemainder def: The least nonnegative remainder used by numerator descent, for an actual labelled prime sample and an actual Boolean choice. · OAI.Problem337.RandomProducts.sampledRemainder
sampledResidueKernel def: Exact progression j=h+kP; no divisibility restriction on the passage integer. · OAI.ThreeTorus.sampledResidueKernel
sampledResponse def: The response obtained from an ordered sample of columns. · OAI.JointDickman.sampledResponse
sampledSelection def: The three-seed selection in the concrete matrix of the prescribed size. · OAI.Problem348.OrderedHost.sampledSelection
sampledTrial def: The zero-target wrapper supplies only sampling spaces. The execution uses D.input, including its original target. · OAI.LowSpaceSubsetSum.RepresentationData.sampledTrial
sampledWindow def: The actual r-child window at an internal offset of a specified parent. · OAI.SmoothObstruction.WordHierarchy.sampledWindow
SampleEqualityPattern abbrev: A pattern records exactly which sampled positions coincide. Its class values are distinct; independence is imposed only on their dominating product prior. · OAI.Ostmann.SampleEqualityPattern
sampleEqualityPatternEquiv def: Every sample has a unique equality pattern and a unique injective assignment of values to its classes. · OAI.Ostmann.sampleEqualityPatternEquiv
sampleEquiv def: Relabel every sampled vertex without identifying distinct replica paths. · OAI.DilutedSpinGlass.PrescribedTree.sampleEquiv
SampleInput abbrev: Phase index, relative tolerance, and failure allowance. · OAI.ContingencyTables.CountingOracleScheduleProgram.SampleInput
sampleLaw def: Every atom records exactly the finite random choices made by the long-code Max-Cut test. In particular no real random numbers occur in the construction. · OAI.OptimalMaxCut.LongCode.Game.sampleLaw
sampleOffset def: Offset and slope are rational values of polynomial bit length, evaluated using the arithmetic nearest-bin rule rather than any transcendental oracle. · OAI.ExactQuantumFactoring.OrderTrial.sampleOffset
samplePatternLabels def: The class coordinates used by the finite independent-prior comparison. · OAI.Ostmann.samplePatternLabels
sampleProgram def: Retain the modular value register while interfering the exponent register; no copy of the exponent or hidden measurement is inserted. · OAI.ExactQuantumFactoring.sampleProgram
sampler def: Compression draws one permutation and one prime. Its residue counter is enumerated deterministically by compressionTrial. · OAI.LowSpaceSubsetSum.CompressionData.sampler
samplerCandidateBudget def: Upper tally of the random candidates read, including all four possible prime calls and one direct Boolean draw for each sign coordinate. · OAI.LowSpaceSubsetSum.RepresentationSetup.samplerCandidateBudget
SamplerCoupling structure: An explicit common probability space for an ideal result and a bounded result. Every nonfailure agrees exactly; failure remains part of the law. · OAI.LowSpaceSubsetSum.SamplerCoupling
samplerDivisorBudget def: Each term is the all-attempts bound for the counted primeRun. · OAI.LowSpaceSubsetSum.RepresentationSetup.samplerDivisorBudget
samplerLaw def: The real pushforward of the uniform law on sampler outcomes. · OAI.MaxCutGames.Decoder.NoiseSampler.samplerLaw
samplerLaw def: The real pushforward of the uniform law on sampler outcomes. · OAI.UniqueGamesTheorem.Decoder.NoiseSampler.samplerLaw
sampleRoot def: The root element selected by the independent coordinate-cube sample. · OAI.PolycyclicRecognition.CartanFactorModel.sampleRoot
samplerPrimeWidths def: The low factors and the extra prime, in their sampling order. Factors whose width is below four use the constant one and incur no prime test. · OAI.LowSpaceSubsetSum.RepresentationSetup.samplerPrimeWidths
samplerSelectedNet def: Full selection compares retained read-only inputs and every sampler scratch bit, not only the apparent phase and modular residue fields. · OAI.ExactQuantumFactoring.OrderTrial.samplerSelectedNet
SampleSelected def: Selected phase block, with every workspace bit accounted for. The input copies are compared to their retained read-only values. · OAI.ExactQuantumFactoring.OrderTrial.SampleSelected
sampleWords def: All words of a fixed length over a finite alphabet. · OAI.Problem337.RandomProducts.sampleWords
SamplingConflict def: Failure of the full symmetric bounded-frequency separation condition. · OAI.FiniteFactor.SamplingConflict
samplingDistance def: Distance of a representative to the two ends of the unit interval. · OAI.FiniteFactor.samplingDistance
samplingHeightDomain def: The open profile-height domain retains the strict facet inequalities. · OAI.PolycyclicRecognition.samplingHeightDomain
samplingIdealSecond def: Actual averaged density of the uncut diagonal transport. · OAI.FiniteFactor.samplingIdealSecond
samplingKernel def: The finite Fejer kernel, as a nonnegative real function. · OAI.FiniteFactor.samplingKernel
samplingOrientedWeight def: Both source orientations and both coordinate types, with exact x/y tests. · OAI.FiniteFactor.samplingOrientedWeight
samplingParameter def: One admissible positive parameter sequence. Later diagram restrictions may be imposed by choosing smaller parameters before each finite selection. · OAI.HyperbolicObstruction.IncidenceTemplate.samplingParameter
samplingPolytope def: The actual normalized halfspaces used to sample the root ideal. · OAI.PolycyclicRecognition.CartanWordModel.samplingPolytope
samplingPrecision def: Safely locate the precision field after the two margin vectors. · OAI.ContingencyTables.ProfilePrograms.samplingPrecision
samplingSecondLevelLeft def: The two closed intervals representing a positive superlevel of the second-oriented tested reciprocal. Endpoint changes are dealt with separately. · OAI.FiniteFactor.samplingSecondLevelLeft
SamplingSequence structure: Only combinatorial sampling assertions occur here, independent of the action that may be chosen after the random tables have been exposed. This is not a replacement for the random-table producer. · OAI.HyperbolicObstruction.SamplingSequence
SamplingStage structure: All finitely many requested tests and the actual six-letter array at one stage. This is selected below, not an assumption about a desired action. · OAI.HyperbolicObstruction.IncidenceTemplate.SamplingStage
samplingTau def: One concrete choice satisfying all three weight margins. · OAI.JointDickman.samplingTau
samplingTestedSecondWeight def: Exact x- and y-tests in the second diagonal orientation. · OAI.FiniteFactor.samplingTestedSecondWeight
samplingTransport def: The uncut diagonal model moves z by v times its endpoint distance. · OAI.FiniteFactor.samplingTransport
samplingTransportDensity def: Exact density of the two affine branches of the uncut transport. · OAI.FiniteFactor.samplingTransportDensity
sandwichShuffle def: Exact rearrangement, not a resampling assumption. · OAI.CubeShuffle.sandwichShuffle
SardSurfaceInput def: Sard, The measure of the critical values of differentiable maps, Bull. AMS 48 (1942), Theorem 4.1, p.885: the smooth equal-dimensional surface case in coordinates. This is the sole critical-value input needed for the manuscript's … · OAI.ClosedSurfaceR4.PublishedInputs.SardSurfaceInput
Satisfiable def: `F.Satisfiable` means there exists a total Boolean assignment making each clause occurring in `F.clauses` evaluate to `true`. · OAI.UniqueGamesTheorem.BinaryFormula.Formula.Satisfiable
satisfied def: `constraint.satisfied labeling` is the Boolean decision that the target vertex's label equals the forward image of the source vertex's label. · OAI.UniqueGamesTheorem.Foundations.Target.Constraint.satisfied
Satisfies def: The coupling is on actual master matches. Its branches agree on exactly the prescribed named fields, with no assumptions about numerical cross-field values. · OAI.HyperbolicObstruction.IncidenceTemplate.ConditionalTest.Satisfies
Satisfies def: The displayed finite list of quadratic equations, in an arbitrary k-algebra. · OAI.LittleFinitistic.QuadraticEncoding.Relations.Satisfies
Satisfies def: The stability and dispersion claim of Theorem 3.4 for fixed errors and rank. · OAI.MaxCutGames.Gadget.ActualGadget.Satisfies
Satisfies def: Precisely the finite list of merger relations, interpreted in N. Diagonal entries of start are ignored, as in the definition of Rel. Reflected joins, nonjoins or injectivity are not required in the target. · OAI.PartitionConsistency.DiagramForcing.OverlapPrescription.Satisfies
Satisfies def: The stability and dispersion claim of Theorem 3.4 for fixed errors and rank. · OAI.UniqueGamesTheorem.Gadget.ActualGadget.Satisfies
satisfyingTriple def: Two bits encode the four satisfying answers for an occurrence. · OAI.MaxCutGames.Clean.IncidenceGap.satisfyingTriple
satisfyingTriple def: Two bits encode the four satisfying answers for an occurrence. · OAI.UniqueGamesTheorem.Clean.IncidenceGap.satisfyingTriple
SATTimeBound def: An actual execution bound; no computation is hidden in its interpretation. · OAI.ParityWL.Complexity.SATTimeBound
saturationIndex def: The additive index [O_K M : M]. For a full lattice this is finite. As usual, the index is zero if the quotient is infinite. · OAI.DukePrimeDegree.saturationIndex
saturationLattice def: The maximal-order span, viewed again as an integer lattice. · OAI.DukePrimeDegree.saturationLattice
SB structure: support restricted to active set S' · OAI.ExactDerandomization.Hie.SB
scalar def: The scalar of a natural endomorphism on affine module sheaves. · OAI.AffineNaturalScalar.scalar
scalar def: The ONE genuine smooth scalar anisotropy attached to an infinite stream. · OAI.HarmonicCounterexample.AngularStream.scalar
scalarAEPeriods def: Almost-everywhere periods form an actual additive subgroup. · OAI.KadisonKastler.scalarAEPeriods
scalarAffineDistribution def: Distributional affine pullback, with its change-of-variables Jacobian. · OAI.PositiveEinsteinFour.Geometry.scalarAffineDistribution
scalarAffineTestCLM def: Inverse affine coordinates on test functions. · OAI.PositiveEinsteinFour.Geometry.scalarAffineTestCLM
scalarAlpha def: The actual offset q^(-(D+1)), written using a natural power. · OAI.QuantitativeVanDerWaerden.Parameters.scalarAlpha
scalarBaseline def: The exact parameter-free scalar baseline, produced from the convergent Peano--Baker solution rather than assumed as a Riccati solution. · OAI.HarmonicCounterexample.LinearODE.scalarBaseline
scalarBesselFirstL2 def: First distributional derivative of the normalized inverse, realized in L2. · OAI.PositiveEinsteinFour.Geometry.scalarBesselFirstL2
scalarBesselFirstL4 def: The first derivative of the inverse Bessel operator is a continuous map from L2 to L4 in dimension four. · OAI.PositiveEinsteinFour.Geometry.scalarBesselFirstL4
scalarBesselInverseL2 def: The actual L2 inverse of 1 - (2π)⁻² Δ. · OAI.PositiveEinsteinFour.Geometry.scalarBesselInverseL2
scalarBesselJetL2 def: Every derivative of order at most n of the inverse n-th Bessel operator is continuous from L2 to L2. · OAI.PositiveEinsteinFour.Geometry.scalarBesselJetL2
scalarBesselJetUniform def: Jets leaving at least three Sobolev derivatives are continuous in L∞. · OAI.PositiveEinsteinFour.Geometry.scalarBesselJetUniform
scalarBesselSecondL2 def: Second distributional derivative of the normalized inverse, realized in L2. · OAI.PositiveEinsteinFour.Geometry.scalarBesselSecondL2
scalarBesselThreeUniform def: Three inverse Bessel derivatives give a continuous map from L2 to L∞. · OAI.PositiveEinsteinFour.Geometry.scalarBesselThreeUniform
scalarBlockHom def: The scalar finite-block representation before adjoining the complement. · OAI.KadisonKastler.FullMatrixCorner.scalarBlockHom
ScalarBoundaryMean structure: Scalar equivariant conditional expectation in the section model of L-infinity(G × D). These are the exact structural scalar conclusions of Adams–Elliott–Giordano, Theorem 3.4; existence is not asserted here. · OAI.KadisonKastler.ScalarBoundaryMean
scalarBoundaryModule def: The scalar module action on a bounded family of L-infinity sections. · OAI.KadisonKastler.scalarBoundaryModule
scalarBoundaryStar def: Coordinatewise conjugation on the bounded family of scalar sections. · OAI.KadisonKastler.scalarBoundaryStar
scalarBoundaryTranslate def: Simultaneous inverse translation of the group and boundary coordinates. · OAI.KadisonKastler.scalarBoundaryTranslate
ScalarCharacterIntegrationInput def: The scalar-target specialization of Lee 20.19 and 20.8(g). · OAI.PolycyclicRecognition.ScalarCharacterIntegrationInput
scalarChartReconstruction def: Reconstruction through a compact coordinate set, in actual base volume. · OAI.PositiveEinsteinFour.Geometry.scalarChartReconstruction
scalarClip def: Scalar clipping, equivalently sign(a) * min(|a|,h) when h >= 0. · OAI.ExactModuli.scalarClip
ScalarCoareaIdentity def: Precisely the scalar planar coarea identity, including its required Lebesgue measurability assertion. · OAI.CircleDomainRigidity.ScalarCoareaIdentity
scalarCommutatorNorm def: The scalar commutator seminorm g_P(Y), expressed as the norm of the bounded linear map a ↦ Ya - aY on the represented algebra. · OAI.KadisonSimilarity.scalarCommutatorNorm
scalarComplexAffinePullbackL2 def: Pullback of the complex L2 representative by an invertible affine map. · OAI.PositiveEinsteinFour.Geometry.scalarComplexAffinePullbackL2
scalarCoordinateDerivative def: The actual distributional derivative in one Euclidean coordinate. · OAI.PositiveEinsteinFour.Geometry.scalarCoordinateDerivative
scalarCoordinateJet def: A finite ordered coordinate derivative on tempered distributions. · OAI.PositiveEinsteinFour.Geometry.scalarCoordinateJet
scalarCouplingTerm def: The tested scalar norm equations, trace subtraction and variance term that are added to the two actual Hessian Green integrands. · OAI.PositiveEinsteinFour.QuaternionFrame.scalarCouplingTerm
ScalarCovariantData structure: Raw classical scalar calculus, stated on the actual metric tensors. · OAI.PositiveEinsteinFour.Geometry.ScalarCovariantData
ScalarCutoffNested def: The outer cutoff equals one near the support of the inner cutoff. · OAI.PositiveEinsteinFour.Geometry.ScalarCutoffNested
scalarDiagonal def: Scalar diagonal lift in rank six, exactly the three factors in the source. · OAI.FiniteAlgebraicEnvelopes.Steinberg.scalarDiagonal
scalarDiagonal def: A scalar diagonal on the finite coordinate set. · OAI.KervaireMatrix.scalarDiagonal
ScalarDimensionalBochner def: The scalar weak BE(K,N) inequality on the compact regular class used in this application. The mixed source is interpreted by the actual weak Laplacian; no Sobolev derivative of that source is required. · OAI.RegularPointCoordinates.ScalarDimensionalBochner
scalarDirichletDensity def: The squared norm of the scalar differential on the base. · OAI.PositiveEinsteinFour.Geometry.scalarDirichletDensity
scalarDomain def: The open set of valid prefixes extended by one scalar coordinate. · OAI.Problem358.ConvexTriangularSampler.scalarDomain
scalarDomain def: The two genuinely free scalar parameters are anchor position and drift. · OAI.QuantitativeVanDerWaerden.AnchoredPatterns.scalarDomain
scalarEnergy def: The part of the compression energy fixed by the current block traces. · OAI.Generator.Tracial.CornerPartition.scalarEnergy
scalarEquiv def: The restricted scalar convention in the detector agrees with the original compatible k action by actual tensor maps in both directions. · OAI.LittleFinitistic.PDBarQuotient.scalarEquiv
scalarEquiv def: Recover the exact original scalar module after retaining both actions. · OAI.LittleFinitistic.RetainedBimodule.scalarEquiv
scalarEvaluationIso def: Exact comparison to native module evaluation with its original compatible scalar structure. · OAI.LittleFinitistic.BimoduleTotal.scalarEvaluationIso
scalarExpression def: The scalar expression of an actual polynomial section in the covering frame. · OAI.Nagata.CoefficientSpaces.scalarExpression
scalarFieldIncrements def: The non-root Gaussian increments in the given field step. · OAI.InvariantIsing.scalarFieldIncrements
scalarFlowTranslation def: The L² isometry obtained by translating an actual scalar field along one parallel flow. · OAI.PositiveEinsteinFour.Geometry.scalarFlowTranslation
scalarForm def: Scalar lower bound after controlling the v-term and applying Cauchy–Schwarz. · OAI.SidorenkoCounterexample.scalarForm
ScalarFree def: Scalar freeness, in the alternating-centered-word convention of the paper. · OAI.KadisonSimilarity.ScalarFree
ScalarFreeDiagonal def: Scalar freeness, specialized to the span of each finite diagonal partition. Adjacent copy indices differ; every letter has zero trace. · OAI.KadisonKastler.ScalarFreeDiagonal
scalarH1 def: Intrinsic first-order Sobolev space generated by actual smooth scalars. · OAI.PositiveEinsteinFour.Geometry.scalarH1
scalarH1CutoffGradientL2 def: Assemble the four bounded weak coordinate derivative maps into the ordinary Euclidean vector gradient. · OAI.PositiveEinsteinFour.Geometry.scalarH1CutoffGradientL2
scalarH1CutoffJet def: The compact chart localization of an intrinsic scalar H¹ first jet. · OAI.PositiveEinsteinFour.Geometry.scalarH1CutoffJet
scalarH1LocalizedCoordinate def: The localized weak coordinate derivative, reconstructed in the base volume. · OAI.PositiveEinsteinFour.Geometry.scalarH1LocalizedCoordinate
scalarH1LocalizedFrame def: Actual frame gradient multiplied by a smooth base cutoff. · OAI.PositiveEinsteinFour.Geometry.scalarH1LocalizedFrame
scalarH1Mean def: Integration of the value coordinate, represented by inner product with the constant one and hence continuous on the actual H¹ completion. · OAI.PositiveEinsteinFour.Geometry.scalarH1Mean
scalarH1MeanZero def: The closed subspace on which the actual value coordinate has mean zero. · OAI.PositiveEinsteinFour.Geometry.scalarH1MeanZero
scalarH1MeanZeroProjection def: The mean-subtraction projection with its closed mean-zero codomain. · OAI.PositiveEinsteinFour.Geometry.scalarH1MeanZeroProjection
scalarH1One def: The genuine constant first jet, with zero derivative coordinates. · OAI.PositiveEinsteinFour.Geometry.scalarH1One
scalarH1SubtractMean def: Subtract the constant jet with coefficient equal to the scalar mean. · OAI.PositiveEinsteinFour.Geometry.scalarH1SubtractMean
scalarHalfNestedBumpSchwartz def: Complexification turns the actual compact real bump into a Schwartz cutoff. · OAI.PositiveEinsteinFour.Geometry.scalarHalfNestedBumpSchwartz
scalarHalfNestedRealBump def: A concrete real bump; the gap shrinks geometrically with the index. · OAI.PositiveEinsteinFour.Geometry.scalarHalfNestedRealBump
scalarHom def: Scalars in the block are k b. In particular the identity is b. · OAI.Donovan.BlockIdempotent.scalarHom
scalarInfinityConst def: Constant scalar functions as actual L-infinity classes. · OAI.KadisonKastler.scalarInfinityConst
scalarInfinityGlue def: An actual L-infinity section obtained by bounded countable gluing. · OAI.KadisonKastler.scalarInfinityGlue
scalarInfinityIndicator def: The scalar characteristic function of a measurable set. · OAI.KadisonKastler.scalarInfinityIndicator
scalarInfinityMul def: Pointwise multiplication descends to L-infinity classes. · OAI.KadisonKastler.scalarInfinityMul
ScalarInvariantMean structure: The usual scalar invariant-mean data, with no assertion of existence. · OAI.KadisonKastler.ScalarInvariantMean
ScalarJetL2 abbrev: The value and four derivative coordinates, in a genuine Hilbert product. · OAI.PositiveEinsteinFour.Geometry.ScalarJetL2
ScalarKaplanskyClosure def: The scalar specialization of matrix Kaplansky density and the C*-quotient norm used in the final derivation argument (Blackadar, I.9.1.3). · OAI.KadisonSimilarity.ScalarKaplanskyClosure
scalarKernel def: The bounded sesquilinear kernel of a representative. · OAI.KadisonKastler.WeakOperatorRepresentative.scalarKernel
scalarKernelEquiv def: A kernel which is exactly a regular scalar multiple of its domain is integrally isomorphic to that domain, not just to its fraction-field span. · OAI.KLInvariance.ReverseFiltration.scalarKernelEquiv
scalarKernelMap def: Multiplication by the actual nonzero label identifies the kernel of coordinate reduction with a free module. · OAI.KLInvariance.ReverseFiltration.scalarKernelMap
scalarL2FourierMultiplier def: A bounded Fourier symbol acts continuously on the actual L2 space. · OAI.PositiveEinsteinFour.Geometry.scalarL2FourierMultiplier
scalarL2Multiply def: Physical multiplication by an essentially bounded coefficient. · OAI.PositiveEinsteinFour.Geometry.scalarL2Multiply
scalarL2One def: The constant one in scalar L². · OAI.PositiveEinsteinFour.Geometry.scalarL2One
scalarL2Perturbation def: The lower-order perturbation after composing on the right with Bessel(-2). · OAI.PositiveEinsteinFour.Geometry.scalarL2Perturbation
scalarL2TestPairing def: The bilinear, rather than Hermitian, L2 pairing used by distributions. · OAI.PositiveEinsteinFour.Geometry.scalarL2TestPairing
scalarLaplacianCommutator def: The part created by commuting a smooth cutoff with the Laplacian. · OAI.PositiveEinsteinFour.Geometry.scalarLaplacianCommutator
scalarLinePolynomial def: Polynomial obtained by evaluating a multivariate polynomial on a scalar line. · OAI.Nagata.Workers.W25.scalarLinePolynomial
scalarLocalSourceSchwartz def: A smooth source restricted by a compact Schwartz cutoff is Schwartz, even if it has no regularity or growth control outside the cutoff. · OAI.PositiveEinsteinFour.Geometry.scalarLocalSourceSchwartz
scalarLowJetSubstitution def: The smooth coefficient substitution needs no separate position input because an actual low jet already stores its base point. · OAI.ClosedSurfaceR4.JetPolynomial.scalarLowJetSubstitution
scalarLpRepresentation def: Multiplication by bounded continuous scalar functions on actual L². · OAI.WeakPureInfiniteness.scalarLpRepresentation
scalarMatrixAlgebra def: The scalar matrix algebra, as the commutant of diagonal corner operators. · OAI.KadisonKastler.FullMatrixCorner.scalarMatrixAlgebra
scalarMatrixCopy def: The one-by-one matrix copy of the algebra. · OAI.KadisonKastler.scalarMatrixCopy
scalarMatrixInitialProjection def: The initial diagonal projection in the scalar matrix algebra. · OAI.KadisonKastler.FullMatrixCorner.scalarMatrixInitialProjection
scalarMeanZeroDirichlet def: The bounded Dirichlet bilinear form on the actual mean-zero space. · OAI.PositiveEinsteinFour.Geometry.scalarMeanZeroDirichlet
scalarMeanZeroGradient def: Restrict each of the four actual scalar derivatives to that subspace. · OAI.PositiveEinsteinFour.Geometry.scalarMeanZeroGradient
scalarMeanZeroValue def: Restrict the scalar value map to the mean-zero Hilbert subspace. · OAI.PositiveEinsteinFour.Geometry.scalarMeanZeroValue
scalarMeanZeroWeakPoisson def: The unique mean-zero weak solution, as a continuous operator on L². Its sign convention is the positive Dirichlet operator -Δ. · OAI.PositiveEinsteinFour.Geometry.scalarMeanZeroWeakPoisson
scalarMeshSize def: The actual number of positive radial mesh intervals. · OAI.QuantitativeVanDerWaerden.Parameters.scalarMeshSize
scalarMetricErrorCoefficient def: An explicit coefficient for the cumulative metric error after using the local-mass and weighted-defect budgets. · OAI.RegularPointCoordinates.scalarMetricErrorCoefficient
scalarModule def: The canonical field action on a left D-module. · OAI.DerivedBar.BimoduleTensor.scalarModule
scalarNestedBumpSchwartz def: Complexification turns the actual compact real bump into a Schwartz cutoff. · OAI.PositiveEinsteinFour.Geometry.scalarNestedBumpSchwartz
scalarNestedRealBump def: A concrete real bump; the gap shrinks geometrically with the index. · OAI.PositiveEinsteinFour.Geometry.scalarNestedRealBump
ScalarOperatorKernel structure: The bounded sesquilinear coefficient data from which an operator field is reconstructed. The first vector is the input to the operator. · OAI.KadisonKastler.ScalarOperatorKernel
ScalarParallelData structure: Only differentiation along the chosen parallel flow remains input. Regularity, symmetry and the Ricci commutator follow from the metric. · OAI.PositiveEinsteinFour.Geometry.ScalarParallelData
ScalarPartition structure: A finite set of chart centers with subordinate smooth weights. · OAI.PositiveEinsteinFour.Geometry.ScalarPartition
scalarPerturbationEquiv def: Neumann inversion of Id + K in the actual Banach space. · OAI.PositiveEinsteinFour.Geometry.scalarPerturbationEquiv
scalarPerturbedCutoffCommutator def: The commutator with a cutoff, for the actual differential operator. · OAI.PositiveEinsteinFour.Geometry.scalarPerturbedCutoffCommutator
scalarPerturbedOperator def: The actual variable-coefficient operator on tempered distributions. · OAI.PositiveEinsteinFour.Geometry.scalarPerturbedOperator
scalarPhase def: Generic initial data for the actual scalar second-order propagator. · OAI.HarmonicCounterexample.LinearODE.scalarPhase
scalarPhase def: The unit complex phase of a weight. · OAI.LipschitzHilbert.scalarPhase
scalarPhaseTwist def: Scalar multiplication of a unitary by the chosen finite phase. · OAI.KadisonKastler.scalarPhaseTwist
scalarPhaseUnitary def: The scalar root-of-unity phase, as a unitary in any unital complex C-star algebra. · OAI.KadisonKastler.scalarPhaseUnitary
scalarPowerCharacter def: The characters attached to the first positive powers. · OAI.KadisonKastler.scalarPowerCharacter
scalarPrice def: The common scalar price assigned to a root. The geometric discount is applied at the actual birth scale of each zero-mass error. · OAI.DyadicHats.scalarPrice
scalarPrice def: The common scalar price assigned to a root. The geometric discount is applied at the actual birth scale of each zero-mass error. · OAI.SignedDisk.DyadicHats.scalarPrice
scalarProductAtom def: Every ordered derivative of the scalar product is a finite expression. Repeated formal slow differentiation supplies the exact Leibniz expansion. · OAI.ClosedSurfaceR4.JetPolynomial.scalarProductAtom
scalarProductPullback def: Substitute the complete finite jet polynomial of the fixed scalar product, including positional low-jet coefficients. · OAI.ClosedSurfaceR4.JetPolynomial.Expression.scalarProductPullback
ScalarProfile structure: The scalar profile hypotheses used in Section 6. Only positive indices carry mathematical content; the values on either coordinate axis are unused. · OAI.MatrixMultiplication.AuxiliarySeparation.ScalarProfile
scalarProjectionHom def: Scalars acting on a projection corner form a nonunital star representation. · OAI.KadisonKastler.scalarProjectionHom
scalarRadius def: The logarithmic radius of the exponential mesh. · OAI.QuantitativeVanDerWaerden.Parameters.scalarRadius
scalarRecipe structure: Conditions for one orientation of roles, §5 (scalar recipe). · OAI.HadwigerCounterexample.Volley.scalarRecipe
ScalarRecurrence def: The explicit recurrence obtained from a 15*sqrt embedding error. · OAI.MaxCutGames.Foundations.Repetition.ScalarRecurrence
ScalarRecurrence def: The explicit recurrence obtained from a 15*sqrt embedding error. · OAI.UniqueGamesTheorem.Foundations.Repetition.ScalarRecurrence
scalarRepresentation def: Multiplication on C identifies it with its full bounded-operator algebra. · OAI.KadisonKastler.scalarRepresentation
scalarSchwartzBesselInverse def: The negative-two Bessel multiplier preserves the actual test space. · OAI.PositiveEinsteinFour.Geometry.scalarSchwartzBesselInverse
scalarSchwartzHelmholtz def: The normalized constant-coefficient operator on test functions. · OAI.PositiveEinsteinFour.Geometry.scalarSchwartzHelmholtz
scalarSchwartzMultiplyL2 def: Multiplication by a Schwartz coefficient, as a physical L2 operator. · OAI.PositiveEinsteinFour.Geometry.scalarSchwartzMultiplyL2
scalarSchwartzTranspose def: The transpose includes the derivatives of the coefficients. · OAI.PositiveEinsteinFour.Geometry.scalarSchwartzTranspose
scalarSeries def: Scalar series embedded as scalar matrices at each coefficient. · OAI.RationalHitting.HahnMatrix.scalarSeries
scalarShiftedSolution def: The weak solution of the shifted scalar equation. Its construction is the Hilbert adjoint of value evaluation, so needs no elliptic premise. · OAI.PositiveEinsteinFour.Geometry.scalarShiftedSolution
scalarTensor def: A scalar matrix tensored with the identity of a line fiber. This formula uses only scalar multiplication, not a choice of a vector in that line. · OAI.Kaplansky.scalarTensor
scalarTensor def: Scalar matrices embedded without any finite/countable assumption on the group. · OAI.KervaireMatrix.scalarTensor
scalarTensor def: The direct sum of r scalar multiplication tensors, in diagonal coordinates. · OAI.MatrixMultiplication.AuxiliarySeparation.scalarTensor
scalarTensorEquiv def: Scalar tensor multiplication is right B-linear with the actual external action. · OAI.LittleFinitistic.CornerTensor.scalarTensorEquiv
scalarTensorEquiv def: Scalar restriction coherence for an arbitrary right tensor factor. · OAI.LittleFinitistic.PDResidueAssociativity.scalarTensorEquiv
scalarTensorIso def: Canonical restriction and the native compatible k action give the same B-valued tensor. · OAI.LittleFinitistic.BimoduleTotal.scalarTensorIso
scalarTrace def: The normalized trace on the scalar algebra. · OAI.KadisonKastler.scalarTrace
scalarUnitaryOfNormOne def: The scalar phase supplied by the invariant vector defines a unitary scalar operator. · OAI.KadisonKastler.scalarUnitaryOfNormOne
ScalarVarianceInputs structure: Only the precise classical inputs of the smooth scalar argument. · OAI.PositiveEinsteinFour.Geometry.ScalarVarianceInputs
scalarVonNeumann def: The full operator algebra on the one-dimensional Hilbert space. · OAI.KadisonKastler.scalarVonNeumann
scalarWalkProduct def: Multiply departure-indexed edge coefficients along the actual integer walk. · OAI.TwoPointCorrelations.scalarWalkProduct
scalarWeakResolvent def: The value of the weak shifted solution, as an operator on scalar L². · OAI.PositiveEinsteinFour.Geometry.scalarWeakResolvent
scalarWeightedH1 def: The closed image of actual smooth first jets, with the Sobolev energy norm. · OAI.PositiveEinsteinFour.Geometry.scalarWeightedH1
scalarWeightedH1Equiv def: The weighted energy space is continuously linearly equivalent to actual H¹. · OAI.PositiveEinsteinFour.Geometry.scalarWeightedH1Equiv
scalarWeightedH1ToH1 def: Continuous recovery of the original first jet, including its value. · OAI.PositiveEinsteinFour.Geometry.scalarWeightedH1ToH1
scalarWeightedJetEquiv def: Invertible diagonal rescaling of the actual scalar first jet. · OAI.PositiveEinsteinFour.Geometry.scalarWeightedJetEquiv
scale def: The annular scale s₀ exp(-jη). · OAI.BoltzmannNonuniqueness.DatumBounds.scale
scale def: The bounded scaling map on integer proposals. · OAI.ContingencyTables.FairBits.scale
scale def: The scale at index j has exponent j + 1. · OAI.DoublingHilbert.scale
scale def: Nonnegative scalar multiplication of a positive finite matrix. · OAI.Kaplansky.CuntzPositiveMatrix.scale
scale def: Every positive dilation is again the exact relative gauge. · OAI.PettyProjection.RelativeGauge.scale
scale def: The square of the first-moment factor at a layer. · OAI.Problem335.LocalMomentWeights.scale
scale def: The integer number of prime factors used at scale S. · OAI.Problem337.ResidueLevels.scale
scale def: Fiber scaling: the positive-weight column is multiplied by a unit and the negative-weight row by its inverse. · OAI.StableCoordinate.FrameLift.Frame.scale
scaleA def: The exact real scale exponents in the scale inequalities; natural indices are zero-based. · OAI.TraceReconstruction.scaleA
scaleBetween def: Scale frame vectors when the metric matrices have the inverse square relation. · OAI.PositiveEinsteinFour.Geometry.OrthonormalFrame.scaleBetween
Scaled def: Scaled S f = f is at most a constant multiple of S+1 at every argument. · OAI.Fifths.Scaled
scaled def: All structural properties survive the actual scalar homothety. · OAI.ReflexiveFixedPoints.MinimalInvariantData.scaled
scaledBranchKernel def: The manuscript kernel K(s,y), including both signs of s. · OAI.ThreeTorus.scaledBranchKernel
scaledCoordinateEquiv def: Include the physical inverse radius in the invertible coordinate change. · OAI.RegularPointCoordinates.scaledCoordinateEquiv
scaledDensity def: Density under multiplication by a nonzero real scale, including negative scales. · OAI.ThreeTorus.scaledDensity
ScaledGraph structure: The positive integer scale Q is part of the output. It need not equal the number of graph edges. · OAI.OptimalMaxCut.ScaledGraph
scaledGridPoint def: Grid centers with mesh s⁻¹. A common fixed translation cancels in all pair differences and so does not enter the counting argument. · OAI.Falconer.scaledGridPoint
ScaledImages def: Equality of embedded images with a scalar on the upper image. · OAI.KLInvariance.MomentGraph.Sheaf.ScaledImages
scaledKL def: The manuscript's q^{-d/2} P(q), written with t^2=q so no square-root branch is hidden. This is an evaluation of the already constructed genuine KL polynomial, not a new combinatorial invariant. · OAI.KLInvariance.scaledKL
scaledMap def: Conjugate an actual selfmap by the scalar bijection. · OAI.ReflexiveFixedPoints.scaledMap
scaledOperator def: The scaled amplitude of the literal differential expression; the PD operators are already identified with true scaled first/second derivatives. · OAI.ElasticityBoundarySmoothOperator.scaledOperator
scaledPolynomialMap def: Homogeneous rescaling of the literal section polynomial. · OAI.SectionCompletion.scaledPolynomialMap
scaledPotential def: Compact radial potential centered at a with positive radius r. Its scale derivative produces an averaging kernel by the radial Laplacian. · OAI.Bernoulli.HarmonicKernel.scaledPotential
scaledReproducingKernel def: The normalized spatial dilation has the correct scale and mass factor for frequency cutoffs at scale 1/r. · OAI.LipschitzHilbert.scaledReproducingKernel
scaledSeparatorPolynomial def: Polynomial before translation, with the scaling denominator cleared. · OAI.DefocusingNLS.SeparatorArithmetic.scaledSeparatorPolynomial
ScaledSliceWidth def: Width after positive dilation and arbitrary vertical translation. · OAI.Nagata.W04.ScaledSliceWidth
scaledSubtypeEquiv def: The actual homothety between a set and its nonzero scalar image. · OAI.ReflexiveFixedPoints.scaledSubtypeEquiv
scaledThetaPositive def: One half of the theta product, after substituting a shrinking center. · OAI.Nagata.W22.scaledThetaPositive
scaledTransportWeight def: Multiplying the annular derivative by the physical radius preserves coercivity when the working balls shrink. · OAI.RegularPointCoordinates.scaledTransportWeight
scaledYRep def: The actual scale used by the second representative system. · OAI.QuantitativeVanDerWaerden.scaledYRep
scaleE def: Multiplication of the argument of a character. · OAI.PlaneFiveColor.Spectral.scaleE
scaleH1 def: Scalar multiplication in the weak form domain, without choosing pointwise representatives. · OAI.SharpLiebThirring.scaleH1
ScaleInvariant def: Positive scalar invariance is a dimension property, not spectral homogeneity. · OAI.Kaplansky.PositiveDimensionFunctional.ScaleInvariant
ScaleLarge def: Exactly the two largeness requirements stated at the start of06, converted from log R to bit exponent r. Both constants remain the source constants. · OAI.CoordinateSweeps.ScaleLarge
scaleLoop def: Each input tally emits the fixed block of g tallies. The delimiter starts the new power and the level is incremented by an actual push. · OAI.MaxCutGames.Foundations.Complexity.MachineCeilingPower.scaleLoop
scaleLoop def: Each input tally emits the fixed block of g tallies. The delimiter starts the new power and the level is incremented by an actual push. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineCeilingPower.scaleLoop
scaleNonzeroVector def: Multiplication by a nonzero scalar on actual nonzero vectors. · OAI.Nagata.Workers.W11.scaleNonzeroVector
scalePath def: The actual capacity-normalized scaling on continuous trace paths. · OAI.SLEExactGauge.ScaleTraceLaw.scalePath
scalePoly def: The affine map taking [r,m] onto [1,-1]. · OAI.QAC.scalePoly
scaleSelfAdjoint def: Real scaling written through the complex scalar action. · OAI.KadisonKastler.scaleSelfAdjoint
scaleSum def: The geometric sum appearing in the admissibility condition. · OAI.TalagrandThreshold.scaleSum
scan def: Only the finite head and final-digit registers are inspected by branches. · OAI.MaxCutGames.BinaryNameMachine.scan
scan def: Only the finite head and final-digit registers are inspected by branches. · OAI.UniqueGamesTheorem.BinaryNameMachine.scan
scanArches def: Deterministic stack matching of one colour; none skips the other colour. · OAI.FKCRT.MapCycles.scanArches
scanBuckets def: A counter loop, with no array of all residues. Each completed iteration releases its parent workspace before the next iteration. · OAI.LowSpaceSubsetSum.RepresentationScheduleData.scanBuckets
scanCost def: Number of real finite-stack macrosteps for scanning a given prefix. · OAI.RationalHitting.StackLanguage.scanCost
scanInTime def: A real timed execution; the instruction equation is the only program premise. · OAI.MaxCutGames.BinaryNameMachine.scanInTime
scanInTime def: A real timed execution; the instruction equation is the only program premise. · OAI.UniqueGamesTheorem.BinaryNameMachine.scanInTime
scanned def: The finite word including both endmarkers. · OAI.OneWayLiveness.scanned
scannedBits def: This counts the bits through the selected word, capped at the whole table when the index is invalid. It is not a machine step count. · OAI.MaxCutGames.Foundations.Complexity.MachineLookupSpec.scannedBits
scannedBits def: This counts the bits through the selected word, capped at the whole table when the index is invalid. It is not a machine step count. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineLookupSpec.scannedBits
scanPairs def: A bounded scan; either the sole valid label, or the distinguished zero pair. · OAI.ExactQuantumFactoring.OrderTrial.scanPairs
scanRunTransitions def: Scan the remaining positions, recording precisely the starts of new runs. · OAI.TwoPointCorrelations.scanRunTransitions
scanSpec def: A specification with the same one- or two-symbol cases as the program. · OAI.MaxCutGames.BinaryNameMachine.scanSpec
scanSpec def: A specification with the same one- or two-symbol cases as the program. · OAI.UniqueGamesTheorem.BinaryNameMachine.scanSpec
scanTapes def: Normal form for the changing counter and consumed work-copy prefix. · OAI.MaxCutGames.Foundations.Hastad.SourceClauseLookup.scanTapes
scanTapes def: Normal form for the changing counter and consumed work-copy prefix. · OAI.MaxCutGames.Reduction.SourceEquationLookup.scanTapes
scanTapes def: Normal form for the changing counter and consumed work-copy prefix. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceClauseLookup.scanTapes
scanTapes def: Normal form for the changing counter and consumed work-copy prefix. · OAI.UniqueGamesTheorem.Reduction.SourceEquationLookup.scanTapes
scanWitness def: Executable first-match search, with a proof that a match is present. · OAI.MaxCutGames.Foundations.PCP.PoweringLabels.scanWitness
scanWitness def: Executable first-match search, with a proof that a match is present. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringLabels.scanWitness
scatterEuclideanLinearEquiv def: The collision linear equivalence transported to the Euclidean pair norm. · OAI.BoltzmannNonuniqueness.scatterEuclideanLinearEquiv
scatterLinearEquiv def: A unit-normal collision is a linear involution. · OAI.BoltzmannNonuniqueness.scatterLinearEquiv
scatterLinearIsometryEquiv def: Conservation of total kinetic energy makes the collision map an isometry. · OAI.BoltzmannNonuniqueness.scatterLinearIsometryEquiv
scatterLinearMap def: Linearity holds even before imposing the unit-normal condition. · OAI.BoltzmannNonuniqueness.scatterLinearMap
scatterPair def: The pre/post-collisional map, with the collision normal fixed. · OAI.BoltzmannNonuniqueness.scatterPair
schattenMoment def: Unnormalized Schatten 2q moment, without division by the dimension. · OAI.CoordinateSweeps.schattenMoment
schattenMoment def: Ordinary, unnormalized even Schatten moment. For a matrix K this is Tr((K* K)^q), NOT divided by its dimension. · OAI.CoordinateSweeps.schattenMoment
schauderBasis def: Finite prefix normers yield a basic sequence. · OAI.SeparableQuotient.PrefixNormers.schauderBasis
Schedule structure: Parameters of the two-order search. · OAI.FourierLLogL.Compress.Schedule
Schedule def: A schedule specifies one allowed height for each edge of a fixed word. This recursive finite product carries its usual compact topology when the band is closed. · OAI.PlanarL1.Columns.Model.Schedule
schedule def: Chronological insertion history after t requests. qualifies will be computed from the actual true-prefix posterior and the fixed tier parameter. · OAI.KServer.Chronological.schedule
scheduleAtomFullCoprimeValid def: The original requirements at every reversed transfer: current atoms are pairwise coprime, and the inserted pivot is coprime to every current atom. · OAI.Ostmann.scheduleAtomFullCoprimeValid
scheduledActualSourceScheme def: In v H_R - w H_L, the left coefficient is the right H source product. · OAI.Ostmann.scheduledActualSourceScheme
scheduleData def: All fields are determined by the input and the random draw history. In particular, the second layout coordinate is the actual sign mask. · OAI.LowSpaceSubsetSum.RepresentationData.scheduleData
scheduledCacheParameters def: The same cache parameters used by the executed approximate sampler. · OAI.ContingencyTables.ProfilePrograms.scheduledCacheParameters
ScheduledControl structure: The two precise outputs of the shared physical reservoir selection used by the covariance induction. A witness is produced below from the model, with its low-increment and parameter bounds retained alongside it. · OAI.DilutedSpinGlass.UniversalDictionary.ScheduledControl
scheduledCountInputMeasure def: Numeric loop lengths are charged explicitly, in addition to binary input size. · OAI.ContingencyTables.ProfilePrograms.scheduledCountInputMeasure
scheduledCounts def: All first-success numerators, computed from the public binary margins. · OAI.ContingencyTables.ProfilePrograms.scheduledCounts
scheduledCountsLaw def: No extra table oracle: this is the integer law of the ordinary public branch. · OAI.ContingencyTables.SmallGraphProfiles.scheduledCountsLaw
ScheduledData structure: A deformation toward a retract which waits until time a. · OAI.Thomason.Construction.TopologicalDeformation.ScheduledData
scheduledFinalGraph def: The final graph is obtained by restricting the computed graph to the surviving words, anchors, and remaining slots. · OAI.Ostmann.scheduledFinalGraph
scheduledFinalPerm def: Only actual word coordinates are reassigned. Every other H coordinate, including future pivots and fresh anchors, is fixed. · OAI.Ostmann.scheduledFinalPerm
scheduledGlobalShift def: Literal net global displacement immediately before a child window: a pre block at v+1 remains active through its matched post region N-v. The paired displacement has even t trials in each affected region. · OAI.TraceReconstruction.scheduledGlobalShift
scheduledIdentity def: The identity can wait for the entire interval. · OAI.Thomason.Construction.TopologicalDeformation.scheduledIdentity
scheduledLoad def: Exact number of calls of a given type at a given time in the cyclic schedule. · OAI.ExactFourier.Packing.scheduledLoad
scheduledNextAmplitude def: The two old frequency histories become the two children of the new root. Only the forced positive integer pivot is inserted into the inherited weights. · OAI.Ostmann.scheduledNextAmplitude
scheduledOutputVertex def: No unused copied or retained alternative occurs in this map. · OAI.Ostmann.scheduledOutputVertex
scheduledPrefixMean def: Conditional scheduled-tree mean after a genuine sampled prefix. The terminal reserve n is held fixed and the evaluation depth ranges over the first r transitions. No unobserved ancestral variable is resampled. · OAI.DilutedSpinGlass.ReducedTopology.scheduledPrefixMean
scheduledProductMean def: Ranged proper-child conditional product in the actual scheduled shape. · OAI.DilutedSpinGlass.ReducedTopology.scheduledProductMean
ScheduledRegularRow def: The row assertion is restricted to surviving slots, as required by the actual H/Y partition; unused ambient labels are irrelevant. · OAI.Ostmann.ScheduledRegularRow
ScheduledResidualCache abbrev: Ordinary count vector, original margins, and reciprocal branch probability. · OAI.ContingencyTables.ProfilePrograms.ScheduledResidualCache
scheduledResidualCounts def: This computes the complete correction vector; it asks for no probability oracle. · OAI.ContingencyTables.ProfilePrograms.scheduledResidualCounts
scheduledRetainedGraph def: The input graph retains only the total pivot and the actual H/Y slots. · OAI.Ostmann.scheduledRetainedGraph
scheduledShapeEnergy def: The actual single-copy conditional covariance of a reduced topology at a specified evaluation depth, with all unary grid transitions retained. Outside the available evaluation horizon it is zero; all induction applications are inside the … · OAI.DilutedSpinGlass.ReducedTopology.scheduledShapeEnergy
scheduledSourceScheme def: Every ancestor is a genuine proper prefix of this node. The source products are constructed from H slots, rather than assumed coefficients. · OAI.Ostmann.scheduledSourceScheme
scheduledTableDraw def: The mathematical draw whose code is the output of the single public program. · OAI.ContingencyTables.SmallGraphProfiles.scheduledTableDraw
scheduledTailProjection def: A branching-schedule projector evaluated on the SAME actual full path. Its initial p coordinates are retained, not independently resampled. · OAI.DilutedSpinGlass.ReducedTopology.scheduledTailProjection
scheduledTrialWidth def: The all-small branch uses only the proposal bits of each walk. · OAI.ContingencyTables.ProfilePrograms.scheduledTrialWidth
ScheduledVariable abbrev: Actual surviving atoms and a finite supply of labelled pivot positions. Unused pivot positions have no sampling prior. · OAI.Ostmann.ScheduledVariable
scheduledWordH def: An actual copied word coordinate belongs to H at every depth. · OAI.Ostmann.scheduledWordH
scheduledWordMultiplier def: This multiplier is constructed from the role character, original Gauss multiplier, and copy signs alone; it contains no frequency parameters. · OAI.Ostmann.scheduledWordMultiplier
scheduledWordTemplate def: Each internal address names the product of the pivot atom erased at that node. Both child templates read the same Y and their own H copies. · OAI.Ostmann.scheduledWordTemplate
scheduleQueries def: Proof-only total query count when early acceptance is ignored. · OAI.LowSpaceSubsetSum.RepresentationSetup.scheduleQueries
scheduleScans def: The analysis records the actual scans of the full hypothetical loop; early acceptance only omits a suffix of this work. · OAI.LowSpaceSubsetSum.RepresentationSetup.scheduleScans
Scheme structure: Each statistical unit offers several opaque services. · OAI.ExactDerandomization.Chime.Scheme
schemeAmplitudeSlots def: Each slot remembers either a fixed amplitude or one inserted coordinate. · OAI.PolycyclicRecognition.schemeAmplitudeSlots
schemeIso def: A slice-site line trivialization is an trivialization on the corresponding open subscheme; no extra local freeness hypothesis is added. · OAI.ActualSheafTensor.LineTrivialization.schemeIso
schemePresheafRestrict def: The concrete presheaf restriction underlying open-scheme restriction. · OAI.ActualSheafTensor.schemePresheafRestrict
schemePresheafTensorRestrictIso def: Tensor presheaves commute with restriction to an open subscheme. · OAI.ActualSheafTensor.schemePresheafTensorRestrictIso
schemeRingRestrictHom def: The coefficient-ring map for open-scheme restriction. · OAI.ActualSheafTensor.schemeRingRestrictHom
schemeSheafifyRestrictMap def: Canonical open restriction comparison for module sheafification. · OAI.ActualSheafTensor.schemeSheafifyRestrictMap
schrammGauge def: A deterministic completion of the exponent-one log-log gauge proposed by Schramm. · OAI.SLEExactGauge.schrammGauge
schrodingerFlow def: The formalization constructs stable self-similar finite-time blowup for defocusing nonlinear Schrödinger equations on the twelve-dimensional torus. For every prescribed lower bound on the power, it selects an odd power at least that large … · OAI.DefocusingNLS.schrodingerFlow
schrodingerFlow def: Free evolution preserves the norm of every weighted Fourier vector. · OAI.DefocusingNLS.schrodingerFlow
schrodingerForm def: The polarized form of -d²/dx² - W, on exactly H¹. · OAI.SharpLiebThirring.schrodingerForm
schrodingerInteractionField def: The derivative field for v(t) = S(-t)u(t) in the defocusing equation. · OAI.DefocusingNLS.schrodingerInteractionField
schrodingerMultiplier def: The exact free Schrödinger phase for a Fourier frequency. · OAI.DefocusingNLS.schrodingerMultiplier
schrodingerOperator def: The unbounded self-adjoint operator associated with exactly the H¹ form. · OAI.SharpLiebThirring.OperatorProof.schrodingerOperator
schrodingerUnitary def: Free evolution is a complex-linear isometric equivalence, with inverse time -t. · OAI.DefocusingNLS.schrodingerUnitary
SchurWeylMultilinearSpanning def: The finite-dimensional consequence of Schur--Weyl duality used in the paper. Source: Grinko--Ozols, Linear programming with unitary-equivariant constraints, Comm. Math. Phys. 405 (2024), article 278, Theorem 3.1, combined with the … · OAI.KadisonSimilarity.SchurWeylMultilinearSpanning
schwartzRadialRestriction def: Restrict a Schwartz function to a radial line through a unit vector. · OAI.DefocusingNLS.schwartzRadialRestriction
schwarzGram def: The positive Gram matrix used for the Schwarz inequality. · OAI.KadisonKastler.schwarzGram
Scoped def: Every free index is below the context length. · OAI.PureTypeSystem.Expr.Scoped
scopedBit def: Projection notation for conditioning on a specified (possible) diagnostic or diagonal only. · OAI.HadwigerCounterexample.scopedBit
score def: Expected arctangent after centering at a deterministic real number. · OAI.MetricMeasureLimits.DiameterCenter.score
score def: The affine score whose slope is y and whose intercept is c. · OAI.Problem358.TripleCell.score
scoreCell def: The closed cell of an affine score, before restriction to the source set. · OAI.Problem358.scoreCell
scoredArgmaxWithWork def: A scored scan keeps the winning score instead of recomputing it. It retains the first maximizer, including ties. · OAI.EditApproximation.scoredArgmaxWithWork
scoreError def: Literal full-root globally centered score for the physical iid-uniform Poisson reservoir, with bounded representatives equal to the original bounded physical variables almost surely. Every old dictionary term remains present. · OAI.DilutedSpinGlass.ConcreteReservoir.scoreError
scoreScale def: The scale prescribed by the manuscript; no enlarged moment hypothesis is used in the bounded-model concentration step. · OAI.DilutedSpinGlass.scoreScale
Script inductive: An unrestricted script of exactly n unit edits. · OAI.EditDistortion.Script
Script abbrev: within-endpoint L/R bit tests. Read only a x=a y, x≠y. (tests between ordered points; no double counting since L,R and both orientations are in the mixer hypothesis). · OAI.HadwigerCounterexample.Script
Seam def: Boundary vertices glued with opposite side parametrizations. · OAI.ArtinCAT0.TriangleMesh.Replacement.Seam
seamColumnsBasis def: The whole ordered child fiber, retaining every old occurrence. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.seamColumnsBasis
SeamCompatible def: Seam gluing in the two planar sheets: reflection of base and velocity changes the sheet. This is a local equality, not an assumption of transport invariance or of an analytic estimate. · OAI.TriangularBilliards.SeamCompatible
seamGrade def: Ordered flag grade for the direct sum of the child fibers. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.seamGrade
seamGradient def: Tangential matching gives the gradient of the one-homogeneous seam height. · OAI.Bernoulli.DeSilvaJerison.seamGradient
seamLeft def: Seam-parallel transport from false to true uses the inverse CHILD transport, since each child's boundary orientation is opposite to the parent's. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.seamLeft
search def: Probe a pure function of (info, cursor) before any descent. · OAI.ExactDerandomization.Arena.Flash.search
search def: This finite algorithm never calls the simulated partial function. · OAI.FiniteAlgebraicEnvelopes.Compiler.search
searchInTime def: Explicit polynomial bound for the actual successful run. · OAI.MaxCutGames.BinaryNameSearch.searchInTime
searchInTime def: Explicit polynomial bound for the actual successful run. · OAI.UniqueGamesTheorem.BinaryNameSearch.searchInTime
searchPrime def: Bounded deterministic prime search. Its bound follows from Bertrand, not a prime oracle or an unbounded assumed operation. · OAI.OptimalMaxCut.Unweighted.searchPrime
searchRight def: Each task regenerates its restricted leaves and streams. An overflowing full prefix is dropped; a completed array is passed to the actual checker. The first YES returns immediately. · OAI.LowSpaceSubsetSum.RepresentationParentData.searchRight
searchTapes def: Cursor, permanent stream, key, search stream, candidate, key-copy, copy scratch, index counter, reversed output. · OAI.MaxCutGames.BinaryLiteralMachine.searchTapes
searchTapes def: Cursor, permanent stream, key, search stream, candidate, key-copy, copy scratch, index counter, reversed output. · OAI.UniqueGamesTheorem.BinaryLiteralMachine.searchTapes
searchUniverse def: Filtering by bases and boundary flags gives exactly the source's fixed internal-search state space. This is computable from the literal sector. · OAI.FiniteAlgebraicEnvelopes.Compiler.Scan.searchUniverse
seatfare def: sector start width sums (need cost bound on packed call). · OAI.PowerSaving.Grove.seatfare
sechErrorPositive def: Negative of the exact Cauchy-model transform residual at v=-s. The 2N common nodes are not symmetrized with the final distinct F/G nodes. · OAI.Honeycomb370.sechErrorPositive
sechExpectation def: The probability average under dp/cosh(cp), pushed forward by p ↦ p². The transform below also proves its normalization. · OAI.CriticalHoneycomb.FiniteFunctional.StripAsymptotic.sechExpectation
Second def: A module supported on the second factor of the actual product algebra. · OAI.LittleFinitistic.BlockModule.Second
second def: The entire second arrow space inside the encoding algebra. · OAI.LittleFinitistic.EncodingGenerators.second
Secondary inductive: Occurrence labels for the lower strips; the maximal parent is deliberately not a constructor. Equal ranks or isomorphic modules do not identify labels. · OAI.IntegralCharacterVarieties.TwoFlagBand.Secondary
SecondaryBatch structure: All kept groups with one old identifier and one common orientation. Their closed domains may overlap in regions where several clippings act. · OAI.PlanarL1.Columns.Model.BState.SecondaryBatch
secondaryChart def: Exact shallow-secondary ordinary chart: c+η(pc+R/20−Q_F). · OAI.PlanarL1.Columns.Model.Point.RadialTree.AtomSource.secondaryChart
secondaryInstall def: The source ordered-support branch, including the old affine chart conversion. The auxiliary field is zero and the circular order reverses exactly when η is negative. · OAI.PlanarL1.Columns.Model.OrderedData.secondaryInstall
SecondaryLayer structure: The simultaneous secondary update, grouped by its actual old identifier. Within an identifier we take a maximum, not disjoint patching. Distinct old identifiers have disjoint domains because the old chart name is a function. · OAI.PlanarL1.Columns.Model.BState.SecondaryLayer
SecondaryNames def: All nonprincipal facet names remain explicit and may repeat. The new disk parent is a separate name, not a replacement or quotient of old facet names. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.Kind.SecondaryNames
SecondaryNode def: One oriented SECONDARY corner of an actual allowed vertex. Removing the principal parent corner is essential: this is the graph of the lower strips. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.Kind.SecondaryNode
SecondaryStep structure: One kept group's literal secondary clipping. Only the geometric admission, seed and eligibility data enter; every new chart below is constructed from the winning formula, not postulated. · OAI.PlanarL1.Columns.Model.BState.SecondaryStep
secondBlock def: The last ten coordinates in the standard orthogonal decomposition of R²⁰. · OAI.ProjectionCounterexample.secondBlock
secondBlockComparison def: The second actual tensor application selects Q_T and retains first-block support. · OAI.LittleFinitistic.OrdinarySimulation.XResolution.secondBlockComparison
secondBox def: The actual coordinatewise second-system box label. · OAI.QuantitativeVanDerWaerden.secondBox
secondCoeff def: Scalar diagonal Hessian coefficient in a unit radial/angular frame. · OAI.Bernoulli.Axisymmetric.secondCoeff
secondCoordinate def: Orthogonal diagonal coordinate recovered directly from the entries. · OAI.PositiveEinsteinFour.Bianchi.secondCoordinate
secondExponent def: Exponents of a genuine term in the second-order polynomial field. · OAI.HarmonicCounterexample.Control.secondExponent
secondField def: The genuine second-order contribution, with every ordered differentiation and replacement included. · OAI.HarmonicCounterexample.Control.secondField
secondForm def: The Euclidean second fundamental form obtained by normal projection. · OAI.ClosedSurfaceR4.SphericalJets.secondForm
SecondInput structure: Sum.inl o records a full occurrence and Sum.inr n a single named bit. The summand tag preserves the set J without exposing any retained index. · OAI.MaxCutGames.Soundness.ZeroInformation.SecondInput
SecondInput structure: Sum.inl o records a full occurrence and Sum.inr n a single named bit. The summand tag preserves the set J without exposing any retained index. · OAI.UniqueGamesTheorem.Soundness.ZeroInformation.SecondInput
secondLower def: The second branch's lower interval endpoint. · OAI.Nagata.FiniteExponents.secondLower
secondNeighbors def: `secondNeighbors r v` contains a vertex `w` when some `u` gives a two-step path `v → u → w`, while explicitly excluding `v` itself and every first out-neighbor of `v`. This is exact distance two, not merely reachability within two steps. · OAI.SeymourSecondNeighborhood.secondNeighbors
secondOutput def: Only rotating coordinates can produce a dummy scalar output. · OAI.QuantitativeVanDerWaerden.AnchoredPatterns.secondOutput
SecondPoint def: At single positions the partner space has a free bit and no occurrence or rhs. Unused ambient coordinates are fixed to zero, giving a canonical embedding of the position-dependent coordinate space rather than extra degrees of freedom. · OAI.MaxCutGames.Soundness.ZeroInformation.SecondPoint
SecondPoint def: At single positions the partner space has a free bit and no occurrence or rhs. Unused ambient coordinates are fixed to zero, giving a canonical embedding of the position-dependent coordinate space rather than extra degrees of freedom. · OAI.UniqueGamesTheorem.Soundness.ZeroInformation.SecondPoint
secondPrimitive def: The second scalar primitive used for the matrix trace potential. · OAI.SharpLiebThirring.ScalarProof.secondPrimitive
secondProjectionMap def: The second local map knows the sector through its original coordinate. · OAI.MatrixMultiplication.AuxiliarySeparation.secondProjectionMap
secondProjectionMap def: The second simplicial projection records the time ordinal of each chart. · OAI.Thomason.Construction.PrismBarycentric.secondProjectionMap
SecondRow abbrev: A raw row function on ambient partner coordinates. · OAI.MaxCutGames.Soundness.ZeroInformation.SecondRow
SecondRow abbrev: A raw row function on ambient partner coordinates. · OAI.UniqueGamesTheorem.Soundness.ZeroInformation.SecondRow
secondSeparationMap def: The second local Fourier map followed by its possibly negative weight. · OAI.MatrixMultiplication.AuxiliarySeparation.secondSeparationMap
secondSourceThetaEquiv def: Positive high blocks have the exact centered theta multiplier with epsilon1. · OAI.Nagata.Workers.W10.secondSourceThetaEquiv
secondSumGraph def: Graph coefficient vector for the second block's sum facet. · OAI.ProjectionCounterexample.secondSumGraph
secondTotalIso def: Actual tensor with the off-diagonal complex on a second-block input is the first-block T-total. · OAI.LittleFinitistic.OrdinarySimulation.BlockTensor.secondTotalIso
secondWeight def: Weight on the second tensor leg. · OAI.MatrixMultiplication.AuxiliarySeparation.secondWeight
secondWeight def: Weight one on the middle second-leg interval, zero elsewhere. · OAI.MatrixMultiplication.AuxiliarySeparation.Sector.secondWeight
secretaryAcceptedThrough def: Irrevocable accepted labels through an arbitrary number of arrivals. · OAI.MatroidProphet.secretaryAcceptedThrough
SecretaryChallenge def: The full order-oblivious conclusion of cor:secretary, including a measurable history-only rule, every-prefix feasibility, prefix rejection, arbitrary full-seed suffix adversaries, and the unchanged 2⁻²⁹³ ratio. · OAI.MatroidProphet.SecretaryChallenge
secretaryDecisionAt def: Execution forces rejection of every observation-prefix arrival. · OAI.MatroidProphet.secretaryDecisionAt
SecretaryFeasible def: Feasibility is pointwise for every seed, vector, order, and arrival prefix. · OAI.MatroidProphet.SecretaryFeasible
secretaryFinSeed def: Canonical finite-index representation, with no nonemptiness restriction. · OAI.MatroidProphet.secretaryFinSeed
secretaryFinSeedLaw def: Push the complete finite seed law forward along the same canonical bijection. · OAI.MatroidProphet.secretaryFinSeedLaw
SecretaryGuarantee def: Performance against every full-seed-aware suffix adversary, on an arbitrary ambient probability space. The seed law is fixed before the weight vector. · OAI.MatroidProphet.SecretaryGuarantee
SecretaryRule structure: A measurable secretary rule with an initially fixed observation-prefix length. · OAI.MatroidProphet.SecretaryRule
section4Iteration def: Section 4, final corollary, with only objectwise splitting isomorphisms. · OAI.ExternalTensor.Binomial.section4Iteration
sectionA def: Restriction to the section a=1 of the determinant bundle on the first base chart. · OAI.StableCoordinate.BundleCharts.sectionA
sectionAPolynomialMap def: The section with a=1 over D(z+1), expressed on polynomial generators. · OAI.StableCoordinate.BundleCharts.sectionAPolynomialMap
sectionAt def: The section at a chosen interval point. · OAI.Thomason.Construction.TopologicalInterval.sectionAt
SectionBalance def: Source equation uniform-section-balance, in unshifted coordinates; equivalently, -ρ(closure S-a) ⊆ closure S-a. · OAI.AffineBernstein.SectionBalance
sectionBasisOfLaurentData def: A basis of the actual section space follows once actual Laurent uniqueness, recurrence, and the theta coefficient identities have been proved. · OAI.Nagata.Workers.W10.sectionBasisOfLaurentData
sectionBody def: The remaining compact body after fixing a valid prefix. · OAI.Problem358.ConvexTriangularSampler.sectionBody
sectionCoefficient def: The eighteen degree-one sections in the chosen rational trivialization of H. The original y-indexing is shifted by one, so z₀=1 is the sixth line. · OAI.ExplicitCone.sectionCoefficient
sectionCoverMap def: Literal multiplication by a section on the genuine line-bundle cover. · OAI.Nagata.CoefficientSpaces.sectionCoverMap
sectionD def: Restriction to the section d=1 of the determinant bundle on the second base chart. · OAI.StableCoordinate.BundleCharts.sectionD
sectionDeformation def: A straight-line deformation to the section, stationary on that section. · OAI.Thomason.Construction.TopologicalInterval.sectionDeformation
sectionDensity def: The concrete section-density ratio is jointly measurable even at invalid prefixes. · OAI.Problem358.ConvexTriangularSampler.sectionDensity
sectionDeriv def: Actual complex derivative at a point of the punctured plane. · OAI.Nagata.W07.sectionDeriv
sectionDPolynomialMap def: The section with d=1 over D(z), expressed on polynomial generators. · OAI.StableCoordinate.BundleCharts.sectionDPolynomialMap
sectionDynkinConstant def: The product constant depends on dimension, nilpotence degree, and a uniform bracket bound, rather than on the chosen basis. · OAI.PolycyclicRecognition.sectionDynkinConstant
sectionEval def: Evaluation in the actual covering-space frame. · OAI.Nagata.W07.sectionEval
sectionEvaluation def: Evaluation of the complex coefficients of a finite family of sections. · OAI.Kaplansky.sectionEvaluation
sectionEvaluationCoordinates def: The evaluation family in an actual bundle frame, with complex parameter space and real-smooth dependence on the base. · OAI.Kaplansky.sectionEvaluationCoordinates
sectionFirstJet def: Value and actual first derivative in one fixed local frame. · OAI.Nagata.W07.sectionFirstJet
sectionHomologyMap def: The section-induced map into the homology of the one-point total space. This is an actual singular homology morphism, not a postulated characteristic class. · OAI.Kaplansky.sectionHomologyMap
sectionKernel def: The normalized section laws form a genuine measurable kernel. · OAI.Problem358.NormalizedSections.sectionKernel
sectionLinePolynomial def: The literal projective linear equation corresponding to a section. · OAI.Nagata.W21.sectionLinePolynomial
sectionMap def: Iterated realization prisms give a continuous section of the cube coordinates. · OAI.Thomason.Construction.RealizedCube.sectionMap
sectionMeasure def: The original product measure for any prescribed continuous section. · OAI.PolycyclicRecognition.CartanWordModel.MeasuredCone.sectionMeasure
sectionNonzeroLocus def: Nonvanishing locus of this section, with the actual base topology. · OAI.Kaplansky.sectionNonzeroLocus
sectionNonzeroLocus def: The genuine nonvanishing locus of a section in the punctured covering plane. · OAI.Nagata.CoefficientSpaces.sectionNonzeroLocus
sectionOfFunction def: The resulting element of the genuine section space. · OAI.Nagata.W08.sectionOfFunction
sectionProbeCoordinates def: Coordinates of a point of the actual quotient/kernel product cell. · OAI.PolycyclicRecognition.sectionProbeCoordinates
sectionQuotientMap def: The actual globally defined map between the two multiplier quotients. · OAI.Nagata.CoefficientSpaces.sectionQuotientMap
sectionRatio def: The literal affine coordinate functions for the degree-one section z₀z₁z₄ in the fixed H=3J−E linear system. · OAI.ExplicitCone.sectionRatio
sectionRelativeHomologyMap def: The actual map of pairs (B, B minus zeros) to (Thom, Thom minus zero section). · OAI.Kaplansky.sectionRelativeHomologyMap
sectionRestriction def: Restriction of sections regarded as a linear map over the base ring R. · OAI.AffineDerivation.sectionRestriction
sectionRestriction def: Forget the forced zero value and restrict to the actual punctured-plane domain. · OAI.Nagata.W08.sectionRestriction
sections def: The actual module of sections on a vertex set. Coordinates outside the set are harmless and ignored by all restriction maps. · OAI.KLInvariance.MomentGraph.Sheaf.sections
sectionToInfinity def: Push a nowhere-zero section out along each fiber, ending at infinity. · OAI.Kaplansky.sectionToInfinity
SectionTrivialization abbrev: A trivialization of the section module of the ideal. · OAI.ActualCartier.SectionTrivialization
sectionTwoEval def: Values at two specified covering points. · OAI.Nagata.W07.sectionTwoEval
Sector structure: A list zipper for the same finite-scan semantics. Left cells are nearest first. This representation permits induction on literal scan paths. · OAI.FiniteAlgebraicEnvelopes.Compiler.Scan.Sector
Sector structure: A headed sector in zipper notation. The left list is in reverse order, so its head is exactly the cell adjacent to the validator. · OAI.FiniteAlgebraicEnvelopes.Compiler.Validator.Sector
Sector def: Both closed angular inequalities, including a nonnegative radial coordinate. · OAI.TriangularCovering.Sector
sectorCoordinates def: The two tensor axes choose block-copy labels independently, even for repeated species. · OAI.ExactFourier.Packing.sectorCoordinates
sectorGroup def: Permutations preserving a position sector. · OAI.Saxl.sectorGroup
sectorList def: Only control heads in the supplied literal support are enumerated. The ambient coded head type need not be finite. · OAI.FiniteAlgebraicEnvelopes.Compiler.Scan.Uniform.sectorList
sectorNumber def: The paper numbers sectors from one, while Fin M numbers them from zero. · OAI.MatrixMultiplication.AuxiliarySeparation.sectorNumber
SectorSplit abbrev: Choices of the first input number in the total-number sector of (k,e). · OAI.EntropyPhotonNumber.SectorSplit
Seed def: An actual finitely presented source for the finite-table group. The two copies of Γ map to the root and one half-cube, and the free factor maps to the four explicit untwisted generators. Its quotient kernel is a construction, not an extra … · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Generators.Seed
Seed abbrev: A single seed supplies a full local table to each side. · OAI.MaxCutGames.Foundations.Games.KernelSampling.Seed
Seed abbrev: The independent original seed and two full local completion tables. · OAI.MaxCutGames.Foundations.Repetition.CompletedSampling.Seed
Seed structure: Literal geometric data retained after both descent tests and the low-tip exclusion. These records are finite witnesses of those tests, not estimates or success/persistence assumptions. · OAI.PlanarL1.Columns.Model.SurvivingGroups.Seed
Seed abbrev: A single seed supplies a full local table to each side. · OAI.UniqueGamesTheorem.Foundations.Games.KernelSampling.Seed
Seed abbrev: The independent original seed and two full local completion tables. · OAI.UniqueGamesTheorem.Foundations.Repetition.CompletedSampling.Seed
seedBase def: The vertex of the rose common to its two circles. · OAI.EilenbergGanea.seedBase
SeedCorrect def: Exact compressed initialization, to be instantiated by singletons plus at most d+1 threshold domains below. · OAI.WLIdentification.CellularCircuit.SeedCorrect
seedExponentMatrix def: The source exponent-sum boundary, with rows indexed by its two relators. · OAI.EilenbergGanea.seedExponentMatrix
seedGraph def: Comparability graph of the seven vertices, 51 edges and 45 sectors. · OAI.EilenbergGanea.seedGraph
seedLaw def: The fixed weighted seed law; it does not depend on the sampled questions. · OAI.MaxCutGames.Foundations.Games.KernelSampling.seedLaw
seedLaw def: The fixed weighted seed law; it does not depend on the sampled questions. · OAI.UniqueGamesTheorem.Foundations.Games.KernelSampling.seedLaw
Seeds abbrev: The three seeds are the common leaf, an axis-one seed and an axis-two seed. · OAI.Problem348.TreeSampling.Seeds
seedToActualDraw def: Convert the actual two-stage seed into the full hidden experiment. The second summand of the seed remains available for the target-side observation. · OAI.MaxCutGames.Decoder.AdviceLaw.seedToActualDraw
seedToActualDraw def: Convert the actual two-stage seed into the full hidden experiment. The second summand of the seed remains available for the target-side observation. · OAI.UniqueGamesTheorem.Decoder.AdviceLaw.seedToActualDraw
SeedValues structure: Finitely many seed values. Diagonal W seeds will be killed. · OAI.LittleFinitistic.FinitePresentation.SeedValues
seedWord def: The signed words encode the relators x²y⁻⁵ and x²(yx⁻¹)³. · OAI.EilenbergGanea.seedWord
seedWord def: false names x; true names y. The second Boolean is the positive sign. · OAI.EilenbergGanea.seedWord
Seek structure: Pick the zero-based kth nonzero answer; the pair of index and answer is positive, or 0 if none. · OAI.ExactDerandomization.Fret.Seek
seekCycle def: Inspect one proposed length-(r+1) cycle. Every queried input must be in S and must not have appeared in an earlier slot or earlier in this cycle. · OAI.CubeShuffle.QueryTree.seekCycle
seekCycle def: Inspect one proposed length-(r+1) cycle. Every queried input must be in S and must not have appeared in an earlier slot or earlier in this cycle. · OAI.CubeShuffle.RoutingNetwork.QueryTree.seekCycle
seekCycles def: Distinguishable cycle slots, with each requested length encoded as r+1. Later factorial-moment counting divides by precisely these rotation counts. · OAI.CubeShuffle.QueryTree.seekCycles
seekCycles def: Distinguishable cycle slots, with each requested length encoded as r+1. Later factorial-moment counting divides by precisely these rotation counts. · OAI.CubeShuffle.RoutingNetwork.QueryTree.seekCycles
seen def: Chronological node heads, in reverse recording order. Zero denotes a dummy slot after the pending queue has become empty. Duplicate occurrences remain. · OAI.ExactQuantumFactoring.AuxiliaryTree.seen
Segment def: A geodesic parametrized proportionally to length on the closed unit interval. Values outside [0,1] are irrelevant. · OAI.ArtinCAT0.Segment
Segment structure: Uniform probability on a nondegenerate real interval. · OAI.DoublingHilbert.Segment
Segment structure: Segment of n flow moves. Suspended-frame total at each nonfinal point bounded by B. Active frame exists at such points. · OAI.Fifths.Flow.Segment
segment def: The comparison segment chosen from the CAT(0) comparison condition. · OAI.HyperbolicObstruction.CATZero.segment
segment def: Coordinate segments are clamped outside the parameter interval, so the boundedness obligation never relies on the arbitrary extension of a segment. · OAI.HyperbolicObstruction.MetricLimit.segment
segment def: Coordinate segments stay uniformly bounded. · OAI.HyperbolicObstruction.OrbitLimit.segment
segment def: Later times multiply on the left, as in the manuscript. · OAI.SmoothObstruction.RelativeProduct.segment
segment def: A segment inherits the original labels, including shared prime classes. · OAI.TwoPointCorrelations.LabeledPrimeWord.segment
SegmentChain structure: A finite segment chain, represented by its value at every integer; only the first length + 1 values are used. · OAI.PlanarL1.Topology.SegmentChain
SegmentComparison abbrev: There are at most K² P³ ordered choices of two codeword labels, a positive segment length at most P, and two starting positions. The validity conditions only remove comparisons from this ambient index type. · OAI.TreeEdit.BinaryLower.CodeUnionBound.SegmentComparison
segmentIndex def: Shift an index known to lie in a segment into that segment's index type. · OAI.TreeEdit.BinaryLower.segmentIndex
SegmentInside def: All points of the closed segment between the base point 0 and r remain in the open line layer. · OAI.HenonLaneEmden.LineComponents.SegmentInside
segmentIntegral def: Integral over the segment of half-length r centered at x. · OAI.HenonLaneEmden.IntervalGeometry.segmentIntegral
segmentIntersection def: Directed open-segment intersection, with the ordinary determinant sign. · OAI.Paper170.segmentIntersection
segmentMatchingEvent def: The event that two fixed windows admit a matching of size s. · OAI.TreeEdit.BinaryLower.RandomMatching.segmentMatchingEvent
segmentOccurrences def: Group translates whose two endpoints occur in a specified finite path. · OAI.HarmonicArtin.ParabolicIntersections.TreeAction.segmentOccurrences
segmentPoint def: The fixed-interval parametrization of the entire vertical segment. · OAI.SymmetricMahler.segmentPoint
segments def: Nonempty finite intervals for an ancestral order. · OAI.DiamondDistortion.Ancestry.segments
segments def: Durations followed by a terminal first-hitting segment. · OAI.FKCRT.SimpleWalk.segments
segments def: Ordered product of independent alternating column segments. · OAI.Saxl.Columns.segments
segmentSeminorm def: The disjoint-segment seminorm on finite coordinate vectors. · OAI.ForestSpace.segmentSeminorm
SegmentSeparated def: Segment separation in terms of increasing position lists. · OAI.TreeEdit.BinaryLower.SegmentCode.SegmentSeparated
segmentSupport def: The support exponent of the segment from zero to w. · OAI.PolycyclicRecognition.SupportDatum.segmentSupport
segmentSupportTime def: A linear weight on the segment chooses the final time exactly when its value on the direction is nonnegative. · OAI.PolycyclicRecognition.SupportDatum.segmentSupportTime
segmentThreshold def: The segment threshold used in the prescribed-length code construction. · OAI.TreeEdit.BinaryLower.Code.segmentThreshold
segmentVisitPath def: The specified segment visit, retained for measurable path sampling. · OAI.PolycyclicRecognition.CartanWordModel.segmentVisitPath
segSum def: The sum of coordinates over a segment. · OAI.ForestSpace.segSum
select def: Check that a field exists before seeding its destination delimiter. · OAI.MaxCutGames.Foundations.Complexity.MachineLookup.select
select def: The subset is formed from literal children of the bounding name. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.UnaryRelation.select
select def: Ordered routing associated to the spatial selector values. · OAI.Problem310.StableRouting.select
select def: Option-valued form of a partial transition. Only the single allowed element is selected. · OAI.TwoWayComplementation.CompanionBridge.select
select def: Check that a field exists before seeding its destination delimiter. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineLookup.select
selectActive def: The deterministic tie rule chooses the first closed cell containing a point. On all main-cell overlaps every proposal is the original activity. · OAI.PlanarL1.Columns.Model.SetPatch.selectActive
Selected def: A working code and the assertion that no smaller code can ever become a working representative of the same value. · OAI.PartitionConsistency.ConditionCodeSelection.Selected
Selected abbrev: Persistent least codes, defined over the fixed ground wellordering. · OAI.PartitionConsistency.ProductForcing.ChainCode.Selected
Selected def: The downward-persistent relation that can be used as a name recipe. · OAI.PartitionConsistency.TailSelector.TailData.Selected
Selected def: Whether both positions of a fixed raw cell are selected. · OAI.Problem348.Construction.HostSampling.Selected
selected def: `selected` removes repetitions from a decoded edge list and maps each dependent edge copy through `edgeEquiv` into the `Fin` indices used by `x.graph`. · OAI.AlgorithmicThinTrees.BinaryInput.selected
selected def: A canonical enumeration of a selected list of auxiliary cards. · OAI.CoordinateSweeps.Grid.Holes.selected
selected def: Selected simplex-facet incidences in a finite invariant domain. · OAI.SharpLiebThirring.PLParity.FinitePairing.selected
selected_bounded def: Realizes the selected component by exactly its literal retained paths and interval sides, with the same signs, noncoincident lifts, and prescribed root. · OAI.TorsionFreeZeroDivisors.SampleGraph.PathSystem.selected_bounded
selectedAdaptiveMesh def: The actual exponential mesh for the selected progression length and base. · OAI.QuantitativeVanDerWaerden.Parameters.selectedAdaptiveMesh
selectedAlphabet def: The selected unknowns remain separate from coefficients, even when two letters have the same image in the group quotient. · OAI.EilenbergGanea.selectedAlphabet
selectedChanges def: Remove duplicate changed sets only when forming the containment cover. · OAI.TalagrandThreshold.Selector.selectedChanges
selectedCols def: Selected anchor columns after the full axis has been enumerated by Fin n. · OAI.Problem348.AnchorFrames.selectedCols
SelectedCycle abbrev: The genuine cycles retained completely in S; fixed points are included. · OAI.CubeShuffle.RoutingNetwork.SelectedCycle
SelectedCycle abbrev: The genuine cycles retained completely in S; fixed points are included. · OAI.CubeShuffle.SelectedCycle
selectedDensity def: A coordinate may be omitted altogether, and every included coordinate may choose its endpoint independently of every other coordinate. · OAI.FiniteFactor.CouplingParameters.selectedDensity
selectedDescription def: A total visible selector. Bad advice has no selected witness. · OAI.MaxCutGames.Inverse.RowErasure.SliceFamily.selectedDescription
selectedDescription def: A total visible selector. Bad advice has no selected witness. · OAI.UniqueGamesTheorem.Inverse.RowErasure.SliceFamily.selectedDescription
selectedEvent def: Good visible advice and the selected column restrictions on the hidden matrix. The selector receives only its observed row value. · OAI.MaxCutGames.Inverse.RowErasure.SliceFamily.selectedEvent
selectedEvent def: Good visible advice and the selected column restrictions on the hidden matrix. The selector receives only its observed row value. · OAI.UniqueGamesTheorem.Inverse.RowErasure.SliceFamily.selectedEvent
selectedExtension def: The actual extension used at this stage. · OAI.ReflexiveFixedPoints.CandidateContext.selectedExtension
selectedFactor def: Vary only the selected type, not the entire countable reservoir. · OAI.DilutedSpinGlass.HeterogeneousMarks.selectedFactor
selectedGraphPolynomial def: The actual normal polynomial with univariate graph-restricted coefficients. · OAI.Nagata.Workers.W28.selectedGraphPolynomial
selectedH def: The exact least successful outer input, not the convenient existence bound 217. · OAI.KaplanskyCounterexample.Prescription.selectedH
selectedIndex def: The global target number computed from an arbitrary returned cloud index. · OAI.MaxCutGames.Foundations.PCP.PreprocessingInternalRows.selectedIndex
selectedIndex def: The global target number computed from an arbitrary returned cloud index. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingInternalRows.selectedIndex
selectedIndicator def: A Bernoulli indicator, for counting selected edges. · OAI.AsymptoticallyMinimalLittlewood.Nibble.selectedIndicator
selectedInterval def: The actual ith selected dyadic cell at common depth 2D. · OAI.ThompsonNonamenability.selectedInterval
selectedLabel def: Modulo makes the semantic recursion total even for unsuccessful machines. The main statement separately requires the output to be < k at every reachable outcome, so no invalid output is accepted or silently repaired. · OAI.UniformKServer.selectedLabel
selectedLabels def: The labelled coordinates selected by a Boolean word on a given coordinate set. · OAI.Problem337.RandomExposure.selectedLabels
selectedNodeSources def: Restricting the source type does not change the value or identify sources. · OAI.Ostmann.selectedNodeSources
selectedOriginals def: Actual bad original colorings that select one fixed endpoint. · OAI.TalagrandThreshold.Selector.selectedOriginals
selectedPair def: The first successful pair in the literal finite lexicographic search. · OAI.KaplanskyCounterexample.Prescription.selectedPair
selectedPathCoordinates def: Coordinates of one path belonging to a fixed Boolean variable group, viewed in that group's own coordinate type. · OAI.Problem335.selectedPathCoordinates
selectedPrimeLogDensity def: The reference density is the actual selected Page-corrected density. It stays inside the integral throughout weighted summation. · OAI.Ostmann.selectedPrimeLogDensity
selectedPrivateEquiv def: The actual selected private coordinates are in bijection with the complete projected affine slice. The supporting point is used only to state its column equations; the canonical visible base still defines the private coordinates. · OAI.MaxCutGames.Decoder.SelectedFiberLaw.selectedPrivateEquiv
selectedPrivateEquiv def: The actual selected private coordinates are in bijection with the complete projected affine slice. The supporting point is used only to state its column equations; the canonical visible base still defines the private coordinates. · OAI.UniqueGamesTheorem.Decoder.SelectedFiberLaw.selectedPrivateEquiv
selectedProfiles def: Only the full profiles that occur for this endpoint. · OAI.TalagrandThreshold.Selector.selectedProfiles
selectedQuestionMarginal def: The coordinate pair under the actual repeated question law conditioned on the selected wins, rather than a separately postulated posterior law. · OAI.MaxCutGames.Foundations.Repetition.selectedQuestionMarginal
selectedQuestionMarginal def: The coordinate pair under the actual repeated question law conditioned on the selected wins, rather than a separately postulated posterior law. · OAI.UniqueGamesTheorem.Foundations.Repetition.selectedQuestionMarginal
selectedRankFixed def: Rank in the selected suborder: recurse only along the fixed ground wellordering. The predecessor set is small, with no bound on name ranks. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.selectedRankFixed
selectedRestartChildNode def: Embed an immediate child of a relative selected stop into the next restart nodes. · OAI.RieszRectifiability.selectedRestartChildNode
SelectedRestartSurfaceData structure: Plane fits, a limit model, and bounded surface pieces covering the restart set. · OAI.RieszRectifiability.SelectedRestartSurfaceData
selectedRows def: Selected anchor rows after the full axis has been enumerated by Fin n. · OAI.Problem348.AnchorFrames.selectedRows
selectedSampleAmplitude def: Unnormalized joint amplitude after modular powering and the selected phase-transform block. The modular-power register is NOT copied before this interference sum. · OAI.ExactQuantumFactoring.OrderTrial.selectedSampleAmplitude
selectedSet def: A total sequence, constant after all n coordinates have been selected. · OAI.MaxCutGames.Foundations.Repetition.SelectionSequence.selectedSet
selectedSet def: A selector outcome chooses the terminal address read at each point. · OAI.Problem310.RandomSetDensity.selectedSet
selectedSet def: A total sequence, constant after all n coordinates have been selected. · OAI.UniqueGamesTheorem.Foundations.Repetition.SelectionSequence.selectedSet
selectedSubmodule def: The image of the central idempotent, expressed by its fixed vectors. · OAI.LittleFinitistic.selectedSubmodule
selectedSum def: Sum only the positions selected by an arbitrary predicate. · OAI.MaxCutGames.Reduction.WeightRounding.selectedSum
selectedSum def: Sum only the positions selected by an arbitrary predicate. · OAI.UniqueGamesTheorem.Reduction.WeightRounding.selectedSum
SelectedSystem inductive: A fully selected list of actual path components, ending with no occupied wire. This is a structural conclusion, not an axiom enumerating tensor terms. · OAI.CriticalHoneycomb.Transfer.RasterGrid.SelectedSystem
SelectedTop abbrev: ALL selected top polygons incident to this fan occurrence, not only those whose boundary actually visits it. This implements source03's nominal weight. · OAI.HyperbolicObstruction.ConeSkeleton.RelDisk.SelectedTop
SelectedTop abbrev: ALL selected top polygons incident to this fan occurrence, not only those whose boundary actually visits it. · OAI.HyperbolicObstruction.ConeSkeleton.SelectedTop
selectedWalshMean def: The bounded observable on precisely the touched residue coordinates. · OAI.TwoPointCorrelations.selectedWalshMean
Selection structure: Selected positions; no host construction or matching assumptions are hidden in this data structure. Values past the chosen depth are irrelevant. · OAI.Problem348.GuardedPath.Selection
selection def: The actual selected path and four anchor frames at each edge. · OAI.Problem348.Construction.SampledSelection.selection
selectionCount def: The same finite count, packaged opaquely by choice to avoid evaluating the 249-entry certificate during kernel conversion of dependent finite sums. · OAI.HarmonicCounterexample.Construction.selectionCount
SelectionGroupData structure: The relators used for the off-diagonal generators in Section 2. The presentation and its finite-presentation reduction are separate obligations. · OAI.LittleFinitistic.SelectionGroupData
selectionMeasure def: Independent selection of points at a prescribed probability. · OAI.KaplanskyCounterexample.FiniteGeometry.selectionMeasure
SelectionRanges structure: All non-arithmetic hypotheses of the common sampled-Fourier selection, for the actual geometric middle levels and terminal interval. The finite indices retain multiplicity; no residue or cancellation bound is assumed. · OAI.Problem337.GeometricRandomRanges.SelectionRanges
selectionSets def: Ordered monomials, retaining multiplicities, with repeated coordinates collapsed. · OAI.QAC.selectionSets
selectionStep def: Exactly H(Y) = restriction along alpha of eY from the manuscript. · OAI.LittleFinitistic.selectionStep
selectionType def: The ordinal consisting of the ranks of all selected elements. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.selectionType
selector def: The coset selected at the least distinguishing bound. · OAI.PartitionConsistency.TailSelector.TailData.selector
SelectorAddress abbrev: Finite selector-table addresses include the table name as well as its key. · OAI.Problem310.FiniteTableModel.SelectorAddress
selectorAddress def: The center exposure consults exactly one deterministic coordinate per selector table. · OAI.Problem310.FiniteTableModel.selectorAddress
selectorAssignment def: Set the first two rows of the indeterminate matrix to coordinate selectors. The third row is zero. · OAI.MaxCutGames.Quadratic.selectorAssignment
selectorAssignment def: Set the first two rows of the indeterminate matrix to coordinate selectors. The third row is zero. · OAI.UniqueGamesTheorem.Quadratic.selectorAssignment
selectorCenteredRawVariation def: The fixed main coefficient is the only part requiring cancellation. · OAI.LipschitzHilbert.SourceLabeledLattice.selectorCenteredRawVariation
selectorChoice def: The actual first-success/default child choice made by the finite selector tables. · OAI.Problem310.RoutedSetDensity.selectorChoice
selectorEdge def: The nondefault child edge used by a selector table. · OAI.Problem310.FiniteTableModel.selectorEdge
selectorEndpoints def: Canonical selector resolutions are the incoming window endpoints. · OAI.Problem310.FiniteTableModel.selectorEndpoints
selectorLaw def: A finite table of independent fair selectors. · OAI.Problem310.ConcreteLabels.selectorLaw
Selectors def: Ordered finite selectors, expressed by a finite support and distinguished entry. · OAI.FiniteAlgebraicEnvelopes.Contracted.Selectors
selectorValue def: Selector values are extended to all lists; routing uses only valid nodes. · OAI.Problem310.FiniteTableModel.selectorValue
selectParent def: The actual anchor discharges all analytic inputs to adaptive selection. · OAI.ReflexiveFixedPoints.CandidateContext.selectParent
selectPhase def: The fixed selected-slot tuple is loaded in one actual transition. · OAI.MaxCutGames.Foundations.Hastad.SourceTupleBody.selectPhase
selectPhase def: The fixed selected-slot tuple is loaded in one actual transition. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceTupleBody.selectPhase
selectRows def: Select, independently at every row, one of each pair of resampled coordinates. · OAI.ExactModuli.KW.selectRows
selectState def: Only the chosen-slot/query registers change; the computed context profile and original signature remain available across all local traversals. · OAI.MaxCutGames.Foundations.Hastad.SourceTupleBody.selectState
selectState def: Only the chosen-slot/query registers change; the computed context profile and original signature remain available across all local traversals. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceTupleBody.selectState
selfAdjointConjugation def: Conjugation on the real self-adjoint part. · OAI.WeakPureInfiniteness.selfAdjointConjugation
selfAdjointSymmetryDilation def: The two-by-two symmetry associated to a selfadjoint contraction. · OAI.KadisonSimilarity.selfAdjointSymmetryDilation
selfAdjointSymmetryUnitary def: The symmetry as an actual unitary, with no existence assumption. · OAI.KadisonSimilarity.selfAdjointSymmetryUnitary
selfAdjointUnitizationSnd def: The self-adjoint second coordinate is real linear and continuous, although it need not preserve positivity. Positivity is supplied later by domination. · OAI.WeakPureInfiniteness.selfAdjointUnitizationSnd
selfAdjointVectorImage def: A fixed real-linear transform of the vector ball. · OAI.KadisonKastler.selfAdjointVectorImage
SelfQuasiIsometry structure: The upper, lower, and coarse-surjectivity bounds of a self quasi-isometry. · OAI.PolycyclicRecognition.SelfQuasiIsometry
SelfQuasiIsometry structure: The usual upper, lower and coarse-surjectivity inequalities, with the lower inequality multiplied by K to avoid division. For 1 ≤ K this is exactly the standard (K,C) definition. · OAI.PolycyclicRecognition.SelfQuasiIsometry
selfSlice def: Integer positions of one actual comparison within a single block. · OAI.TorsionFreeZeroDivisors.IntervalSelfLinks.selfSlice
semanticContent def: The whole semantic tape, read in physical order starting at cell zero. · OAI.FiniteAlgebraicEnvelopes.Compiler.Simulator.semanticContent
SemanticGapReduction structure: Only the semantic part of a reduction. This is not a definition of an efficient reduction or an NP-hardness certificate. · OAI.MaxCutGames.Foundations.Target.SemanticGapReduction
SemanticGapReduction structure: Only the semantic part of a reduction. This is not a definition of an efficient reduction or an NP-hardness certificate. · OAI.UniqueGamesTheorem.Foundations.Target.SemanticGapReduction
SemanticReturns def: Standard literal, first-blank-delimited machine output. · OAI.FiniteAlgebraicEnvelopes.Compiler.Simulator.SemanticReturns
semantics def: Decode validated finite data into the actual constraint-graph semantics. · OAI.MaxCutGames.Foundations.PCP.GenericGraphTables.semantics
semantics def: Decode validated finite data into the actual constraint-graph semantics. · OAI.MaxCutGames.Foundations.PCP.GraphTables.semantics
semantics def: Decode validated finite data into the actual constraint-graph semantics. · OAI.UniqueGamesTheorem.Foundations.PCP.GenericGraphTables.semantics
semantics def: Decode validated finite data into the actual constraint-graph semantics. · OAI.UniqueGamesTheorem.Foundations.PCP.GraphTables.semantics
semanticStep def: The literal machine command interpreted on an infinite right-hand tape. · OAI.FiniteAlgebraicEnvelopes.Compiler.Simulator.semanticStep
SemanticTape structure: Semantic one-sided tape: its right end is an infinite blank tail, not an extra halting edge. Auxiliary simulator tags do not enter this semantics. · OAI.FiniteAlgebraicEnvelopes.Compiler.Simulator.SemanticTape
semanticToAddress def: The corresponding injection from the original semantic vertex space. · OAI.MaxCutGames.Reduction.AddressGame.semanticToAddress
semanticToAddress def: The corresponding injection from the original semantic vertex space. · OAI.UniqueGamesTheorem.Reduction.AddressGame.semanticToAddress
semidirect_inl_properCocompact def: The rotation subgroup is one fixed compact right-coset cover. · OAI.PolycyclicRecognition.semidirect_inl_properCocompact
semidirectRootExp def: The root exponential in the first coordinate of the semidirect cover. · OAI.PolycyclicRecognition.CartanFactorModel.semidirectRootExp
SemigroupDifference def: Formal differences, before identifying order-equivalent pairs. · OAI.Kaplansky.SemigroupDifference
SemigroupOrderedGroup abbrev: Ordered group completion: order-equivalent differences are identified. This need not embed the original noncancellative monoid injectively. · OAI.Kaplansky.SemigroupOrderedGroup
semigroupOrderedGroupLift def: Every monotone additive map to an ordered group descends to the ordered completion. This constructs the map; it does not postulate a state. · OAI.Kaplansky.semigroupOrderedGroupLift
semigroupToOrderedGroup def: The actual additive map from the monoid into its ordered completion. · OAI.Kaplansky.semigroupToOrderedGroup
semilinearEquiv def: exterior powers commute with semilinear coordinate isomorphisms, including all open-restriction coefficient-ring isomorphisms. · OAI.ActualExterior.semilinearEquiv
seminorm def: The proved seminorm; no bounded-vector convergence is asserted. · OAI.Tingley.UltraSequence.seminorm
seminormUnit def: Closed unit set of the actual seminorm. It is allowed to be a cylinder. · OAI.PettyProjection.seminormUnit
semisimpleOperator def: The full semisimple part, before its imaginary part is removed. · OAI.PolycyclicRecognition.ComplexCartan.semisimpleOperator
SemPort def: The intended input-port semantics; only scalar gates can be seeds. · OAI.WLIdentification.CellularCircuit.SemPort
send def: Law of a function (random variable). · OAI.FourierLLogL.Law.send
sendFirst def: Move first component of frame k to empty y normal order, leaving second on k. · OAI.Fifths.SM.sendFirst
sensitiveEdgeCount def: Number of ordered sensitive edges, indexed by their initial vertex and direction. · OAI.LeanBlast.GotsmanLinial.sensitiveEdgeCount
sensitiveVertices def: Vertices at which reversing coordinate i changes the function. · OAI.LeanBlast.GotsmanLinial.sensitiveVertices
sensitivity def: Maximum ordinary sensitivity over all Boolean inputs. · OAI.Paper320.sensitivity
sensitivityAt def: The ordinary number of sensitive individual input coordinates. · OAI.Paper320.sensitivityAt
separableInvariantStage def: The successive closed convex hulls used in the separable reduction. · OAI.ReflexiveFixedPoints.separableInvariantStage
separableReducingTestSpace def: The canonical separable reducing space of a finite test family. · OAI.KadisonSimilarity.separableReducingTestSpace
SeparableStrongStability def: The separable-Hilbert-space intermediate conclusion of §9. · OAI.KadisonKastler.SeparableStrongStability
SeparableTwoVectorKaplanskyInput def: Blackadar I.9.1.3 and C-star quotient norm lifting, only on two test vectors in a separable represented Hilbert space. This is the exact published closure input used in the reduction below. · OAI.KadisonSimilarity.SeparableTwoVectorKaplanskyInput
SeparableUniformCommutatorEstimate def: The separable version of the actual universal von Neumann estimate. · OAI.KadisonSimilarity.SeparableUniformCommutatorEstimate
separate def: Separate this new disk from any old label shared by its working strip coordinates. No occurrence or existing corner transport is changed. · OAI.IntegralCharacterVarieties.FiniteCycles.Polygon.separate
separate def: The Separation name for an arbitrary formula with an arbitrary finite parameter tuple. Its entries are taken only from the original name's domain. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.separate
Separated def: Pairwise separation of the coordinates of a triple. · OAI.Problem356.CoulombEstimates.Separated
separatedChildren def: Children whose prescribed period does not divide the relative coefficient. · OAI.TreeEdit.BinaryLower.FiniteTree.separatedChildren
separatedFamilyBoundary def: The self-contained construction includes its support proofs. · OAI.CoarseAssembly.DiscreteRoe.separatedFamilyBoundary
separatedJetMatrix def: Genuine mixed-derivative matrix on the source's fixed row and column types. · OAI.Nagata.Workers.W11.separatedJetMatrix
separatedTreeEnergy def: The separated portion of the original coefficient family, as in Eq. (size) of the manuscript. · OAI.LipschitzHilbert.SourceLabeledLattice.separatedTreeEnergy
separateUnitaryProducts def: Products of an algebra unitary and a commutant unitary. · OAI.KadisonKastler.separateUnitaryProducts
separatingSubsets def: Eligible subsets that contain the left mask and avoid the right mask. · OAI.LowSpaceSubsetSum.separatingSubsets
separation def: The instance of Separation for an entirely arbitrary formula p(params,x). All parameters are universally closed. Neither the quantifier complexity of p nor the number of parameters is bounded. · OAI.PartitionConsistency.SetLanguage.separation
separationFirstEquiv def: The first-leg relabeling adds the trivial first coordinate of the dot product. · OAI.MatrixMultiplication.AuxiliarySeparation.separationFirstEquiv
separationPolynomial def: The polynomial after the Fourier projection and square weighting. · OAI.MatrixMultiplication.AuxiliarySeparation.separationPolynomial
separationPolynomialApproximation def: The explicit separation polynomial, viewed as a polynomial approximation to its constant tensor. Its rank bound is supplied by a coordinate expansion. · OAI.MatrixMultiplication.AuxiliarySeparation.separationPolynomialApproximation
separationSideEquiv def: The other two relabelings merely reassociate their coordinates. · OAI.MatrixMultiplication.AuxiliarySeparation.separationSideEquiv
separationTarget def: The surviving tensor: sector labels match on all three legs, while the additional second/third coordinates form a dot product. · OAI.MatrixMultiplication.AuxiliarySeparation.separationTarget
separator def: The source's separator is the seventh letter, distinct from both code alphabets, all four selector markers, and the zero letter. · OAI.FiniteAlgebraicEnvelopes.Bootstrap.Source.separator
separatorCoordinate def: A separator gives one bounded real function for each translation. · OAI.ThompsonNonamenability.separatorCoordinate
separatorDefect def: The finite sum whose mean vanishes by left invariance. · OAI.ThompsonNonamenability.separatorDefect
SeparatorObligation def: The exact remaining sharp-separator obligation for the literal prepared interaction graphs of disk pictures. This is NOT proved or inserted as a hypothesis into the main target: it records the unclosed dependency. · OAI.TorsionFreeZeroDivisors.SampleGraph.Picture.SeparatorObligation
separatorRationalAlpha def: The floor-computed rational slope used in the sampling rule. · OAI.LowSpaceSubsetSum.separatorRationalAlpha
SeparatorSample abbrev: One sample is an indexed, fixed-size subset of the block. · OAI.LowSpaceSubsetSum.SeparatorSample
separatorScanBudget def: A real-valued threshold for the executable integer watchdog. · OAI.LowSpaceSubsetSum.RepresentationSetup.separatorScanBudget
separatorScanLimit def: An integer watchdog threshold computed using only multiplication and natural exponentiation. · OAI.LowSpaceSubsetSum.RepresentationSetup.separatorScanLimit
separatorWorkingCells def: A cell count for every indexed bit of each live separator record, including a conservative node/header multiplier. · OAI.LowSpaceSubsetSum.RepresentationSetup.separatorWorkingCells
sequence def: Sequentially composed bounded logical updates compile by retained history. · OAI.ExactQuantumFactoring.BooleanNetwork.sequence
sequence def: The actual length-two relation subspace, all paths, and their quotient. · OAI.LittleFinitistic.RelationSequence.sequence
sequenceAction def: A concrete star-algebra action, constructed from the trace quotient. · OAI.Generator.Tracial.sequenceAction
sequenceGeneratedCStarAlgebra abbrev: The concrete norm-closed C-star algebra generated by a sequence. · OAI.KadisonSimilarity.sequenceGeneratedCStarAlgebra
sequenceHom def: Contractive coordinate homomorphisms preserve boundedness, with no dimension bound. · OAI.NuclearUltrapower.VariableUltra.sequenceHom
sequenceInclusion def: The continuous Hilbert realization used by the analytic remainder theorem. · OAI.Generator.Tracial.sequenceInclusion
sequenceIso def: The actual independent relation columns and first-arrow path columns are the canonical exact sequence. · OAI.LittleFinitistic.RelationMatrixSequence.sequenceIso
sequenceMatrixEquiv def: The actual star-algebra equivalence between the two bounded-sequence constructions. · OAI.KadisonKastler.sequenceMatrixEquiv
sequenceMatrixHom def: Entrywise transposition of the sequence index and matrix indices. · OAI.KadisonKastler.sequenceMatrixHom
sequenceNorm def: A sequence norm is written as a real power of its finite power sum. · OAI.LipschitzHilbert.PhaseCombinatorics.sequenceNorm
sequenceQuotient def: The actual star-algebra quotient map on complete bounded sequences. · OAI.Generator.Tracial.sequenceQuotient
sequenceRealExtension def: No interpolation or regularity of the envelope is required. · OAI.PolycyclicRecognition.sequenceRealExtension
sequenceScalarHom def: Diagonal amplification of a bounded sequence, as an actual star homomorphism. · OAI.KadisonKastler.sequenceScalarHom
sequenceSplitting def: Over the base field this exact sequence has an actual splitting. · OAI.LittleFinitistic.RelationSequence.sequenceSplitting
sequenceStandardGNSData def: A dense sequence gives a standard representation with a separating vector. · OAI.WeakPureInfiniteness.sequenceStandardGNSData
sequenceTapes def: Exact successive stack updates, in the original field order. · OAI.MaxCutGames.Foundations.Hastad.SourceFieldArray.sequenceTapes
sequenceTapes def: Exact successive stack updates, in the original field order. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceFieldArray.sequenceTapes
sequenceUnitaryEval def: The unitary represented by one coordinate of a bounded unitary sequence. · OAI.KadisonKastler.sequenceUnitaryEval
SequentialApproximation def: Sequential approximation on the whole exponent cube, allowing the approximating sequence to lie outside the cube. · OAI.Problem326.Affine.SequentialApproximation
SequentialHistory inductive: Validity records chronology, not merely membership in an unordered set. · OAI.ReflexiveFixedPoints.SequentialHistory
series def: The product of the graded pieces with graded Cauchy multiplication. · OAI.SectionCompletion.series
seriesAlgebra def: The natural scalar-series algebra structure, derived through an injective matrix realization rather than assuming commutative matrix coefficients. · OAI.RationalHitting.HahnMatrix.seriesAlgebra
seriesProbabilityLaw def: The finite-support law of the hidden motif anchor, not an additional distributional assumption or a changed channel parameter. · OAI.TraceReconstruction.seriesProbabilityLaw
seriesWeight def: The infimum of the real weights of nonzero coefficients. · OAI.CartierSections.seriesWeight
serve def: This declaration is an exact comparator-selected statement. Read the Lean statement below for its mathematical content. · OAI.KServer.serve
serve def: Serve one request by moving exactly the indicated label to the request. · OAI.KServer.serve
serviceCost def: This declaration is an exact comparator-selected statement. Read the Lean statement below for its mathematical content. · OAI.KServer.serviceCost
serviceCost def: Deterministic movement of a label string on a fixed finite request sequence. · OAI.KServer.serviceCost
set def: Semantics of a write into fresh storage; out-of-range writes are ignored. Only used during construction, never a whole-tape functional update instruction. · OAI.PowerSaving.Tape.set
setDensity def: Indicator density of a measurable region in a product space. · OAI.Problem358.NormalizedSections.setDensity
setDigits def: Set the active low coordinates; higher digits and every other tape retain their original contents. Unary digit stacks have no unread suffix. · OAI.MaxCutGames.Foundations.Hastad.SourceOdometerSchedule.setDigits
setDigits def: Odometer coordinate zero is the low coordinate; physical saved positions are reversed to match the actual occurrence-tuple enumeration. · OAI.MaxCutGames.Reduction.AddressOdometerSchedule.setDigits
setDigits def: Set the active low coordinates; higher digits and every other tape retain their original contents. Unary digit stacks have no unread suffix. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceOdometerSchedule.setDigits
setDigits def: Odometer coordinate zero is the low coordinate; physical saved positions are reversed to match the actual occurrence-tuple enumeration. · OAI.UniqueGamesTheorem.Reduction.AddressOdometerSchedule.setDigits
SetHierarchy structure: A reversed predecessor chain, with its first term empty and last term the given set. The relation records the actual rule used to select predecessors. · OAI.LeanBlast.SecondKahnKalai.SetHierarchy
SetPatch structure: Local update data on closed full-column pieces. Cross-domain switches must be unchanged, as ensured by the strict frontier margins in the source. · OAI.PlanarL1.Columns.Model.SetPatch
sets def: Rectangles: free past if e=true, fixed zero otherwise; select S in the current axis; future fixed zero. · OAI.PowerSaving.Cluster.sets
setStar def: Set just the new children and the new park of one star. · OAI.KServer.TreeRounding.Tree.setStar
SetStep def: The multiplicative scalar inequality for one actual selected-set update. · OAI.MaxCutGames.Foundations.Repetition.SelectionSequence.SetStep
SetStep def: The multiplicative scalar inequality for one actual selected-set update. · OAI.UniqueGamesTheorem.Foundations.Repetition.SelectionSequence.SetStep
SetSystem structure: The exact hypotheses in the component-refinement lemma, put on an actual measure space. The vertical bound is the invariant of the cut construction. · OAI.PlanarL1.Columns.Model.SetSystem
settings def: Only natural/rational arithmetic selects the phase precision and all finite loops. The supplied grid point is the evaluator's fixed start. · OAI.ContingencyTables.RootedCountingSchedule.settings
SetUp structure: Uniform hypotheses supplied for the whole finite stage range. · OAI.ExactDerandomization.Net.Scene.SetUp
setupCharge def: This reserve covers the two initializations, the scalar loop limits, and the fixed wrapper bookkeeping. · OAI.LowSpaceSubsetSum.BudgetedPreparedParent.setupCharge
setupInTime def: Actual context preparation and the shared right base are computed once. · OAI.MaxCutGames.Foundations.Hastad.SourceTupleBody.setupInTime
setupInTime def: The driver executes its actual setup statements from its actual input configuration. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Initialization.setupInTime
setupInTime def: Actual program placement is the only execution premise. Header values and unread row bits specify the input, and every auxiliary role starts empty. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Setup.setupInTime
setupInTime def: Actual context preparation and the shared right base are computed once. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceTupleBody.setupInTime
setupInTime def: The driver executes its actual setup statements from its actual input configuration. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Initialization.setupInTime
setupInTime def: Actual program placement is the only execution premise. Header values and unread row bits specify the input, and every auxiliary role starts empty. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Setup.setupInTime
setWeight def: The component weight, including the zero continuation when the set is empty or the length condition fails. · OAI.HenonLaneEmden.LineComponents.setWeight
sew def: Sewing consumes exactly each designated parent interface once. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.CyclicInterface.sew
sgn def: Directed pair couplings. Both directions are counted; diagonal terms are constants. This intermediate presentation allows exact restriction without re-enumerating edges. · OAI.BufferedIsing.DSS.sgn
shade def: aligned spells with coordinates u and v · OAI.PowerSaving.Slant.shade
shades def: quality trip coupons that cover all high nonbad jumps before activation. · OAI.FourierLLogL.Dyadic.Procedure.Paths.Troupe.shades
shadowCoordLinear def: Orthogonal coordinates with the last coordinate in a specified unit direction. · OAI.PettyProjection.shadowCoordLinear
shadowMeasurableEquiv def: The coordinate change preserves the actual standard Euclidean measures. · OAI.PettyProjection.shadowMeasurableEquiv
shadowVolume def: Volume in the induced Euclidean structure on the orthogonal hyperplane. · OAI.PettyProjection.shadowVolume
Shape inductive: The eight choices of non-intersecting arcs in one actual triangular pair. W,S meet the up triangle; E,N meet the down triangle. The forbidden second two-arc choice would use both centers twice and is absent. · OAI.CriticalHoneycomb.Transfer.PhysicalTiles.Shape
Shape inductive: A D/X pattern with length and X-count in its type, avoiding any degree casts. · OAI.DerivedBar.Multibar.Indexed.Shape
Shape structure: analytic clocks and polynomial choices are separated from the running program. · OAI.ExactDerandomization.Chime.Dome.Shape
Shape inductive: Boundary type of a word; unit belongs only to the empty word. · OAI.FiniteAlgebraicEnvelopes.Compiler.Path.Shape
Shape structure: Construction parameters that affect dimensions (except n,N). j = selector degree; r = number of ordered tester pairs in a tester. J, h mix and channel indices. · OAI.HadwigerCounterexample.Shape
Shape abbrev: Table types and the incidence of all seven fan slots of every top. Fans are indexed once, including those used by more than one top. · OAI.HyperbolicObstruction.IncidenceTemplate.Occurrences.Shape
Shape structure: The normalized diagonal spectral shape. · OAI.PositiveEinsteinFour.Bianchi.Shape
shapeAction def: Exact chosen physical lozenge connectivity, before restoring κ^visits. · OAI.CriticalHoneycomb.Transfer.PlanarNormalization.shapeAction
shapeCharge def: A shape-only constant, independent of every assigned depth and system size. · OAI.DilutedSpinGlass.PrescribedTree.shapeCharge
shapeEnergyAt def: Conditional covariance of the actual shape with d preceding unary transitions, averaged over their genuine kernel prefix. · OAI.DilutedSpinGlass.PrescribedTree.shapeEnergyAt
shapeEquiv def: In each degree the resolution is a disjoint union of evaluations of X. · OAI.Thomason.Construction.FaceSimplexResolution.shapeEquiv
shapeHistory def: Literal signed extension history with an arbitrary prescribed split-depth weight. In particular indicator weights encode exact consistent constraints. · OAI.DilutedSpinGlass.PrescribedTree.shapeHistory
shapeIndexEquiv def: The recursive total index is exactly the manuscript's full-degree D/X shape. · OAI.DerivedBar.RecursiveTotal.shapeIndexEquiv
shapeOfBlocks def: Construct the degree-indexed tensor shape of a tuple of block lengths. · OAI.DerivedBar.Multibar.Indexed.shapeOfBlocks
shapeRectangle def: A specified lozenge-shape configuration in a literal finite rectangle, scanned east-to-west with unmarked north/south boundaries. · OAI.CriticalHoneycomb.Transfer.PlanarNormalization.shapeRectangle
shapeRoleConstant def: The manuscript's finite shape/role constant, now a literal natural number independent of the actual child-label sizes and all channel data. · OAI.TraceReconstruction.OccupiedParityTree.shapeRoleConstant
Shapes structure: The small category of strict globe parameters and all strict maps. · OAI.Thomason.Construction.StrictOmegaCategory.Globe.Shapes
shapeSubrep def: Chosen regular constituent. · OAI.CoordinateSweeps.YoungCorner.shapeSubrep
shapeTensor def: Pure orientation kernel for a local connectivity. Indices are W,N,S,E, respectively: output order W,N, input order S,E in the braided drawing. · OAI.CriticalHoneycomb.Transfer.PhysicalTiles.shapeTensor
shared abbrev: A same-wire interface, even when the input and output word encodings differ. · OAI.ExactQuantumFactoring.BitStackProgram.Procedure.shared
sharedCost def: Only truncated subtraction duplicates its operands in the actual compiler. In particular a priority-search tail in an ite is compiled ONCE, not twice. · OAI.ExactQuantumFactoring.NatExpr.sharedCost
sharedFirstDependentTensor def: Branches sharing their first input but with independently sized other legs. · OAI.MatrixMultiplication.AuxiliarySeparation.sharedFirstDependentTensor
sharedFirstTensor def: Branches sharing X, with matching and disjoint Y and Z sectors. · OAI.MatrixMultiplication.AuxiliarySeparation.sharedFirstTensor
sharedOutputLaw def: The law of the two local outputs, driven by a seed chosen independently of the question pair. Local maps receive only their own question. · OAI.MaxCutGames.Foundations.Repetition.sharedOutputLaw
sharedOutputLaw def: The law of the two local outputs, driven by a seed chosen independently of the question pair. Local maps receive only their own question. · OAI.UniqueGamesTheorem.Foundations.Repetition.sharedOutputLaw
sharedT def: b_t read on the full ambient profile. · OAI.HadwigerCounterexample.sharedT
sharingCoverage def: Covered particles for the actual symmetric sharing relation on path data. · OAI.CoordinateSweeps.Coverage.sharingCoverage
sharpAtoms def: The two fixed outer slopes and the variable central slope. · OAI.Problem358.sharpAtoms
sharpAxis0 def: The first coordinate used in the sharpness construction. · OAI.Problem358.sharpAxis0
sharpAxis1 def: The transverse coordinate used in the sharpness construction. · OAI.Problem358.sharpAxis1
SharpCertificate def: A normalized sharp certificate for disks of radius one half. · OAI.SharpPlanar.SharpCertificate
sharpHom def: Taking adjoints in both source and target preserves the algebra map. · OAI.KadisonSimilarity.sharpHom
SharpnessBoundsExample structure: Transport data sufficient for sharpness, needing bounds rather than exact distances. · OAI.Problem358.SharpnessBoundsExample
sharpnessBoundsExample def: The explicit cube maps satisfy a conservative sharpness certificate. This construction uses a small rectangle of disagreement, not the exact L² formula. · OAI.Problem358.sharpnessBoundsExample
SharpProjectionFamily structure: The precise analytic data used in the sharp-projection contradiction. The residual is the actual orthogonal projection residual. · OAI.RegularPointCoordinates.SharpProjectionFamily
sharpRectIntervals def: A thin rectangle contained in the region where the two sharpness maps disagree. · OAI.Problem358.sharpRectIntervals
SharpScalarSobolev def: The sharp scalar inequality in normalized base volume, with the actual Dirichlet pairing. Its geometric proof is independent of this transfer. · OAI.PositiveEinsteinFour.Geometry.SharpScalarSobolev
sharpSliceLaw def: The uniform probability law on the interval [-1,1]. · OAI.Problem358.sharpSliceLaw
sharpTarget def: The common three-atom target family; use b = 0 and b = a / 2. · OAI.Problem358.sharpTarget
sharpWeights def: All three target masses are independent of the transverse slope. · OAI.Problem358.sharpWeights
sheaf def: The Zariski cotangent sheaf. · OAI.ActualCotangent.sheaf
sheafFunctor def: The sheaf of exterior powers. · OAI.ActualExterior.sheafFunctor
sheafificationCompOver def: Sheafification and restriction are naturally isomorphic. · OAI.ActualSheafTensor.sheafificationCompOver
sheafifyOverMap def: The canonical comparison between sheafification and restriction. · OAI.ActualSheafTensor.sheafifyOverMap
sheafPullbackUnitIso def: Canonical identification of the pullback of the unit sheaf. · OAI.ActualSheafTensor.sheafPullbackUnitIso
sheafTensorHomEquiv def: The tensor-Hom equivalence for actual module sheaves. The domain is the sheafification of the genuine sectionwise tensor product, and the codomain is the sheaf of local linear transformations proved to satisfy descent. · OAI.CoherentDual.sheafTensorHomEquiv
shear def: The determinant-one shear that supplies the linear part of a facet extrusion. · OAI.DiagonalZonotope.shear
shear def: Shear of cone coordinates for translations. · OAI.GeneralMahler.Body.shear
shear def: The branch change of variable H ↦ H+ψ(H) is a genuine increasing bijection on the WHOLE real line, not just on the current chart heights. · OAI.PlanarL1.Rebase.shear
shearEquiv def: The independent-variable shear is a polynomial algebra automorphism. · OAI.StableCoordinate.Linearization.shearEquiv
sheetColumn def: Derivative column, extended by zero off the open sheet domain. · OAI.DoublingHilbert.sheetColumn
shell def: Rows above the lower cutoff, inside one half-open power-of-two interval. · OAI.Problem337.VaughanDyadic.shell
shellMajorant def: A radius-independent Borel majorant combining separated shell pairs and ray jumps. · OAI.DiskGeometry.shellMajorant
shellMajorant def: A radius-independent Borel majorant combining separated shell pairs and ray jumps. · OAI.SignedDisk.DiskGeometry.shellMajorant
shift def: A finite exponential shift fixing p, x, y, z and translating u by -p²w. · OAI.ComplexCancellation.Cylinder.Point.shift
shift def: Adding a nonnegative scalar identity to an actual positive finite matrix. · OAI.Kaplansky.CuntzPositiveMatrix.shift
shift def: Rank-one translation in the row kernel. · OAI.MaxCutGames.Inverse.AffineWitness.shift
shift def: Translation acts on all bounded functions because the domain is discrete. · OAI.PlaneFiveColor.InvariantMean.shift
shift def: The directional shift as a linear endomorphism. · OAI.Problem346.PolynomialShift.shift
shift def: The source map: bilateral symbol shift and addition of one in every phase. Its inverse subtracts one, with no modulo-null restriction. · OAI.SmoothObstruction.PhaseDynamics.shift
shift def: Every shift up to the minimum coordinate remains a barycentric point. · OAI.Thomason.Construction.HornBarycentric.shift
shift def: Rank-one translation in the row kernel. · OAI.UniqueGamesTheorem.Inverse.AffineWitness.shift
shiftAggregate def: The same aggregate, evaluated on actual child shifts. · OAI.MaxCutGames.Gadget.shiftAggregate
shiftAggregate def: The same aggregate, evaluated on actual child shifts. · OAI.UniqueGamesTheorem.Gadget.shiftAggregate
shiftColumnIso def: The tensor shift has the Koszul sign (-1)^(s*p) on its pth column. · OAI.LittleFinitistic.BimoduleTotal.shiftColumnIso
shiftComparison def: Full left B-valued derived tensor commutes with every integer shift. · OAI.LittleFinitistic.BimoduleTotal.BoundedProjective.shiftComparison
shiftD def: Constant complex phase shift, still an actual differential operator on Schwartz maps. · OAI.ElasticityPhysicalAlgebra.shiftD
shifted def: The prime-power cyclotomic polynomial translated to put the root at ξ - 1. · OAI.CirculantHadamard.RamifiedEisenstein.shifted
shifted def: A complex constant phase shifts an actual coordinate derivative. · OAI.ElasticityAugmented.shifted
shifted def: The shift by c(0), essential because c need not preserve either 0 or 1. · OAI.FiniteAlgebraicEnvelopes.Contracted.shifted
shifted abbrev: The actual shifted selecting bimodule, with its full B actions. · OAI.LittleFinitistic.SelectedShiftedRealization.shifted
shifted def: Opposite-index shifts are bijections of the block coordinates. · OAI.Release075.CartesianBlock.shifted
shifted def: Evaluation on the shifted interval summand. · OAI.Thomason.Construction.DirectedComplex.RightPath.shifted
ShiftedBochnerLowerBound def: The scalar consequence of M >= |Hess f|² m and tr Hess f = Δf needed to subtract a constant multiple of the metric. · OAI.RegularPointCoordinates.ShiftedBochnerLowerBound
shiftedBochnerMeasure def: Subtracting the trace correction from the full Bochner measure makes a positive bilinear measure on the augmented space. · OAI.RegularPointCoordinates.shiftedBochnerMeasure
ShiftedBound def: Localized norms whose first q derivatives are unweighted. · OAI.ClosedSurfaceR4.FiniteOrderSmoothing.SmoothingAtlas.ShiftedBound
ShiftedCoulombCertificate structure: A nonnegative dual certificate after adding a finite constant to the cost. The shift permits a signed bounded real-valued supporting potential. · OAI.Problem356.ShiftedCoulombCertificate
ShiftedCurvaturePositive def: Pointwise shifted-matrix positivity is the elementary sectional condition consumed by the local tensor estimate. Its geometric derivation from the curvature decomposition is separate from the weighted estimate. · OAI.PositiveEinsteinFour.Geometry.ShiftedCurvaturePositive
shiftedDerivation def: Multiplication by a Laurent monomial changes the filtration shift. · OAI.StableCoordinate.FilteredRees.shiftedDerivation
shiftedDerivative def: Differentiation of the polynomial amplitude after factoring out exp(-t). · OAI.ElasticityBoundarySymbol.shiftedDerivative
shiftedDerivedIso def: The ordinary shifted-derived API agrees with any native model of the identical shifted complex on [-length,0]. No shift-left tensor theorem is reproved or assumed. · OAI.LittleFinitistic.BimoduleTotalPresentation.shiftedDerivedIso
shiftedDerivedTensor def: The native derived tensor functor computed by the source's actual bounded P[b]. · OAI.LittleFinitistic.OrdinarySimulation.shiftedDerivedTensor
shiftedFamily def: The finite union of shifted local families; duplicate labels are harmless. · OAI.Problem326.Affine.shiftedFamily
shiftedInput def: The sole nondeterministic-preparation branch bit precedes the retained N; all remaining physical input coordinates are clean. · OAI.ExactQuantumFactoring.shiftedInput
shiftedIso def: The source's shifted P[b] tensor is the concrete selecting shifted model. · OAI.LittleFinitistic.SelectedOrdinaryOrbit.shiftedIso
shiftedIterateDiagram def: The actual discrete diagram of genuine shifted finite derived tensor iterates. · OAI.DerivedBar.ParityBridge.shiftedIterateDiagram
shiftedIterateInclusion def: The genuine shifted iterate included into actual left-derived base change. · OAI.DerivedBar.ParityBridge.shiftedIterateInclusion
shiftedIterateIsColimit def: Both existence and uniqueness in the derived coproduct universal property hold for the shifted iterates, without assuming a central splitting theorem. · OAI.DerivedBar.ParityBridge.shiftedIterateIsColimit
shiftedLabel def: The cumulative shift assigned to a local family at level j. · OAI.Problem326.Affine.shiftedLabel
shiftedLayer def: The family that observes a single real layer relative to a moving basepoint. · OAI.HenonLaneEmden.LineComponents.shiftedLayer
shiftedMaskedBand def: The shifted finite sum before reindexing its cyclic variable. · OAI.WeakPureInfiniteness.shiftedMaskedBand
shiftedMean def: The affine-noise image of the physical spin mean. · OAI.InvariantIsing.FieldSecondFamily.shiftedMean
shiftedMoment def: The exact linear functional for a fixed subset of the non-good paths. Translation by -1 retains all correlations between repeated uses of a line. · OAI.CoordinateSweeps.MomentExpansion.shiftedMoment
shiftedPieceEquiv def: Degree pieces of a shifted finite free module, as a genuine linear equivalence. · OAI.KLInvariance.Hilbert.shiftedPieceEquiv
shiftedPotential def: A bounded-below signed potential is made nonnegative by a constant shift. · OAI.Problem356.shiftedPotential
shiftedPowerIso def: The same comparison on every actual shifted-derived iterate. · OAI.LittleFinitistic.BimoduleTotalPresentation.shiftedPowerIso
shiftedPrefixDepths def: Charged depths after retaining the genuine shared prefix above d. · OAI.DilutedSpinGlass.PrescribedTree.shiftedPrefixDepths
shiftedPrimeSieve def: Every density and support here is concrete; only the local discrepancy and bounding sum remain to be estimated in the subsequent counting argument. · OAI.TotientAsymptotic.shiftedPrimeSieve
ShiftedPrimitiveHeckeAnalyticData structure: Phragmen--Lindelof derives strip growth for the shifted completion from its entire order-one bound. · OAI.CubicFirstMoment.ShiftedPrimitiveHeckeAnalyticData
shiftedResolutionModel def: The actual P[b] itself is a finite right-projective tensor model with exactly the source support. · OAI.LittleFinitistic.OrdinarySimulation.shiftedResolutionModel
shiftedStaircase def: The smooth staircase whose transition centers form θ + hℤ. · OAI.CircleDomainRigidity.FiniteTransfer.shiftedStaircase
shiftedTangent def: The raw covariance tangent of the affine-noise image of the value. · OAI.InvariantIsing.FieldSecondFamily.shiftedTangent
shiftedThetaProduct def: The actual theta product evaluated at the shrinking center. · OAI.Nagata.W22.shiftedThetaProduct
shiftedTotalIso def: A genuine ordinary bimodule in degree zero has exactly its ordinary tensor complex as total. · OAI.LittleFinitistic.SingleBimoduleTensor.shiftedTotalIso
shiftedTriplingIndex def: The indices obtained by repeatedly applying the affine tripling map. · OAI.MatrixMultiplication.AuxiliarySeparation.shiftedTriplingIndex
shiftedWeights def: Coordinates after shifting a nonnegative amount of mass to the cone vertex. · OAI.Thomason.Construction.HornBarycentric.shiftedWeights
shiftEquiv def: The explicit exponential substitution, with inverse obtained by negating its parameter. · OAI.StableCoordinate.ExponentialShift.shiftEquiv
shiftIndex def: Integer height separates words of different lengths; the binary coordinate partitions every level into the two isometric ranges. · OAI.NuclearUltrapower.CuntzModel.shiftIndex
shiftIso def: Naturality of the full tensor shift, in every input complex and chain map. · OAI.LittleFinitistic.BimoduleTotal.shiftIso
shiftIso def: The additive extension respects the native formal shift. · OAI.LittleFinitistic.SourceImageEvaluation.shiftIso
shiftKernel def: The logical kernel viewed as a submodule of the ambient input space. · OAI.MaxCutGames.Gadget.Quotient.shiftKernel
shiftKernel def: The logical kernel viewed as a submodule of the ambient input space. · OAI.UniqueGamesTheorem.Gadget.Quotient.shiftKernel
shiftLetter def: False letters are generators; true letters are their actual adjoints. · OAI.Kaplansky.shiftLetter
shiftOffset def: Shift an affine label without changing its slope. · OAI.Problem326.Affine.Label.shiftOffset
shiftQuotientEquiv def: The explicit exponential substitution identifies the two hypersurface rings. · OAI.StableCoordinate.Stabilization.shiftQuotientEquiv
ShiftRelation def: A valid source-occupation shift arising from two paths at a single layer. · OAI.Problem335.LocalMoments.ShiftRelation
ShiftScheme structure: Data of either of the two small-angle links. At V use the first block coordinate; at W use the second and swap u,v. The membership and uniqueness fields are exactly the partition of I(z) into blocks, not geometric conclusions. · OAI.Release075.ShiftScheme
shiftSignIso def: The source's degree-n sign (-1)^(b*n), for all integers b and n. · OAI.LittleFinitistic.OrdinarySimulation.shiftSignIso
shiftTotalIso def: The complete signed total commutes with every integer shift, with the actual horizontal maps. · OAI.LittleFinitistic.BimoduleTotal.shiftTotalIso
shiftTy abbrev: count halving powers · OAI.PowerSaving.RAM.Bench.shiftTy
shiftVector def: Translation of a tensor vector by the stable-letter coordinate. · OAI.NuclearUltrapower.shiftVector
shoots def: List from endpoint j to endpoint k of the other unit. · OAI.HadwigerCounterexample.Volley.shoots
ShortBypass def: A geodesic arc through o, closed by a short bypass outside the open T-ball. · OAI.Release075.Discrete.ShortBypass
shortcodeAcceptance def: Uniform shortcode sampling; neither nonzero-factor conditioning nor rank-one-matrix-uniform sampling is substituted for its two independent factors. · OAI.MaxCutGames.Decoder.SurrogateFibers.shortcodeAcceptance
shortcodeAcceptance def: Uniform shortcode sampling; neither nonzero-factor conditioning nor rank-one-matrix-uniform sampling is substituted for its two independent factors. · OAI.UniqueGamesTheorem.Decoder.SurrogateFibers.shortcodeAcceptance
ShortColumnGeometry def: The geometric conclusions required on every actual short perfect block. · OAI.TwoPointCorrelations.ShortColumnGeometry
shortCombination def: The uncentered interpolated short average. · OAI.JointDickman.PublishedInputs.shortCombination
shortComplexTildeIso def: The affine module/sheaf comparison respects short complexes. · OAI.CoherentGlobal.shortComplexTildeIso
shortcut def: Replace several successive edges by one segment already known to lie in the permitted set. · OAI.PlanarL1.Topology.SegmentChain.shortcut
shortcutAt def: Delete the part between two intersecting, nonadjacent segments. · OAI.PlanarL1.Topology.SegmentChain.shortcutAt
ShortDirect def: Direct sum for distinct selections of at most m members of a family. · OAI.KaplanskyCounterexample.FiniteGeometry.ShortDirect
shortEquiv def: The canonical presentation isomorphism fixes every named generator. · OAI.KaplanskyCounterexample.GluedGroup.shortEquiv
ShortestAugmentation def: An actual saturating augmentation along a shortest residual path. The bottleneck is positive, bounded by every residual capacity, and attained by an arc. The displayed update adds it forward and subtracts it backward on precisely the … · OAI.ContingencyTables.PublishedEdmondsKarp.ShortestAugmentation
shortExponentialIntegral def: The mean over interval origins, with a single fixed frequency. · OAI.TwoPointCorrelations.shortExponentialIntegral
shortExponentialSum def: Sum over the integers y < n ≤ y+H (for the nonnegative starting points used in the integral). · OAI.TwoPointCorrelations.shortExponentialSum
ShortFactorFamily structure: The arithmetic data of the actual short factors, before choosing a set of characters on which their values are large. · OAI.CubicFirstMoment.ShortFactorFamily
shortLengthError def: The short-length error; it is fixed in the inner scale limit. · OAI.JointDickman.PublishedInputs.shortLengthError
shortlex abbrev: The shortlex order of literal encodings, fixed independently of a budget. The alphabet order may be the concrete compiler's numerical symbol order. · OAI.FiniteAlgebraicEnvelopes.Compiler.Path.shortlex
shortPart def: The part of an arithmetic function supported on integers at most U. · OAI.Problem337.Vaughan.shortPart
shortPresence def: Auxiliary short lifetime: a trial on each positive age increment, compulsory retirement at K. It does not define the actual partition. · OAI.KServer.RosterLifetimes.shortPresence
shortResidues def: The finite residue neighborhood used to detect same-chunk collisions. · OAI.AsymptoticallyMinimalLittlewood.Packing.shortResidues
shortSeamSideValues def: The side value of a short seam, in the coordinate labels of the original seam. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.shortSeamSideValues
shortTransform def: The symmetrized form of the hard-truncated directional integral. · OAI.LipschitzHilbert.shortTransform
shortTransform def: The positive-time, symmetrized short directional Hilbert transform. · OAI.LipschitzHilbert.shortTransform
shortTypeIFour def: The four-factor Type I term, keeping both short Möbius factors explicit. · OAI.CubicFirstMoment.shortTypeIFour
shortTypeITwo def: The two-factor Type I term furnished by the short inverse. · OAI.CubicFirstMoment.shortTypeITwo
shortWindowSum def: The short Fourier polynomial indexed by increments 1,...,D. · OAI.TwoPointCorrelations.shortWindowSum
shoulder def: simpler shoulder for the hierarchy threshold; gap W. · OAI.ExactDerandomization.Analysis.shoulder
shrinkingCircularWeight def: The original weight is recovered at the original radius, even though both circular bumps vanish on and outside its boundary. · OAI.ClosedSurfaceR4.FiniteOrderSmoothing.shrinkingCircularWeight
Side abbrev: A side is an occurrence, not a facet: none denotes the parent and some j the j-th ordered child. The same facet may appear arbitrarily often. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.Side
Side structure: A stage on one prescribed side. Its marked vertices do not include any isolated vertices of inactive chains. The source number of available vertices is n on EACH side, not the cardinality 2n of their disjoint union. · OAI.TorsionFreeZeroDivisors.TwoSideStage.Side
side def: Drop comparisons exactly when either whole path has been deleted. · OAI.TorsionFreeZeroDivisors.RestrictComparisons.side
side def: The open side from vertex i to the next vertex, indexed cyclically. · OAI.TriangularBilliards.Triangle.side
side def: Side length in the original (unrotated) plane, after scaling by exp m. · OAI.WeakPinned.Grid.side
sideAllowed def: Labels allowed on the side selected by b; a label survives precisely when both of its incident edges have the opposite sign. · OAI.EuclideanFiveColor.sideAllowed
sideCircle def: The old boundary circle of an occurrence, read from the exhaustive boundarySide equivalence, not inferred from its rank or its facet color. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.sideCircle
SideCompatible def: Both actual one-sided strip restrictions on a palette. · OAI.EuclideanFiveColor.SideCompatible
sideCongr def: Changing the ambient permutation off a retained invariant dart set does not change its actual rotation vertices. · OAI.FKCRT.PermutationWedge.sideCongr
sideEmbedding def: A side-preserving graph isomorphism transports the actual rooted embedding, not just its edge or vertex counts. · OAI.TorsionFreeZeroDivisors.PortSubdivision.Iso.sideEmbedding
sideEntries def: Separate parents from their individual ordered child occurrences. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.PortWiring.sideEntries
sideEquiv def: Number the left side first and the right side second. · OAI.MaxCutGames.Reduction.Encoding.sideEquiv
sideEquiv def: Number the left side first and the right side second. · OAI.UniqueGamesTheorem.Reduction.Encoding.sideEquiv
sideGlueRaw def: Glue the prescribed initial map and boundary homotopy on the closed bottom-and-sides subspace. · OAI.HyperbolicObstruction.CubicalHomotopyExtension.sideGlueRaw
sideGraphCongr def: The all-edge graph is invariant under restriction/normalization on a closed set of darts; only the two literal permutation actions are used. · OAI.FKCRT.PermutationWedge.sideGraphCongr
SideHeight def: The stronger side-height information used when tensoring the basis. It is supplied by the already proved signed simplex boundary order. · OAI.Thomason.Construction.DirectedComplex.OrderedAtomBasis.SideHeight
sideInput def: Actual side totals and reset times are derived from the epoch recurrence. · OAI.KServer.CoarseSide.sideInput
sideMap def: Each original globe gives its actual positive-table cylinder inclusion. · OAI.Thomason.Construction.PastingGlobeCylinderBasis.sideMap
sideMap def: The image of an original vertex under the cut prism map. · OAI.Thomason.Construction.PrismBarycentric.sideMap
sideNeighbor def: The neighbor of a center represented by one of its three actual edges. · OAI.CriticalHoneycomb.Transfer.FiniteConfigurations.sideNeighbor
sideNumber def: Numerical side labels are injective because the finite arity is retained by the seam itself. Used only to discharge dependent casts at sewing. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.sideNumber
sideOf def: The side belonging to one local germ after whole-port wiring. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.PortWiring.sideOf
Sides structure: Three marked, equally subdivided sides of a triangular patch. The marks are combinatorial; the vertices and faces need not be mapped injectively. · OAI.ArtinCAT0.TrianglePatch.Sides
sides def: Oriented interval sides; the manuscript's I counts pairs. · OAI.TorsionFreeZeroDivisors.PlanarParameters.sides
sideSensitivity def: Maximum sensitivity over all indices and inputs with the indicated output. · OAI.Paper320.sideSensitivity
sideStripRoot def: Root of the literal doubled strip containing this side, if any. Parent sides receive none. Selection is unique by strip disjointness. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.RealizedBand.sideStripRoot
SideThin def: Each vertex of p is within δ of a vertex of q or r. · OAI.Release075.SideThin
sieve def: The actual multiplicity-preserving sieve for an even target. · OAI.Problem337.GoldbachSelberg.sieve
sieveCharacteristic def: The monotone zero/one characteristic map of a downward-closed subset. · OAI.Thomason.Construction.sieveCharacteristic
SieveDimension def: Ford's interval condition (Ω). Primes outside P have local density zero. · OAI.JointDickman.SieveDimension
sieveIntersection def: Mass of the intersection of the events indexed by D. · OAI.JointDickman.sieveIntersection
sieveIntersectionError def: Total error in the finitely many intersection counts used by a sieve truncation. Each subset appears just once. · OAI.TwoPointCorrelations.sieveIntersectionError
sieveRemainder def: Ford's remainder r_d, with D the prime factors of d. · OAI.JointDickman.sieveRemainder
SieveRetraction structure: A downward-closed full subposet whose inclusion has a right adjoint. · OAI.Thomason.Construction.SieveRetraction
siftedPairs def: The original integer coordinates left after sieving both prime factors. · OAI.Problem337.PrimePairSieve.siftedPairs
Sigma abbrev: Exactly the maps whose cone lies in S. · OAI.LittleFinitistic.TwoConeVerdier.ConePair.Sigma
sigma def: Normalized cubic spectral moment. · OAI.PositiveEinsteinFour.Bianchi.Shape.sigma
sigma def: An injective code greater than every entry of its final metadata. · OAI.SeparableQuotient.PathCoding.sigma
sigma def: Reported metadata is retained, so the squared moment of a normalized score assembles without a mixing penalty. No conditioning of the original latent metadata distribution occurs. · OAI.TraceReconstruction.ClippedOracle.sigma
sigmaCoeff def: The unreduced Lott–Sturm–Villani distortion coefficient σ. · OAI.WeakHessian.sigmaCoeff
sigmaCycle def: A countable family of normalized cycles gives an actual locally finite cycle. · OAI.CoarseAssembly.Analytic.sigmaCycle
sigmaFinSucc def: Separating the first output block, without inserting extra coordinates. · OAI.ExactFourier.sigmaFinSucc
sigmaHomologyBasis def: The integral H1 basis of an arbitrary disjoint union of copies of a space with H1=Z; all chains and decompositions have genuinely finite support. · OAI.EilenbergGanea.CubicalSingular.sigmaHomologyBasis
sign def: The manuscript's ±1 spin, represented by Bool. · OAI.DilutedSpinGlass.Pattern.sign
sign def: A real-valued binary character, with the uniform Fourier normalization. · OAI.MaxCutGames.Decoder.PrivateFourier.sign
sign def: The real-valued character of the two-element field. · OAI.SmoothObstruction.BinaryCharacter.sign
sign def: A real-valued binary character, with the uniform Fourier normalization. · OAI.UniqueGamesTheorem.Decoder.PrivateFourier.sign
signalCircleModel def: The exact three-mode index signal, in the source's positive-moment convention. · OAI.ThreeTorus.signalCircleModel
signalJitter def: Density of J_*=(R/B+h/P)/c; the finite representatives are h.val. · OAI.ThreeTorus.signalJitter
signalJoint def: Joint law; its marginal over the residue coordinate has total mass one. · OAI.ThreeTorus.signalJoint
signalKernel def: The transmitting kernel when the old coefficient modulus is independent of the spare coordinate and its jitter law has density rho. · OAI.ThreeTorus.signalKernel
signalPrimeProduct def: The manuscript's finite sieve modulus, unchanged (not a factorial proxy). · OAI.ThreeTorus.signalPrimeProduct
signalRamanujanC def: The exact Ramanujan coefficient in the joint law of Section 3. · OAI.ThreeTorus.signalRamanujanC
Signature structure: All local control data have sizes determined solely by the repetition parameter. Global variable and clause indices are absent. · OAI.MaxCutGames.Foundations.Hastad.SourceLocalSignature.Signature
Signature structure: All local control data have sizes determined solely by the repetition parameter. Global variable and clause indices are absent. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceLocalSignature.Signature
signatureCanonicalSlot def: The first slot matching the visible selected name, computed using only the equality matrix. The selected slot guarantees visible support. · OAI.MaxCutGames.Foundations.Hastad.SourceLocalSignature.signatureCanonicalSlot
signatureCanonicalSlot def: The first slot matching the visible selected name, computed using only the equality matrix. The selected slot guarantees visible support. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceLocalSignature.signatureCanonicalSlot
signatureKeyMap def: In the finite-key application this map is the subtype's injective inclusion into the ambient literal box/band pairs. · OAI.QuantitativeVanDerWaerden.signatureKeyMap
SignatureMatch def: Complete ordered names on each internal whole seam, before closing the four short-end interfaces. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.PortPatch.SignatureMatch
signatureSatisfies def: Local clause truth is computed from signs and the three answer bits. · OAI.MaxCutGames.Foundations.Hastad.SourceLocalSignature.signatureSatisfies
signatureSatisfies def: Local clause truth is computed from signs and the three answer bits. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceLocalSignature.signatureSatisfies
signC def: The complex-valued sign character. · OAI.Saxl.signC
signedAlphabet def: A finite alphabet of signs and coefficients. · OAI.Generator.Tracial.signedAlphabet
signedAssembly def: Summation of genuine tensor blocks followed by the source's sign rescaling. · OAI.DerivedBar.Multibar.Indexed.signedAssembly
signedAssignment def: Signs remain separate at every position, even inside one unsigned class. · OAI.HyperbolicObstruction.Coincidence.signedAssignment
signedAt def: Indexed input lookup, one explicit sign membership test, and negation. · OAI.LowSpaceSubsetSum.RepresentationKeys.signedAt
signedAtomRole def: The first squared-amplitude branch supplies the copied roles. The other branch is retained outside throughout the transfer schedule. · OAI.Ostmann.signedAtomRole
signedAverage def: Signed normalized average on a centered Euclidean disk in the complex plane. · OAI.DiskMoments.signedAverage
signedAverage def: Signed normalized average on a centered Euclidean disk in the complex plane. · OAI.SignedDisk.DiskMoments.signedAverage
signedBand def: Signed envelope over all real radii in the closed band, including its endpoints. · OAI.SignedDisk.CenteredDiskEndpoint.signedBand
signedBlockComplex def: Signed block chain model, still indexed by the full homological degree r+Σiⱼ. · OAI.DerivedBar.Multibar.Indexed.signedBlockComplex
signedBlockIso def: Chain isomorphism between the signed block model and the preserved count subcomplex. · OAI.DerivedBar.Multibar.Indexed.signedBlockIso
SignedCandidate def: Changing coordinate order changes no candidate sets. · OAI.PureTypeSystem.Candidates.SignedCandidate
signedCoordinateHalfspace def: A finite signed halfspace cover, using actual continuous linear coordinates. · OAI.Kaplansky.signedCoordinateHalfspace
signedCStarRepresentation def: The signed matrix representation with the C-star matrix type explicit. · OAI.KadisonKastler.FullMatrixCorner.signedCStarRepresentation
signedDifference def: Signed difference of arbitrary naturals, in canonical signed magnitude. · OAI.RationalHitting.StackLanguage.signedDifference
signedDyadicUnion def: The paired pullbacks used to cover all signed scales. · OAI.Problem310Support.signedDyadicUnion
signedGroupUnion def: The locally finite group carried by the countable matrix units. · OAI.KadisonKastler.FullMatrixCorner.signedGroupUnion
signedInitialDegreeBound def: Explicit low-degree majorant from the signed finite-difference estimate. · OAI.TraceReconstruction.signedInitialDegreeBound
signedLtInvocations def: Signed comparison may compare both words with zero. Its total recursive Boolean-comparison frames are bounded by the explicitly counted work. · OAI.EditApproximation.BinaryFraction.signedLtInvocations
signedMap def: Signed slot action in the orthonormal word basis. It reorders the odd subsequence, and does not introduce a sign for even slots. · OAI.RowColumn.Signed.signedMap
signedOccupation def: Signed occupations, retaining the layer coordinate explicitly. · OAI.Problem335.LayeredFourPath.signedOccupation
signedOccupation def: An occupation vector viewed as a signed exponent vector. · OAI.Problem335.signedOccupation
signedOddKernelCharacter def: The odd-part character, with the parity twist prescribed by reciprocity. · OAI.Ostmann.signedOddKernelCharacter
signedPart def: The positive or negative part of an integral chain, coefficient by coefficient. · OAI.Thomason.Construction.SimplexComplex.signedPart
SignedPath inductive: Finite directed walks, with repeated edges permitted and parity retained. · OAI.PureTypeSystem.SignedPath
signedPermutation def: A determinant-one signed permutation with the requested underlying cycle. One sign correction at i0 is enough and does not alter any coordinate norm. · OAI.HarmonicCounterexample.Cycling.signedPermutation
signedPermutationGroup def: The actual finite subgroup used in the matrix separability homotopy. · OAI.HochschildMatrixSystems.MatrixSystem.signedPermutationGroup
signedPermutationHom def: The finite signed-permutation group acts by actual unitary matrices. · OAI.KadisonKastler.signedPermutationHom
signedPower def: Power of a balanced signed integer without the exponentially expanding positive/negative multiplication tree. The sign test is static in parity and uses only absolute value and linear-length natural-power expressions. · OAI.ExactQuantumFactoring.IntExpr.signedPower
signedPower def: Exact rational power, with polynomial syntax in k, including zero unreduced denominators. Existing pow and its semantics are not changed. · OAI.ExactQuantumFactoring.RatExpr.signedPower
signedProjection def: Projection to count r, rescaled by one minus sign per base/ideal inversion. For block lengths i₀,...,iᵣ the exponent is Σⱼ (r-j)iⱼ. · OAI.DerivedBar.SquareZero.signedProjection
signedRankCoefficient def: Exact degree-r coefficient difference of the actual randomized motif in the marked central run, prior to any rank clipping. · OAI.TraceReconstruction.signedRankCoefficient
signedRankProduct def: Product coefficients of signed gap series, including an arbitrary translation of each observed rank gap. · OAI.TraceReconstruction.signedRankProduct
signedRep def: The literal signed action, as a linear representation. · OAI.CoordinateSweeps.YoungCorner.signedRep
SignedRepresents def: The signed combinatorial reflection action is the actual root sign under the faithful Tits representation. · OAI.KLInvariance.TitsSpace.SignedRepresents
signedShift def: Every path chooses exactly one coordinate in every layer. · OAI.Problem335.LayeredFourPath.signedShift
signedShiftComm def: The standard signed commutation structure for a cochain shift. · OAI.LittleFinitistic.BimoduleTotal.signedShiftComm
signedSource def: The integer occupation after subtracting one path shift from another. · OAI.Problem335.ShiftedOccupation.signedSource
signedSourceWeight def: Multiply a finite signed source by a bounded continuous weight. · OAI.RegularPointCoordinates.signedSourceWeight
signedValue def: The same positive construction gives a canonical signed value by positive and negative parts. Additivity on general commuting elements is not assumed. · OAI.Kaplansky.CuntzDimensionState.signedValue
signEquiv def: Actual D-linear involution giving the manuscript's component rescaling. This alone is not the chain comparison or the derived tensor identification. · OAI.DerivedBar.SquareZero.signEquiv
signLayoutEquiv def: Independent uniform signs can be expressed in original or permuted coordinates. · OAI.LowSpaceSubsetSum.signLayoutEquiv
signMatrix def: Multiplication by a real function on the cube, represented in the point-mass basis. · OAI.LeanBlast.GotsmanLinial.signMatrix
signMultiplier def: The same diagonal action on the counting-inner-product Euclidean space. · OAI.LeanBlast.GotsmanLinial.signMultiplier
Signs def: Literal sign hypotheses for the four blocks, without a norm premise. · OAI.CirculantHadamard.BinaryCoefficients.Signs
Signs structure: Coordinate reflections form an elementary two-group. · OAI.SharpLiebThirring.CubeFlags.Signs
signScalar def: Ordinary permutation sign, embedded into the complex scalars. · OAI.CoordinateSweeps.SparseDimension.signScalar
signScalar def: Ordinary source sign. · OAI.CoordinateSweeps.YoungCorner.signScalar
signStateEquiv def: Reassociate finite registers without adding a transition. · OAI.MaxCutGames.Foundations.Hastad.SourceSignaturePrepare.signStateEquiv
signStateEquiv def: Reassociate finite registers without adding a transition. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceSignaturePrepare.signStateEquiv
signTwist def: Tensoring with the ordinary sign character, on exactly the same space. · OAI.CoordinateSweeps.SparseDimension.signTwist
signTwist def: Twisting by the actual sign character, on the same complex vector space. · OAI.Saxl.signTwist
silentMarkExpectation def: Independent silent block erasures and conditionally emitted primitive marks. An emitted empty mark is retained in the list and differs from a silent erasure. · OAI.TraceReconstruction.silentMarkExpectation
Sim abbrev: An instance of the simultaneous kernel program. · OAI.PowerSaving.RAM.Sim
similarityCenterValue def: The value at the blowup center after restoring amplitude and phase. · OAI.DefocusingNLS.similarityCenterValue
SimilarityTheorem def: The exact unrestricted similarity statement, Theorem thm:similarity. · OAI.KadisonSimilarity.SimilarityTheorem
SimilarToStar def: Similarity by a bounded invertible operator to an adjoint-preserving homomorphism. · OAI.KadisonSimilarity.SimilarToStar
SimpleBipartite structure: A canonical orientation from the false side to the true side. Injectivity is on occurrence indices, so two distinct list entries cannot be parallel even when their constraint tables are different. · OAI.MaxCutGames.Explicit.MachineOutputContract.SimpleBipartite
SimpleBipartite structure: A canonical orientation from the false side to the true side. Injectivity is on occurrence indices, so two distinct list entries cannot be parallel even when their constraint tables are different. · OAI.UniqueGamesTheorem.Explicit.MachineOutputContract.SimpleBipartite
SimpleCycle structure: A simple combinatorial cycle with a chosen direction of traversal. · OAI.Paper170.SimpleCycle
simpleFixedAction def: The reflection is linear over its actual invariant subalgebra. · OAI.KLInvariance.TitsSpace.simpleFixedAction
simpleGraph def: Forgetting multiplicities, with loops contributing only the already present vertex to a closed neighborhood. · OAI.TorsionFreeZeroDivisors.MatchingDiameter.simpleGraph
simpleGraph def: Forgetting loops and multiplicities, only for measuring component distances. · OAI.TorsionFreeZeroDivisors.ShortClosure.simpleGraph
simpleLengths def: Lengths realized by words with no repeated column. · OAI.PlanarL1.Columns.Model.Point.simpleLengths
simpleRoot def: Canonical simple positive roots, indexed by Bool just like the system. · OAI.KLInvariance.TitsSpace.PlaneBasis.simpleRoot
SimpleStable def: Stable joint colors, transported to actual finite vertex types. · OAI.WLIdentification.SimpleStable
simpleTorIsoTerm def: The derived tensor with every actual simple right module is the tensor of that simple module with the n-th minimal projective-resolution term. · OAI.LittleFinitistic.PDTensor.simpleTorIsoTerm
SimpleWordRelator inductive: Exactly the source03 finite presentation: all simple cycles of each included fan and compatible included top graph, plus the two decorations. The underlying template graph is not quotiented by its random labels. · OAI.HyperbolicObstruction.LetterPresentation.SimpleWordRelator
simplex def: The convex hull of the origin and the d standard unit vectors. · OAI.ProjectionCounterexample.simplex
simplexAffineSSet def: Interpolation into actual singular simplices, in every degree. · OAI.Kaplansky.simplexAffineSSet
simplexAtoms def: All atom data, including generation, are constructed from normalized simplex identities. · OAI.Thomason.Construction.simplexAtoms
SimplexAtomsInput structure: Precisely the published atom facts used in the full-face argument. · OAI.Thomason.Construction.SimplexAtomsInput
simplexCarrier def: The subcomplex generated by the restriction to the nonzero barycentric coordinates. · OAI.Thomason.Construction.RealizationPrism.simplexCarrier
simplexChainMaximum def: Greatest vertex as a natural transformation on the finite ordered sets indexing simplicial sets. · OAI.Thomason.Construction.simplexChainMaximum
simplexCocone def: Reflect the actual oriental presentation into the chosen finite dimension. · OAI.Thomason.Construction.FiniteReflectionInput.simplexCocone
simplexCocone def: The oriental presentation, including degenerate simplices. · OAI.Thomason.Construction.NormalizedSimplexIdentities.simplexCocone
simplexCocone_isColimit def: Its colimit property is preserved by the reflection adjunction. · OAI.Thomason.Construction.FiniteReflectionInput.simplexCocone_isColimit
simplexCocone_isColimit def: This is the actual colimit presentation of categorification. · OAI.Thomason.Construction.NormalizedSimplexIdentities.simplexCocone_isColimit
simplexCombination def: Affine interpolation of vertices in an actual standard simplex. · OAI.Kaplansky.simplexCombination
simplexCombinationMap def: Interpolation is continuous in the barycentric coordinates. · OAI.Kaplansky.simplexCombinationMap
simplexComparison def: The ordered simplex cylinder compares naturally with the global right tensor. · OAI.Thomason.ConstructedInterval.simplexComparison
simplexCoordinates def: Canonical homeomorphism to a closed simplex with distinct vertices e. · OAI.HyperbolicObstruction.FinitePolyhedron.simplexCoordinates
simplexCoordinates def: Omit the first barycentric coordinate. Its value is recovered by the sum. · OAI.Thomason.Construction.SimplexPolytopes.simplexCoordinates
simplexCubeClamp def: Truncate the last vertex of a standard simplex to its preceding one. · OAI.Thomason.Construction.simplexCubeClamp
simplexCubeLast def: The final Boolean coordinate records exactly the final simplex vertex. · OAI.Thomason.Construction.simplexCubeLast
simplexCylinderIso def: In particular, the actual cylinder of each oriental is its literal normalized-chain right cylinder. · OAI.Thomason.Construction.StrictOmegaCategory.GrayChainExtension.simplexCylinderIso
simplexDensityCocone abbrev: The standard-simplex density presentation, with the harmless universe lift already included in SSet.stdSimplex. · OAI.Thomason.Construction.simplexDensityCocone
simplexFacetLinear def: Linear part of each simplex facet chart. · OAI.ProjectionCounterexample.simplexFacetLinear
simplexFiberCocone def: Take actual finite-dimensional full endpoint fibres of the pieces. · OAI.Thomason.Construction.FiniteReflectionInput.simplexFiberCocone
simplexFiberCocone def: Take one full endpoint fibre of every labelled oriental piece. · OAI.Thomason.Construction.NormalizedSimplexIdentities.simplexFiberCocone
simplexFiberCocone_isColimit def: Their colimit is the finite full fibre. · OAI.Thomason.Construction.FiniteReflectionInput.simplexFiberCocone_isColimit
simplexFiberCocone_isColimit def: The actual full fibre is the colimit of the actual full simplex fibres. · OAI.Thomason.Construction.NormalizedSimplexIdentities.simplexFiberCocone_isColimit
simplexFinalContraction def: The explicit monotone contraction from a simplex to its final vertex. · OAI.Thomason.Construction.simplexFinalContraction
simplexFunctional def: Outward defining functionals: the sum facet, then the coordinate facets. · OAI.ProjectionCounterexample.simplexFunctional
simplexHomeomorph def: Standard barycentric coordinates on the realization of a representable. · OAI.Thomason.Construction.HornBarycentric.simplexHomeomorph
simplexHomotopyFiberEquiv def: The projection of the genuine homotopy fibre over a geometric simplex is a homotopy equivalence, for any continuous map into that simplex. · OAI.Thomason.Construction.simplexHomotopyFiberEquiv
simplexIndex abbrev: The jth maximal simplex of the standard simplicial prism. · OAI.Thomason.Construction.PrismBarycentric.simplexIndex
simplexInitialContraction def: The ordered simplex contracts simplicially to its initial vertex. · OAI.Thomason.Construction.simplexInitialContraction
simplexInitialVertexChain def: A degreewise section, not claimed to be natural in the ordinal. · OAI.Thomason.Construction.simplexInitialVertexChain
simplexLastVertex def: The last-vertex map on the subdivisions of the standard simplices. · OAI.Thomason.Construction.simplexLastVertex
simplexLastVertexSection def: The section on the nerve of each ordinal. · OAI.Thomason.Construction.simplexLastVertexSection
simplexLeg def: The canonical oriental attached to a simplex of a simplicial set. · OAI.Thomason.Construction.NormalizedSimplexIdentities.simplexLeg
simplexLoopContraction def: The explicit straight-line nullhomotopy of a based simplex-valued loop. · OAI.Thomason.Construction.HomotopyMaps.simplexLoopContraction
simplexMaximumMap def: The ordinal map given by the maxima of the vertices of a simplex in the chain-poset nerve. · OAI.Thomason.Construction.simplexMaximumMap
simplexNaturality def: The lifted ordinal map is an actual map between the corresponding simplices of the simplicial prism, including degenerate ordinal maps. · OAI.Thomason.Construction.PrismBarycentric.simplexNaturality
simplexNerveIso def: The standard cosimplicial simplex is the nerve of its ordered vertex set. · OAI.Thomason.Construction.simplexNerveIso
simplexOrder def: A poset simplex as its monotone list of vertices. · OAI.Thomason.Construction.NormalizedSimplexIdentities.simplexOrder
simplexPoint def: Barycentric coordinates on one original realized simplex. · OAI.Thomason.Construction.HornBarycentric.simplexPoint
simplexPoint def: The continuous map from the standard barycentric simplex specified by a simplex of K. · OAI.Thomason.Construction.RealizationPrism.simplexPoint
simplexPointInterpolation def: Convex interpolation in the barycentric coordinates of a standard simplex. · OAI.Thomason.Construction.simplexPointInterpolation
SimplexPoints abbrev: The disjoint union of all simplex parameter spaces, with their given topologies. · OAI.Thomason.Construction.RealizationPrism.SimplexPoints
simplexPosetLabel def: The ordered label on one oriental in the nerve of a poset. · OAI.Thomason.Construction.NormalizedSimplexIdentities.simplexPosetLabel
simplexPullbackCocone def: The canonical cocone is compatible with the original total space. · OAI.Thomason.Construction.simplexPullbackCocone
simplexPullbackDesc def: The induced map is defined through the actual density colimit. · OAI.Thomason.Construction.simplexPullbackDesc
simplexPullbackDescCocone def: Every cocone on base-simplex pullbacks induces a compatible cocone on the standard simplices of the original total space. · OAI.Thomason.Construction.simplexPullbackDescCocone
simplexPullbackDiagram abbrev: Pull back the original fibration along every simplex of its base, including degenerate simplices and every ordinal map between them. · OAI.Thomason.Construction.simplexPullbackDiagram
simplexPullbackIsColimit def: The canonical pullback cocone really reconstructs the original total simplicial set. No Kan or weak-equivalence assumption is needed here. · OAI.Thomason.Construction.simplexPullbackIsColimit
simplexPullbackSection def: A simplex in the total space lifts its own base simplex canonically. · OAI.Thomason.Construction.simplexPullbackSection
simplexSet abbrev: Nonnegative weights of total mass one in the ambient function space. · OAI.GeneralMahler.simplexSet
simplexSubdivisionChains def: Barycentric subdivision followed by actual affine singular interpolation. · OAI.Kaplansky.simplexSubdivisionChains
simplexSubdivisionPrism def: The corresponding interpolated chain homotopy components. · OAI.Kaplansky.simplexSubdivisionPrism
simplexToCube def: Embed the ordered simplex as the chain of cube vertices with increasing numbers of final coordinates. · OAI.Thomason.Construction.simplexToCube
simplexToFace def: Every original horn simplex factors through any face it omits. · OAI.Thomason.Construction.HornBarycentric.simplexToFace
simplexTupleCenter def: Barycenter of a tuple in the ambient standard simplex. · OAI.Kaplansky.simplexTupleCenter
simplexTupleHull def: The ambient convex hull of a tuple's vertices. · OAI.Kaplansky.simplexTupleHull
simplexVertex def: Read the original vertex carried by an object of one presentation simplex. · OAI.Thomason.Construction.NormalizedSimplexIdentities.simplexVertex
simplexVertexCoordinates def: Repeated vertices receive the sum of their barycentric coordinates. · OAI.Thomason.Construction.RealizationPrism.simplexVertexCoordinates
simplicialCofan_pullback def: Canonical pullback pieces above an arbitrary coproduct form a coproduct of the original total space. · OAI.Thomason.Construction.simplicialCofan_pullback
simplicialCofan_pullback_isColimit def: An arbitrary coproduct of simplicial sets remains a coproduct after pullback. The proof uses the literal disjoint partition in each degree. · OAI.Thomason.Construction.simplicialCofan_pullback_isColimit
simplicialComponentsSpace def: Edge components carry the discrete topology for this comparison. · OAI.Thomason.Construction.RealizationPrism.simplicialComponentsSpace
SimplicialLabeling structure: Simplicial labels permit a face to map onto an edge or a vertex. The mate conditions retain the labelled side, not merely its group endpoints. · OAI.ArtinCAT0.TriangleMesh.SimplicialLabeling
simplicialMappingPathContraction def: The pullback contraction is an actual simplicial homotopy from the inclusion followed by retraction to the identity, over the source. · OAI.Thomason.Construction.simplicialMappingPathContraction
simplicialMappingPathShrinkingMap def: Shrinking the path preserves its start, so it restricts to every mapping-path pullback, even when the base is not Kan. · OAI.Thomason.Construction.simplicialMappingPathShrinkingMap
simplicialPointFiber def: The actual simplicial fibre over a vertex, in all degrees. · OAI.Thomason.Construction.simplicialPointFiber
simplicialPointFiberBaseChangeIso def: Fibres of the actual simplicial pullback identify exactly with the original point fibres. · OAI.Thomason.Construction.simplicialPointFiberBaseChangeIso
simplicialPointFiberLift def: A map with constant projection factors through the literal point fibre. · OAI.Thomason.Construction.simplicialPointFiberLift
SimplyConnectedSolvableLieModel structure: A connected simply connected finite-dimensional real solvable Lie group. · OAI.PolycyclicRecognition.SimplyConnectedSolvableLieModel
SimplyConnectedSolvableLieModel structure: A connected simply connected finite-dimensional real solvable Lie group. No metric or lattice is part of this record. · OAI.PolycyclicRecognition.SimplyConnectedSolvableLieModel
simulatedMovement def: Simulated movement measured by occurrence matching, exactly Section02. · OAI.KServer.simulatedMovement
simulatedStrategy def: Local question reconstruction and answer decoding never expose the other player's question to a strategy. Any fixed advice is an ordinary fixed parameter of these functions. · OAI.MaxCutGames.Foundations.Games.Game.simulatedStrategy
simulatedStrategy def: Local question reconstruction and answer decoding never expose the other player's question to a strategy. Any fixed advice is an ordinary fixed parameter of these functions. · OAI.UniqueGamesTheorem.Foundations.Games.Game.simulatedStrategy
SimulatesTrace def: A finite auxiliary observation, decoded by a fixed function, simulates the usual deletion channel on a literal binary word. Test functions range over the observable traces only, never latent boundaries. · OAI.TraceReconstruction.SimulatesTrace
simulation def: Simulate the identical countable probe presentation over the larger base; this is a homomorphism of the actual reduced-word extensions. · OAI.PartitionConsistency.ProbeAlgebra.Path.simulation
SimulationContract def: Full source contract for lem:simulation, with arbitrary measurable seed and probability spaces. The simulator is constructed before the distributions and adversary are supplied. Its decision function is specified concretely; the … · OAI.MatroidProphet.General.SimulationContract
simultaneousHornData def: A coproduct of generating horns has a stationary realized deformation. · OAI.Thomason.Construction.simultaneousHornData
sincMasked def: Only the synthesis waves change. Coefficients and all lattice geometry are retained literally. · OAI.LipschitzHilbert.LinePacketFamily.sincMasked
sincProductSchwartz def: Multiplication by the actual even mask is a Schwartz operation. · OAI.LipschitzHilbert.sincProductSchwartz
sineMeasure def: The actual probability measure on the unit time interval, not a moment axiomatization of that measure. · OAI.PositiveEinsteinFour.Averages.sineMeasure
single abbrev: A one-coordinate grid, the base of06's induction. · OAI.CoordinateSweeps.Grid.single
single def: The partition with its corner as its only piece. · OAI.Generator.Tracial.CornerPartition.single
single def: A single cyclic Cesàro average, always with a positive denominator. · OAI.PlaneFiveColor.InvariantMean.single
single def: A single populated stack: the exact Mathlib initial/final convention. · OAI.Superstring.BitCode.single
singleChoice def: Every line-choice array in a one-coordinate grid is one permutation. · OAI.CoordinateSweeps.Grid.singleChoice
singleEvaluation def: Agreement with the canonical native degree-zero evaluation, including its full B action. · OAI.LittleFinitistic.BimoduleTotal.BoundedProjective.singleEvaluation
singleEvaluationIso def: The ordinary signed total on an input module in degree zero is the exact native P tensor_B N complex. · OAI.LittleFinitistic.BimoduleTotal.singleEvaluationIso
singleInput def: Linear insertion into one physical input coordinate. · OAI.MaxCutGames.Gadget.singleInput
singleInput def: Linear insertion into one physical input coordinate. · OAI.UniqueGamesTheorem.Gadget.singleInput
singleIso def: An object concentrated in one degree is the stalk of its actual cohomology. · OAI.LittleFinitistic.DerivedTruncation.singleIso
singleIso def: The off-diagonal direct sum of two ordinary modules is their ordinary bimodule in degree zero. · OAI.LittleFinitistic.OrdinarySimulation.OffDiagonal.singleIso
singleLabelVector def: A coordinate has one possible nonzero label at each question. · OAI.MaxCutGames.Repetition.ProjectionKernel.singleLabelVector
singleLabelVector def: A coordinate has one possible nonzero label at each question. · OAI.UniqueGamesTheorem.Repetition.ProjectionKernel.singleLabelVector
singleLeaf def: The unique labeled position in a unary tree. · OAI.DilutedSpinGlass.PrescribedTree.singleLeaf
singleLocal def: A physical one-qubit gate, as a local layer. · OAI.QAC.singleLocal
singleResidualEquiv def: Exact bijection of conditioned one-block choices with the remaining symmetric group, not an assumption of uniformity of that residual law. · OAI.CoordinateSweeps.Grid.Holes.singleResidualEquiv
singleSlot def: The single coordinate slot identification is explicit. · OAI.CoordinateSweeps.Grid.singleSlot
SingleSource def: Every image which can replace a neutral head originates at the chosen outer binder profile. Ordinary one-variable substitution, and every lifting of it through additional binders, have this property. · OAI.PureTypeSystem.SingleSource
singleStalk def: The single-support part as an actual submodule of its stalk. · OAI.KLInvariance.MomentGraph.Sheaf.singleStalk
singleSweepEquiv def: No hidden fibers in the one-block base: all permutations are possible, and each one determines the unique line-choice array. · OAI.CoordinateSweeps.Grid.singleSweepEquiv
singletMatrixElement def: Matrix element in the singlet of squared norm two. · OAI.ContinuumCoulomb.singletMatrixElement
singleton def: For programs with dedicated scratch Sq. · OAI.Fifths.Comp.singleton
singleton def: Depth zero consists of the original projection itself. · OAI.Generator.Tracial.EquivalentPartition.singleton
singletonBranchCount def: The number of one-vertex branches. · OAI.HadwigerMatching.singletonBranchCount
singletonFamily def: Singleton families ensure that the segment seminorm separates points. · OAI.DiamondDistortion.SegmentSystem.singletonFamily
singletonIntercept def: Only the third position changes the homogeneous intercept. · OAI.UniqueGamesTheorem.Clean.LocalReplay.singletonIntercept
singletonPairWeights def: The average of the laws of (a,0), (0,a), and (a,a) for uniform a. · OAI.MaxCutGames.Decoder.SparseLaw.singletonPairWeights
singletonPairWeights def: The average of the laws of (a,0), (0,a), and (a,a) for uniform a. · OAI.UniqueGamesTheorem.Decoder.SparseLaw.singletonPairWeights
singletonSamplerWeights def: Explicit finite pushforward of the three equally likely singleton choices. The same uniform slope supplies both entries of the diagonal choice. · OAI.MaxCutGames.Decoder.SparseLaw.singletonSamplerWeights
singletonSamplerWeights def: Explicit finite pushforward of the three equally likely singleton choices. The same uniform slope supplies both entries of the diagonal choice. · OAI.UniqueGamesTheorem.Decoder.SparseLaw.singletonSamplerWeights
singletonSlopeLaw def: Independent singleton choice and uniform advice slope. · OAI.MaxCutGames.Clean.singletonSlopeLaw
singletonSlopeLaw def: Independent singleton choice and uniform advice slope. · OAI.UniqueGamesTheorem.Clean.singletonSlopeLaw
singletonSlope₀ def: Pullback of a singleton column to the first free equation coordinate. · OAI.UniqueGamesTheorem.Clean.LocalReplay.singletonSlope₀
singletonSlope₁ def: Pullback of a singleton column to the second free equation coordinate. · OAI.UniqueGamesTheorem.Clean.LocalReplay.singletonSlope₁
singletProjector def: Orthogonal projection onto the one-dimensional singlet space. · OAI.ContinuumCoulomb.singletProjector
SingleValuedF def: Ordinary internal single-valuedness, expressed via forcing at every stronger condition. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.SingleValuedF
singleZeroNet def: Copy the runtime modulus into each literal sampler block; all other input cells are zero. Only fixed wire placements and constants are used. · OAI.ExactQuantumFactoring.PhysicalListSlots.singleZeroNet
SinglyGenerated def: Generation of the same algebra by one of its bounded operators. · OAI.Generator.SinglyGenerated
singularAnodyneExtension def: Every singular map on an anodyne subobject extends by the constructed retraction. · OAI.Thomason.Construction.singularAnodyneExtension
singularCubeBoundaryProductEquiv def: The same exact adjunction holds for the COMPLETE cube boundary. · OAI.Thomason.Construction.singularCubeBoundaryProductEquiv
singularCubeProductEquiv def: Higher cubes with arbitrary simplicial parameter spaces admit the exact adjunction; the inverse retains all continuous family coordinates. · OAI.Thomason.Construction.singularCubeProductEquiv
singularCylinderEquiv def: Continuous cylinders and singular-valued simplicial cylinders are exactly equivalent, with the constructed evaluation formula as inverse. · OAI.Thomason.Construction.singularCylinderEquiv
singularCylinderMap def: The simplicial adjoint of an arbitrary continuous cylinder. · OAI.Thomason.Construction.singularCylinderMap
singularHomotopy def: A continuous homotopy applied simplexwise, with the affine interval coordinate. · OAI.Thomason.Construction.singularHomotopy
singularHornFiller def: An explicit filler of any horn in the singular simplicial set. · OAI.Thomason.Construction.singularHornFiller
singularMap def: A continuous cube has an exact lift through singular evaluation after finite triangulation. · OAI.Thomason.Construction.RealizedCube.singularMap
singularPathCylinderEquiv def: Singular path families are exactly singular-valued simplicial cylinders. · OAI.Thomason.Construction.singularPathCylinderEquiv
singularPathInternalEquiv def: The hom-set form of the path-space/internal-hom identification. · OAI.Thomason.Construction.singularPathInternalEquiv
singularPathMapArrowIso def: Naturality gives an isomorphism of the actual induced path-map arrows. · OAI.Thomason.Construction.singularPathMapArrowIso
singularPathSpaceIso def: The actual compact-open path space and the simplicial internal hom have isomorphic singular complexes. · OAI.Thomason.Construction.singularPathSpaceIso
singularPathTranspose def: The strict Gray homotopy transposed into the actual compact-open path space of the Street realization. · OAI.Thomason.Construction.RightGrayInput.singularPathTranspose
singularProductEquiv def: Product realization gives an exact singular adjunction for a whole parameterized family, using the actual canonical product comparison. · OAI.Thomason.Construction.singularProductEquiv
singularRealizedPointIso def: The singular simplicial set of this point is the actual zero simplex. · OAI.Thomason.Construction.singularRealizedPointIso
singularSimplexOfMap def: A continuous simplex as an element of the library singular set. · OAI.Kaplansky.singularSimplexOfMap
singularSimplexTime def: The affine time coordinate of a simplex of the simplicial interval. · OAI.Thomason.Construction.singularSimplexTime
singularSimplexTopMap def: The actual map represented by a singular simplex, with unlifted domain. · OAI.Kaplansky.singularSimplexTopMap
singularSubdivision def: Constructed chain endomorphism, with no subdivision or excision hypothesis. · OAI.Kaplansky.singularSubdivision
singularSubdivisionHomotopy def: An explicit homotopy between the identity and barycentric subdivision. · OAI.Kaplansky.singularSubdivisionHomotopy
singularSubdivisionMap def: Barycentric subdivision on the existing integral singular chains. · OAI.Kaplansky.singularSubdivisionMap
singularSubdivisionPrism def: The universal prism transported along each actual singular simplex. · OAI.Kaplansky.singularSubdivisionPrism
singularSubset def: Singular simplices whose entire image lies in the specified subspace. · OAI.Kaplansky.singularSubset
singularSubsetIso def: Identification with singular simplices of the actual subspace, including its inherited topology, rather than a replacement chain theory. · OAI.Kaplansky.singularSubsetIso
singularVertex def: A point of a space gives the corresponding vertex in its singular realization. · OAI.Thomason.Construction.singularVertex
singularVertexPath def: Every actual path gives a path between the corresponding singular vertices. · OAI.Thomason.Construction.singularVertexPath
sinkCopy def: Sink normalization also replaces missing transitions at accepting states. · OAI.OneWayLiveness.Machine.sinkCopy
site def: Single-site factor of a product projection. · OAI.QAC.Count.site
siteColumnResponse def: The unnormalized residual column response for fixed single-site tests. · OAI.JointDickman.siteColumnResponse
siteFlux def: The mass variation caused by motion of the sites, in face coordinates. · OAI.Problem358.CellLaplacian.siteFlux
SiteKernelGood def: The one common event used before taking a finite union of concentration tails. Its complement is charged once, not once per column sample. · OAI.JointDickman.SiteKernelGood
siteProductMass def: Independent site mass, retaining the actual coordinate laws. · OAI.JointDickman.PublishedInputs.siteProductMass
siteRowMean def: Its conditional mean after fixing the row site. · OAI.JointDickman.siteRowMean
siteRowSquareMass def: Integrated sum of squared row entries, with the row site also averaged. · OAI.JointDickman.siteRowSquareMass
siteRowSum def: A row evaluated on the independent site configuration. · OAI.JointDickman.siteRowSum
sites def: The three site vectors, ordered negative/positive/central. · OAI.Problem358.Sharpness.sites
SixExtendToSeven def: Precise completion statement, including preservation of the original bases. · OAI.MUB6.SixExtendToSeven
sixFieldsInTime def: The six natural fields of a three-literal clause use seven actual stacks. · OAI.MaxCutGames.Foundations.Hastad.SourceFieldArray.sixFieldsInTime
sixFieldsInTime def: The six natural fields of a three-literal clause use seven actual stacks. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceFieldArray.sixFieldsInTime
sixfoldProduct def: The product of character values in the six orders of possibly different legs. · OAI.MatrixMultiplication.AuxiliarySeparation.Character.sixfoldProduct
sixLine def: AB, AC, AD, BC, BD, CD in the fixed linear-functional indexing. · OAI.HyperbolicObstruction.CoreGeometry.sixLine
SixthInput def: The exact finite-head sixth input. The S=0 convention is represented by real division, for which the entire right side is zero when S=0. · OAI.DiamondDistortion.RecursiveApplications.SixthInput
size def: Cardinality of powers of two not exceeding an integer, hence its bit length. · OAI.OptimalMaxCut.CounterMachine.Expr.size
size def: Number of nodes, including the root, in the complete K-ary tree of depth d. · OAI.Problem310.TreePreorderRank.size
size def: Admissible sizes are exactly the multiplicities of the empirical table. · OAI.TorsionFreeZeroDivisors.TypedGraphs.size
size def: Gates plus input occurrences. · OAI.TwoPointCorrelations.AC0Circuit.size
SizedCode abbrev: Carrier counts distinguish inactive padding from actual graph indices. The four extra counts are uniformly bounded, independent of H and n. · OAI.TorsionFreeZeroDivisors.PatternCoding.SizedCode
sizedWordEquiv def: The actual successful words and compatible independently marked tree codes are equivalent at each fixed total edge size. · OAI.FKCRT.Inventory.CoreTree.sizedWordEquiv
sizePolynomial def: The actual padded vertex count is linear in the original encoded size. · OAI.MaxCutGames.Foundations.Complexity.MachinePaddedOverlay.sizePolynomial
sizePolynomial def: A fixed quadratic in the full serialized input length. · OAI.MaxCutGames.Foundations.PCP.FinalTableFormula.sizePolynomial
sizePolynomial def: The actual padded vertex count is linear in the original encoded size. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachinePaddedOverlay.sizePolynomial
sizePolynomial def: A fixed quadratic in the full serialized input length. · OAI.UniqueGamesTheorem.Foundations.PCP.FinalTableFormula.sizePolynomial
sizeRank def: The broader size function, literally as in §03. · OAI.KServer.RankFunctions.sizeRank
Sizes abbrev: package extracted charges · OAI.PowerSaving.RAM.Bench.Sizes
sizes def: Each new size is past the entire preceding potential envelope, not merely past those actually selected. The index and current body bounds are included. · OAI.FiniteAlgebraicEnvelopes.Bootstrap.Selectors.sizes
skeletalMap def: The induced map of skeletal subcomplexes, with its literal inclusion square rather than an unspecified comparison isomorphism. · OAI.Thomason.Construction.RealizationPrism.skeletalMap
skeleton def: Ordered cells and both flags are preserved by every internal validator step. · OAI.FiniteAlgebraicEnvelopes.Compiler.Validator.skeleton
skeletonBoundary def: The actual attaching maps of a classical CW skeleton. · OAI.GroupRingTopology.Cellular.skeletonBoundary
skeletonHomeomorph def: A classical CW skeleton is literally the topological adjunction of the previous skeleton and its disks, including arbitrary infinite cell families. · OAI.GroupRingTopology.Cellular.skeletonHomeomorph
skew def: The skew rank-two operator associated with an ordered pair. · OAI.ThompsonNonamenability.Analytic.skew
SkewAt def: A real-skew field is simultaneously tangent to every sphere and complex quadric. · OAI.ArnoldCounterexample.Field.SkewAt
skewBlock11 def: The first diagonal block of a skew-Hermitian matrix. · OAI.KervaireFixedSpace.skewBlock11
skewBlock22 def: The second diagonal block of a skew-Hermitian matrix. · OAI.KervaireFixedSpace.skewBlock22
skewConj def: Conjugation is a real linear automorphism of the skew-Hermitian space. · OAI.KervaireIncidenceCurrent.skewConj
skewLogDet def: The skew-shift log-determinant deficit, including non-real determinants. · OAI.GAD.skewLogDet
skewMoment def: Rational form of ρ / √3. · OAI.PositiveEinsteinFour.Bianchi.Shape.skewMoment
skewPart def: The ambient projection onto the skew-Hermitian tangent at the identity. · OAI.KervaireIncidenceCurrent.skewPart
skipBits def: A fixed finite chain skips Boolean unary words. A true head causes one additional delimiter pop; a false head is already the complete zero word. Only the one-bit head register is examined. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Lookup.skipBits
skipBits def: A fixed finite chain skips Boolean unary words. A true head causes one additional delimiter pop; a false head is already the complete zero word. Only the one-bit head register is examined. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Lookup.skipBits
skipInput def: Skip exactly the prescribed number of attempted reads, recording only real bits on the backup. EOF is handled by a genuine branch. · OAI.OptimalMaxCut.CounterMachine.Command.skipInput
skipWord def: Saturating removal of one zero-delimited word. · OAI.MaxCutGames.Foundations.Complexity.MachineLookupSpec.skipWord
skipWord def: Saturating removal of one zero-delimited word. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineLookupSpec.skipWord
skirt def: 2*b slots. k < b for positive or zero displacement. Leave non-required displacements 0 for convenience -- coefficients beyond the supplied span need not be evaluated. · OAI.PowerSaving.Slant.skirt
slab def: A literal half-sign label rectangle, pulled back through the actual old chart variables. No independence of different old charts is asserted. · OAI.PlanarL1.Threshold.Variables.slab
slabCost def: Sum over a slab, padded by its actual zero costs when its levels exceed d. · OAI.CubeShuffle.InputTree.slabCost
slabIndex def: A finite label for the cell partition and its pushforward law. · OAI.Problem358.Sharpness.slabIndex
slabLoadBound def: A convenient literal finite arithmetic load bound, prior to specializing q. · OAI.KaplanskyCounterexample.FiniteGeometry.DirectionFrame.slabLoadBound
slabMap def: A transport map selected from the maximum of -xᵢ, xᵢ and a+b*xⱼ. · OAI.Problem358.Sharpness.slabMap
slabMeasure def: Independent choices of uniformly distributed slab direction planes. · OAI.KaplanskyCounterexample.FiniteGeometry.slabMeasure
slabSelector def: The selected branch, with central branches taking precedence on ties. · OAI.Problem358.Sharpness.slabSelector
slack def: Positive slack used by the uniform finite real perturbation. · OAI.CoordinateSweeps.SparseScalar.slack
Slice abbrev: The actual simultaneous row/column slice, including its affine offsets. · OAI.MaxCutGames.Inverse.AffineWitness.Slice
Slice structure: A description includes possibly dependent equations; the row and column counts need not equal their ranks. · OAI.MaxCutGames.Inverse.Shortcode.Slice
Slice abbrev: The actual simultaneous row/column slice, including its affine offsets. · OAI.UniqueGamesTheorem.Inverse.AffineWitness.Slice
Slice structure: A description includes possibly dependent equations; the row and column counts need not equal their ranks. · OAI.UniqueGamesTheorem.Inverse.Shortcode.Slice
slice def: The image of U on a NAMED old graded fiber. This definition does not choose an unrelated subspace with the same dimension. · OAI.IntegralCharacterVarieties.OldContactSlice.slice
slice def: Actual covariance passes to a scalar cross-multiplicity slice. · OAI.KadisonSimilarity.CrossMultiplicityMatrixCovariance.slice
slice def: Taking a scalar multiplicity coefficient preserves actual word covariance. · OAI.KadisonSimilarity.MultiplicityMatrixCovariance.slice
slice def: A slice is a proof view, not a newly allocated row array. · OAI.LowSpaceSubsetSum.SliceCursor.slice
slice def: The restriction of the presentation to the i-th Cartesian slice. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.SliceData.slice
slice def: Prefix a continue frame. Its slicing operation commutes with relabeling. · OAI.PureTypeSystem.FreeObservations.Table.slice
slice def: The positions of one comparison side in a pair of occurrence blocks. · OAI.TorsionFreeZeroDivisors.OccurrenceMatching.Data.slice
sliceCorrelation def: Correlation on a genuine affine slice. N₀ is retained: its phase cannot be discarded when the witness has a nonzero affine intercept. · OAI.MaxCutGames.Decoder.PrivateFourier.sliceCorrelation
sliceCorrelation def: Correlation on a genuine affine slice. N₀ is retained: its phase cannot be discarded when the witness has a nonzero affine intercept. · OAI.UniqueGamesTheorem.Decoder.PrivateFourier.sliceCorrelation
SliceData structure: Parameters of the ordinal slicing of a presentation and a surjection. There are no correctness assumptions in this structure. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.SliceData
sliceEnergy def: An arbitrary actual Fourier slice; in the mixed estimate P fixes selected frequency coordinates. · OAI.MaxCutGames.Inverse.KMSFourthMoment.sliceEnergy
sliceEnergy def: An arbitrary actual Fourier slice; in the mixed estimate P fixes selected frequency coordinates. · OAI.UniqueGamesTheorem.Inverse.KMSFourthMoment.sliceEnergy
SliceFamily structure: Each description includes its row map, row value, column equations and affine target. points_row is the necessary relationship to actual row advice. · OAI.MaxCutGames.Inverse.RowErasure.SliceFamily
SliceFamily structure: Each description includes its row map, row value, column equations and affine target. points_row is the necessary relationship to actual row advice. · OAI.UniqueGamesTheorem.Inverse.RowErasure.SliceFamily
SliceLower def: Left endpoint of the affine horizontal slice. · OAI.Nagata.W04.SliceLower
sliceOffset def: Signed distance of the transverse collision line from the velocity axis. · OAI.BoltzmannNonuniqueness.GeometricSlice.sliceOffset
slicesHessian def: The actual four-index array associated to differentiated Bianchi tensors. · OAI.PositiveEinsteinFour.TraceHessian.slicesHessian
slicesTrace def: Trace of the two derivative slots, for a family of Bianchi gradients. · OAI.PositiveEinsteinFour.TraceHessian.slicesTrace
sliceUnitIso def: Convert an open-subscheme unit trivialization to the exact slice-site unit trivialization used by the tensor construction. · OAI.ActualSheafTensor.sliceUnitIso
SliceUpper def: Right endpoint of the affine horizontal slice. · OAI.Nagata.W04.SliceUpper
slide def: Local Huber function on a window of length 10 b. · OAI.FourierLLogL.Dyadic.Procedure.Paths.slide
slight def: Upper target thresholds, without claiming nonnegativity. · OAI.HadwigerCounterexample.slight
slope def: The oriented slope of a stage edge in the normalized metric. · OAI.DiamondDistortion.Diamond.slope
slope def: The logarithmic derivative of the actual center-regular value operator. · OAI.HarmonicCounterexample.LinearODE.slope
slope def: The free-coordinate block slope (x,y) ↦ x • a + y • d. · OAI.MaxCutGames.Decoder.TableKeys.slope
slope def: The free-coordinate block slope (x,y) ↦ x • a + y • d. · OAI.UniqueGamesTheorem.Decoder.TableKeys.slope
SlopeBox structure: A top box in the manuscript's slope coordinates. · OAI.LipschitzHilbert.SlopeBox
slopeShear def: The coordinate map (x,y) ↦ (x,y+s*x), with its explicit inverse. · OAI.LipschitzHilbert.slopeShear
Slot abbrev: The positions of a binary deck of dimension d. · OAI.BinaryCoordinateSweeps.Slot
Slot abbrev: A slot is a disk number, a disk kind, and a position in the cyclic word. · OAI.KervaireMixing.Slot
Slot abbrev: true directs the prospective successor to the right. · OAI.OneWayLiveness.CellNetwork.Slot
Slot inductive: The occurrence of a payload along a raw finite path. Certificates of a particular domain introduce no computational information. · OAI.PureTypeSystem.PlainSpaces.Schema.Slot
slot def: The slot of the containing selected child inside the containing parent. · OAI.SmoothObstruction.WordHierarchy.slot
slotAtRank def: Slot at an observed return rank, with the original total order. · OAI.TraceReconstruction.slotAtRank
slotCenteredMaskedBand def: The slot functions use the original alternate packet and original mask at each retained item. · OAI.LipschitzHilbert.SourceLabeledLattice.slotCenteredMaskedBand
slotDecode def: The common, bit-blind projection simply forgets flags and concatenates in the publicly fixed ID order. · OAI.TraceReconstruction.PairedMechanisms.slotDecode
SlotItem def: The spatial items retained in one label slot at a fixed scale. · OAI.LipschitzHilbert.SourceSlopeLabels.SlotItem
slotItemsEquiv def: Every original item occurs in exactly one slot. · OAI.LipschitzHilbert.SourceLabeledLattice.slotItemsEquiv
slotLaw def: Independent occurrence and uniform retained-position sampling. · OAI.MaxCutGames.Clean.IncidenceGap.slotLaw
slotLaw def: Independent occurrence and uniform retained-position sampling. · OAI.UniqueGamesTheorem.Clean.IncidenceGap.slotLaw
slotLeft def: Site immediately to the left of a slot. The actual sites are left+1,...,left+L_i. Each full chunk keeps R-1 translated periods. · OAI.AsymptoticallyMinimalLittlewood.Packing.slotLeft
slotParity def: The parity operator is +1 on a D slot and -1 on an X slot. · OAI.DerivedBar.SquareZero.slotParity
slotPartitionCode def: A partition is recorded by a representative slot for each slot. · OAI.TwoPointCorrelations.slotPartitionCode
slots def: Contiguous finite label fibers give a genuine ordered slot chart. · OAI.CoordinateSweeps.FiberBlocks.slots
slots def: The actual tree has 2n² nodes, 2n split clocks per node, one list slot and n^5 padded base/order slots per split. This is not the n^10 padding cap. · OAI.ExactQuantumFactoring.PhysicalTree.slots
slots def: Each scale has its own label assignment, but the same finite set of slots suffices for all scales. Inactive slots may be represented by an empty spatial lattice. · OAI.LipschitzHilbert.SourceSlopeLabels.slots
slotted def: One genuine translated-lattice family per slot. No hypotheses on a selector are introduced for missing labels or empty scales. · OAI.LipschitzHilbert.SourceLabeledLattice.slotted
slotValue def: The value is used only when the corresponding slot has an item. Zero is a harmless geometric default when that slot is empty. · OAI.LipschitzHilbert.SourceLabeledLattice.slotValue
slow def: The chain rule differentiates smooth low-jet coefficients and introduces one additional actual jet factor. · OAI.ClosedSurfaceR4.JetPolynomial.Expression.slow
slowBoundaryColumn def: The derivative-shift formula gives an explicitly analytic boundary column. · OAI.DefocusingNLS.slowBoundaryColumn
slProductToUnits def: The ordinary product of REAL SL groups has a continuous genuine embedding into units of the product matrix algebra. · OAI.HarmonicCounterexample.FiniteControl.slProductToUnits
SM structure: Usage as bounded computations will have X,J,K and Y finite. · OAI.Fifths.SM
Small def: Pointwise polynomial bound with absolute constants. The bound relevant to word sizes will use n as B. Local computations may also have polynomial work in a different B. · OAI.PowerSaving.RAM.Small
Small def: An actual finite containment cover with total cost at most one half. · OAI.TalagrandDiscreteConvexity.Small
Small def: Smallness is witnessed by an actual containment cover of cost at most one half. · OAI.TalagrandDiscreteConvexity.Small
Small def: Every retained vertex lies in a small set closed under all retained edges. · OAI.TorsionFreeZeroDivisors.RecursiveSeparator.Small
smallActivity def: Points at which a label is active for some positive parameter below δ. · OAI.Problem326.AffineApproximation.smallActivity
smallAssignment def: Only the small ordinal of the shrunk relation indexes the table. · OAI.PartitionConsistency.OrderAssignment.smallAssignment
smallCoeff def: The Fourier array of the small full-rank component. · OAI.MaxCutGames.Inverse.KMSAnalytic.smallCoeff
smallCoeff def: The Fourier array of the small full-rank component. · OAI.UniqueGamesTheorem.Inverse.KMSAnalytic.smallCoeff
smallCoefficientQuotientSection def: An integer-linear section of the quotient by the constant functions. · OAI.PolycyclicRecognition.smallCoefficientQuotientSection
smallCoefficientRepresentation abbrev: The coefficient ring with its actual translation representation. · OAI.PolycyclicRecognition.smallCoefficientRepresentation
smallCoefficientTransversal def: The indicator of the chosen transversal of a finite subgroup. · OAI.PolycyclicRecognition.smallCoefficientTransversal
smallComponent def: A genuine function on the small primal matrix space, synthesized in its own orthonormal trace-character basis. · OAI.MaxCutGames.Inverse.KMSAnalytic.smallComponent
smallComponent def: A genuine function on the small primal matrix space, synthesized in its own orthonormal trace-character basis. · OAI.UniqueGamesTheorem.Inverse.KMSAnalytic.smallComponent
SmallEmbedding abbrev: The actual injective maps from the comparison space. · OAI.UniqueGamesTheorem.Inverse.KMSAnalytic.SmallEmbedding
smallFrameCompressionInput def: The elementary compression model on the actual half corner. · OAI.KadisonSimilarity.TracialCornerGeometry.smallFrameCompressionInput
SmallFreeCornerData structure: The exact geometric data used in one application of the free-corner estimate. None of the fields mentions the operator being estimated. · OAI.KadisonSimilarity.SmallFreeCornerData
smallFrequencyEquiv def: Reindex the actual dual frequency maps by an equivalence of their targets. · OAI.MaxCutGames.Inverse.KMSAnalytic.smallFrequencyEquiv
smallFrequencyEquiv def: Reindex the actual dual frequency maps by an equivalence of their targets. · OAI.UniqueGamesTheorem.Inverse.KMSAnalytic.smallFrequencyEquiv
smallIso def: The shrinking isomorphism of wellorders uses no new ordering choice. · OAI.PartitionConsistency.OrderAssignment.smallIso
smallMayerVietorisBoundaryIso def: For two contractible subspaces, the actual small-chain connecting map is an isomorphism in degrees at least two. Full singular homology still requires the small-to-full comparison theorem. · OAI.Kaplansky.smallMayerVietorisBoundaryIso
smallMayerVietorisContinuousMap def: Naturality includes the intersection term and the signs of the Mayer–Vietoris differential. · OAI.Kaplansky.smallMayerVietorisContinuousMap
smallMayerVietorisOneCoordinate def: The coefficient of the first intersection component after the actual Mayer–Vietoris boundary. This fixes an integral sign convention. · OAI.Kaplansky.smallMayerVietorisOneCoordinate
smallMayerVietorisSequence def: Mayer–Vietoris chain sequence for genuine subspaces, with the complex of small chains at the right. No open-cover comparison is assumed. · OAI.Kaplansky.smallMayerVietorisSequence
smallMayerVietorisSequenceMap def: Naturality of the actual small-chain gluing sequence under inclusions of both members of the cover. · OAI.Kaplansky.smallMayerVietorisSequenceMap
SmallNormalizerFactorization def: The local statement consumed by normalizer averaging. · OAI.KadisonKastler.SmallNormalizerFactorization
smallQuaternionArc def: A globally smooth path that remains in the closed chord ball of its endpoint, avoiding any branch choice of a quaternion logarithm. · OAI.EilenbergGanea.smallQuaternionArc
smallRel def: Shrinking only changes the carrier; the selected relation is retained. · OAI.PartitionConsistency.OrderAssignment.smallRel
smallRelativeChains def: Small relative chains modulo the full chains of the second subspace. · OAI.Kaplansky.smallRelativeChains
smallRelativeComparison def: Comparison with the full relative chains of (X,V). · OAI.Kaplansky.smallRelativeComparison
smallRelativeExcisionIso def: Actual chain-level excision into small relative chains. This needs no openness or covering hypothesis because the target only uses small simplices. · OAI.Kaplansky.smallRelativeExcisionIso
smallRelativeMayerVietorisSequence def: Relative Mayer–Vietoris at chain level. The removed set is restricted to each chart and to their intersection; at the right it uses small chains in the removed subspace. Comparison with full chains is a separate theorem. · OAI.Kaplansky.smallRelativeMayerVietorisSequence
smallSamplerWordBudget def: Each outer trial contains a small walk and a fresh completion word. · OAI.ContingencyTables.smallSamplerWordBudget
SmallScaleVolumeUpper def: A weak form of an upper density bound: arbitrarily small radii approach the given upper bound. Bishop--Gromov propagates it to all radii. · OAI.RegularPointCoordinates.SmallScaleVolumeUpper
smallSingularChainsContinuousMap def: The induced map on small chains for a continuous map respecting covers. The construction uses the proved pushout of actual singular-chain complexes. · OAI.Kaplansky.smallSingularChainsContinuousMap
smallSingularChainSequence def: Proven gluing sequence for actual small singular chains. A quasi-isomorphism from small to full chains is a separate subdivision theorem. · OAI.Kaplansky.smallSingularChainSequence
smallSingularSet def: The complex of simplices small for the two-set family. Its simplices lie wholly in one member, which need not hold for an arbitrary singular simplex. · OAI.Kaplansky.smallSingularSet
smallSphereFlux def: Radius-r sphere flux in the positive radial parametrization. The surface Jacobian is r^(2k+1); the normal is z, directed out of the ball. · OAI.Mahler.smallSphereFlux
SmallStandardRegion def: Exactly the closed phase ranges, so adjacent types are BOTH included at ties. · OAI.CubicTorus.SmallStandardRegion
smallStar def: Intersection of a straight-contraction star with an arbitrarily small ambient ball. · OAI.HyperbolicObstruction.FinitePolyhedron.smallStar
smallStarHomeomorph def: Flatten the two subtypes without changing their subspace topology. · OAI.HyperbolicObstruction.FinitePolyhedron.smallStarHomeomorph
smallSubdivisionMap def: Subdivision restricted to the actual small chain groups. · OAI.Kaplansky.smallSubdivisionMap
smallSubdivisionPrism def: The prism restricted to the same small chain groups. · OAI.Kaplansky.smallSubdivisionPrism
smallWeightFamily def: The remaining nonzero weights; this finite family is partitioned into bins. · OAI.TalagrandDiscreteConvexity.smallWeightFamily
smooth def: Smoothing at scale s, using the kernel whose moments vanish through r. · OAI.ClosedSurfaceR4.FiniteOrderSmoothing.smooth
smooth def: The actual global operator; no choice in its definition depends on f. · OAI.ClosedSurfaceR4.FiniteOrderSmoothing.SmoothingAtlas.smooth
smooth def: Positive-height smoothing after removing the scalar modulation. · OAI.KadisonKastler.FrequencySmoothing.smooth
smoothAbs def: Smooth absolute-value approximation with zero preserved. · OAI.WeakSobolev.smoothAbs
SmoothBranchData structure: The construction data expected by the measure-level nonattainment and separated-support approximation arguments. · OAI.Problem356.SmoothBranchMarginal.SmoothBranchData
smoothCapTest def: A smooth convex cap test, quadratic below the fixed transition interval and identically zero above it. · OAI.AffineBernstein.smoothCapTest
SmoothCircleOn def: Smoothness in every real quotient chart whose cut point is absent. This uses the actual ℝ/ℤ charts, without requiring a separate manifold instance for AddCircle. · OAI.CircleDomainRigidity.FiniteTransfer.SmoothCircleOn
smoothCorrection def: The finite Euler-factor inverse supported on integers whose prime factors all belong to the excluded set. · OAI.JointDickman.smoothCorrection
SmoothDynamics structure: An actual smooth, volume preserving integer action on the physical torus. This is limit DATA constructed below, not an assumption on a stage. · OAI.ThreeTorus.SmoothDynamics
smoothExplicitFormula_of_individual def: Only the ordinary individual count and the explicit formula are needed. The family count, Mellin decay and absolute convergence are already proved. · OAI.Ostmann.smoothExplicitFormula_of_individual
smoothExplicitFormula_of_multiplicity def: For multiplicity-respecting enumerations, only the actual explicit formula remains in this analytic package. The zero count, Mellin decay and absolute convergence are all derived. · OAI.Ostmann.smoothExplicitFormula_of_multiplicity
SmoothExplicitFormulaCore structure: The fixed-test explicit formula. · OAI.Ostmann.SmoothExplicitFormulaCore
smoothFreshCore def: The fresh block on Y_{j+1}, consisting of exactly a_{j+1,j+1} occurrences of the hyperplane line of its last factor. · OAI.Kaplansky.smoothFreshCore
smoothH1 def: Classical smooth L² functions with L² derivative belong to the exact weak H¹ domain. · OAI.SharpLiebThirring.SobolevProof.smoothH1
SmoothingAtlas structure: The outer cutoff is one on the support of the partition weight and has its own closed support inside the same chart. · OAI.ClosedSurfaceR4.FiniteOrderSmoothing.SmoothingAtlas
SmoothKahlerMetric structure: A smooth Kähler metric, in the torsion-free metric-connection formulation. No topology, flow, coordinate-map, or uniformization conclusions occur among these fields. · OAI.KahlerUniformization.SmoothKahlerMetric
SmoothL2TwoForm def: The L2 topology of the completion differs from the pointwise topology on the smooth function submodule. · OAI.PositiveEinsteinFour.Geometry.SmoothL2TwoForm
smoothLineSumCore def: The ordinary smooth Hermitian direct sum of the specified hyperplane pullbacks and trivial lines on the actual projective product. · OAI.Kaplansky.ComplexGrassmann.smoothLineSumCore
SmoothMetric structure: A genuine smooth positive definite covariant metric tensor on R^n. · OAI.HarmonicCounterexample.SmoothMetric
SmoothMetric abbrev: A positive definite smooth real inner product on each tangent space. · OAI.KahlerUniformization.SmoothMetric
SmoothMetric abbrev: Ordinary smooth positive-definite Riemannian metrics, with the existing tangent topology. · OAI.YauCounterexamples.SmoothMetric
SmoothNear def: No boundary hypothesis is included in this neighborhood regularity predicate. · OAI.RuledApproximation.SmoothNear
smoothNextCore def: E_{j+1} presented as the direct sum of its replacement and fresh blocks. · OAI.Kaplansky.smoothNextCore
smoothPhase def: The regularization used in the trace inequality. · OAI.KervaireScalarPhase.smoothPhase
SmoothPhysicalDiskCover def: Concrete finite smooth Jordan disks covering the physical fiber of x. The indexing set is a finite subset of the regular level's components. · OAI.CircleDomainRigidity.SmoothPhysicalDiskCover
SmoothPlanarSard def: Published input: Sard (1942), Theorem 7.2, restricted to the smooth compactly supported scalar planar case needed for contour selection. This is a proposition parameter, not a global axiom. · OAI.CircleDomainRigidity.SmoothPlanarSard
SmoothPositiveDensity def: Positive smooth local coordinate densities. In a half-space chart, ContDiffOn includes smoothness up to the boundary. Absolute Lebesgue density is used; no orientation or volume form is part of the data. · OAI.SmoothObstruction.SmoothPositiveDensity
smoothPrimeCenters def: The integer translates that can occur in a finite prime set. · OAI.Ostmann.smoothPrimeCenters
smoothPrincipalValue def: The symmetric quotient extended continuously at zero for smooth inputs. · OAI.LipschitzHilbert.smoothPrincipalValue
smoothPrincipalValue def: The continuous extension at zero is integrated on the closed short scale. The value at zero has no effect on the integral. · OAI.LipschitzHilbert.smoothPrincipalValue
SmoothRegularJordanCollars def: Explicit published-input parameter: the regular-level planar Jordan specialization of the collar-neighborhood theorem described above. · OAI.CircleDomainRigidity.SmoothRegularJordanCollars
smoothRestriction def: Restriction to integers supported on a prescribed finite prime set. · OAI.Problem337.DivisorMoment.smoothRestriction
smoothRotationalModificationLieEquiv def: The actual graph has the real-shadow bracket in its original tangent coordinates. · OAI.PolycyclicRecognition.ComplexWeightFactorModel.smoothRotationalModificationLieEquiv
smoothRotationalModificationMetric def: The metric has exactly the existing graph topology. · OAI.PolycyclicRecognition.ComplexWeightFactorModel.smoothRotationalModificationMetric
smoothScalarConst def: A constant as an actual smooth scalar function. · OAI.PositiveEinsteinFour.Geometry.smoothScalarConst
smoothScalarDirichlet def: The actual scalar Dirichlet pairing in the frame measure. · OAI.PositiveEinsteinFour.Geometry.smoothScalarDirichlet
smoothScalarGradient def: Each actual frame component of the scalar differential is linear. · OAI.PositiveEinsteinFour.Geometry.smoothScalarGradient
smoothScalarModule def: Smooth real scalar functions, with their pointwise vector-space structure. · OAI.PositiveEinsteinFour.Geometry.smoothScalarModule
SmoothScalarPoincare def: The estimate needed here, stated solely for actual smooth mean-zero scalars. The geometric Poincaré theorem supplies this hypothesis. · OAI.PositiveEinsteinFour.Geometry.SmoothScalarPoincare
smoothScalarSubtractMean def: Subtracting the actual integral preserves smoothness. · OAI.PositiveEinsteinFour.Geometry.smoothScalarSubtractMean
smoothScalarToWeightedH1 def: A smooth function, regarded as an element of the weighted Hilbert space. · OAI.PositiveEinsteinFour.Geometry.smoothScalarToWeightedH1
smoothScalarValue def: Viewing a smooth scalar on the orthonormal frame bundle. · OAI.PositiveEinsteinFour.Geometry.smoothScalarValue
SmoothSobolev def: Smooth dependence in the genuine Hilbert Sobolev topology, with the representative tied to the given distribution family. · OAI.ElasticityRegularity.SmoothSobolev
smoothStaircase def: A smooth staircase with transitions near the half-shifted h lattice. · OAI.CircleDomainRigidity.FiniteTransfer.smoothStaircase
SmoothSwitching def: The literal quantitative smooth switching property, with one fixed sequence of original finite cell laws, summable sixteenth powers, and uniform two-sided shadowing over the full dyadic windows. · OAI.SmoothObstruction.SmoothSwitching
smoothTestBundleCore def: E_n on the first b projective factors, retaining the full-R K_i. · OAI.Kaplansky.smoothTestBundleCore
smoothTestBundleZeroTotalHomeomorph def: A global linear chart for the actual full-product E₀, identifying it with Y times its Hilbert fiber, with exactly d₀ coordinates. · OAI.Kaplansky.smoothTestBundleZeroTotalHomeomorph
smoothTrace def: A smooth trace approximation to the fractional spectral phase. · OAI.KervaireTraceCalculus.smoothTrace
snag def: snag only denotes a nonbad high depth. · OAI.FourierLLogL.Dyadic.Procedure.Paths.Troupe.snag
snapExp def: Snap parameter l associated with a node and h. · OAI.FourierLLogL.Dyadic.Thermal.snapExp
snappedLatticeSite def: Quantize a translated, sheared lattice without identifying distinct sites of that lattice. The shear is measured in the reference coordinates. · OAI.LipschitzHilbert.snappedLatticeSite
snapshot def: Deep-copy the mutable tournament; the immutable right array is shared. · OAI.LowSpaceSubsetSum.CountedParentJoin.snapshot
snapshotCells def: Only the mutable tournament and its cached cursors are copied. The immutable stored arrays are shared by every replay snapshot. · OAI.LowSpaceSubsetSum.CompressionData.snapshotCells
snoc def: Adjoin one final switch at height H. · OAI.PlanarL1.Columns.Model.Path.snoc
sobolevBilinearProduct def: The usual torus product is a continuous complex-bilinear operation in H^k. · OAI.DefocusingNLS.sobolevBilinearProduct
sobolevCenterEvaluation def: Evaluation at the torus center, in the Fourier model of H^k. · OAI.DefocusingNLS.sobolevCenterEvaluation
sobolevColumnMap def: Linear combinations of the scalar coordinate functions. · OAI.RegularPointCoordinates.sobolevColumnMap
sobolevConjugation def: Conjugation of torus functions is a continuous real-linear map on H^k. · OAI.DefocusingNLS.sobolevConjugation
sobolevFourierCoefficient def: The unweighted Fourier coefficient of an element of the H^k model. · OAI.DefocusingNLS.sobolevFourierCoefficient
sobolevGram def: Riesz representation of the weighted Sobolev Gram form. · OAI.RegularPointCoordinates.sobolevGram
sobolevGramForm def: The weighted mean energy as a continuous symmetric bilinear form. · OAI.RegularPointCoordinates.sobolevGramForm
sobolevInjection def: H^s is represented isometrically by L2 via J^{-s}; its injection is into the actual space of tempered distributions. · OAI.ElasticityRegularity.sobolevInjection
SobolevInteractionPatch structure: A classical solution patch on an open interval containing the initial time. · OAI.DefocusingNLS.SobolevInteractionPatch
sobolevMeanGram def: The Gram average on a measurable set, with its actual reference measure. · OAI.RegularPointCoordinates.sobolevMeanGram
sobolevNonlinearRemainder def: The actual odd-power nonlinearity with its constant and linear parts removed. · OAI.DefocusingNLS.sobolevNonlinearRemainder
sobolevNorm def: The sum-of-component semiclassical Fourier norm, written using the Euclidean norm on the target; this is exactly that sum under the square root. · OAI.ElasticityCarleman.sobolevNorm
sobolevObservationVector def: The inverse Sobolev weights form an ℓ² vector exactly above dimension/2. · OAI.DefocusingNLS.sobolevObservationVector
sobolevOddPower def: The formalization constructs stable self-similar finite-time blowup for defocusing nonlinear Schrödinger equations on the twelve-dimensional torus. For every prescribed lower bound on the power, it selects an odd power at least that large … · OAI.DefocusingNLS.sobolevOddPower
sobolevOddPower def: The actual odd-power map on H^k, built from its bounded algebra operations. · OAI.DefocusingNLS.sobolevOddPower
sobolevPhase def: Multiplication by a constant unit complex phase is an isometry. · OAI.DefocusingNLS.sobolevPhase
sobolevPointEvaluation def: Point evaluation is a continuous complex-linear functional on the Fourier Sobolev model. · OAI.DefocusingNLS.sobolevPointEvaluation
sobolevPointObservationVector def: The modulated inverse weights representing evaluation at x. · OAI.DefocusingNLS.sobolevPointObservationVector
sobolevProduct def: The formalization constructs stable self-similar finite-time blowup for defocusing nonlinear Schrödinger equations on the twelve-dimensional torus. For every prescribed lower bound on the power, it selects an odd power at least that large … · OAI.DefocusingNLS.sobolevProduct
sobolevProduct def: Multiplication closes in the original Sobolev Fourier space. · OAI.DefocusingNLS.sobolevProduct
sobolevProductCoefficient def: The physical coefficients of the product are the discrete convolution. · OAI.DefocusingNLS.sobolevProductCoefficient
sobolevProductMajorant def: The two unweighted Young convolutions which dominate the weighted product. · OAI.DefocusingNLS.sobolevProductMajorant
sobolevProductWeight def: The positive Sobolev weight, inverse to the coefficient normalization. · OAI.DefocusingNLS.sobolevProductWeight
sobolevRealBilinearProduct def: Pointwise multiplication viewed as a continuous real-bilinear operation. · OAI.DefocusingNLS.sobolevRealBilinearProduct
sobolevTorusEmbedding def: The Fourier representation is a bounded linear map into continuous functions. · OAI.DefocusingNLS.sobolevTorusEmbedding
sobolevTorusFunction def: The actual continuous periodic function represented by a Fourier Sobolev vector. · OAI.DefocusingNLS.sobolevTorusFunction
sobolevTorusTerm def: A weighted Fourier coefficient times its periodic character. · OAI.DefocusingNLS.sobolevTorusTerm
sobolevTranslation def: Translation by x multiplies the Fourier coefficients by exp(i n·x). · OAI.DefocusingNLS.sobolevTranslation
sobolevVariance def: The deterministic expected energy weight. · OAI.DefocusingNLS.sobolevVariance
Solution abbrev: A split-frame solution of the finite incidence equations over S. · OAI.IntegralCharacterVarieties.FiniteMatrixPresentation.Solution
Solution abbrev: The split-frame solution functor of the incidence diagram. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.Solution
solution def: The unique solution of the short-time linear Volterra equation. · OAI.ForcedComputation.JetVolterra.solution
solution def: All finite-order solutions determine this one scalar function. · OAI.ForcedComputation.PlaneScalarMild.CompatibleData.solution
solutionJet def: The unique finite-jet solution used to represent the common scalar function. · OAI.ForcedComputation.PlaneScalarMild.CompatibleData.solutionJet
solutionOfEquations def: Assemble a solution from its surface, seam, and vertex equations. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.solutionOfEquations
Solvable def: The decision predicate includes the empty subset. · OAI.LowSpaceSubsetSum.Solvable
someLayout def: A layout on any fixed finite indexing type; only used for fixed operations. · OAI.PowerSaving.Layout.someLayout
sort def: sort f is the permutation that orders Fin n according to the order of the outputs of f. · OAI.Nagata.Workers.W10.LocalTupleSort.sort
sortAtoms def: Height sorting is a permutation, so it retains every positive table. · OAI.Thomason.Construction.SimplexComplex.sortAtoms
SortedAt def: Sorting at a profile records all source sorting judgments, not an exact profile assertion; wrappers intentionally permit profile growth. · OAI.PureTypeSystem.Labelled.SortedAt
sortedExit def: Sorting independently at every coordinate gives the monotone layer function. · OAI.Superstring.Hierarchical.Threading.sortedExit
sortedSequence def: Append the terminal zero used in the telescoping flag formula. · OAI.Thomason.Construction.SubdivisionBarycentric.sortedSequence
sortPerm def: Sort a finite injective natural-valued list without changing its indices. · OAI.CoordinateSweeps.YoungCorner.sortPerm
Sound def: A semantic bound on this input game, not a hypothesis asserting NP-hardness or an assumed Max-Cut reduction. · OAI.OptimalMaxCut.LongCode.Game.Sound
SoundAt def: Every success declared along this execution is certified by positives queried within that transaction alone. · OAI.MatroidProphet.MarkedQueryProgram.SoundAt
SoundAt def: Every labeling satisfies a fraction at most error. · OAI.MaxCutGames.Foundations.Target.SoundAt
SoundAt def: Every labeling satisfies a fraction at most error. · OAI.UniqueGamesTheorem.Foundations.Target.SoundAt
Source structure: Source machines with supplied polynomial clock parameters (need not be positive for building the graph) and controller space coefficient. Semantics will justify these externally. · OAI.ExactDerandomization.Base.Source
Source abbrev: The manuscript's five-generator single-relation presentation. · OAI.StableCoordinate.NonzeroFiber.Source
source def: The literal source algorithm: a singleton does not invoke f; every longer maximal raw run invokes f once on its unmodified concatenation. · OAI.FiniteAlgebraicEnvelopes.Compiler.RawRuns.source
source def: The source coefficient tensor, with its actual shared first-input space. · OAI.MatrixMultiplication.AuxiliarySeparation.Sector.source
source def: The actual nonempty, numbered occurrence list at the fixed parameters. · OAI.MaxCutGames.Foundations.Hastad.SourceGap.source
source def: Conversion to natural occupations; exact whenever all signed entries are nonnegative. · OAI.Problem335.ShiftedOccupation.source
source def: The actual marked initial vertex of the designated interval, if any. · OAI.TorsionFreeZeroDivisors.StageRoot.source
source def: The actual nonempty, numbered occurrence list at the fixed parameters. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceGap.source
sourceActiveProduct def: Inactive indices have factor one, exactly as in the source path expansion. The threshold is tested against the original forward masses. · OAI.LipschitzHilbert.sourceActiveProduct
sourceAlgebra def: The actual generated algebra of the source regular representation. Factor, trace, finiteness and predual properties are separate obligations. · OAI.FiniteFactor.sourceAlgebra
sourceAngleLaw def: This law is selected before any invariant projection is considered. · OAI.FiniteFactor.sourceAngleLaw
sourceAngleMarginal def: Normalized Lebesgue measure on the actual source interval. · OAI.FiniteFactor.sourceAngleMarginal
sourceAngleUnit def: The source angle interval has length 1/N_s, where N_s=m_s K_s. · OAI.FiniteFactor.sourceAngleUnit
sourceAngularShellMeasure def: The radial shell measure shared by the planar and all-dimensional arguments. · OAI.PlanarFalconer.sourceAngularShellMeasure
SourceArithmeticStage structure: Literal length and overlap-count state of one nested source block. copies and trials refer to the transition into this state. · OAI.TraceReconstruction.SourceArithmeticStage
sourceAutomaton def: The explicit source machine: none starts, some none accepts, and some (some h) is the state labelled by h. · OAI.TwoWayComplementation.sourceAutomaton
sourceBasedDriverPath def: The same continuous Brownian path, retaining its pointwise zero start. · OAI.SLEExactGauge.sourceBasedDriverPath
SourceBaseFunction structure: A bounded Borel function on the source cube. Values outside the cube are irrelevant because the representation evaluates rotation representatives. · OAI.FiniteFactor.SourceBaseFunction
sourceBaseIndicator def: Bounded measurable indicator on the base. · OAI.FiniteFactor.sourceBaseIndicator
sourceBaseMap def: The existing bundle base map with the canonical source quotient aliases. · OAI.StableCoordinate.FrameLift.sourceBaseMap
sourceBaseStar def: Conjugate of a bounded source base function, with the same bound. · OAI.FiniteFactor.sourceBaseStar
sourceBasis def: Constant coordinate vector, realized in the original source L2 space. · OAI.FiniteFactor.sourceBasis
sourceBiasedPair def: The mixed pair has the original x endpoint and selects x or y coordinatewise. · OAI.FiniteFactor.sourceBiasedPair
sourceBiasedPrefixMask def: At prefix n, use y in the first n source blocks and x in the tail. · OAI.FiniteFactor.sourceBiasedPrefixMask
sourceBimoduleTotalComparison def: The source signed total comparison for every Z, with the actual outside B-action retained. · OAI.LittleFinitistic.OrdinarySimulation.sourceBimoduleTotalComparison
sourceBlock def: The genuine source section space before high-branch multiplication by Pi. · OAI.Nagata.Workers.W10.sourceBlock
sourceBlockWindows def: Levels outside the finite source block are irrelevant. Their harmless positive extension permits the global-index API of the finite tree lemmas. · OAI.TraceReconstruction.sourceBlockWindows
sourceBlockWord def: The actual recursively substituted source block words. Indices in P are shifted down by one, but physical binary strings are unchanged. · OAI.TraceReconstruction.sourceBlockWord
sourceBottomIndicator def: The unsigned bottom constraint at one literal complete-tree leaf. · OAI.TraceReconstruction.sourceBottomIndicator
sourceBoundaryForm def: The finite shuffle form used in both Stokes equations. · OAI.Mahler.sourceBoundaryForm
sourceBoundarySlots def: The leading one-form slot precedes the interleaved two-form slots. · OAI.Mahler.sourceBoundarySlots
SourceBundle abbrev: One localBits bundle at each child, and one globalBits bundle at each vertex in the first quarter, paired with the reflected last-quarter post region. · OAI.TraceReconstruction.SourceBundle
sourceBundleBoundary def: The prefix-defined boundary uses the same conjugated bundle increments as the Haar-preserving state-coordinate map. · OAI.PolycyclicRecognition.sourceBundleBoundary
sourceCappedPairZeroSet def: The capped equation uses the previously constructed field of the original operator, on the actual sampling torus. · OAI.FiniteFactor.sourceCappedPairZeroSet
sourceCell def: The original quotient and kernel cell underlying the tile probes. · OAI.PolycyclicRecognition.CartanWordModel.TiledProbeChainData.sourceCell
sourceCenterWord def: The actual center before recursively cutting its pure side regions. · OAI.TraceReconstruction.sourceCenterWord
SourceCentral def: All actual motif support lies in the chosen central interval at every vertex of the full window tree. This is a predicate of the complete bits, not an event on which any posterior law is conditioned. · OAI.TraceReconstruction.SourceCentral
sourceCentralCoefficient def: The scalar coefficient is the actual field's 00 entry. · OAI.FiniteFactor.sourceCentralCoefficient
sourceChains def: Pull the stored chains back along the actual source coordinates. No selection of new generator words is needed on the probability cell. · OAI.PolycyclicRecognition.CartanWordModel.TiledProbeChainData.sourceChains
sourceCoefficient def: The coefficient is defined directly from the two literal leaf-type patterns under the common complete posterior bit tree. · OAI.TraceReconstruction.sourceCoefficient
sourceCoefficientEquiv def: Both positive-degree branches and the degree-zero tip use the literal source space and the actual injective marked-product multiplication. · OAI.Nagata.Workers.W10.sourceCoefficientEquiv
sourceConjugation def: Explicit conjugate-linear isometry; no modular or tracial property assumed. · OAI.FiniteFactor.sourceConjugation
sourceConormalFrame def: Line trivializations induced by these global sections. · OAI.SourcePullbackChart.sourceConormalFrame
SourceContract def: Full named source obligation, including well-definedness for every interleaving. The test function is injective because the manuscript tests a set of n nonloops. The conclusion actually remains true without that injectivity assumption. · OAI.MatroidProphet.MarkedPivots.SourceContract
sourceCoordinates def: Reparametrization uses the actual Haar product cell, so the selected probability measure can be chosen there before concentration is applied. · OAI.PolycyclicRecognition.CartanWordModel.TiledProbeChainData.sourceCoordinates
sourceCoordinateSections def: Actual coordinate sections, with the source's unreduced τ^η multiplier type. · OAI.Nagata.W06.TorusConfiguration.sourceCoordinateSections
sourceCorner def: The unique outgoing boundary side at the corner containing a germ. This uses the corner involution, not the boundary-circle color. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.sourceCorner
sourceCornerHom def: The original inclusion restricted to matching corners. · OAI.KadisonSimilarity.sourceCornerHom
sourceCover def: An affine open cover of the literal source by the tensor rings on which coneEvaluation was constructed. · OAI.SourcePullbackChart.sourceCover
sourceDCOperator def: Linear jet operator for the normalization -D(u)(iv)/4. · OAI.Mahler.sourceDCOperator
sourceDegree def: Source degree in a second coefficient block. · OAI.Nagata.FiniteExponents.sourceDegree
sourceDegreeBudget def: Full unabsorbed bound, now with no placeholder anchor-energy hypothesis. · OAI.TraceReconstruction.sourceDegreeBudget
sourceDensePair def: Literal dense substitution of the old auxiliary laws, used only for overlap comparison. Statistical cost is paid only for differing motif leaves, not the filler atoms. · OAI.TraceReconstruction.sourceDensePair
sourceDiagonal def: The actual diagonal entry f(sigma^k x) of pi(f). · OAI.FiniteFactor.sourceDiagonal
sourceDilation def: The continuous source dilation integrates the same Lie automorphism as the algebraic dilation used by the stored linked chains. · OAI.PolycyclicRecognition.CartanWordModel.sourceDilation
sourceDimension def: Dimension of the source of the actual mixed operator. · OAI.Problem335.LowerParameters.sourceDimension
sourceDimension def: Literal source block dimensions, with zero-based block number. · OAI.TraceReconstruction.sourceDimension
sourceDual def: The source term of the quadratic Kantorovich certificate. · OAI.Problem358.FiniteAffine.sourceDual
sourceEffectiveHiddenTail def: Exact finite-union metadata cost; it contains no postulated typical event and no independence assertion between different collections. · OAI.TraceReconstruction.sourceEffectiveHiddenTail
sourceEndpoint def: The negative endpoint in the boundary of b ⊗ [0,1]. · OAI.Thomason.Construction.RightCylinderBasis.sourceEndpoint
sourceEpigraph def: The manuscript's epigraph; the domain is not silently replaced by all of space. · OAI.AffineBernstein.sourceEpigraph
sourceEquiv def: Changing the source realization by an equivalence preserves Hom. · OAI.Representation.IntertwiningMap.sourceEquiv
sourceEuler def: Euler operator of the first coordinate block. · OAI.Problem346.PolynomialShift.sourceEuler
sourceExclusionWidth def: The source margin 2K_j/N_s. At s=j it is exactly 2/m_j. · OAI.FiniteFactor.sourceExclusionWidth
SourceExpUpper def: Literal finite-block factors are subpower uniformly over the joint deletion regime when their logarithms grow polynomially in the depth. · OAI.TraceReconstruction.SourceExpUpper
sourceFiberEnergy def: Squared fiber norm of an actual source L2 vector. Fubini proves finiteness and identifies it with the square-summable coordinate norm almost everywhere. · OAI.FiniteFactor.sourceFiberEnergy
sourceFiberEquiv def: Every nonzero fiber of the manuscript presentation is a polynomial algebra. · OAI.StableCoordinate.NonzeroFiber.sourceFiberEquiv
sourceFiberVector def: A fiber vector with an irrelevant zero value on the exceptional null set. · OAI.FiniteFactor.sourceFiberVector
sourceFieldGoodSet def: A Borel good set, removing just one null set for algebra elements. · OAI.FiniteFactor.sourceFieldGoodSet
sourceFillerWord def: Fixed pure fillers, including the deterministic empty bottom fillers. · OAI.TraceReconstruction.sourceFillerWord
sourceFiniteReferenceKernel def: Independent diagonal-or-Haar replacement at a finite set of coordinates. · OAI.FiniteFactor.sourceFiniteReferenceKernel
sourceFiniteVector def: Finite coordinate vectors as actual constant-base L2 vectors. · OAI.FiniteFactor.sourceFiniteVector
sourceFinsupp def: A natural shifted occupation, represented in the canonical finite-exponent type. · OAI.Problem335.ShiftedOccupation.sourceFinsupp
sourceFirstRank def: Source letters precede all test letters in the orientation of a pair. · OAI.KadisonSimilarity.sourceFirstRank
sourceFitTail def: Explicit finite tree union bound. The source's scale estimates will make this number exponentially small; no independence of sibling shifts is used. · OAI.TraceReconstruction.sourceFitTail
sourceFluxForm def: The bundled complex form, with its specified boundary slot order. · OAI.Mahler.sourceFluxForm
sourceFrame def: The original determinant frame on SL₂ × A¹. · OAI.StableCoordinate.DeterminantLiftWeights.sourceFrame
sourceFrame def: The canonical determinant frame in the exact quotient. · OAI.StableCoordinate.FrameLift.sourceFrame
sourceFullWord def: Full literal unconditioned root partition, including both members of every outer paired side. Its top trial count is the source's J, even or odd. · OAI.TraceReconstruction.sourceFullWord
sourceGap def: The neighbor radius at the final transition, in child-width units. · OAI.LipschitzHilbert.AngularGrid.sourceGap
sourceGeneratedVonNeumann def: Double commutant of a concrete family, including its adjoints. · OAI.FiniteFactor.sourceGeneratedVonNeumann
sourceGlobalPre def: Global trial positions are ID functions, not positions inferred from a realized output or orientation. · OAI.TraceReconstruction.sourceGlobalPre
sourceGlobalTrials def: The actual complete-tree global schedule has the exact plateau identity needed by the already proved correlated anchor energy theorem. · OAI.TraceReconstruction.sourceGlobalTrials
sourceGroundEnergy def: Variational source energy over all normalized spin states. · OAI.ContinuumCoulomb.sourceGroundEnergy
sourceGroupedCoefficient def: The scalar factor at one node is constant on every top/length fiber. · OAI.LipschitzHilbert.AngularGrid.sourceGroupedCoefficient
sourceHalfMoment def: Source moment order is twice this integer. · OAI.TraceReconstruction.sourceHalfMoment
sourceHamiltonian def: Complete finite Hamiltonian, with no magnetic or one-site terms. · OAI.ContinuumCoulomb.sourceHamiltonian
sourceHeisenbergAction def: The local Heisenberg action on the complete n-spin Hilbert space. · OAI.ContinuumCoulomb.sourceHeisenbergAction
sourceHeisenbergForm def: The real quadratic form of the local Heisenberg interaction. · OAI.ContinuumCoulomb.sourceHeisenbergForm
sourceHessian def: Complex Hessian indexing: row k is z_k, column l is bar z_l. · OAI.Mahler.sourceHessian
sourceHiddenTail def: Exact finite-union metadata cost; it contains no postulated typical event and no independence assertion between different collections. · OAI.TraceReconstruction.sourceHiddenTail
sourceHomogeneousPrimitive def: The literal homogeneous primitive, continuously bundled by exact finite basis expansion. No exterior differentiation identity is assumed here. · OAI.Mahler.sourceHomogeneousPrimitive
sourceIdealLine def: Line trivialization of the kernel ideal sheaf on W. · OAI.SourceZeroSections.sourceIdealLine
sourceIncidence def: The ordinary game uses actual occurrence IDs and actual variable names. · OAI.MaxCutGames.Decoder.ActualAdviceUpper.sourceIncidence
sourceIncidence def: The ordinary game uses actual occurrence IDs and actual variable names. · OAI.UniqueGamesTheorem.Decoder.ActualAdviceUpper.sourceIncidence
sourceIndex def: The unique shifted source as an element of the original bidegree domain. · OAI.Problem335.ShiftedOccupation.sourceIndex
sourceInitial def: A source polynomial of weight at most d has a degree-d symbol. · OAI.StableCoordinate.GradedPresentationMap.sourceInitial
sourceInitialAnchor def: One-based manuscript anchors translated once to zero-based list indices. · OAI.TraceReconstruction.sourceInitialAnchor
SourceInitialGeometry structure: Only arithmetic predicates on the literal source integers. · OAI.TraceReconstruction.SourceInitialGeometry
sourceInitialM def: Literal integer choices of the initial-pair construction, including the outer floors. · OAI.TraceReconstruction.sourceInitialM
sourceInitialOffset def: The manuscript's one-based middle defect, translated to a list index. · OAI.TraceReconstruction.sourceInitialOffset
sourceInitialWeight def: Actual primitive-score weight at the base scale. · OAI.TraceReconstruction.sourceInitialWeight
sourceInteriorExponent def: The interior scale mass excludes the bottom window, as required by the sparse role count rather than the much larger total leaf count. · OAI.TraceReconstruction.sourceInteriorExponent
sourceKappa def: The source's kappa_s, with natural indices shifted to start at one. · OAI.FiniteFactor.sourceKappa
sourceKernelCell def: The scale-dependent kernel part of the original source cell. · OAI.PolycyclicRecognition.CartanWordModel.TiledProbeChainData.sourceKernelCell
SourceLabeledLattice structure: Common packet bounds with a finite set of translated lattice labels at each scale. All bounds are imposed only on labels that are present. · OAI.LipschitzHilbert.SourceLabeledLattice
sourceLanguage def: The source language consists of the words with nonempty relation product. · OAI.TwoWayComplementation.sourceLanguage
SourceLatticePacket structure: Concrete local packet data. These are geometric, seminorm and dimensionless scale bounds, with no coefficient or concentration premise. · OAI.LipschitzHilbert.SourceLatticePacket
SourceLayoutMargins structure: Numerical layout conditions from the source. These contain no word or statistical conclusions, only the genuine integer margins. · OAI.TraceReconstruction.SourceLayoutMargins
sourceLayoutReserve def: One-bundle reserved nominal count is integral because the number of paired trials is even. The actual filler reserves both local and global bundles when both are present in a region. · OAI.TraceReconstruction.sourceLayoutReserve
SourceLevelPair abbrev: The literal induction invariant, including the depth-dependent moment, actual complete-trace simulator, distinct words, length, and power overlap. · OAI.TraceReconstruction.SourceLevelPair
sourceLineScale def: The manuscript line-window length at one angular generation. · OAI.LipschitzHilbert.sourceLineScale
sourceLinkContraction def: Relabeling uses the original seed graph and merely chooses an order on its vertices for the signed integral chain convention. · OAI.EilenbergGanea.sourceLinkContraction
sourceLocalTrials def: At a nonroot node the source trial count is an even integer. · OAI.TraceReconstruction.sourceLocalTrials
sourceLog def: The source logarithmic cocycle, using the actual weighted floor sum. · OAI.FiniteFactor.sourceLog
sourceMarkedCell def: The exact two inequalities in the source's cell marking rule. · OAI.LipschitzHilbert.sourceMarkedCell
sourceMask def: The observation mask of the concrete source rule, before any values are seen. · OAI.MatroidProphet.Secretary.sourceMask
sourceMatrixEntry def: Actual matrix coefficients obtained by applying H to constant e_n. · OAI.FiniteFactor.sourceMatrixEntry
sourceMeasure abbrev: The regular representation written as L2(X × Z), canonically equivalent to the source's L2(X; l2(Z)). · OAI.FiniteFactor.sourceMeasure
sourceMixedOperator def: The full operator restricted to a finite-bidegree source, as a linear family in the polynomial defining the operator. · OAI.Problem335.sourceMixedOperator
sourceMultiplication def: A bounded multiplication operator from the actual source formula. · OAI.FiniteFactor.sourceMultiplication
sourceNoncentralMass def: The discarded noncentral role contributions are bounded by the actual conditional probability of a failed full-tree layout. Typical metadata may bound this mass; the averaging distribution itself remains unconditional. · OAI.TraceReconstruction.sourceNoncentralMass
sourceNormalEvaluation def: Evaluation by each actual algebra element on the constructed space. · OAI.FiniteFactor.sourceNormalEvaluation
sourceNormalFunctionalSpace abbrev: The norm-closed span of actual vector functionals. This constructs a Banach space; identifying its whole dual with the source algebra is a separate surjectivity theorem, not part of this definition. · OAI.FiniteFactor.sourceNormalFunctionalSpace
SourceNormalPredualSpace def: A named carrier keeps the inherited Banach-space structures coherent in iterated continuous duals. Its underlying set is exactly the closed span. · OAI.FiniteFactor.SourceNormalPredualSpace
sourceNormalRieszOperator def: Every functional on the constructed Banach space gives an actual bounded operator by Riesz representation. Membership in the source algebra is proved separately by commutation, not assumed here. · OAI.FiniteFactor.sourceNormalRieszOperator
sourceNormValues def: The testing set with m≥1, nonempty segments, and the square root taken before the supremum. · OAI.ForestSpace.sourceNormValues
sourceOffsets def: The exponents in Θ(τ^(x₀−x_i)e^x), not the point exponents x_i themselves. · OAI.Nagata.W29.sourceOffsets
sourceOmittedBadEvent def: The source omitted-index masks. Near tagged indices and retained terms have empty masks. Far assignments depend on n,v and fixed angles only. · OAI.FiniteFactor.sourceOmittedBadEvent
sourceOperator def: T=U pi(f), constructed as a bounded operator, with no assumed factor or invariant-subspace properties. · OAI.FiniteFactor.sourceOperator
sourceOperatorField def: The actual bounded operator field, extended by zero on a Borel null set. · OAI.FiniteFactor.sourceOperatorField
sourceOracleDecode def: A fixed public decoder. It does not know the hidden orientations or the input hypothesis, and it drops rather than reveals all pairing flags. · OAI.TraceReconstruction.sourceOracleDecode
sourceOracleDensity def: Literal density of the source's unconditioned finite canonical oracle. · OAI.TraceReconstruction.sourceOracleDensity
sourceP def: The distinguished element p in the manuscript presentation. · OAI.StableCoordinate.NonzeroFiber.sourceP
sourcePairingTest def: Independent label choices on source-source pairs. · OAI.KadisonSimilarity.sourcePairingTest
sourcePairLabels def: The source tuple uses a variable label on each source-source pair and a fixed label on each source-test pair. The cycle class separates the pools. · OAI.KadisonSimilarity.sourcePairLabels
sourcePathAmplitude def: The explicit path amplitude on the canonical fixed-bidegree source indices. · OAI.Problem335.LocalMoments.sourcePathAmplitude
SourcePhaseAssignment structure: A deterministic source phase assignment, chosen from the fixed angles. · OAI.FiniteFactor.SourcePhaseAssignment
sourcePhaseFailureMajorant def: An explicit majorant for the positive-power union bound. · OAI.FiniteFactor.sourcePhaseFailureMajorant
sourcePolynomial def: One explicit output-length polynomial for each fixed u,D. · OAI.MaxCutGames.Foundations.Hastad.SourceBounds.sourcePolynomial
sourcePolynomial def: One explicit output-length polynomial for each fixed u,D. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceBounds.sourcePolynomial
SourcePolyUpper def: A convenience for the manuscript's fixed-B polynomial losses. This predicate is only used to prove bounds on the literal finite constants. · OAI.TraceReconstruction.SourcePolyUpper
sourcePredictionValue def: The exact source prediction in mass-normalized child variables. · OAI.LipschitzHilbert.sourcePredictionValue
sourcePredualIdentification def: The full dual identification, with the exact conjugate-linear convention required by Target.HasSeparablePredual. · OAI.FiniteFactor.sourcePredualIdentification
sourcePresentation def: The exact canonical source hypersurface is the four-variable polynomial ring. · OAI.StableCoordinate.Obstruction.sourcePresentation
sourcePrimitiveSlots def: The order alpha, beta, followed by k ordered d-alpha factors. · OAI.Mahler.sourcePrimitiveSlots
sourceProductive def: Configurations from which the explicit source machine has a suffix witness. · OAI.TwoWayComplementation.sourceProductive
sourceProjectionEvent def: The actual three conditions defining E_j in the source, with j+1 indexing. · OAI.FiniteFactor.sourceProjectionEvent
sourceQualifying def: The finite deterministic qualifying list at one selected block. · OAI.FiniteFactor.sourceQualifying
sourceQuotientCell def: The fixed quotient part of the original source probability cell. · OAI.PolycyclicRecognition.CartanWordModel.TiledProbeChainData.sourceQuotientCell
sourceQuotientCoordinates def: The actual quotient chart associated to the section retained by the tiles. · OAI.PolycyclicRecognition.CartanWordModel.TiledProbeChainData.sourceQuotientCoordinates
sourceR def: Verbatim source local R entries. · OAI.CriticalHoneycomb.Transfer.PlanarNormalization.sourceR
SourceRationalIndependent def: Exact integer formulation of rational independence of 1 and all angles. · OAI.FiniteFactor.SourceRationalIndependent
sourceReferenceToPhysical def: Reference coordinates are ordered (transverse, horizontal). · OAI.LipschitzHilbert.sourceReferenceToPhysical
sourceReflectionIndex def: The reflection for the source regular representation. · OAI.FiniteFactor.sourceReflectionIndex
sourceReportFlags def: The actual report flags, in the literal nested trial coordinates. · OAI.TraceReconstruction.sourceReportFlags
sourceReverseLineGrid def: A single root line grid contains every source block window, in reverse angular order. Its depth includes the shorter initial source block exactly. · OAI.LipschitzHilbert.sourceReverseLineGrid
sourceRoleTerm def: A fixed real coefficient term evaluated on the actual complete bit tree. The selected role choice is fixed independently of these bits. · OAI.TraceReconstruction.sourceRoleTerm
sourceRootTrials def: The root schedule count: the literal floor after subtracting its window. · OAI.TraceReconstruction.sourceRootTrials
sourceSamplingCylinder def: Finite cylinders in the full real product: coordinates outside the finite support are unrestricted. On the conull cube these are the original rational tests. The ambient real product is not asserted to be compact. · OAI.FiniteFactor.sourceSamplingCylinder
sourceSamplingDiagonalContribution def: The self-pair part of the empirical squared norm, with its actual P^-2 weight. · OAI.FiniteFactor.sourceSamplingDiagonalContribution
sourceSamplingLimitRectangleMass def: Explicit finite-coordinate rectangle mass of the triangular/diagonal model. The triangular term retains the actual source weight integral. · OAI.FiniteFactor.sourceSamplingLimitRectangleMass
sourceSamplingPowers def: A deterministic finite set fixed from the numerical stages alone. · OAI.FiniteFactor.sourceSamplingPowers
SourceSamplingRationalTest abbrev: A finite rational cylinder test, with two closed intervals per tested coordinate. Whole-circle tests fill the remaining coordinates. · OAI.FiniteFactor.SourceSamplingRationalTest
sourceSamplingSquaredError def: The full sampling error, including every distinct-pair and mixed term. · OAI.FiniteFactor.sourceSamplingSquaredError
sourceSamplingTrimmed def: The actual ordered restriction of the already constructed average. · OAI.FiniteFactor.sourceSamplingTrimmed
sourceSamplingUntrimmed def: Finite positive untrimmed measures averaged over the constructed powers. No weak limit or uniform marginal bound is asserted by this definition. · OAI.FiniteFactor.sourceSamplingUntrimmed
sourceScalar def: Scalar base multiplication, independent of the fiber coordinate. · OAI.FiniteFactor.sourceScalar
sourceScheduledTrials def: Global trials at a parent use its children's local-trial count. This fixes the index shift between source level labels and zero-based arrays. · OAI.TraceReconstruction.sourceScheduledTrials
sourceSection def: The tile construction's fixed linear section as a continuous group section. · OAI.PolycyclicRecognition.CartanWordModel.TiledProbeChainData.sourceSection
sourceSharpDegreeBudget def: Below the genuine forced binary leaf threshold there is NO forced contribution; retaining even its coarse bound would lose the main exponent. · OAI.TraceReconstruction.sourceSharpDegreeBudget
sourceShiftComparison def: Exact source indexing: b=a+l, with only the manuscript's support hypothesis. · OAI.LittleFinitistic.OrdinarySimulation.sourceShiftComparison
sourceShiftIndex def: The shift S e_k=e_(k+1), so its action on coordinate functions uses k-1. · OAI.FiniteFactor.sourceShiftIndex
sourceSnapshotPrediction def: The same snapshot, with only its prescribed diagonal carriers moving. · OAI.LipschitzHilbert.sourceSnapshotPrediction
sourceSpinSwap def: Swap the two selected spin coordinates, leaving every spectator fixed. · OAI.ContinuumCoulomb.sourceSpinSwap
SourceStageChoice structure: Complete numerical output of one source stage. The polynomial bound is proved before L, Gnext and m are selected. Stages use s=0 for source j=1. · OAI.FiniteFactor.SourceStageChoice
SourceStageState structure: State already fixed before the new block length is chosen. B is the previous weighted block-size sum; d is the previous coordinate count. · OAI.FiniteFactor.SourceStageState
sourceStageState def: The infinite numerical construction is fixed without an operator or a candidate invariant projection as an input. · OAI.FiniteFactor.sourceStageState
sourceSubgraph def: The literal positive source subgraph; no integrability is implicit. · OAI.HenonLaneEmden.LayerTonelli.sourceSubgraph
sourceSubsetCode def: A monomial is precisely its set of actual leaf addresses. · OAI.TraceReconstruction.sourceSubsetCode
sourceSwapExpectation def: The expectation of coordinate exchange, taking its real part. · OAI.ContinuumCoulomb.sourceSwapExpectation
sourceTagEndpointEvent def: Endpoint consequence of a source tag; the label implication and positive probability are separate obligations, not encoded into this predicate. · OAI.FiniteFactor.sourceTagEndpointEvent
sourceTargetScale def: The backwards target scales as an exact product of source exponents. · OAI.TraceReconstruction.sourceTargetScale
sourceTensor def: Polynomial multiplication in its original monomial coordinates. · OAI.MatrixMultiplication.AuxiliarySeparation.DeterminantFiltration.sourceTensor
sourceTopSlots def: Explicit order of the differentiated top form: first the two slots created from derivative and one-form, then all existing interleaved pairs. · OAI.Mahler.sourceTopSlots
sourceTorusOperatorField def: The proved source field on the actual compact sampling torus. · OAI.FiniteFactor.sourceTorusOperatorField
sourceTracialRepresentation def: Actual instantiation of the certified tracial-representation API. Both central obligations are proved for this source algebra. · OAI.FiniteFactor.sourceTracialRepresentation
sourceTransportIndex def: Combined integer base rotation and fiber translation. · OAI.FiniteFactor.sourceTransportIndex
sourceTreeAnchors def: Displacements from the jointly generated bit tree. The ancestor global map may vary off its plateau. No independent jitter per child is introduced. The fixed center includes the deterministic half-trial corrections. · OAI.TraceReconstruction.sourceTreeAnchors
SourceTreeFit def: The actual fit event uses a joint test of all displacements. It does not assert independence and will only be charged probabilistically afterwards. · OAI.TraceReconstruction.SourceTreeFit
sourceTreeLeafBool def: All actual leaf Booleans on the complete deterministic window tree. · OAI.TraceReconstruction.sourceTreeLeafBool
sourceTreeLeafSign def: Exact recursive leaf signs for the complete bit tree, with one joint parent-global shift supplied to each sibling's local mechanism. · OAI.TraceReconstruction.sourceTreeLeafSign
sourceTreeWord def: The unconditional literal surrogate partition: local and common-global pairs contribute their real selected pure lengths, on every bit assignment. · OAI.TraceReconstruction.sourceTreeWord
sourceTrialWord def: Two actual deterministic pure binary side words for each paired trial ID. · OAI.TraceReconstruction.sourceTrialWord
sourceTwoRows def: Source U_A U_B in its original, unswapped spin labels A,B,sites. · OAI.CriticalHoneycomb.Transfer.PlanarNormalization.sourceTwoRows
SourceTypical def: One metadata event, fixed before choosing a coefficient, controls local and global smoothing and all internal central-third tests. · OAI.TraceReconstruction.SourceTypical
sourceVacuum def: The constant e_0 field in the actual L2(X times Z) source model. · OAI.FiniteFactor.sourceVacuum
sourceValue def: The bounded source is continued constantly in time outside the finite slab. · OAI.ForcedComputation.PlaneScalarMild.sourceValue
sourceVectorFunctional def: An actual vector functional restricted to the original source algebra. · OAI.FiniteFactor.sourceVectorFunctional
sourceWidth def: Both source widths, derived from the stages already constructed. · OAI.FiniteFactor.sourceWidth
sourceWidth def: Number of second-leg coordinates in the original convolution. · OAI.MatrixMultiplication.AuxiliarySeparation.Sector.sourceWidth
sourceWindowSize def: The source integer window size, with zero-based nesting index. · OAI.TraceReconstruction.sourceWindowSize
sourceZeroFiberEquiv def: The zero fiber of the five-variable hypersurface presentation is a three-variable polynomial algebra. · OAI.StableCoordinate.ZeroFiber.sourceZeroFiberEquiv
sow def: Build Tape len by count writes (initialized with a blank z). · OAI.PowerSaving.RAM.Bill.sow
Space abbrev: The full Banach dual of the incomparable-segment primal completion. · OAI.DiamondDistortion.IncomparableModel.Space
Space abbrev: The Banach space of bounded continuous spatial jets through order k. · OAI.ForcedComputation.BoundedSpatialJets.Space
Space def: The points of an adjunction are the old points and the interiors of the new disks. · OAI.GroupRingTopology.Attachment.Space
Space def: Finitely supported probability weights on a face, with the weak topology. · OAI.GroupRingTopology.Simplicial.Space
Space abbrev: The separated metric ultralimit. Completeness is not needed for the finite obstruction; all segment and CN comparisons are established below. · OAI.HyperbolicObstruction.MetricLimit.Space
Space abbrev: The standard free module of the actual Coxeter group. · OAI.KLInvariance.Hecke.Space
Space def: The actual (not postulated) plain observation space of every normalized plain actual type. The base parameter B is the set of candidate assignments when used by the component interpretation. · OAI.PureTypeSystem.ActualPlain.Space
Space abbrev: The two coordinate halves of the source's F₂^(2s); this product is the canonical coordinate bijection, not a different graph or bilinear form. · OAI.SmoothObstruction.WordAlphabet.Space
space def: The actual fixed submodule eN, for an idempotent e. · OAI.ExternalTensor.Section4.ArrowVertex.space
spaceEquiv def: Identify the vertex functor with the actual fixed subspace of scalar restriction. · OAI.LittleFinitistic.LeftVertexHomology.spaceEquiv
spaceThrough 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.Machine.spaceThrough
span def: Total span occupied by the windows in a complete tree of the given height. At a bottom-level child there is no following subtree gap; at higher levels the incoming window is followed by one gap and the complete child subtree. · OAI.Problem310.WindowRecursion.span
spanCoord def: Bounded algebraic coordinate functionals. · OAI.SeparableQuotient.PrefixNormers.spanCoord
spanned def: The Fourier coefficients are supported within a bandwidth of the signed span. This form (arbitrary thickening of our unthickened case) will also be used for broad capture. · OAI.FourierLLogL.Dyadic.spanned
spanningGraph def: The spanning graph with precisely the edges in A, with loops omitted only from adjacency: they still exist as edge orbits, have FK weight, and contribute two darts to the degree measure. Omitting loops does not change distance or k. · OAI.FKCRT.spanningGraph
SpanningTree def: `G.SpanningTree T` means `T` connects all vertices and deleting any one edge of `T` destroys that connectivity. · OAI.StrongThinTree.MultiGraph.SpanningTree
spanPiece def: Multiplicative homogeneous monomials describing every degree-n coefficient. · OAI.HomogeneousGeneration.spanPiece
spanSolve def: The vector in span(Y,C) with prescribed pairings with Y and C. · OAI.ClosedSurfaceR4.PeriodicCorrector.spanSolve
spanStarAlgebra def: The span itself is the required unital star subalgebra. · OAI.KadisonSimilarity.DihedralWordAlgebra.spanStarAlgebra
spark def: Shear strong when f ≠ 0. · OAI.PowerSaving.Slant.spark
sparseNode def: A bad node is active but has fewer than b active children. · OAI.TreeEdit.BinaryLower.FiniteActivity.sparseNode
Spart def: The source pieces are B_i intersected with the preimages of Y_i. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.SliceData.Spart
spatial def: Spatial inclusion in the cylinder. · OAI.ArnoldCounterexample.Forms.spatial
spatialAmplitude def: Store a fixed function of position in the existing low-jet coefficient API. · OAI.ClosedSurfaceR4.JetPolynomial.spatialAmplitude
spatialAssociator def: The canonical associator on completed Hilbert tensor products. · OAI.KadisonKastler.spatialAssociator
spatialAssocLeftEmbedding def: The algebraic threefold tensor product in the left-associated completion. · OAI.KadisonKastler.spatialAssocLeftEmbedding
spatialAssocRightEmbedding def: The algebraic threefold tensor product in the right-associated completion. · OAI.KadisonKastler.spatialAssocRightEmbedding
SpatialBadPath def: The manuscript's bad simple-path event at a fixed scale. The path starts at the origin; the exact first-exit conditions and empty remainder are kept. · OAI.GammaFPP.CrossingBridge.SpatialBadPath
spatialDescentTolerance def: The auxiliary tolerance used before choosing any representation or algebra. · OAI.KadisonKastler.spatialDescentTolerance
spatialEval def: Spatial evaluation of the algebraic tensor product on separate Hilbert spaces. Its operator norm is unchanged on the Hilbert completion. · OAI.NuclearUltrapower.spatialEval
spatialGradient def: Spatial gradient of a smooth one-parameter family, as a joint field. · OAI.Bernoulli.Foliation.spatialGradient
spatialMatrixAlgebraHom def: The finite matrix image with coefficients restricted to M. · OAI.KadisonKastler.spatialMatrixAlgebraHom
spatialMatrixAmplification def: Entrywise amplification of the finite spatial matrix representation. · OAI.KadisonKastler.spatialMatrixAmplification
spatialMatrixHom def: The actual finite spatial matrix representation. · OAI.KadisonKastler.spatialMatrixHom
spatialMatrixUnits def: Matrix units on the second tensor leg. · OAI.KadisonKastler.spatialMatrixUnits
spatialParameterDerivative def: Restrict the total derivative to spatial directions. · OAI.ForcedComputation.spatialParameterDerivative
spatialProductAlgebra def: The spatial von Neumann tensor product in the given representations. · OAI.KadisonKastler.spatialProductAlgebra
spatialRow def: The actual locally finite transition row on the entire spatial grid. · OAI.FiniteLabels.spatialRow
spatialRow def: The actual locally finite transition row on the entire spatial grid. · OAI.SignedDisk.FiniteLabels.spatialRow
spatialScheduleChange def: Actual normalized spatial squared norm of the conditional-mean change. Both schedules use the same ancestral kernel and spin function. · OAI.DilutedSpinGlass.ReducedTopology.spatialScheduleChange
spatialSlice def: Zero extension of a spatial slice; no value at a seam is used. · OAI.TriangularBilliards.spatialSlice
spatialSplitChange def: The literal spatial squared norm of the change when this one split moves from d to d+1. · OAI.DilutedSpinGlass.PrescribedTree.spatialSplitChange
spatialSq def: Squared normalized spatial norm, with the original zero-size convention. · OAI.DilutedSpinGlass.FiniteLaw.spatialSq
spatialSupport def: The manuscript's essential spatial color support. · OAI.EuclideanFiveColor.spatialSupport
spatialTensorEquiv def: The completed tensor product of two Hilbert-space identifications. · OAI.KadisonKastler.spatialTensorEquiv
spatialTransport def: Transport of all the matrix units, including the strong sum condition. · OAI.KadisonKastler.FullMatrixCorner.spatialTransport
spatialTransportAlgebra def: The concrete algebra transported by a Hilbert-space isometry. · OAI.KadisonKastler.spatialTransportAlgebra
spatialWitnessFamily def: Exact labelled path witnesses for the two defining spatial rare events. A support-defect witness retains which strict bound it satisfies. · OAI.GammaFPP.CrossingBridge.spatialWitnessFamily
Spec abbrev: Two bounded mark tables and the choice of spin or marker anchor. · OAI.DilutedSpinGlass.UniversalDictionary.Spec
spechtCharacter def: Character of the actual Specht module as an actual class function. · OAI.Saxl.spechtCharacter
spechtRep def: The genuine symmetric-group representation on the complex Specht module. · OAI.Saxl.spechtRep
spechtSelfDual def: Actual self-duality of every complex Specht representation. · OAI.Saxl.spechtSelfDual
spechtSub def: The Specht module, defined as the orbit span of a polytabloid in the row-tabloid permutation module, realized inside the word space. · OAI.Saxl.spechtSub
specialGramDet def: Real determinant of the actual holomorphic derivative's Hermitian Gram matrix. · OAI.SymmetricMahler.specialGramDet
SpecialImages structure: Exactly the individual scalar-image hypotheses of special-twist sufficiency. No simultaneous choice satisfying the wires is required. · OAI.WLIdentification.ComparisonSchemas.SpecialImages
specializationEquiv def: Specialize p to zero in the hypersurface presentation. · OAI.StableCoordinate.ZeroFiber.specializationEquiv
specialize def: Embed t into a coefficient field, leaving p,r polynomial indeterminates. · OAI.RationalHitting.ThreeLift.specialize
specialize def: Evaluate the additive action at a scalar parameter. · OAI.StableCoordinate.CoactionLND.PolynomialAdditiveAction.specialize
specializeEquiv def: An algebraic additive coaction gives genuine algebra automorphisms at all scalar parameters, with inverse parameter -t. · OAI.StableCoordinate.CoactionLND.PolynomialAdditiveAction.specializeEquiv
specialPlaneRotation def: The same matrix exponential, now as an element of the special orthogonal group. · OAI.InvariantIsing.specialPlaneRotation
SpecialRegularMassBound def: The regular-level mass bound for the finite-strip map. · OAI.SymmetricMahler.SpecialRegularMassBound
specialRotation def: Restricting from O(N) to SO(N) preserves the same spectral action. · OAI.InvariantIsing.specialRotation
specialTau def: Squared Euclidean norm of the actual holomorphic vector, in coordinates. · OAI.SymmetricMahler.specialTau
specialToOrthogonal def: Forget only the determinant-one condition. · OAI.InvariantIsing.specialToOrthogonal
specIsoOfAlgEquiv def: Contravariance takes an algebra equivalence to an actual scheme isomorphism. · OAI.StableCoordinate.AffineGeometry.specIsoOfAlgEquiv
speck def: b=1 small cross · OAI.PowerSaving.Slant.speck
SpectatorDiagram structure: The data of one actual diagram at a spectator prime. · OAI.Ostmann.SpectatorDiagram
spectralCandidate def: A single compatible base/M2 candidate, built from one dimension state. Only the abelian linearity constraints are still missing. · OAI.Kaplansky.CuntzDimensionState.spectralCandidate
spectralCLM def: Finite spectral synthesis as a real continuous linear map. · OAI.SharpLiebThirring.MatrixProof.spectralCLM
spectralCurve def: A positive-definite geodesic in an eigenbasis for its Hermitian direction. · OAI.Kaplansky.spectralCurve
spectralCutFunction def: The positive resolvent approximation to the open cut (c,∞). · OAI.KadisonSimilarity.spectralCutFunction
spectralDensityIntertwinerData def: Retain the intertwining property with the chosen measure-change isometry, so later concrete instantiations do not unfold its extension construction. · OAI.KadisonSimilarity.spectralDensityIntertwinerData
spectralEnergy def: Diagonal spectral quadratic form. · OAI.MaxCutGames.Fourier.MatrixEnergy.spectralEnergy
spectralEnergy def: Diagonal spectral quadratic form. · OAI.UniqueGamesTheorem.Fourier.MatrixEnergy.spectralEnergy
spectralFreeCoreBoundary def: The free core has zero first real channel and a regular harmonic second channel. · OAI.DefocusingNLS.spectralFreeCoreBoundary
spectralFunctional def: Equation (1.7) at temperature parameter one. · OAI.InvariantIsing.spectralFunctional
spectralLog def: The Mercator logarithm is continuous on the unitary spectrum. · OAI.KervaireSpectral.spectralLog
spectralMeasure def: The single finite Borel measure associated with a commutative representation. · OAI.Kaplansky.CuntzDimensionState.spectralMeasure
spectralMoment def: The trace: sum of every negative eigenvalue of the form operator, counted once for each member of a full eigenspace basis. · OAI.SharpLiebThirring.OperatorProof.spectralMoment
spectralMultiplier def: A complex spectral multiplier for a finite family of projections. · OAI.LeanBlast.GotsmanLinial.spectralMultiplier
spectralPartitionNonzeroIndex abbrev: Discard the zero spectral pieces without changing the spectral sum. · OAI.KadisonKastler.spectralPartitionNonzeroIndex
spectralPermutation def: A permutation of spectral coordinates, with determinant corrected to one. · OAI.InvariantIsing.spectralPermutation
spectralPerturbationCoefficients def: The concrete Haar-dependent tensor coefficient map. · OAI.InvariantIsing.spectralPerturbationCoefficients
SpectralProjectionRoundingInput def: Blackadar, I.6.2.4 and I.9.2.1(iv): bounded Borel functional calculus with the cutoff at 1/2. The scalar estimate is |x - 1_[1/2,∞)(x)| ≤ 2 |x²-x|. This predicate is used only for the fixed finite von Neumann algebra, not for arbitrary … · OAI.KadisonSimilarity.SpectralProjectionRoundingInput
spectralProjector def: Actual normalized Fourier projector onto the indices satisfying P. · OAI.MaxCutGames.Appendix.Derivatives.spectralProjector
spectralProjector def: Actual normalized Fourier projector onto the indices satisfying P. · OAI.UniqueGamesTheorem.Appendix.Derivatives.spectralProjector
spectralRelativeRadius def: Uniform relative error allowed by half-unit logarithmic brackets. · OAI.DukePrimeDegree.spectralRelativeRadius
spectralSmooth def: The regularized scalar phase is continuous on the unitary spectrum. · OAI.KervaireSpectral.spectralSmooth
spectralSpinPlaneDerivative def: A normalized spectral coordinate has a uniformly bounded plane derivative. · OAI.InvariantIsing.spectralSpinPlaneDerivative
spectralStep def: A scalar offset plus a finite linear combination of spectral projections. · OAI.KadisonSimilarity.spectralStep
SpectralStepApproximation def: Spectral approximation with a bound on the sum of absolute coefficients. · OAI.KadisonSimilarity.SpectralStepApproximation
spectralTensorPlaneDerivative def: The ordinary product-rule derivative of one spectral tensor feature. · OAI.InvariantIsing.spectralTensorPlaneDerivative
spectralValue def: Finite layer-cake integral. This definition does not assert the quasitrace axioms. · OAI.Kaplansky.CuntzDimensionState.spectralValue
speed def: The normal velocity of an affine cut, in a fixed transverse coordinate. · OAI.Problem358.AffineHalfspaceSpeed.speed
spernerGoodFacets def: The codimension-one subsets carrying every label except the last. · OAI.Tingley.spernerGoodFacets
spernerRestrictedLabel def: Remove the last coordinate from labels on an embedded boundary face. · OAI.Tingley.spernerRestrictedLabel
spernerSmallLabels def: The labels on the distinguished boundary face. · OAI.Tingley.spernerSmallLabels
Sphere abbrev: The Riemann sphere as the one-point compactification of the complex plane. · OAI.WeakPinned.Bounded.Sphere
sphereAnnulus def: A fixed closed annulus about the unit sphere. · OAI.Mahler.sphereAnnulus
sphereArea def: Ordinary surface area, rather than normalized probability on the sphere. · OAI.BoltzmannNonuniqueness.sphereArea
sphereArea def: The concrete area measure induced by Euclidean volume on the unit sphere. · OAI.Mahler.sphereArea
sphereCoe def: The canonical ambient completion embedding restricted to the unit sphere. · OAI.Tingley.sphereCoe
sphereCompletionEquiv def: The metric completion of the old sphere is the sphere of the ambient completion. · OAI.Tingley.sphereCompletionEquiv
sphereDensity def: Surface density of a top degree form: evaluate normal^flat wedge omega on the positively oriented Euclidean coordinate frame. · OAI.Mahler.sphereDensity
sphereDilation def: Positive dilation sends the unit sphere to the radius-r sphere. · OAI.Mahler.sphereDilation
sphereDirectOperator def: Evaluation of the exact direct polynomial at a linear angular operator. · OAI.Falconer.sphereDirectOperator
sphereDirectPolynomial def: The direct stationary coefficient, as a polynomial in the sphere Laplacian. Here n is the sphere dimension, so the ambient dimension is n+1. · OAI.Falconer.sphereDirectPolynomial
sphereEquatorEuclideanHomeomorph def: The equator has the expected Euclidean dimension. · OAI.Kaplansky.sphereEquatorEuclideanHomeomorph
sphereEquatorHomologyIso def: Actual integral homology suspension isomorphism for a unit sphere, obtained from its pole cover and the actual equator. · OAI.Kaplansky.sphereEquatorHomologyIso
sphereEuclideanHomologyStep def: Degree recursion to a sphere in one less real dimension. The base H₁ calculation is not included in this statement. · OAI.Kaplansky.sphereEuclideanHomologyStep
SphereFillable def: Extension property with respect to the actual sup-norm disks. · OAI.GroupRingTopology.Cellular.SphereFillable
sphereFlux def: The integral uses Euclidean area and the cofactor surface density with outward normal x on the unit sphere. · OAI.Mahler.sphereFlux
sphereFlux def: The sphere integral uses the outward orientation from the ball. · OAI.MahlerStokes.sphereFlux
sphereFluxFrame def: The positive frame is exactly x_1,y_1,...,x_n,y_n. · OAI.Mahler.sphereFluxFrame
sphereFrameSlots def: Reassociate the two leading real slots to preserve the interleaved order. · OAI.Mahler.sphereFrameSlots
sphereGradient def: The ambient extension of the spherical gradient of a degree-one support function. · OAI.AffineBernstein.sphereGradient
sphereGradient def: Genuine tangential gradient on the unit sphere, expressed ambiently. · OAI.PettyProjection.Spherical.sphereGradient
sphereHessian def: The spherical Hessian evaluated on tangent vectors. · OAI.AffineBernstein.sphereHessian
sphereInversePolynomial def: The finite inverse coefficients from osc:B-operators. · OAI.Falconer.sphereInversePolynomial
sphereIsometry def: The action of a genuine orthogonal coordinate change on the unit sphere. · OAI.AffineBernstein.sphereIsometry
sphereIsometry def: The actual restriction of a real linear isometry to the unit sphere. · OAI.MahlerStokes.sphereIsometry
sphereIsometryEquiv def: An ordinary onto sphere isometry is a genuine sphere isometry equivalence. · OAI.Tingley.sphereIsometryEquiv
sphereMap def: Orthogonal changes of variables on the actual unit sphere. · OAI.PettyProjection.Spherical.sphereMap
sphereMeasure def: The normalized spherical measure is fixed, not an arbitrary probability measure satisfying the desired curvature averages. · OAI.PositiveEinsteinFour.Angular.sphereMeasure
sphereMetric abbrev: A fixed compatible sphere metric, pulled back from the Euclidean unit sphere. Its topology is the original one-point topology. · OAI.CircleDomainRigidity.sphereMetric
sphereMobius def: The sphere Möbius map has its pole at infinity and its value at infinity is b; no finite-valued division convention is used there. · OAI.WeakPinned.Bounded.sphereMobius
sphereNeg def: Literal negation of a unit vector. · OAI.Tingley.sphereNeg
sphereNormal def: The real metric dual of the outward unit normal, with complex-valued output. · OAI.Mahler.sphereNormal
spherePlaneOverlapRealHomeomorph def: In a real plane, the overlap of the two pole charts is the punctured real line. · OAI.Kaplansky.spherePlaneOverlapRealHomeomorph
spherePoleHomeomorph def: The sphere with one pole removed is the actual orthogonal hyperplane. · OAI.Kaplansky.spherePoleHomeomorph
spherePoleOverlapHomeomorph def: Stereographic projection identifies the overlap of the two pole charts with the punctured orthogonal hyperplane. · OAI.Kaplansky.spherePoleOverlapHomeomorph
spherePoleOverlapHomotopyEquiv def: The overlap of opposite pole charts is homotopy equivalent to the equator. · OAI.Kaplansky.spherePoleOverlapHomotopyEquiv
sphereRealFunction def: A real-coordinate function, with an arbitrary unused value at infinity. · OAI.CircleDomainRigidity.sphereRealFunction
sphereTangent def: The tangent hyperplane of a Euclidean sphere, with its usual real metric. · OAI.Mahler.sphereTangent
sphericalPreferredNormal def: The inward radial section is a global preferred normal, without an orientation or a global choice of a second normal vector. · OAI.ClosedSurfaceR4.sphericalPreferredNormal
sphericalSecondForm def: Removing the radial component gives the second fundamental form within the unit sphere. · OAI.ClosedSurfaceR4.SphericalJets.sphericalSecondForm
Spin abbrev: The cube, with exactly 2^N configurations. · OAI.InvariantIsing.Spin
spinCoordinate def: Spectral coordinate of a rotated Ising spin. · OAI.InvariantIsing.spinCoordinate
spinFlipEquiv def: Global spin reversal is a permutation of configurations. · OAI.BufferedIsing.spinFlipEquiv
spinIndex def: The manuscript's spin order is +,0,-; vacancies are not weighted. · OAI.CriticalHoneycomb.Transfer.PlanarNormalization.spinIndex
spinMatrixOperator def: The ordinary Euclidean operator represented by a finite spin matrix. · OAI.ContinuumCoulomb.spinMatrixOperator
spinRoot def: Literal finite perturbed spin root with one independent mark prior for each realized perturbation. The final spin prior is uniform; earlier spin kernels are dummy point masses. No physical or mark law is replaced. · OAI.DilutedSpinGlass.SizeCoupling.spinRoot
spinsFromColumns def: Literal grid spins after fixing the west and north outer boundaries. The remaining horizontal and vertical wires are genuine independent coordinates, not redundant copies at the centers. · OAI.CriticalHoneycomb.Transfer.RasterGrid.spinsFromColumns
spinSinglet def: The unnormalized singlet, of squared norm two. · OAI.ContinuumCoulomb.spinSinglet
spinTensorPlaneDerivative def: The independent linear enrichment has zero rotation derivative. · OAI.InvariantIsing.spinTensorPlaneDerivative
spinTripletZero def: The unnormalized zero-magnetization triplet, of squared norm two. · OAI.ContinuumCoulomb.spinTripletZero
spinValue def: Real spin coordinates. · OAI.InvariantIsing.spinValue
splice def: Splicing oriented boundary polygons at their outgoing corners. This is connected sum at the outside faces when the two corners are in distinct components; no edge or side map is changed. · OAI.HyperbolicObstruction.PictureSwitch.PictureMap.splice
splicePermutationWord def: Splicing two adjacent slots multiplies their matrix arguments in cycle order. · OAI.KadisonSimilarity.splicePermutationWord
Split structure: A split stores its actual successor states, progress and the exact semantic branching equation. Its numerical inequality is proved for the three cases. · OAI.Sparsification.Split
Split structure: A genuine one-step separator, including its actual components/blocks. · OAI.TorsionFreeZeroDivisors.RecursiveSeparator.Split
split def: The coordinate split is first axial, then transverse. · OAI.BoltzmannNonuniqueness.VelocityWeight.split
split def: Move one distinguished mark out of the dependent row. · OAI.DilutedSpinGlass.HeterogeneousMarks.split
split def: Split at marked letters, retaining every letter and the possibly empty prefix before the first mark. The marks themselves label the corners. · OAI.FKCRT.CornerSplit.split
split def: An split vertex, retaining its branch with its fine ordered grades. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.OrderedStep.split
split def: A split or its orientation-reversed merge, with all three port lists. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.split
split def: Apply the proved B-linear truncation splitting to the actual fixed tensor, using full module cohomology models and source-diagram support. · OAI.LittleFinitistic.SelectedTensorSplitting.split
split def: A fixed linear complement to the determinant kernel on unrestricted polynomials. · OAI.MatrixMultiplication.AuxiliarySeparation.DeterminantKernel.split
split def: The split weights form a genuine standard simplex. · OAI.Thomason.Construction.PrismBarycentric.split
splitAssignment def: Replace the two disjoint selected coordinate families, keeping all other coordinates fixed. · OAI.TwoPointCorrelations.splitAssignment
splitBasis def: Reindex a full computational basis by the gate wires and all other wires. This is only a proof equivalence; it introduces no new gate in the model. · OAI.ExactQuantumFactoring.Instruction.splitBasis
splitBefore def: Ordered source decomposition of before and after lists. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.LocalRanks.splitBefore
splitBeforeBasis def: Coherent changes in the six fiber types of a split, transported through the exact named corner equalities rather than through rank alone. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.LocalRanks.splitBeforeBasis
splitBidegreeExponents def: Split a finite exponent vector into its derivative and multiplication coordinates. · OAI.Problem335.splitBidegreeExponents
splitBoth def: Two independent axis seeds, with the common-leaf coordinate unused. · OAI.Problem348.TreeSampling.splitBoth
SplitCategories abbrev: The three labels are common, first-exclusive, second-exclusive. · OAI.JointDickman.SplitCategories
splitChild def: Ordered child lists: prefix, the split block, suffix; and then prefix, its ordered subblocks, suffix. The third seam carries just those subblocks. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.splitChild
splitCoinEquiv def: The actual uniform coin law splits as a product; this is an equivalence of finite sample spaces, rather than an independence assumption. · OAI.CubeShuffle.splitCoinEquiv
splitColumn def: An anchor/dummy row and a variable column, with the unused coordinates. · OAI.Problem348.TreeSampling.splitColumn
splitConstant def: A single constant works for every member of the finite partition. · OAI.ClosedSurfaceR4.PhasePartitions.CompactPhasePartition.splitConstant
splitCoordinate def: Move the distinguished coordinate first without changing the relative order of any other coordinates. The same reindexing is used on both matrix axes. · OAI.MahlerStokes.splitCoordinate
splitCoordinates def: Restriction to the clean and nonclean coordinates is a genuine bijection. · OAI.MaxCutGames.Clean.ProductLaw.splitCoordinates
splitCoordinates def: Restriction to the clean and nonclean coordinates is a genuine bijection. · OAI.UniqueGamesTheorem.Clean.ProductLaw.splitCoordinates
splitDepth def: Split depth compares positions, never the sampled spin values. Equal positions have split depth n; distinct positions split strictly before n. · OAI.DilutedSpinGlass.PrescribedTree.splitDepth
splitEquiv def: A chosen linear right inverse splits a surjection with its kernel. · OAI.ExactFourier.RowCompression.splitEquiv
splitEquiv def: An explicit linear equivalence for a split surjection, with the kernel as the second summand. It is used below to produce the local sheaf summand. · OAI.KLInvariance.LocalCover.splitEquiv
splitEquiv def: Linear equivalence of the full restriction domain with the constrained component and the free complementary component. · OAI.MaxCutGames.Inverse.KMSAffineRestrictionSplit.splitEquiv
splitEquiv def: Split off the coordinate measured by a, using the prescribed vector t. The kernel coordinate is exactly x - a x • t. · OAI.MaxCutGames.Inverse.KMSAnalyticFunctionalSplit.splitEquiv
splitEquiv def: Linear equivalence of the full restriction domain with the constrained component and the free complementary component. · OAI.UniqueGamesTheorem.Inverse.KMSAffineRestrictionSplit.splitEquiv
splitEquiv def: Split off the coordinate measured by a, using the prescribed vector t. The kernel coordinate is exactly x - a x • t. · OAI.UniqueGamesTheorem.Inverse.KMSAnalyticFunctionalSplit.splitEquiv
splitFamily def: A literal family with one ranged split. The child shapes and all their later branching depths are fixed. Moving d delays only this split. · OAI.DilutedSpinGlass.PrescribedTree.splitFamily
splitFamily def: Splitting one part of a finite list partition into two consecutive, nonempty pieces adds exactly one block, without discarding fixed points. · OAI.FKCRT.MapCycles.splitFamily
splitFieldMatrix def: Real-linear transformation of any actual coordinate field into the source's independent complex z/bar-z coordinates. · OAI.HarmonicCounterexample.Berger.ComplexStructure.splitFieldMatrix
splitFrame def: The complete old experiment: two entire sides separating at depth d, with their first internal split after another r unary transitions. · OAI.DilutedSpinGlass.PrescribedTree.splitFrame
splitFromProfile def: Recover the complete split-depth matrix from interior sharing bits. The terminal diagonal is known from colors, not from an endpoint mark. · OAI.DilutedSpinGlass.PrescribedTree.splitFromProfile
splitFunctionalEquiv def: The literal split along a normalized support vector. The second component is the kernel of the support; no quotient or degenerate coordinate is used. · OAI.AffineBernstein.splitFunctionalEquiv
SplitGadget structure: Noise indices retain multiplicity even if several indices have equal vectors. · OAI.MaxCutGames.Reduction.ActualSource.SplitGadget
SplitGadget structure: Noise indices retain multiplicity even if several indices have equal vectors. · OAI.UniqueGamesTheorem.Reduction.ActualSource.SplitGadget
splitHead def: Exposes the first dart and the independent remaining port word. · OAI.MaxCutGames.Foundations.PCP.PoweringWalks.splitHead
splitHead def: Exposes the first dart and the independent remaining port word. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringWalks.splitHead
SplitIncidence abbrev: The actual incidence subset, with the topology induced from parameters and V. · OAI.Kaplansky.SmoothKernel.SplitIncidence
splitIso def: Actual B-linear two-degree formality, with the manuscript's shift by three. · OAI.LittleFinitistic.TwoDegreeFormality.splitIso
splitIsoWithModels def: Substitute actual B-module identifications of the two cohomology modules. · OAI.LittleFinitistic.TwoDegreeFormality.splitIsoWithModels
splitKaroubiIso def: An actual splitting realizes the corresponding formal Karoubi object. · OAI.LittleFinitistic.QuadraticEncoding.splitKaroubiIso
splitKernelHom def: The homomorphism on the actual kernels. · OAI.PolycyclicRecognition.splitKernelHom
splitLeft def: The constrained component of an actual quotient parameter. · OAI.MaxCutGames.Inverse.KMSAffineRestrictionSplit.splitLeft
splitLeft def: Retain the left caps and straighten the left transversal endpoints. · OAI.OneWayLiveness.Pairing.splitLeft
splitLeft def: The left half of an actual basic interval. · OAI.ThompsonNonamenability.BasicInterval.splitLeft
splitLeft def: The constrained component of an actual quotient parameter. · OAI.UniqueGamesTheorem.Inverse.KMSAffineRestrictionSplit.splitLeft
splitLists def: One split/merge slot in the source's ordered flag gallery. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.splitLists
SplitMachine structure: A bit-level finite-state controller. Both transition functions are literal Boolean networks; this structure does not grant any classical computation oracle. · OAI.ExactQuantumFactoring.SplitMachine
SplitMap structure: Inclusion of an exact prescribed splitting matrix into the actual history geometry. This is an inclusion of leaf POSITIONS, not of sampled configurations. · OAI.DilutedSpinGlass.PrescribedTree.SplitMap
splitMate def: Germ matching at a split. In particular the child W and the parent W are different occurrences; each subblock is also used twice, not collapsed. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.splitMate
splitMatrix def: Pullback matrix in the independent complex coordinates x_i ± i y_i. This is the exact substitution x=(z+zbar)/2, y=(z-zbar)/(2i), not a spectral model replacing the real vector field. · OAI.HarmonicCounterexample.Berger.ComplexStructure.splitMatrix
splitMeasure def: The central measure together with finitely many weighted branch pushforwards. · OAI.Problem356.Splitting.splitMeasure
splitMix def: The output of the mixer and the two unused coordinates determine both seeds. · OAI.Problem348.TreeSampling.splitMix
splitPairEquiv def: Pack the prefix, special pair and suffix into the cell split. · OAI.ContingencyTables.SmallContextCoordinates.splitPairEquiv
splitPortWord def: Expose the first actual port in a finite port word. · OAI.MaxCutGames.Foundations.PCP.CayleySpectral.splitPortWord
splitPortWord def: Expose the first actual port in a finite port word. · OAI.UniqueGamesTheorem.Foundations.PCP.CayleySpectral.splitPortWord
splitProjection def: Both halves belong to the original algebra. · OAI.Generator.Tracial.splitProjection
splitProjection def: eA is a retract of the rank-one free right A-module. · OAI.LittleFinitistic.IdempotentTensor.splitProjection
splitProjector def: A later-split projector lifted through an arbitrary actual ancestral prefix, without flattening the dependent tuple representation of paths. · OAI.DilutedSpinGlass.PrescribedTree.splitProjector
splitRefinement def: Refinement: leave every old prefix/suffix fiber unchanged, and in exactly the old W-block insert its named third-seam frame. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.LocalRanks.splitRefinement
splitRequests def: Count every local splitting request, including the final failed one. · OAI.ExactQuantumFactoring.DataProducer.splitRequests
splitRight def: The shared coordinate is explicitly a single PUnit port, not an unspecified padding block. · OAI.ExactFourier.OnePort.splitRight
splitRight def: The free component of an actual quotient parameter. · OAI.MaxCutGames.Inverse.KMSAffineRestrictionSplit.splitRight
splitRight def: Retain the right caps and all the original transversals. · OAI.OneWayLiveness.Pairing.splitRight
splitRight def: The right half of an actual basic interval. · OAI.ThompsonNonamenability.BasicInterval.splitRight
splitRight def: The free component of an actual quotient parameter. · OAI.UniqueGamesTheorem.Inverse.KMSAffineRestrictionSplit.splitRight
splitRow def: Move the distinguished Poisson-rooted mark out of the old mark row. · OAI.DilutedSpinGlass.splitRow
splitRow def: A variable row and an anchor/dummy column, together with unused coordinates. · OAI.Problem348.TreeSampling.splitRow
splitRows def: Source rows which contain such a transition. · OAI.TreeEdit.BinaryLower.Tagged.splitRows
splitRun def: Split off the entire initial raw run, without changing raw contents. · OAI.FiniteAlgebraicEnvelopes.Compiler.RawRuns.splitRun
Splits def: Hypothesis in uses of a tuple index (omit entirely any singleton axes). · OAI.PowerSaving.Grove.Splits
splitSample def: Reindex all labelled samples into two free blocks and their exposed complement. · OAI.Problem337.RandomExposure.splitSample
splitSampleExposedFirst def: The same exposure equivalence with exposed samples first, ready for conditional averages. · OAI.Problem337.RandomExposure.splitSampleExposedFirst
splitSeriesTerm def: A finite prefix has the small analytic error; only a fixed bound is used beyond the last controlled scale. · OAI.RegularPointCoordinates.splitSeriesTerm
splitShift def: Shift a binary splitting, using the actual shift functors and their addition isomorphism. · OAI.ExternalTensor.Binomial.splitShift
splitShift def: Shift a split from an exterior brick to the selected one. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Partition.splitShift
splitShiftEnergy def: The sum of squared changes of the actual descendant mean when one split is ranged over ALL available depths, with arbitrary fixed later child shapes. · OAI.DilutedSpinGlass.PrescribedTree.splitShiftEnergy
splitSlots def: Literal Cartesian identification for the split, retaining all bit labels. · OAI.CoordinateSweeps.Grid.splitSlots
splitSupported def: The piece is constructed by ordinary scalar multiplication of the target. · OAI.ClosedSurfaceR4.PhasePartitions.CompactPhasePartition.splitSupported
splitSupportedLM def: With the partition fixed, splitting is linear in the target. · OAI.ClosedSurfaceR4.PhasePartitions.CompactPhasePartition.splitSupportedLM
splitSwitch def: Disjoint partition of all switch locations into the exposed low butterflies and the remaining high butterflies. · OAI.CubeShuffle.splitSwitch
Splitting structure: An invariant complementary splitting. Empty groups are permitted. · OAI.SmoothObstruction.FiniteGreen.Splitting
splitting def: Every module over a product ring splits into its two actual central-idempotent components. · OAI.LittleFinitistic.ProductModules.splitting
splitting def: The concrete matrix sequence splits over the base field. · OAI.LittleFinitistic.RelationMatrixSequence.splitting
splittingDirectionBound def: A uniform bound for all constant directions of the splitting map. · OAI.RegularPointCoordinates.splittingDirectionBound
splitTransitions def: Consecutive matching entries that cross a target-row boundary without crossing a source-row boundary. · OAI.TreeEdit.BinaryLower.Tagged.splitTransitions
splitTrueAux def: Split a list at false positions, retaining a possibly empty initial block. · OAI.Problem335.WordRuns.splitTrueAux
splitUnit def: The binary partition selecting p and its complement. · OAI.Generator.Tracial.CornerPartition.splitUnit
splitWeights def: Duplicate vertex j, assigning mass a to its first occurrence. · OAI.Thomason.Construction.PrismBarycentric.splitWeights
splitWord def: The explicit unitary reindexing between a concatenated tensor word and its two block words. The Koszul signs are retained in phase_blockPerm. · OAI.CoordinateSweeps.SignedTensor.splitWord
SpokeOrders structure: The linear local orders are witnessed by the actual counterclockwise angles, with a cut in a gap. No intersection identity is part of this data. · OAI.Paper170.SpokeOrders
spotDebt def: Total entropy cost of non-spatial branching. · OAI.FourierLLogL.Dyadic.Procedure.Wire.spotDebt
spotted def: A property measurable in depth l. Does not require l≤M. · OAI.FourierLLogL.Dyadic.Procedure.spotted
spout def: logical lower Toeplitz between offsets l (source), m (target), in different physical chunks. · OAI.PowerSaving.Slant.spout
Spray structure: Inputs to the compression estimate. · OAI.ExactDerandomization.Est.Spray
Spring structure: use in array work · OAI.PowerSaving.RAM.Bench.Spring
SQ def: The separable quotient assertion at an arbitrary ambient universe. · OAI.SeparableQuotient.SQ
sqrtWeightEquiv def: Multiplication by the square root of a strictly positive weight, in the counting Euclidean Hilbert space. · OAI.LeanBlast.GotsmanLinial.sqrtWeightEquiv
Square abbrev: Shifted by T into natural-address range; then the block partition can be computed with unsigned arithmetic. · OAI.Fifths.Square
Square structure: A commutative square of actual inclusions which is a pushout among all monoids, not merely among conservative maps. · OAI.PartitionConsistency.ConservativeMonoids.Subobject.Square
square def: The actual graph square. The two return ports appear in reverse order. · OAI.MaxCutGames.Foundations.PCP.ZigzagGraphs.square
square def: A repeated annihilation coordinate retains its extra first-order term. · OAI.Problem335.FactorialExpansion.square
square def: The actual graph square. The two return ports appear in reverse order. · OAI.UniqueGamesTheorem.Foundations.PCP.ZigzagGraphs.square
squareComparison def: The complete actual chain-level square calculation, with the source support and outside action. · OAI.LittleFinitistic.OrdinarySimulation.XResolution.squareComparison
squareCurvatureContraction def: The actual matrix-slot contraction with N = (T²)₀. · OAI.PositiveEinsteinFour.Bianchi.squareCurvatureContraction
squareCurvatureForm def: The analogous form of the trace-free square of the Weyl block. · OAI.PositiveEinsteinFour.HessianAlgebra.squareCurvatureForm
squareD def: The square and its first two derivatives, with the second derivative as an iterated continuous real linear map. · OAI.SymmetricMahler.squareD
squaredDistance def: The square of the pretentious distance; the second sequence can be any twist. · OAI.TwoPointCorrelations.squaredDistance
squaredHarmonicSchwartz def: The actual Schwartz test used in the Cartesian weak equation. · OAI.DefocusingNLS.squaredHarmonicSchwartz
squareDivisorTruncation def: The concrete norm truncation used by corrected dispersion. · OAI.CubicFirstMoment.squareDivisorTruncation
squaredNorm def: Squared Hilbert--Schmidt norm of the symmetric vector-valued matrix. · OAI.PositiveEinsteinFour.Bianchi.Tensor.squaredNorm
squareFirst def: Cross an original edge using the first port; retain its return port. · OAI.MaxCutGames.Foundations.PCP.ZigzagGraphs.squareFirst
squareFirst def: Cross an original edge using the first port; retain its return port. · OAI.UniqueGamesTheorem.Foundations.PCP.ZigzagGraphs.squareFirst
SquarefreeCharacterEstimateInput def: Only the two exponents and the arbitrary fixed logarithmic saving needed for the moving-cutoff argument are retained. · OAI.JointDickman.PublishedInputs.SquarefreeCharacterEstimateInput
squarefreeCharacterSum def: Twist the squarefree coefficients, before removing any small primes. · OAI.JointDickman.squarefreeCharacterSum
squarefreeDivisorTruncation def: Terms at nonsquarefree divisors have zero Möbius weight, so this finite truncation gives exactly the original truncated divisor sum. · OAI.CubicFirstMoment.squarefreeDivisorTruncation
SquarefreeExponent def: Exponents zero or one represent parity classes. · OAI.MaxCutGames.Quadratic.SquarefreeExponent
SquarefreeExponent def: Exponents zero or one represent parity classes. · OAI.UniqueGamesTheorem.Quadratic.SquarefreeExponent
SquarefreeFrequency structure: The squarefree kernel is divided into the part supported on L and the part coprime to L. Positivity excludes the zero frequency. · OAI.Ostmann.SquarefreeFrequency
squarefreeLeadingConstant def: The convergent Euler-product normalization of the first LSD coefficient. · OAI.JointDickman.squarefreeLeadingConstant
squarefreeLocalHankel def: The real local contour contribution, with the factor exp L removed. · OAI.JointDickman.squarefreeLocalHankel
squarefreePairSupport def: The true squarefree product support, without a coprimality replacement. · OAI.CubicFirstMoment.squarefreePairSupport
squarefreeRieszLocalHankel def: The real local contour contribution, with the factor exp L removed. · OAI.JointDickman.squarefreeRieszLocalHankel
SquarefreeSelbergDelangeInput def: Only the two positive exponents used by the paper, with all fixed expansion orders and the actual leading coefficient. · OAI.JointDickman.PublishedInputs.SquarefreeSelbergDelangeInput
squarefreeSingularFactor def: Removing the principal power singularity of ζ(s)^z and the Perron denominator. · OAI.JointDickman.squarefreeSingularFactor
squarefreeWeight def: The squarefree coefficient μ(n)^2 z^ω(n). On squarefree integers, the number of prime factors with multiplicity equals the number of distinct prime factors. · OAI.JointDickman.squarefreeWeight
squareIntegral def: Squared L² size; using the nonnegative integral also covers intermediate expressions. · OAI.LipschitzHilbert.squareIntegral
SquareLatticeHeisenberg structure: The signed rational field-free square-lattice source input. Polynomial size/weight bounds are properties of the published reduction, not extra one-site terms or restrictions on the variational spin states. · OAI.ContinuumCoulomb.SquareLatticeHeisenberg
squareLiftBad def: All forbidden prime-square zeros, one for each occurrence of the prime. · OAI.Ostmann.squareLiftBad
SquareLiftPairs abbrev: Uniform pairs over the fixed giant residues modulo p. · OAI.Ostmann.SquareLiftPairs
squareQuotient def: The perspective of the square, with its zero trace value. · OAI.Bernoulli.Halfspace.squareQuotient
squareReduction def: The actual reduction map modulo a prime. · OAI.Ostmann.squareReduction
squareRoot def: The unique square root, obtained by bijectivity of the finite square map. · OAI.MaxCutGames.Quadratic.squareRoot
squareRoot def: The unique square root, obtained by bijectivity of the finite square map. · OAI.UniqueGamesTheorem.Quadratic.squareRoot
squareSupport def: The four coordinates of a square row: two core fields and two private ones. · OAI.HyperbolicObstruction.IncidenceTemplate.squareSupport
squareUnitLiftEquiv def: Above a nonzero residue every lift is a unit; passing to Haar-unit coordinates leaves the same p uniformly weighted lifts. · OAI.Ostmann.squareUnitLiftEquiv
squareWeight def: The normalized eigenvalues of the trace-free square of the Weyl block. · OAI.PositiveEinsteinFour.Bianchi.Shape.squareWeight
squareWord def: The oriented source four-letter decoration, including the two given core sides. · OAI.HyperbolicObstruction.MasterGraph.squareWord
squareZeroResidueEquiv def: The exact quotient-ring isomorphism S in Section 6. · OAI.LittleFinitistic.PDSquareZero.squareZeroResidueEquiv
sRoot def: The original five roots and the two literal section ratios, as functions in the actual affine chart of the eighteen source sections. · OAI.ExplicitCone.sRoot
stabilityError def: Uniform ambient input, independently uniform noise-list index. · OAI.MaxCutGames.Gadget.ActualGadget.stabilityError
stabilityError def: Uniform ambient input, independently uniform noise-list index. · OAI.UniqueGamesTheorem.Gadget.ActualGadget.stabilityError
stabilizationEquiv def: The last of five independent variables becomes the new indeterminate; the first four become coefficient variables. · OAI.StableCoordinate.stabilizationEquiv
stabilize def: Adjoin one independent variable; the original variables retain their indices. · OAI.StableCoordinate.stabilize
stabilizedPresentation abbrev: The same presentation after adjoining one independent variable. · OAI.StableCoordinate.Presentation.stabilizedPresentation
stabilizer def: The symmetric group on the unmarked input sites. · OAI.CoordinateSweeps.Grid.Holes.stabilizer
stabilizer def: Permutations of the free input sites, presented as permutations fixing the prescribed inputs. This is the symmetric group on exactly s-h sites. · OAI.CoordinateSweeps.Grid.Holes.stabilizer
stabilizer def: The stabilizer of the chain beginning with the trivial subgroup and followed by the nodes in l. · OAI.PCoreCancellation.stabilizer
stabilizerEquivalence def: The stabilizer of the distinguished identity point is exactly the linear factor. · OAI.FiniteAlgebraicEnvelopes.AffinePermutation.stabilizerEquivalence
stabilizerEquivalenceAt def: At every point, evaluation determines the translation from the linear component. · OAI.FiniteAlgebraicEnvelopes.AffinePermutation.stabilizerEquivalenceAt
stabilizerFreeEquiv def: Literal target stabilizer isomorphism, not an assumed representation switch. · OAI.CoordinateSweeps.Grid.Holes.stabilizerFreeEquiv
StableAt def: An actual local class whose availability has become permanent. Keeping the literal name as an argument avoids any chosen representative under transport between vertices. · OAI.PartitionConsistency.ProductForcing.StableAt
StableAt def: Stability in precisely the selector-value form supplied by the canonical window stability construction. · OAI.Problem310.WindowExceptionalTransfer.StableAt
stableBackward def: Negative backward solution of the normalized unstable recurrence. · OAI.DefocusingNLS.stableBackward
stableClosure def: Add retraction images once; idempotence makes this stable already. · OAI.Thomason.Construction.SieveRetraction.stableClosure
StableExtraction def: The extraction property proved for sorted stable telescopes. It is not an extra PTS rule: the following constructors explicitly build its proof terms. · OAI.PureTypeSystem.Labelled.StableExtraction
StableGraph structure: A finite connected stable graph of type (g,n), given by vertices, edges, flag incidences, and nonnegative vertex genera. The integer genus equation avoids truncated subtraction in |E| - |V| + 1. · OAI.PixtonChow.StableGraph
stableLetter def: The stable letter t=(0,I,1). · OAI.NuclearUltrapower.Elementary.stableLetter
stableLetter def: The stable letter t=(0,I,1). · OAI.NuclearUltrapower.stableLetter
StableNonlinearDispersion def: The actual mathematical existence target; no theorem asserts it yet. · OAI.MaxCutGames.Gadget.ActualGadget.StableNonlinearDispersion
StableNonlinearDispersion def: The mathematical existence target. · OAI.UniqueGamesTheorem.Gadget.ActualGadget.StableNonlinearDispersion
stablePairSequence def: Package the stable and unstable weighted coordinates into one sequence. · OAI.DefocusingNLS.stablePairSequence
StableParent def: A parent at any of the k replacement stages, with all of its actual children bounded by the specified multiple of its own slope. The recursion restricts to old vertices and therefore includes every stage, not just the last one. · OAI.DiamondDistortion.Diamond.StableParent
stablePresentationEquiv def: The original presented hypersurface, after adjoining w, is a polynomial ring in which the distinguished p is the first coordinate. · OAI.StableCoordinate.Stabilization.stablePresentationEquiv
stableSequenceValue def: The actual value of a perturbation encoded in the weighted sequence space. · OAI.DefocusingNLS.stableSequenceValue
stacc def: allocate main's body by operation tags. · OAI.ExactDerandomization.SiteX.Studio.stacc
stack def: Stack layer sequences, a performed first. · OAI.PowerSaving.Slant.Reel.stack
StackAction inductive: A `StackAction` either leaves a stack alone, removes its head, or pushes one specified Boolean bit. · OAI.CurrentKS.StackAction
StackAt inductive: A single finite prefix supporting every exact typing of a tested stack. · OAI.PureTypeSystem.Candidates.StackAt
stackEncoding def: A fixed-capacity little-endian word stack. No dynamic address or unbounded list is part of the corresponding Boolean network. · OAI.ExactQuantumFactoring.BitArithmetic.stackEncoding
stackL def: Stacked evaluation of all columns on a batch of free directions. · OAI.HadwigerCounterexample.stackL
StackMatching structure: A certificate for the actual two stack matchings of a finite typed word. The matching uniqueness theorem makes its induced mate choice-independent. · OAI.FKCRT.Inventory.StackMatching
stackPop def: Pop the top, padding the empty end with zero. · OAI.ExactQuantumFactoring.BitArithmetic.stackPop
stackPush def: Push a word; the last slot is dropped. Capacity is verified separately in controller invariants, rather than silently growing the physical width. · OAI.ExactQuantumFactoring.BitArithmetic.stackPush
Stage structure: A potential tuple and its envelope size, independent of all class tests. · OAI.FiniteAlgebraicEnvelopes.Bootstrap.Selectors.Stage
Stage structure: The induction stores the original joint law and every already rounded ancestor, so all conditional correlations are retained. · OAI.KServer.TreeRounding.Stage
Stage structure: An equivariant finite stage, including its concrete finite noise sampler. · OAI.MaxCutGames.Gadget.Stage
Stage structure: An equivariant finite stage, including its concrete finite noise sampler. · OAI.UniqueGamesTheorem.Gadget.Stage
stage def: A finite coloring chosen at each insertion using only earlier colors. · OAI.KServer.OnlineLabels.stage
stage def: The actual recursive spaces, shifts, outputs and finite leaf-noise sampler. · OAI.MaxCutGames.Gadget.QuadraticStageNoise.stage
stage def: Skolem-hull stages, starting from a given nonempty countable set. · OAI.PartitionConsistency.CountableModel.stage
stage def: Each stage of the endpoint construction keeps one earlier endpoint and an activation threshold. The next endpoint is the maximum of the old endpoint and the next enumerated predecessor. · OAI.SeparableQuotient.CoherentClosures.stage
stage def: Each stage retains previous stages and adjoins expressions whose Type I children belong to previous stages, with uncropped path data. · OAI.SeparableQuotient.Norming.stage
stage def: Parallel triangular evaluation; the stabilization proof below shows that this computes the exact descending recursion, rather than a different fixed point. · OAI.Superstring.Counts.Data.stage
stage abbrev: The actual n-fold extension of the original simplicial set. · OAI.Thomason.Construction.InfiniteExtension.stage
stage def: A compatible deformation at every finite stage. · OAI.Thomason.Construction.TopologicalDeformation.Sequential.stage
stage def: The actual multiplicity stage has precisely its incident vertices. No inactive vertices are counted or forced to embed. · OAI.TorsionFreeZeroDivisors.StageSubgraphs.stage
stage def: The actual recursive spaces, shifts, outputs and finite leaf-noise sampler. · OAI.UniqueGamesTheorem.Gadget.QuadraticStageNoise.stage
stageBefore def: Before and after the elementary edit at structure j. · OAI.KServer.PriorityKeys.stageBefore
stageComparisonMass def: The comparison mass is the actual averaged warmup at active states and true cost at carried stopped states. · OAI.EditApproximation.stageComparisonMass
stageCost def: The actual label/block part of the stage exponent, before embedding and uniform error terms are added. Stages are indexed from zero. · OAI.TorsionFreeZeroDivisors.LinkComponents.stageCost
stagedCouplingParameters def: The source coupling parameters now have all width and scale hypotheses discharged by the recursive construction. j is the unique tagged block. · OAI.FiniteFactor.stagedCouplingParameters
stageFunctor def: Iterated extension, with precisely the maps used in the colimit tower. · OAI.Thomason.Construction.InfiniteExtension.stageFunctor
StageHistory structure: The finite history is kept at every successor; no density or limit is part of this bookkeeping data. · OAI.ThreeTorus.StageHistory
stageInTime def: One fully constructed copy/copy/compare/store stage, restoring every stack. · OAI.MaxCutGames.Foundations.Complexity.MachineFieldProfile.stageInTime
stageInTime def: One fully constructed copy/copy/compare/store stage, restoring every stack. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineFieldProfile.stageInTime
stageLaw def: Independent line choices for a single coordinate stage. · OAI.CoordinateSweeps.Grid.stageLaw
StageLine abbrev: A line is named by both its stage and its remaining coordinates. · OAI.CoordinateSweeps.Grid.StageLine
stageLog def: Exact logarithm of a numerical stage bound. power is V_j-E_j-k_j; err contains the independently controlled endpoint/block errors. · OAI.TorsionFreeZeroDivisors.StageFusion.stageLog
stageReturn def: The actual parent sequence at a natural stage of the level recursion. · OAI.ReflexiveFixedPoints.CandidateContext.stageReturn
stages def: Ordinary natural recursion constructs all finite levels successively. · OAI.ReflexiveFixedPoints.AnchoredCandidateContext.stages
stages def: Natural-valued iteration, represented by literal unary words. · OAI.WLIdentification.WordExpressions.NatPoly.stages
StageState structure: Literal admissible states in physical coordinates. No successor or basis is part of the data; successors are produced by the unconditional stage theorem. · OAI.ThreeTorus.StageState
stageSup def: Supremum of all edge slopes at one stage; not a maximum. · OAI.DiamondDistortion.Diamond.stageSup
Stair structure: Recipe-independent additive schedule. · OAI.ExactDerandomization.Est.Stair
staircaseSwap def: Transposition is an honest permutation of the self-conjugate staircase. · OAI.Saxl.staircaseSwap
staircaseToTensor def: The genuine cyclic module is embedded in the actual staircase Specht square. · OAI.Saxl.staircaseToTensor
stairHigh def: The prescribed high sector in the actual staircase enumeration. · OAI.Saxl.stairHigh
staleCharge def: The number of forced queries already determined by earlier exposures. · OAI.CubeShuffle.QueryTree.staleCharge
stalk def: Regard a projective as a bounded complex concentrated in degree zero. · OAI.LittleFinitistic.SourceVerdier.stalk
stalk def: Step bench. Indexed by the 'combined' new prefix; on tray are ambient-axis bench, then decoded old prefix, offset, and previous state. · OAI.PowerSaving.RAM.Bench.Fen.stalk
stalkChainIso def: The stalk total on extended chains agrees with the exact tensor functor used by DerivedBar. · OAI.LittleFinitistic.BimoduleTotal.stalkChainIso
stalkCover def: A sheaf satisfying the two standard section properties gives an actual surjection of its stalk onto the boundary module. · OAI.KLInvariance.MomentGraph.Sheaf.stalkCover
stalkIso def: Exact inflation on derived categories agrees with the selected first-block stalk. · OAI.LittleFinitistic.SelectedFirstBlockFamily.stalkIso
stalkMap def: The lower-end stalk map on all upward edges. · OAI.KLInvariance.MomentGraph.Sheaf.stalkMap
stalkSummandImageIso def: The formal evaluated summand is the stalk of the concrete image of the action. · OAI.LittleFinitistic.QuadraticEncoding.stalkSummandImageIso
stalkTotalColumnIso def: Projecting onto its sole nonzero column is a genuine B-linear chain isomorphism. · OAI.LittleFinitistic.BimoduleTotal.stalkTotalColumnIso
stalkTotalColumnNatIso def: The stalk-total comparison is natural in the entire input complex. · OAI.LittleFinitistic.BimoduleTotal.stalkTotalColumnNatIso
stalkTotalIso def: Finite total with the stalk bimodule is the ordinary tensor functor on all integer complexes. The only indexing condition is that degree zero is included. · OAI.LittleFinitistic.BimoduleTotal.stalkTotalIso
Stand structure: B rank roles station. · OAI.ExactDerandomization.SiteX.Line.Fabric.Stand
standardConfiguration def: The ordered distinct tuple [1 : i : 0], for i = 0, ..., r-1. · OAI.Nagata.ProjectiveGeometry.standardConfiguration
standardCrosscap def: Standard umbrella, with the regular coordinate placed last. · OAI.ClosedSurfaceR4.FiniteOrderSmoothing.standardCrosscap
standardDoubleHomotopy def: The comparison on the double subdivision of a standard simplex. · OAI.Thomason.Construction.standardDoubleHomotopy
standardExcursion def: Standard normalized excursion, realized as the 3D Bessel bridge. · OAI.FKCRT.standardExcursion
standardExcursionPath def: A continuous-path version of the standard excursion; the fallback occurs only on a null set under the standing Brownian hypotheses. · OAI.FKCRT.standardExcursionPath
standardHomeomorphEquiv def: The representation forgets no mathematical data: it is equivalent to the usual subtype of actual increasing finite dyadic PL interval homeomorphisms. · OAI.ThompsonNonamenability.F.standardHomeomorphEquiv
StandardNonatomic def: Standard nonatomic in the source's explicit modulo-null-set sense. · OAI.SmoothObstruction.StandardNonatomic
standardPairRootEquiv def: A standard etale equation with localization polynomial one is precisely its ordinary root algebra. · OAI.KummerEtale.standardPairRootEquiv
StandardPLCondition def: The usual increasing PL interval homeomorphisms with finitely many dyadic breakpoints and slopes in the integer powers of two. · OAI.ThompsonNonamenability.StandardPLCondition
standardPLEquiv def: A faithful identification with the standard PL group carrier, with no uniform-grid or transport assumption in the right-hand side. · OAI.ThompsonNonamenability.F.standardPLEquiv
standardPrism def: The actual standard prism, assembled from its maximal simplices. · OAI.Thomason.Construction.PrismBarycentric.standardPrism
StandardRegionAt def: The equality class at every interior volume, including complementation. · OAI.CubicTorus.StandardRegionAt
StandardTracialPosition structure: The minimal concrete data for a standard tracial representation. · OAI.KadisonKastler.StandardTracialPosition
standardTracialPosition_of_relative_central def: Once cyclicity is known, relative centrality supplies all remaining standard tracial position data. · OAI.KadisonKastler.standardTracialPosition_of_relative_central
standardUnit def: The genuine normalized standard-basis action, a unit in the regular endomorphism algebra. No presentation relations are assumed. · OAI.KLInvariance.Hecke.standardUnit
standardUnit def: The genuine normalized standard-basis action, a unit in the regular endomorphism algebra. No presentation relations are assumed. · OAI.KLInvariance.HeckeQ.standardUnit
starAlgEquivOrderIso def: The order isomorphism underlying an algebraic star-isomorphism. · OAI.KadisonKastler.starAlgEquivOrderIso
starAlgHomDescend def: Descent through a surjective star homomorphism with smaller kernel. · OAI.KadisonKastler.starAlgHomDescend
starAlgHomToContinuousLinearMap def: A star homomorphism, viewed as a continuous linear map. · OAI.KadisonKastler.starAlgHomToContinuousLinearMap
StarDart abbrev: Three individually indexed sides for each old side: its old edge, and two radii to the new center of the old face. This is literal face starring. · OAI.HyperbolicObstruction.PictureSwitch.PictureMap.StarDart
starEvent def: The sharing event of one directed system under the full independent data. Forbidden parent choices simply give an empty event. · OAI.CoordinateSweeps.Coverage.starEvent
starExponent def: Exact critical exponent of a star-coordinate subset. Central coordinates are counted once, and outside coordinates separately in each branch. · OAI.HyperbolicObstruction.SparsePattern.Pattern.starExponent
StarkInput def: The zero-free specialization of Stark's Lemmas 3 and 8, together with his real Hadamard identity (9). Only fields of the fixed degree and without quadratic subfields are used. · OAI.DukePrimeDegree.StarkInput
starPath def: The radial path from a star-center to a point in its open star. · OAI.HyperbolicObstruction.FinitePolyhedron.starPath
starPorts def: After processing a frontier vertex, the old remaining ports and all its new children are the frontier. · OAI.PlanarL1.Topology.starPorts
starShift def: Completing the cut layout to rooted stars gives a uniform affine shift. The real vectors may be arbitrary; internal values are constant on each arm. · OAI.WLIdentification.ComparisonSchemas.starShift
starSubalgebraCommutant def: The commutant of an operator star subalgebra, bundled as a von Neumann algebra by the elementary triple-commutant identity. · OAI.KadisonSimilarity.starSubalgebraCommutant
starSystem def: A finite directed star system. Distinct tails are indexed by the subtype A; the parent lies outside A or is a hole. · OAI.CoordinateSweeps.Coverage.starSystem
start def: The canonical zero grid point is valid even when there are no free coordinates; its construction does not search the geometric state space. · OAI.ContingencyTables.ComputedCountingParameters.start
start def: Concrete entry configuration carries two physical counters and the body. · OAI.MaxCutGames.BinaryHeaderMachine.start
start def: Consume the finite cloud coordinate and input port, and perform the first lookup in the fixed table. Initial register values need no inhabitance choice. · OAI.MaxCutGames.Foundations.PCP.ExpanderRowControl.start
start def: Place windows in preorder, reserving g unused indices after each window. · OAI.Problem310.WindowPlacement.start
start def: Concrete entry configuration carries two physical counters and the body. · OAI.UniqueGamesTheorem.BinaryHeaderMachine.start
start def: Consume the finite cloud coordinate and input port, and perform the first lookup in the fixed table. Initial register values need no inhabitance choice. · OAI.UniqueGamesTheorem.Foundations.PCP.ExpanderRowControl.start
startArray def: Start masses indexed by the literal (component,prefix) array. There is one entry per actual segment, not one entry per path or per chain. · OAI.BoundedTreePotentials.Segments.startArray
startCost def: The start-mass cost, literally the upward recursion, including infinite values. The field m is restricted by the remaining height of recursion. · OAI.BoundedTreePotentials.Segments.startCost
startInTime def: Exact handoff from the genuine machine input to the actual logarithm machine, with the original graph word protected in the archive. · OAI.MaxCutGames.Foundations.Complexity.GraphCounterModel.startInTime
startInTime def: Exact handoff from the genuine machine input to the actual logarithm machine, with the original graph word protected in the archive. · OAI.UniqueGamesTheorem.Foundations.Complexity.GraphCounterModel.startInTime
startIso def: Bar iteration starts at exactly the same first-block derived stalk as the ordinary simulation, through the actual extension and localization isomorphisms. · OAI.LittleFinitistic.SelectedBarStart.startIso
startNet def: The node's initial pending stack contains only the runtime modulus, enlarged by one bit. Emission and error registers are zero. No factor data is supplied. · OAI.ExactQuantumFactoring.PhysicalNode.startNet
StartSpace def: The segment-start space consists of coordinate fields represented by finite-cost countable signed segment arrays. Its norm equals the infimum over exactly these representations. · OAI.BoundedTreePotentials.Segments.StartSpace
startTapes def: Complete machine frame at the first clause-loop guard. · OAI.MaxCutGames.Foundations.PCP.RawInitialMachineStart.startTapes
startTapes def: Complete machine frame at the first clause-loop guard. · OAI.UniqueGamesTheorem.Foundations.PCP.RawInitialMachineStart.startTapes
startupRamp def: A completely specified fixed startup, zero until time 1/2 and one from time 1. · OAI.ForcedComputation.startupRamp
starVertexClass def: The new vertices are exactly the old vertices plus the separate centers of the old faces. This retains all original vertex identifications. · OAI.HyperbolicObstruction.PictureSwitch.PictureMap.starVertexClass
State structure: Positive trace-one operators. For a positive bounded operator on this Hilbert space, summability of its diagonal is the positive trace-class criterion; HasSum ... 1 includes summability as well as normalization. · OAI.EntropyPhotonNumber.State
State abbrev: none is the initial state. Boolean states are accept and reject. · OAI.OneWayLiveness.PathNFA.State
State abbrev: The type used by adaptive selection contains only valid immutable histories. · OAI.ReflexiveFixedPoints.CandidateContext.State
State abbrev: The manuscript's product of cyclic leaf coordinates. · OAI.TreeEdit.BinaryLower.FiniteTree.State
State structure: The finite random experiment is part of the constructed state, not an assumed randomized-policy interface. · OAI.UniformKServer.ShortRoster.State
State abbrev: The last finite-control value is an overflow sentinel. · OAI.UniqueGamesTheorem.MachineTranslationTable.State
state def: Accumulated annihilation constraints and preceding functionals. · OAI.SeparableQuotient.NormerStep.state
state def: State stores the exponent scale and twice the sum of all earlier lengths. · OAI.SeparableQuotient.Parameters.state
stateCoefficient def: Matrix coefficient in the common initial corner. · OAI.Generator.Tracial.EquivalentPartition.stateCoefficient
StateConstraint inductive: The homogeneous inequalities defining a normalized additive state. Both signs of the additive relation are included. · OAI.MatrixMultiplication.AuxiliarySeparation.StateConstraint
stateConstraintDetector def: The multiplier of the distinguished detector difference. · OAI.MatrixMultiplication.AuxiliarySeparation.stateConstraintDetector
stateConstraintLower def: The ordinary monotonicity difference in a state constraint. · OAI.MatrixMultiplication.AuxiliarySeparation.stateConstraintLower
stateConstraintVector def: Integer coefficient vector of a homogeneous state constraint. · OAI.MatrixMultiplication.AuxiliarySeparation.stateConstraintVector
stateCoordinates def: Passing from sum coordinates to the product coordinates used by the implicit function theorem. · OAI.Problem356.LocalHessian.stateCoordinates
stateDifferential def: The linear form represented by the two components of the stationary map. · OAI.Problem356.StationaryMap.stateDifferential
stateDual def: The sum of the two component inner products, bundled for the usual (max-norm) product. No inner-product-space instance on the product is used. · OAI.Problem356.stateDual
stateJointProjection def: Forget the redundant small-profile coordinate while retaining both full margin views. This projection is injective on all actual joint pairs. · OAI.ContingencyTables.SmallGraphProfiles.stateJointProjection
StateJointSuccess def: The success test reads precisely balance and the lower padding bound. · OAI.ContingencyTables.SmallGraphProfiles.StateJointSuccess
Statement def: Compatible extensions whose relative cores retain the inner automorphisms have a unique common extension, and the common core retains the inner automorphisms. · OAI.FiniteLattice.CommonCore.Statement
Statement def: A relative normal core maps onto a nonabelian simple quotient. The subgroups T i lie in R₀; their ambient images are used for normalization and commutators. The normal core inside the generated subgroup W is transported back to R₀. … · OAI.FiniteLattice.RetainedCore.Statement
statement def: Every branch is a concrete finite TM2 statement. The fixed table H is evaluated only on finite control values, never on an unbounded tape value. · OAI.MaxCutGames.Foundations.Complexity.MachineExpanderRow.statement
statement def: Place one of finitely many actual statements into an arbitrary ambient Boolean-tape program, preserving the ambient state component. · OAI.MaxCutGames.Foundations.Complexity.MachineOdometerInit.statement
statement def: Tape roles: 0 original table, 1 unary index, 2 working copy, 3 output, 4 copy/restore scratch. The embedded halt enters the explicit cleanup loop. · OAI.MaxCutGames.Foundations.Complexity.MachinePreservingLookupClean.statement
statement abbrev: Translate instructions using the checked recursive translator, keeping every tape and destination label unchanged. · OAI.MaxCutGames.Foundations.Complexity.MachineStateEquiv.statement
statement def: Optional continuation and labels use the existing subroutine semantics. · OAI.MaxCutGames.Foundations.Complexity.MachineStateFrame.statement
statement def: Slots 0 and 1 contain the operands, 2 the output, 3 the counter, and 4 the reusable empty scratch. The code is independent of operand values. · OAI.MaxCutGames.Foundations.Complexity.MachineUnaryMultiply.statement
statement def: All instructions act on physical stacks. The final load only resets the finite internal register; no input-dependent integer is stored in it. · OAI.MaxCutGames.Foundations.Hastad.SourceRuntimeFinish.statement
statement def: A composable block with exactly seven internal control labels. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.CompareLoop.statement
statement def: Six fixed control labels implement row selection. q fixes the finite predicate parser; no data-dependent number is placed in the program syntax. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Lookup.statement
statement def: The fixed arithmetic program never captures input counts in its code. · OAI.MaxCutGames.Reduction.MachineAddressHeaders.Arithmetic.statement
statement def: Every branch is a concrete finite TM2 statement. The fixed table H is evaluated only on finite control values, never on an unbounded tape value. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineExpanderRow.statement
statement def: Place one of finitely many actual statements into an arbitrary ambient Boolean-tape program, preserving the ambient state component. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineOdometerInit.statement
statement def: Tape roles: 0 original table, 1 unary index, 2 working copy, 3 output, 4 copy/restore scratch. The embedded halt enters the explicit cleanup loop. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachinePreservingLookupClean.statement
statement abbrev: Translate instructions using the checked recursive translator, keeping every tape and destination label unchanged. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineStateEquiv.statement
statement def: Optional continuation and labels use the existing subroutine semantics. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineStateFrame.statement
statement def: Slots 0 and 1 contain the operands, 2 the output, 3 the counter, and 4 the reusable empty scratch. The code is independent of operand values. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineUnaryMultiply.statement
statement def: All instructions act on physical stacks. The final load only resets the finite internal register; no input-dependent integer is stored in it. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceRuntimeFinish.statement
statement def: A composable block with exactly seven internal control labels. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.CompareLoop.statement
statement def: Six fixed control labels implement row selection. q fixes the finite predicate parser; no data-dependent number is placed in the program syntax. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Lookup.statement
statement def: The fixed arithmetic program never captures input counts in its code. · OAI.UniqueGamesTheorem.Reduction.MachineAddressHeaders.Arithmetic.statement
stateOfFeasible def: The successful integer tests recover an actual state without searching the completion space. · OAI.ContingencyTables.SmallGraphProfiles.stateOfFeasible
stateOn def: A literal prescribed spin word; the vacant positions are not summed. · OAI.CriticalHoneycomb.Transfer.FluxWard.Diagram.stateOn
stateOperator def: The regularized state derivative as a continuous linear map on product space. · OAI.Problem356.LocalHessian.stateOperator
statePairing def: Evaluation of a finite integer coefficient vector on a real-valued function. · OAI.MatrixMultiplication.AuxiliarySeparation.statePairing
statePotential def: The state potential with the explicit common central quadratic, K = 20. · OAI.Problem356.StationaryMap.statePotential
StateTracks def: The linked support displacement of an actual next state. The old support point is shared by the index and every temporal comparison. · OAI.ThreeTorus.StateTracks
stateTrial abbrev: Read balance and padding, then return the unique unpadded table. · OAI.ContingencyTables.SmallGraphProfiles.stateTrial
stateVertex def: State configurations embed without identifying any added leaves. · OAI.OneWayLiveness.CellNetwork.stateVertex
staticPowerStep def: Static natural exponent, with retained accumulator and no repeated construction of earlier expressions. This circuit has O(e w²) Boolean nodes. · OAI.ExactQuantumFactoring.BitArithmetic.staticPowerStep
stationary def: The concrete stationary equation, with parameter first and state second. · OAI.Problem356.StationaryMap.stationary
stationary def: Equivalent to reduced denominator one when h > 0; no drift restriction is part of stationarity. · OAI.QuantitativeVanDerWaerden.AnchoredPatterns.stationary
stationaryCutoffLift def: The actual cutoff old-frequency Jacobian in real coordinates. The first coordinate is the new slow coordinate; the second is the old frequency. · OAI.ThreeTorus.stationaryCutoffLift
stationaryDifferential def: The full parameter/state derivative of the concrete stationary equation. · OAI.Problem356.StationaryMap.stationaryDifferential
stationaryEquation def: The state gradient of the regularized three-body Coulomb cost. The input is (parameter, (central point, negative outer point)). · OAI.Problem356.GeometryJets.stationaryEquation
stationaryImage def: A stationary image of an amplitude support. The pair records frequency and the unchanged slow coordinate. · OAI.ThreeTorus.stationaryImage
stationaryL2Action def: The isometric additive action on the genuine L² space. · OAI.PolycyclicRecognition.stationaryL2Action
stationaryL2Mean def: Averaging as an actual bounded linear operator on L². · OAI.PolycyclicRecognition.stationaryL2Mean
stationaryLeading def: Actual stationary leading term: zero outside the derivative branch. · OAI.ThreeTorus.stationaryLeading
stationaryMean def: The average is a genuine homomorphism; invariance removes the translated second summand in the cocycle equation. · OAI.PolycyclicRecognition.stationaryMean
stationaryPullback def: Inverse pullback is the left representation associated to a left action. · OAI.PolycyclicRecognition.stationaryPullback
stationaryWeight def: The normalized physical-frequency mass used in (W-definition). · OAI.ThreeTorus.stationaryWeight
stats def: average of forward statistics under uniform output path · OAI.FourierLLogL.Dyadic.Procedure.Paths.stats
stem def: Add an actual unary initial stem, preserving all descendant geometry. · OAI.DilutedSpinGlass.PrescribedTree.stem
stemCarrier def: The untouched raw carrier together with the actual exterior stubs. · OAI.PlanarL1.Topology.stemCarrier
stemContractionSq def: The true L2 contraction projection error, retaining the entire sampled prefix to the designated branching node. The projector is the literal conditional descendant mean evaluated on the chosen leaf's path. · OAI.DilutedSpinGlass.PrescribedTree.stemContractionSq
stemIncrementSq def: The actual squared L2 martingale increment at depth d. · OAI.DilutedSpinGlass.PrescribedTree.stemIncrementSq
stemLeaf def: Retain a specified descendant leaf through the actual unary prefix. · OAI.DilutedSpinGlass.PrescribedTree.stemLeaf
stemMeanAt def: On the SAME full sampled path, the true conditional mean of a fixed later tree at each depth of its initial unary stretch. · OAI.DilutedSpinGlass.PrescribedTree.stemMeanAt
Step inductive: A single local cancellation or braid replacement, with fixed contexts. · OAI.ArtinCAT0.WordDiagram.Step
Step inductive: A single unit-cost insertion, deletion, or (when enabled) substitution. · OAI.EditDistortion.Step
Step inductive: Delete an inverse pair after commuting its endpoints together. · OAI.EilenbergGanea.TraceWords.Step
Step inductive: One directed rule in an arbitrary two-sided context. · OAI.FiniteAlgebraicEnvelopes.Bootstrap.Step
Step inductive: The internal graph consists of literal validator equations and simulator edges. These edges contain neither quota insertion nor exit to cleanup. · OAI.FiniteAlgebraicEnvelopes.Compiler.Validator.Extended.Step
Step inductive: Literal internal equations compiler-validation, oriented as in the source. There is no entry from P and no transition from the machine back here. · OAI.FiniteAlgebraicEnvelopes.Compiler.Validator.Step
Step def: A row replaces exactly its three specified prefixes and preserves every tail. · OAI.ForcedComputation.Balanced.Step
Step structure: An oriented traversal of one of the fixed master fields. · OAI.HyperbolicObstruction.MasterGraph.Step
Step structure: Stage-indexed iteration. The second factor records the actual stage; limits take its supremum, rather than skipping pending operations. · OAI.PartitionConsistency.ConstructionIteration.Step
step def: The implicit midpoint step. The actual step size is 2τ. · OAI.ArnoldCounterexample.Midpoint.step
step def: All forcing expressions are formed before the full correction is applied. · OAI.ClosedSurfaceR4.JetPolynomial.MetricPolynomial.step
step def: The realized step evaluates the current and proposed weights from separate words, then uses a third independent word for its rational coin. · OAI.ContingencyTables.CompanionBooleanPhase.step
step def: One iteration retains the transition table unchanged. · OAI.ContingencyTables.IntegerMatrixPrograms.step
step def: The realized step evaluates the current and proposed weights from separate words, then uses a third independent word for its rational coin. · OAI.ContingencyTables.RootedBooleanPhase.step
step def: One physical shuffle: rotate after independently switching each pair. · OAI.CubeShuffle.step
step def: A halted configuration stays fixed. Otherwise the current state and all visible stack heads select one instruction, whose next state and stack actions produce the next configuration. · OAI.CurrentKS.Machine.step
step def: All four right-hand sides use the old state, as in the supplement. · OAI.DefocusingNLS.ProfileCertificate.step
step def: Positive interpolation coefficients, indexed by coarse and fine grid indices. · OAI.DyadicHats.step
step 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.Machine.step
step def: The left sweep deletes every consumed tagged cell, and only marks on anchors produce a clean right-hand anchor. The boundary test is literal. · OAI.FiniteAlgebraicEnvelopes.Compiler.Cleanup.step
step def: The finite state transition on sectors with fixed ordered base sequence. The two flags record whether alpha and omega are present. There is no insertion or deletion among internal steps. · OAI.FiniteAlgebraicEnvelopes.Compiler.step
step def: One elementary matrix 1+T E_xy, acting without choosing an enumeration of the finite vertex set. · OAI.KLInvariance.SchedulePaths.step
step def: Choose a strict improvement when one exists; otherwise keep the tree. Existence of improvements will be supplied by the actual good-tree reduction. · OAI.LeanBlast.SecondKahnKalai.ProbabilityTree.CoverState.step
step def: Materialize every row of the next table in the declared order. · OAI.MaxCutGames.Foundations.PCP.ExpanderTables.step
step def: Close a set once under all formula witnesses with finitely many parameters. · OAI.PartitionConsistency.CountableModel.step
step def: Every kept, available cell becomes an actual MainStep, including its old chart and exact interpolation tree. All parameters are the fixed constants. · OAI.PlanarL1.Columns.Model.MainFamily.step
step def: A nonempty admitted component now produces the literal, fully proved secondary clipping operation. No admission, support, name-gluing or movement conclusion is added as an assumption of the step. · OAI.PlanarL1.Columns.Model.SecondaryGeometry.Group.step
step def: Positive interpolation coefficients, indexed by coarse and fine grid indices. · OAI.SignedDisk.DyadicHats.step
step def: The QR frame of the image of an orthonormal frame under an invertible map. · OAI.SmoothObstruction.QR.step
step def: Extend a scheduled deformation across one additional stationary retract. · OAI.Thomason.Construction.TopologicalDeformation.ScheduledData.step
step def: A threshold viewed as a real-valued zero-one step. · OAI.TreeEdit.BinaryLower.BoolCuts.step
step def: One legal step: both head positions stay within the marked input. · OAI.TwoWayComplementation.TwoNFA.step
step def: Materialize every row of the next table in the declared order. · OAI.UniqueGamesTheorem.Foundations.PCP.ExpanderTables.step
step def: One read/write/head-movement transition. A halted configuration is frozen. · OAI.WLIdentification.step
StepAlphabet abbrev: Finite numerical alphabet: an orientation and an admissible tuple/padding pair. · OAI.TwoPointCorrelations.StepAlphabet
stepBody def: The same body in prefix/next-coordinate/remainder coordinates. · OAI.Problem358.FinitePrefix.stepBody
stepCell def: The chosen cell at an active stage is independent of the queried word. · OAI.PartitionConsistency.ConservativeMonoids.CellTower.stepCell
stepEquiv def: Translation of the sampled matrix is a permutation while the two sampled factors remain fixed; this also covers either factor being zero. · OAI.MaxCutGames.Inverse.RowErasureMatrix.stepEquiv
stepEquiv def: Translation of the sampled matrix is a permutation while the two sampled factors remain fixed; this also covers either factor being zero. · OAI.UniqueGamesTheorem.Inverse.RowErasureMatrix.stepEquiv
StepEstimates structure: All estimates propagated from one stage to the next. · OAI.ExactModuli.KW.StepEstimates
stepGraph def: The actual square/zigzag graph, expressed in the table's integer coordinates. · OAI.MaxCutGames.Foundations.PCP.ExpanderTables.stepGraph
stepGraph def: The actual square/zigzag graph, expressed in the table's integer coordinates. · OAI.UniqueGamesTheorem.Foundations.PCP.ExpanderTables.stepGraph
stepLaw def: A finite kernel is extended and its output written into one star. · OAI.KServer.TreeRounding.stepLaw
stepLookup def: A row uses exactly the displayed four lookups in already stored tables. · OAI.MaxCutGames.Foundations.PCP.ExpanderTables.stepLookup
stepLookup def: A row uses exactly the displayed four lookups in already stored tables. · OAI.UniqueGamesTheorem.Foundations.PCP.ExpanderTables.stepLookup
stepOccupation def: Occupation update for one derivative or multiplication variable. Natural subtraction is harmless when the derivative coefficient vanishes. · OAI.Problem335.stepOccupation
stepPosition def: The outside has no occurrence position. Its letters are genuine boundary penalties, and cannot be silently identified with a random row field. · OAI.HyperbolicObstruction.ConeSkeleton.RelDisk.stepPosition
Steps inductive: Execution with an exact finite number of local transitions. · OAI.ForcedComputation.Recorder.Steps
Steps inductive: Exact execution in the finite-control graph, stopping before a halt label is stepped. This is a finite trace, not an oracle for a computation. · OAI.Superstring.BitCode.Steps
steps def: Exact sum of actual row-and-increment costs. This recurrence is matched to the program by suffixTrace, rather than postulated as a runtime. · OAI.MaxCutGames.Foundations.Complexity.MachineDummyRows.steps
steps def: Exact copy, lookup, return, and residual-cleanup transition count. · OAI.MaxCutGames.Foundations.Complexity.MachinePreservingLookupClean.steps
steps def: Sum of the actual arithmetic, row-emission, and cleanup transition counts. · OAI.MaxCutGames.Foundations.Complexity.MachineRegularDummyRow.steps
steps def: Exact cleanup cost is computed on the actual post-copy tape contents. · OAI.MaxCutGames.Foundations.Complexity.PoweringMachineFinish.steps
steps abbrev: Proof-side cost on the shared arena. Its arithmetic-header tape dimensions use the fixed polynomial coefficient representation in PolynomialMachine. The executable drain list, entry, and instructions do not evaluate this cost. · OAI.MaxCutGames.Integration.AddressTupleCleanup.steps
steps def: Execute f 0 through f (n-1), iterating an accumulator. · OAI.PowerSaving.RAM.Bill.steps
steps def: Matrix of Newton polynomial values, by its product formula. · OAI.PowerSaving.Spectral.steps
steps def: Exact sum of actual row-and-increment costs. This recurrence is matched to the program by suffixTrace, rather than postulated as a runtime. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineDummyRows.steps
steps def: Exact copy, lookup, return, and residual-cleanup transition count. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachinePreservingLookupClean.steps
steps def: Sum of the actual arithmetic, row-emission, and cleanup transition counts. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRegularDummyRow.steps
steps def: Exact cleanup cost is computed on the actual post-copy tape contents. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringMachineFinish.steps
steps abbrev: Proof-side cost on the shared arena. Its arithmetic-header tape dimensions use the fixed polynomial coefficient representation in PolynomialMachine. The executable drain list, entry, and instructions do not evaluate this cost. · OAI.UniqueGamesTheorem.Integration.AddressTupleCleanup.steps
stepTrack def: Join the new deformation to the old one at their common stationary value. · OAI.Thomason.Construction.TopologicalDeformation.ScheduledData.stepTrack
stereographic def: Rational parametrization of a genuine radius-r circle. · OAI.EuclideanRamsey.CircleConsequence.stereographic
stinespringEmbedding def: The bounded map sending a vector to its unit-labelled kernel vector. · OAI.BoundedHochschild.KadisonRingrose.stinespringEmbedding
stinespringKernel def: The coefficient kernel on formal algebra-vector pairs. · OAI.BoundedHochschild.KadisonRingrose.stinespringKernel
stinespringOperator def: The induced action on the separated Hilbert completion. · OAI.BoundedHochschild.KadisonRingrose.stinespringOperator
stinespringPreOperator def: The bounded left action before completing the kernel space. · OAI.BoundedHochschild.KadisonRingrose.stinespringPreOperator
stinespringRepresentation def: The genuine unital star representation on the positive-kernel Hilbert space associated to a completely positive map. · OAI.BoundedHochschild.KadisonRingrose.stinespringRepresentation
StinespringSpace abbrev: The auxiliary Hilbert space of a completely positive map. · OAI.BoundedHochschild.KadisonRingrose.StinespringSpace
stock def: Deterministic park of all k units at the root, solely for initialization. · OAI.KServer.TreeRounding.stock
stoppedAcc def: Return immediately at exhaustion or callback halt. The fuel counter is only a bound; its unused suffix is never traversed. · OAI.LowSpaceSubsetSum.StreamJoin.stoppedAcc
stoppedAtPath def: Stop at a possibly infinite time, leaving the path running if no front is ever reached. · OAI.SLEExactGauge.FrontSuccessMeasurability.stoppedAtPath
stoppedImageEvent def: Physical containment of the entire trace segment through a finite stop. · OAI.SLEExactGauge.EarlyPassageLocalization.stoppedImageEvent
StoppedInput def: The independent stopped-flow intrinsic input, with no change to its energy average. · OAI.DiamondDistortion.PathApplications.StoppedInput
stoppedNarrowCertificate def: This concrete certificate depends only on the stopped trace. Its height and horizontal-strip conditions can therefore be tested in the stopped past. · OAI.SLEExactGauge.FirstHeightSupport.stoppedNarrowCertificate
stoppedTracePath def: Stop a continuous trace at a finite stopping time, retaining a continuous path on the common nonnegative-time domain. · OAI.SLEExactGauge.FirstHeightSupport.stoppedTracePath
stoppingRemainingTest def: The exact remaining-side predicate in the stopping split. · OAI.CubicFirstMoment.stoppingRemainingTest
storage def: Eight arrays of incidence lists, their headers, and the four sampled families with each local mask represented by its indexed bits. · OAI.LowSpaceSubsetSum.SeparatorStoredMasks.storage
storageCells def: A conservative simultaneous footprint, including the preliminary single-block enumeration and three copies of every group state. · OAI.LowSpaceSubsetSum.CompressionData.storageCells
StoredSubstitution abbrev: Substitution carries stored trees at all declaration occurrences. · OAI.PureTypeSystem.TypingTree.StoredSubstitution
storeLoop def: Store the result chosen by the actual comparison's equal/different exit. This does not compare tape contents in a register transition. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.RuntimeModel.storeLoop
storeLoop def: Store the result chosen by the actual comparison's equal/different exit. This does not compare tape contents in a register transition. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.RuntimeModel.storeLoop
stow def: operation in packed addresses · OAI.PowerSaving.Grove.stow
Straddle def: The source's strict min/max straddling region, in equivalent witness form. · OAI.EuclideanFiveColor.Straddle
straight def: The affine path when its entire segment lies in the actual carrier. · OAI.HyperbolicObstruction.FinitePolyhedron.straight
Stream structure: A single constructed infinite history, including the thresholds and exact whole-period equalities, for any prescribed determinant-one simultaneous target. · OAI.HarmonicCounterexample.Construction.Stream
street def: The deleted street for one of N translated grids of the same mesh Nδ. · OAI.SmoothObstruction.GridMargin.street
streetNerve def: Restrict Yoneda to the actual oriental cosimplicial object. · OAI.Thomason.Construction.NormalizedSimplexIdentities.streetNerve
streetWeakEquivalences def: The prescribed weak equivalences for strict omega-categories, defined through their Street nerves and geometric realization. · OAI.Thomason.Construction.NormalizedSimplexIdentities.streetWeakEquivalences
StrictChain def: The nodes after the initial bottom in a strict chain. The empty list gives the chain containing only bottom. · OAI.FiniteChain.StrictChain
StrictColimitsInput class: The small-colimit specialization of local presentability of strict omega-categories. · OAI.Thomason.Construction.StrictColimitsInput
strictCollisionPairs def: Ordered collisions between distinct inputs. · OAI.Problem337.strictCollisionPairs
StrictDisjointPaths def: Original strict witnesses with pairwise disjoint physical edge supports. · OAI.GammaFPP.BK.StrictDisjointPaths
StrictEditScript inductive: Genuine changes only: this excludes the harmless identity substitutions permitted by EditStep. · OAI.TreeEdit.BinaryLower.StrictEditScript
strictLabelHomotopy def: Ordered object labels give a constructed homotopy of the actual Street nerves. · OAI.Thomason.Construction.NormalizedSimplexIdentities.strictLabelHomotopy
strictLabelRealizedHomotopyRel def: The realized ordered-label homotopy fixes the entire realized image of the equal-label subdiagram, with its actual topology. · OAI.Thomason.Construction.NormalizedSimplexIdentities.strictLabelRealizedHomotopyRel
StrictlySeparated def: Strict separation permits either orientation and always has a positive gap. · OAI.ThompsonNonamenability.StrictlySeparated
StrictlySingular def: A strictly singular operator is not bounded below on any closed infinite-dimensional subspace. · OAI.SeparableQuotient.Singular.StrictlySingular
StrictOmegaCategory structure: A strict globular omega-category with identities represented by the same cell at all higher dimensions. All compositions, units and interchange laws are strict equalities. · OAI.Thomason.Construction.StrictOmegaCategory
StrictPathEvent def: Existence of a strict path witness with support in a prescribed finite edge set. · OAI.GammaFPP.BK.StrictPathEvent
StrictPathWitness structure: A labelled strict finite path. Different bounds on the same path are retained as different witnesses. · OAI.GammaFPP.BK.StrictPathWitness
strictPosetNerveIso def: The canonical comparison is an isomorphism on the Street nerve, including every degenerate simplex and every ordinal map. · OAI.Thomason.Construction.NormalizedSimplexIdentities.strictPosetNerveIso
strictPrism def: Strict realization of the explicitly constructed positive prism map. · OAI.Thomason.Construction.SimplexComplex.strictPrism
strictRegion def: The interior region on which one affine score is strictly maximal. · OAI.Problem358.TripleTieAvoidance.strictRegion
StrictSubsetGrowth def: `StrictSubsetGrowth r` is the amplified inequality `|S|+|F²(S)| < 2|F(S)|` for every nonempty proper finite subset. It compares ordinary relation images, which may overlap `S`, rather than exact-distance neighborhoods. · OAI.SeymourSecondNeighborhood.StrictSubsetGrowth
strictSupportExtensionIso def: The genuine chain isomorphism, including all boundary differentials. · OAI.DerivedBar.strictSupportExtensionIso
strictTieGraph def: Edges are interior ties with no third maximizing branch. For affine families these are precisely the local witnesses of positive-area facets. · OAI.Problem358.PositiveFacet.strictTieGraph
StrictUpper def: Strict upper triangularity, with diagonal vanishing included. · OAI.PolycyclicRecognition.StrictUpper
strictValuationIdeal def: The strict positive-threshold ideal of a nonnegative additive valuation. · OAI.CartierSections.strictValuationIdeal
stride def: First N array columns, then start and k round history columns. · OAI.Fifths.Tour.stride
strip def: The source's strip and tilt-compatible terminal support. · OAI.CriticalHoneycomb.strip
strip def: strips of width 2^-l p about the two edges of both children of p; empty if the strip width needs more bits than available. · OAI.FourierLLogL.Dyadic.Procedure.strip
stripColor def: The measurable periodic selector; boundary values play no role. · OAI.DoublingHilbert.stripColor
stripCoordinateLinear def: The complexification of the real strip matrix, in real pair coordinates. · OAI.SymmetricMahler.stripCoordinateLinear
Stripe inductive: Linear link with internal points explicit; endpoints can coincide. · OAI.ExactDerandomization.Swirl.Stripe
stripFactorsNat def: Literal prime stripping in natural modular arithmetic. This is exactly the scan compiled by stripScan; its inputs come only from the supplied data. · OAI.ExactQuantumFactoring.stripFactorsNat
stripForest def: Produced from all and only the actual internally sewn child-port occurrences. In particular no transverse branch is treated as internal. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.RealizedBand.stripForest
stripForward def: Old transport in independently chosen normalized regular frames. · OAI.IntegralCharacterVarieties.IntrinsicInterval.Transport.stripForward
stripOrderPrime def: One occurrence of the source's prime-stripping loop. A repeated prime factor list supplies exactly as many attempts as its known multiplicity. · OAI.ExactQuantumFactoring.stripOrderPrime
stripRow def: Tallest-column bottom deletion, expressed by interlacing rows. · OAI.Saxl.stripRow
stripShear def: The free extension term is read from old transport and the chosen splittings. It is not set to zero or replaced by a block-diagonal premise. · OAI.IntegralCharacterVarieties.IntrinsicInterval.Transport.stripShear
stripUpdate def: The original input stays read-only; the accumulator is updated once. · OAI.ExactQuantumFactoring.BitArithmetic.stripUpdate
StrongEarly structure: Additional regularity asserted on the common initial interval. The eLpNorm statement is strong L¹ continuity and cannot be made vacuous by a nonintegrable Bochner integral. · OAI.BoltzmannNonuniqueness.StrongEarly
StronglyNormalizing def: Accessibility for the converse of reduction: strong normalization, including reductions in all type annotations. · OAI.PureTypeSystem.StronglyNormalizing
strongOperatorTopology def: Pointwise norm convergence restricted to bounded linear operators. · OAI.Hyperinvariant.strongOperatorTopology
StrongUnitarySpace def: This is the strong-star coordinate model, not the operator-norm topology. · OAI.KadisonKastler.StrongUnitarySpace
structuralBinaryData def: Every algebraic field comes from the finite structural quotient's proved relations. No new relation is postulated in order to invoke coherence. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Generators.structuralBinaryData
structuralLocalization def: The endomorphism exists on the finitely presented structural quotient, not just on its semantic image in the actual table group. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Generators.structuralLocalization
structuralMonoidal def: The actual finite quotient is the positive-object automorphism group of a monoidal category. Its empty object is separate, with trivial automorphism group. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Generators.structuralMonoidal
StructuralQuotient abbrev: Finitely presented structural quotient with genuine evaluation map. The absence of an injectivity assertion is substantive: coloured coherence and label covariance/completeness are still required for the main theorem. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Generators.StructuralQuotient
structuralRelations def: All root-level structural laws are finite after checking naturality and commutation only on a finite generating family of Seed. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Generators.structuralRelations
structuralTensor def: A genuine binary tensor homomorphism on the actual finite structural quotient. Its functoriality follows from the finitely generated disjointness relations, not from semantic equality after evaluation. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Generators.structuralTensor
structureAlgebra def: The controlled pseudolocal algebra used to form the analytic-index quotient. · OAI.CoarseAssembly.DiscreteRoe.structureAlgebra
structureMap def: The map from the ambient structure sheaf to the pushed-forward structure sheaf; its section maps are precisely the scheme comorphism. · OAI.ActualCartier.structureMap
structureTargetRestrictIso def: The pushed-forward structure sheaf has the canonical open base-change identification. · OAI.ActualCartier.structureTargetRestrictIso
Study def: Branches tested by integer registers. · OAI.PowerSaving.RAM.Bench.Study
SturmPowerFamily structure: The exact two-step positive differential system supplied by translating the cosine kernel below its first zero. It does not assume any determinant sign. · OAI.CriticalHoneycomb.FiniteFunctional.StripAsymptotic.SturmPowerFamily
subalgebra def: Elements whose denominators can be cleared with a power of a fixed base element. · OAI.ComplexCancellation.Clearing.subalgebra
subalgebra def: The part of an ambient unital star algebra lying in the corner. · OAI.KadisonSimilarity.CStarCorner.subalgebra
subalgebraTraceVectors def: The trace-vector image of a concrete subalgebra. · OAI.KadisonSimilarity.subalgebraTraceVectors
subcomplexRelativeChainIso def: The chain-level relative excision map for a union of simplicial subcomplexes. The map is an isomorphism before taking homology. · OAI.Kaplansky.subcomplexRelativeChainIso
subcomplexSubdivisionHomeomorph def: The actual realization of the subdivision of any simplex subcomplex is homeomorphic to the original realization. · OAI.Thomason.Construction.SubdivisionBarycentric.subcomplexSubdivisionHomeomorph
subdivide def: A deterministic full-table occurrence list with exactly four rows per input row. · OAI.MaxCutGames.Explicit.SubdivisionTarget.subdivide
subdivide def: A deterministic full-table occurrence list with exactly four rows per input row. · OAI.UniqueGamesTheorem.Explicit.SubdivisionTarget.subdivide
subdivideColumnChunks def: Imperfect positions stay single; perfect intervals are split into full blocks of size s and one possible final shorter block. Empty final blocks are harmless and are included in the upper bound. · OAI.TwoPointCorrelations.subdivideColumnChunks
subdivided def: Exact canonical-rank suppression isomorphisms lift to actual subdivisions. · OAI.TorsionFreeZeroDivisors.RankedSuppressionIso.subdivided
subdividedFace def: The full chain subcomplex supported on the original face vertices. · OAI.Thomason.Construction.subdividedFace
subdividedFaceMap def: The actual inclusion of a subdivided face into the finite-chain nerve. · OAI.Thomason.Construction.subdividedFaceMap
subdivision def: The canonical threefold subdivision, including loops and parallel edges. · OAI.TorsionFreeZeroDivisors.PortSubdivision.PortGraph.subdivision
subdivisionAdjunctHomotopy def: The mates of a simplicial homotopy are simplicially homotopic. · OAI.Thomason.Construction.subdivisionAdjunctHomotopy
subdivisionHornRetraction def: After two subdivisions, a simplex retracts into the once-subdivided horn with restriction exactly Sd(lastVertex_horn). This is the finite-stage extension identity used to fill horns in the extension tower. · OAI.Thomason.Construction.subdivisionHornRetraction
subdivisionInclusion def: The subdivision of a subcomplex, mapped to the standard finite-chain nerve. · OAI.Thomason.Construction.subdivisionInclusion
subdivisionLastVertexHomotopy def: The two canonical last-vertex maps from double subdivision are actually simplicially homotopic, for every small simplicial set. · OAI.Thomason.Construction.subdivisionLastVertexHomotopy
subdivisionMapOfLabelsHomotopy def: The comparison on subdivision has the last-vertex map as its endpoint. · OAI.Thomason.Construction.ChainLabelApproximation.subdivisionMapOfLabelsHomotopy
subdivisionPoint def: The uniform subdivision of the unit interval, constant after its endpoint. · OAI.PolycyclicRecognition.subdivisionPoint
subdivisionUnitHomotopy def: The mate of the double-last-vertex homotopy compares the canonical extension on Sd K with the adjunction unit on K. · OAI.Thomason.Construction.subdivisionUnitHomotopy
subdivisionVertexWords def: The semantic subdivision numbering: originals and middle vertices on the left, then two private right vertices for each occurrence. · OAI.MaxCutGames.Explicit.MachineSubdivisionRows.subdivisionVertexWords
subdivisionVertexWords def: The semantic subdivision numbering: originals and middle vertices on the left, then two private right vertices for each occurrence. · OAI.UniqueGamesTheorem.Explicit.MachineSubdivisionRows.subdivisionVertexWords
subFischer def: The Fischer duality on any actual polynomial subspace. · OAI.HarmonicCounterexample.Angular.subFischer
subgraphFromComplete def: A complete-graph subgraph whose edges occur in G is the same actual subgraph of G. · OAI.LeanBlast.SecondKahnKalai.subgraphFromComplete
subgraphPair def: A chosen simple subgraph is realized by exactly one original paired edge per simple edge, with the original vertex rotations unchanged. · OAI.TorsionFreeZeroDivisors.MapEuler.subgraphPair
subgraphToComplete def: Regard an actual subgraph as a subgraph of the complete graph. Its vertex set, including all isolated vertices, is unchanged. · OAI.LeanBlast.SecondKahnKalai.subgraphToComplete
subgroup def: The genuine subgroup generated by a subset of simple reflections. · OAI.KLInvariance.ParabolicSupport.subgroup
subgroupAlgebraRepresentation def: Regard an ambient unitary subgroup contained in an algebra as internal unitaries. · OAI.KadisonKastler.subgroupAlgebraRepresentation
subgroupCoordinate def: A right-coset transversal gives an equivariant coordinate in the subgroup. · OAI.PolycyclicRecognition.subgroupCoordinate
subgroupFilter def: The actual small-support subgroup filter, rather than a condition asserting that some symmetric interpretation already has the target properties. · OAI.PartitionConsistency.SupportFilter.subgroupFilter
subgroupFunctionExtension def: Extend a bounded subgroup function to the whole group by the equivariant coordinate. · OAI.PolycyclicRecognition.subgroupFunctionExtension
subgroupInsideEquiv def: A subgroup contained in K, viewed inside K, is canonically the same group. · OAI.FiniteLattice.CommonCore.subgroupInsideEquiv
subgroupInvariantMean def: Every subgroup of a group with an invariant mean has an invariant mean. · OAI.PolycyclicRecognition.subgroupInvariantMean
subgroupOf def: The subgroup determined by the class of the identity coset. · OAI.FiniteCongruence.subgroupOf
subgroupResolution def: Restrict an actual integral resolution, retaining its augmentation. · OAI.PolycyclicRecognition.subgroupResolution
subgroupRightCosetMulEquiv def: Right multiplication is well-defined on right cosets without normality. · OAI.PolycyclicRecognition.subgroupRightCosetMulEquiv
sublevel def: The fixed-height radial sublevel in the active column graph. Columns outside the chosen component are absent, as for the source's value +∞. · OAI.PlanarL1.Columns.Model.Point.sublevel
sublevelPatchDomain def: Adjoin the actual sublevel as the interior patch. This finite cover covers the whole closure, so derivative cancellation can be used throughout D_R. · OAI.MahlerStokes.sublevelPatchDomain
submapDefect def: The orientable genus defect, on all spanning ribbon subgraphs. · OAI.FKCRT.submapDefect
submoduleAdjointStabilizer def: The elements whose adjoints preserve a fixed linear subspace form a Lie subalgebra, by the Jacobi identity. · OAI.PolycyclicRecognition.submoduleAdjointStabilizer
Subobject structure: A conservative subobject can equivalently be specified by its carrier submonoid and its two countably-ary closure requirements. · OAI.PartitionConsistency.ConservativeMonoids.Subobject
subpaths def: The finite set of genuine directed paths selected by an edge schedule. · OAI.KLInvariance.SchedulePaths.subpaths
subposetInclusion def: The monotone inclusion of a full subposet. · OAI.Thomason.Construction.subposetInclusion
subposetInclusionWithin def: The inclusion between two full subposets with nested object sets. · OAI.Thomason.Construction.subposetInclusionWithin
subrepAmbient def: Inclusion of a subrepresentation of a subrepresentation into the ambient space. · OAI.Saxl.subrepAmbient
subrepModuleEquiv def: The invariant-subspace and group-algebra subtype constructions agree. · OAI.Saxl.subrepModuleEquiv
subrepPairFromAmbient def: Restriction of the ambient bilinear coordinate pairing to an invariant subspace. · OAI.Saxl.subrepPairFromAmbient
subrepPairMap def: The ordinary bilinear coordinate pairing restricted to any invariant word subspace. · OAI.Saxl.subrepPairMap
subrepProject def: The genuine equivariant orthogonal-coordinate projection. · OAI.Saxl.subrepProject
subrepSelfDual def: Complex conjugation-stable invariant word subspaces are canonically self-dual via bilinear evaluation, yielding equivariant projections. · OAI.Saxl.subrepSelfDual
subroutine def: Any ordinary space-bounded TM with small arguments can be used as a subroutine. The result is a binary stack machine seeing just the public input, expecting the (small) arguments from the caller on designated stacks. No length counter for … · OAI.Fifths.TM.subroutine
subsequence def: Restricting to an increasing subsequence preserves precisely the analytic and geometric hypotheses of local harmonic compactness. · OAI.RegularPointCoordinates.HarmonicCompactnessSequence.subsequence
subset def: a ⊆ b. · OAI.PartitionConsistency.SetLanguage.subset
subsetBound def: All possible immediate pairs over the domain of a particular name form a set. This is a bound on normalizations of subsets, not on all HS names. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.subsetBound
subsetEntry def: The fixed block enumerates all subset products. · OAI.Problem337.ResidueConstruction.subsetEntry
subsetHitEquiv def: The membership indicators of a subset give its finite Boolean coordinates. · OAI.JointDickman.subsetHitEquiv
subsetList def: Each indexed subset is generated in the fixed input-index order. · OAI.LowSpaceSubsetSum.subsetList
subsetRankKernel def: Unhypothesized rank kernel: defined from the actual uniformly retained subset. · OAI.TraceReconstruction.subsetRankKernel
subsets def: The actual finite sample space of all t-subsets of Fin k. · OAI.MaxCutGames.Soundness.SubsetIntersection.subsets
subsets def: The actual finite sample space of all t-subsets of Fin k. · OAI.UniqueGamesTheorem.Soundness.SubsetIntersection.subsets
subspaceAntecedentEquiv def: Transport the canonical antecedent count to arbitrary complementary subspaces. · OAI.MaxCutGames.Appendix.DerivativeInduction.subspaceAntecedentEquiv
subspaceAntecedentEquiv def: Transport the canonical antecedent count to arbitrary complementary subspaces. · OAI.UniqueGamesTheorem.Appendix.DerivativeInduction.subspaceAntecedentEquiv
subspaceEquiv def: The quotient antecedents and the graph-coordinate antecedents are the same maps, with equivalent predicates. · OAI.MaxCutGames.Appendix.AntecedentCount.subspaceEquiv
subspaceEquiv def: The quotient antecedents and the graph-coordinate antecedents are the same maps, with equivalent predicates. · OAI.UniqueGamesTheorem.Appendix.AntecedentCount.subspaceEquiv
subspaceEvaluation def: The actual evaluation operator on a dual subspace. · OAI.SeparableQuotient.subspaceEvaluation
subspaceHomologyEquiv def: Supported homology is the actual singular homology of the subspace. · OAI.EilenbergGanea.CubicalSingular.subspaceHomologyEquiv
subspaceInclusion def: Inclusion of an arbitrary subspace with its inherited topology. · OAI.Kaplansky.subspaceInclusion
subspaceInclusionOfLE def: Inclusion between subspaces of the same ambient space. · OAI.Kaplansky.subspaceInclusionOfLE
subspaceLaw def: Pushforward of uniform sampling in K. For a subspace inclusion this is uniform sampling in that subspace; the definition also permits a linear map with nontrivial fibers without accidentally conditioning on nonzero vectors. · OAI.MaxCutGames.Decoder.PositiveMultiplier.subspaceLaw
subspaceLaw def: Pushforward of uniform sampling in K. For a subspace inclusion this is uniform sampling in that subspace; the definition also permits a linear map with nontrivial fibers without accidentally conditioning on nonzero vectors. · OAI.UniqueGamesTheorem.Decoder.PositiveMultiplier.subspaceLaw
subspaceMap def: Restriction of a continuous map that really carries one subspace into another. · OAI.Kaplansky.subspaceMap
subspaceSubsetHomeomorph def: Remove a redundant enclosing subtype without changing its topology. · OAI.Kaplansky.subspaceSubsetHomeomorph
subst def: Capture-avoiding substitution grafts the given trees at variables. It retains core choices and recurses through the annotation at each binder. · OAI.PureTypeSystem.TypingTree.subst
substitute def: cap invocation program uses a common graph virtual q. · OAI.ExactDerandomization.SiteX.Label.substitute
substitute def: Effective replacement of each generator by a fixed finite word. · OAI.FiniteAlgebraicEnvelopes.WordProblem.substitute
substitute def: Replace each positive or negative input literal by a circuit. · OAI.TwoPointCorrelations.AC0Circuit.substitute
SubstitutionGuard def: Only nonneutral replacements need the activity guard. Variables that remain neutral are harmless even at a skipped clause. · OAI.PureTypeSystem.ObservationValues.SubstitutionGuard
subtract def: Subtract and compare, destructively. Output register 2 has the little-endian fixed-width modular difference; singleton register 3 is exactly the comparison bit (true iff the first number was smaller). · OAI.RationalHitting.StackLanguage.subtract
subtractDigit def: Registers 0,1 input, 2 reverse output, 3 singleton borrow, 4 loop flag. · OAI.RationalHitting.StackLanguage.subtractDigit
subtractWord def: Fixed-width subtraction, returning the overflow borrow. Registers may have leading zeros. Width is exactly the larger input word length. · OAI.RationalHitting.StackLanguage.subtractWord
subtreeAlphabet def: The actual alphabet below a prefix consists of tagged leaf residues. · OAI.TreeEdit.BinaryLower.FiniteTree.subtreeAlphabet
subtreeBlock def: An edge and all descendants under it, in their contiguous preorder block. · OAI.Problem310.TreeSpanBridge.subtreeBlock
subtreeComparisonMass def: The comparison mass is the actual averaged warmup at active states and true cost at carried stopped states. · OAI.EditApproximation.subtreeComparisonMass
subtreeOutput def: The actual outputs between two levels, formed by iterating collapse. · OAI.ReflexiveFixedPoints.TreeEvaluation.subtreeOutput
subtypeSupport def: The ambient support obtained by adjoining free and exterior supports. · OAI.Problem313.BiasedFourier.subtypeSupport
subtypeSupportEquiv def: Free and exterior support sets give each ambient support exactly once. · OAI.Problem313.BiasedFourier.subtypeSupportEquiv
subW def: Restrict weights; not renormalized. · OAI.HadwigerCounterexample.subW
SubwordClosed def: The exact concatenation property of legal source words, expressed at all aligned lower-word positions. · OAI.SmoothObstruction.PhaseDynamics.SubwordClosed
subwords def: Quadratic substring enumeration, with boundaries represented explicitly. · OAI.Superstring.Input.subwords
success def: Sample the Bernoulli mask, a uniform actual map on its partner space, and independent weighted equation/position draws. This is the actual advice law. · OAI.MaxCutGames.Clean.ActualAdviceBridge.success
success def: Exact nested finite expectation of the advice-agreement experiment. · OAI.MaxCutGames.Clean.Experiment.success
success def: The actual law: independent singleton choices followed by one independent uniform coefficient at the intercept and each surviving partner coordinate. · OAI.MaxCutGames.Clean.NativeExperiment.success
success def: Sample the Bernoulli mask, a uniform actual map on its partner space, and independent weighted equation/position draws. This is the actual advice law. · OAI.UniqueGamesTheorem.Clean.ActualAdviceBridge.success
success def: Exact nested finite expectation of the advice-agreement experiment. · OAI.UniqueGamesTheorem.Clean.Experiment.success
success def: The actual law: independent singleton choices followed by one independent uniform coefficient at the intercept and each surviving partner coordinate. · OAI.UniqueGamesTheorem.Clean.NativeExperiment.success
successfulJointEquiv def: Each success produces exactly one original table, and every original table has exactly one successful state/completion preimage. · OAI.ContingencyTables.SmallGraphProfiles.successfulJointEquiv
successfulListEquiv def: The finite functions and literal word lists encode exactly the same successful words, including their fixed length. · OAI.FKCRT.successfulListEquiv
Successive def: Succession refers to the full supports, in both orders in mixed mode. · OAI.SeparableQuotient.Norming.Successive
successor def: The opposite cyclic orders on positive and negative word disks. · OAI.KervaireMixing.successor
successor def: b = a ∪ {a}. · OAI.PartitionConsistency.SetLanguage.successor
successor def: Creation operators contribute the successor occupation. · OAI.Problem335.FactorialExpansion.successor
successorInclusion def: Isometric insertion of a zero level at the start of the sum. · OAI.Kaplansky.successorInclusion
successors def: Possible jump targets; inspecting a head has only three finite cases. · OAI.RationalHitting.StackLanguage.successors
successorSquare def: The repeated creation-coordinate weight has coefficients 1,3,1. · OAI.Problem335.FactorialExpansion.successorSquare
successProbability def: Born probability, summing all final garbage with no cleanliness assumption. · OAI.QAC.successProbability
succFiber def: The two disjoint ways to achieve weight q+1 give Pascal's recurrence. · OAI.ExternalTensor.Binomial.succFiber
suffixEmbed def: Insert any D block before an arbitrary linear suffix, preserving its tensor order. · OAI.DerivedBar.ParityBridge.suffixEmbed
suffixEmbedLinear def: A genuine linear operation on suffix maps, so finite sums distribute exactly. · OAI.DerivedBar.ParityBridge.suffixEmbedLinear
sum def: Literal disjoint union of polygon side-pairings. · OAI.HyperbolicObstruction.PictureSwitch.PictureMap.sum
sum def: Disjoint whole-port union, before the three-port inverse-band surgery. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.PortWiring.sum
sum def: Tensoring two pointed systems constructs their sum-indexed system. · OAI.KadisonKastler.PointedMatrixSites.sum
sum def: The right summand follows the whole left block. · OAI.MaxCutGames.Foundations.Hastad.SourceOccurrences.Encoding.sum
sum abbrev: Pointwise coproduct of the two spans. · OAI.Thomason.Construction.SimplicialSpan.sum
sum def: The right summand follows the whole left block. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceOccurrences.Encoding.sum
sumBit def: Literal full-adder truth tables, not an arithmetic oracle. · OAI.ExactQuantumFactoring.BitStackProgram.sumBit
sumColumns def: Assemble maps from the individual coordinates of a finite Hilbert sum. · OAI.KadisonSimilarity.sumColumns
sumDecidableEq def: Constructor-by-constructor equality for a fixed sum of tape alphabets. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRegularOwnerCleanup.sumDecidableEq
sumDecidableEq_inline_MachineRegularOwnerCleanup def: Constructor-by-constructor equality for a fixed sum of tape alphabets. · OAI.MaxCutGames.Foundations.Complexity.MachineRegularOwnerCleanup.sumDecidableEq_inline_MachineRegularOwnerCleanup
sumDesc def: Distribution over a finite direct sum, defined by its coproduct property. · OAI.DerivedBar.RecursiveTotal.sumDesc
sumDifference def: Sum-and-difference coordinates conjugate a hyperbolic derivative to the square lattice template. · OAI.RuledApproximation.sumDifference
sumE def: Energy of a history against test input. · OAI.FourierLLogL.Dyadic.Thermal.sumE
sumEquiv def: The complete left block precedes the complete right block. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Enumeration.sumEquiv
sumEquiv def: The complete left block precedes the complete right block. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Enumeration.sumEquiv
sumEta def: An admissible positive qualitative choice; outside the possible separation range, any number in (0,1) is allowed. · OAI.DiamondDistortion.RecursiveApplications.sumEta
sumExchange def: The exact future/past exchange in the orthonormal sum coordinates used for the endpoint block matrices. · OAI.SmoothObstruction.EndpointBranch.sumExchange
sumF def: Extend the individual forms by zero outside respective blocks, then add. · OAI.HadwigerCounterexample.sumF
sumFacetMap def: The affine chart on the facet whose coordinate sum is one. · OAI.ProjectionCounterexample.sumFacetMap
sumFinSuccEmbedding def: Put a new extension vector before the previous extension vectors. · OAI.SharpLiebThirring.PLParity.sumFinSuccEmbedding
sumGraph def: Concrete sum-type padding, before converting to integer vertex indices. · OAI.MaxCutGames.Foundations.PCP.PreprocessingPaddingTables.sumGraph
sumGraph def: Concrete sum-type padding, before converting to integer vertex indices. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingPaddingTables.sumGraph
sumHomotopy def: A componentwise specified finite sum of homotopies. · OAI.ExternalTensor.Section4.AssemblyData.sumHomotopy
sumIndex def: Consecutive indexing of two positive blocks, in predecessor-size notation. · OAI.FiniteAlgebraicEnvelopes.BinaryGroupCategory.sumIndex
sumIndices def: All input indices are visited; both membership and signed-input lookup are charged even at indices outside the selected mask. · OAI.LowSpaceSubsetSum.RepresentationKeys.sumIndices
summand def: Literal nested terms, before any coproducts are taken. · OAI.DerivedBar.RecursiveTotal.summand
summand def: Evaluation of a gate on its distinguished (possibly shifted) register, evaluating each factor already at semantic child vectors. · OAI.Fifths.Circuit.summand
summand def: The formal image of the central selecting idempotent. · OAI.LittleFinitistic.FormalSummandAction.summand
summandIsometry def: The inclusion of an individual Hilbert summand. · OAI.CoarseAssembly.Analytic.summandIsometry
summands def: The closed cyclic spaces, as an actual complete orthogonal family. · OAI.KadisonSimilarity.CyclicDecompositionData.summands
summary def: Summarize all finite internal paths between the four boundary pairs. · OAI.TwoWayComplementation.Network.summary
summesh def: Sum of coeffs on multiples equals expectation over annihilator · OAI.FourierLLogL.Dyadic.Thermal.summesh
sumMod def: Linear-length modular accumulation, not a repeated expanded expression. · OAI.ExactQuantumFactoring.BitArithmetic.sumMod
sumNorm def: Componentwise ℓ¹ auxiliary norm; the actual Hermitian Sobolev norm is sobNorm below, and is compared explicitly, not substituted in the statement. · OAI.ElasticityNegativeOrder.sumNorm
sumParameterMap def: Applying the lifted family at the sum of two independent parameters. · OAI.StableCoordinate.FrameLiftCoassoc.sumParameterMap
sumPerm def: The genuine product position permutation under a disjoint-set enumeration. · OAI.Saxl.sumPerm
sumPhase def: The phase measured by a finite family of vectors. · OAI.KervaireSpectral.sumPhase
sumPortAt def: A disjoint union of vertex sets retains every whole ordered port. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.sumPortAt
sumRestriction def: The block-diagonal sum of two restriction matrices. · OAI.MatrixMultiplication.AuxiliarySeparation.sumRestriction
sumSides def: This equivalence retains each individual ordered child occurrence. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.PortWiring.sumSides
sumSite def: The actual site family on the two tensor legs. · OAI.KadisonKastler.PointedMatrixSites.sumSite
sumTensor def: The binary direct sum, with each of its three block labels retained. · OAI.MatrixMultiplication.AuxiliarySeparation.sumTensor
sumTensorPolynomials def: Add the coefficient expressions after padding every member to the same degree. No map or perturbation parameter enters this construction. · OAI.ClosedSurfaceR4.JetPolynomial.Perturbation.sumTensorPolynomials
sumTuples def: Ordered tuples with a prescribed sum in a finite cyclic group. · OAI.Problem337.CompositeSpectral.sumTuples
superColumn def: A column of a superoperator, reshaped as an ordinary matrix. · OAI.MUB6.Weiner.superColumn
superlevel def: Radial superlevels contain exactly the reachable columns whose midpoints have radial value at least the threshold. · OAI.PlanarL1.Columns.Model.Point.superlevel
SupplyCondition def: Exact source criterion, expressed in freshest-first stack order. · OAI.FKCRT.Inventory.SupplyCondition
supplyInteger def: A common supply denominator, before the rational-representation argument. · OAI.Problem337.supplyInteger
supplyIntegerTriples def: All positive ordered triples of fixed sum, without primality restrictions. · OAI.Problem337.supplyIntegerTriples
supplyPrimeCount def: Number of primes used by a logarithmic supply construction. · OAI.Problem337.supplyPrimeCount
supplyPrimeTriples def: Ordered prime triples of sum u, with their coordinates bounded by u. · OAI.Problem337.supplyPrimeTriples
Support def: A coefficient-one term of the original convolution tensor. · OAI.MatrixMultiplication.AuxiliarySeparation.Sector.Support
Support abbrev: Retain only the strictly positive entries of a finite nonnegative weight family. · OAI.Problem358.PositiveFiniteCoupling.Support
support def: The directional support value, before multiplication by the scale. · OAI.DiagonalZonotope.support
support def: All coordinates used by any arc anywhere in the tree. · OAI.LeanBlast.SecondKahnKalai.ProbabilityTree.support
support def: The actual support function of a compact body, not a surrogate gauge. · OAI.PettyProjection.support
support def: Only the actual non-dummy recorded labels are used by an event. · OAI.QuantitativeVanDerWaerden.EligiblePatternTests.support
support def: Literal support of an indexed first-family row test. · OAI.QuantitativeVanDerWaerden.OuterWordTests.support
support def: The labels literally recorded by the pattern. · OAI.QuantitativeVanDerWaerden.RecordedSupport.support
support def: A syntax tree mentions only finitely many registers. · OAI.Superstring.BitCode.Block.support
SupportAt def: A norm-bounded real functional attaining the norm at a vector. · OAI.Tingley.SupportAt
supportAtom def: The diagonal atom labelled by a row and a matrix-unit index. · OAI.KadisonKastler.supportAtom
SupportBelow def: The support of P is order-cofinal below that of Q. · OAI.SeparableQuotient.PathCoding.RawPath.SupportBelow
supportColumn def: The column sends distinct support atoms to distinct diagonal matrix-unit labels. · OAI.KadisonKastler.supportColumn
supportCompact def: The actual compact topological support of one partition coefficient. · OAI.ClosedSurfaceR4.PhasePartitions.CompactPhasePartition.supportCompact
supportCoordinates def: Coordinates on the ordered support. · OAI.Thomason.Construction.HornBarycentric.supportCoordinates
SupportDatum structure: The finite algebraic data of the segment-support formula. · OAI.PolycyclicRecognition.SupportDatum
supportDatum def: The finite support data derived from all word bases. · OAI.PolycyclicRecognition.CartanWordModel.supportDatum
supportDatumOfTrace def: Build the segment-support norm data from homogeneous word counts and the trace-zero matrix identity for the action on the nilpotent ideal. · OAI.PolycyclicRecognition.supportDatumOfTrace
Supported def: A support invariant for a read-only classical input sector. · OAI.ExactQuantumFactoring.Supported
Supported def: A permutation is supported in a set when it fixes its complement pointwise. · OAI.FiniteAlgebraicEnvelopes.DisplacementCommutator.Supported
Supported def: Predicate propagated along a hyperedge path. It explicitly retains the site-open condition, including in the zero-edge case. · OAI.HyperedgeComparison.Site.Supported
Supported def: Vanishing of vertex components below p, for arbitrary modules. · OAI.LittleFinitistic.DirectedBound.DirectedSystem.Supported
Supported def: Inactivity entails an empty core, not merely a zero observed value. · OAI.UniformKServer.SyntheticCore.Supported
supported def: Supported Schwartz vectors, regarded as an actual subspace of L2. · OAI.ElasticityPhysicalDual.supported
supported def: Supported test vectors form the actual L2 domain for the adjoint. · OAI.ElasticityPlanar.supported
supported def: Restrict to actual used darts and their incident vertices. Every original length, comparison and counting hypothesis is transported verbatim. · OAI.TorsionFreeZeroDivisors.AmbientPattern.Data.supported
supportedAssignmentEquiv def: A choice for each large cell extends uniquely by zero on small cells, exactly as used by the manuscript's supported residual completion sum. · OAI.ContingencyTables.FirstPaperProfiles.supportedAssignmentEquiv
supportedCornerColumns def: The same finite range decomposition in the right-supported column. · OAI.KadisonSimilarity.TracialCornerGeometry.supportedCornerColumns
supportedCounitIso def: The canonical coherent counit, inverse to ordinary restriction's unit on a supported sheaf. · OAI.ActualCartier.supportedCounitIso
supportedEquiv def: Actual degreewise B-bimodule comparison throughout the supported range. · OAI.LittleFinitistic.OrdinarySimulation.FullTensorTerm.supportedEquiv
supportedExtensionXIso def: Actual component isomorphisms for extension after restriction of a complex which is already zero outside the index embedding. · OAI.DerivedBar.supportedExtensionXIso
supportedFreeChartOfPhase def: A good phase gives all coordinate data for a supported free chart. · OAI.ClosedSurfaceR4.RealModes.supportedFreeChartOfPhase
supportedHeatEnvelopeConstant def: The supported upper-decrement constant has no dependence on the total Bochner mass outside the working ball. · OAI.RegularPointCoordinates.supportedHeatEnvelopeConstant
supportedIdealEquiv def: In the supported range, the extended ideal is the source's actual right projective. · OAI.LittleFinitistic.OrdinarySimulation.ChainAlgebra.supportedIdealEquiv
supportedIdealEquiv def: Supported extended ideals are the same right ideals as the finite vertex terms. · OAI.LittleFinitistic.OrdinarySimulation.FullTensorTerm.supportedIdealEquiv
supportedInnerIso def: The same literal inner coordinates, with the proved degree-zero input made explicit. · OAI.DerivedBar.ParityBridge.supportedInnerIso
supportedLift def: A supported simplex whose carrier lies in A lifts to an actual point of |A|. · OAI.Thomason.Construction.RealizationPrism.supportedLift
SupportedOn def: A coordinate vector supported on a specified set. · OAI.ForestSpace.SupportedOn
SupportedOnBlock def: The vertex permutation is supported on the block. · OAI.Problem313.SupportedOnBlock
supportedOrbitCoefficient def: Extend the explicit orbit formula by zero outside the residue class of K. · OAI.Nagata.W09.supportedOrbitCoefficient
SupportedPair structure: A pair of supported one-sided inverse elements, with the prescribed commutation with the nonunital coefficient image. · OAI.FiniteAlgebraicEnvelopes.Steinberg.SupportedPair
supportedPairs def: Canonical orientation within each edge; the two edges remain ordered. · OAI.Paper170.supportedPairs
supportedPhaseCutoff def: The supported phase-chart cutoff with its inverse frequency factor. · OAI.ClosedSurfaceR4.JetPolynomial.Perturbation.supportedPhaseCutoff
supportedProbabilityRestriction def: A probability carried by a measurable set, viewed on that subtype. · OAI.PolycyclicRecognition.supportedProbabilityRestriction
supportedSequence def: The same sequence, identified with its supported quotient. · OAI.CartierImageFiltration.supportedSequence
supportedSmoothing def: The supported smoothing used for bounded-mode regularity. · OAI.TriangularBilliards.supportedSmoothing
supportedWeight def: A partition coefficient bundled on its actual compact support. · OAI.ClosedSurfaceR4.PhasePartitions.CompactPhasePartition.supportedWeight
supportedWeights def: The original law restricted to keys with nonzero retained demand. · OAI.UniformSparsestCut.FiniteWeights.supportedWeights
supportEmbedding def: Canonical coordinate injection for a finite support, built from its finite enumeration. · OAI.Problem326.Affine.supportEmbedding
supportExceptionalUnion def: One exceptional union indexed by all positive degrees and all nonnegative multiplicity vectors, retaining exactly the proper support loci. · OAI.Nagata.W13.supportExceptionalUnion
supportFace def: The closed unit ball cut by the level-one hyperplane of one functional. · OAI.Tingley.supportFace
supportField def: Ordered support, with actual vertical and switch proofs. · OAI.PlanarL1.Columns.Model.Point.supportField
supportForm def: A linear piece in the positive-part expression for segment support. · OAI.PolycyclicRecognition.SupportDatum.supportForm
supportForms def: The explicitly finite set of all linear pieces. · OAI.PolycyclicRecognition.SupportDatum.supportForms
supportHom def: The simplex on the ordered support of a point of the geometric horn. · OAI.Thomason.Construction.HornBarycentric.supportHom
supportIn def: Support data in the fixed Euclidean coordinates on height. · OAI.PolycyclicRecognition.CartanWordModel.supportIn
supportIso def: Omitting zero summands is an isomorphism of ordinary direct sums. · OAI.DerivedBar.RecursiveTotal.supportIso
supportIso def: Restriction of a graph isomorphism to its incident vertices. · OAI.LeanBlast.SecondKahnKalai.supportIso
supportJacobian def: Linearization of the actual support equations `F(y)=0`, `DF(y)=lam*ell`. · OAI.AffineBernstein.supportJacobian
supportLegDirection def: The actual Cartan direction of a height displacement. · OAI.PolycyclicRecognition.supportLegDirection
supportLocus def: The actual homogeneous-polynomial incidence condition at a configuration. · OAI.Nagata.W13.supportLocus
supportMap def: The increasing enumeration of the nonzero coordinates. · OAI.Thomason.Construction.HornBarycentric.supportMap
supportNewtonTensor def: Global polynomial tensor whose tangential restriction is the angular cofactor. · OAI.AffineBernstein.supportNewtonTensor
supportOrdinal def: Restriction of an ordinal map to the ordered supports of a barycentric point and its image. · OAI.Thomason.Construction.HornBarycentric.supportOrdinal
supportParam def: Raw Gauss parametrization, before restricting the angular variable to a chart. · OAI.AffineBernstein.supportParam
supportProjection def: The projection onto a set of diagonal row-label atoms. · OAI.KadisonKastler.supportProjection
supportProjection def: Projection to the support, without invoking separability or nonzero spaces. · OAI.NuclearUltrapower.supportProjection
supportRepresentation def: A possibly degenerate representation becomes unital on its actual support. · OAI.NuclearUltrapower.supportRepresentation
Supports def: A globally supporting selection; finite affine maximizers satisfy this directly. · OAI.Problem358.SupportingMapInterpolation.Supports
supportSimple def: The simple support graph forgets only parallel-edge identities. · OAI.MatchingEntropy.LooplessGraph.supportSimple
supportSymmetryGroup def: The linear labels form a subgroup, with the concrete finite support isometry condition as membership. · OAI.PolycyclicRecognition.supportSymmetryGroup
supportTimeBound def: The exact integer lower bound, using log base two. · OAI.CubeShuffle.supportTimeBound
supportToFace def: Support inclusion factors through every zero-coordinate face. · OAI.Thomason.Construction.HornBarycentric.supportToFace
supportValue def: The literal support value, not a freely chosen support coordinate. · OAI.AffineBernstein.supportValue
supportValue def: The literal running-minimum projection of point distance, after cutting at the chosen representative of the root. · OAI.PlanarL1.Columns.Model.Point.supportValue
supportVisitPath def: The specified outward-switch-return path, with no choice of curves. · OAI.PolycyclicRecognition.supportVisitPath
supportWeight def: The pointwise support weight of a trace. It will be identified with the weight of the literal zero ideal, rather than an arbitrary ideal label. · OAI.TraceConeClassification.ExtendedTrace.supportWeight
suppress def: Suppress a two-valent vertex, retaining its two darts as isolated markers. · OAI.KervaireRibbon.suppress
suppressed def: Darts of the actual suppressed multigraph: all outgoing darts at marked vertices. The return map pairs them by the unique maximal degree-two chain. · OAI.TorsionFreeZeroDivisors.DegreeTwoSuppression.suppressed
suppressed def: All suppression data is constructed from the literal paths, with no bounds on a suppressed skeleton included among the raw data. · OAI.TorsionFreeZeroDivisors.RawSourcePattern.Data.suppressed
suppT def: auxiliary and all-purpose witness. · OAI.ExactDerandomization.At.suppT
surf def: Cross term at (p+i) from updated left half of width p. · OAI.PowerSaving.Slant.surf
surfaceFace def: The surface face in a tie hyperplane, translated from an arbitrary point of that plane. This is an intrinsic measure domain, not ambient volume. · OAI.Problem358.PositiveFacet.surfaceFace
surjection def: Internal set surjection f : d ↠ r. · OAI.PartitionConsistency.SetLanguage.surjection
SurjectiveFamily structure: Ordinary continuous, surjective linear families. No kernel bundle is part of the input: the following construction supplies it. · OAI.Kaplansky.SmoothKernel.SurjectiveFamily
surplus def: Euler surplus without counting isolated vertices. It equals the sum of branch excesses minus the number of leaves. · OAI.TorsionFreeZeroDivisors.CoreStripping.surplus
survival def: The fraction of the child capacity occupied by particles already alive. · OAI.IntrinsicMarks.survival
survival def: The fraction of the child capacity occupied by particles already alive. · OAI.SignedDisk.IntrinsicMarks.survival
survive def: Survival through N independent retirement trials of rate q. · OAI.KServer.RosterLifetimes.survive
Survives def: All three corner agreements and all six admissible incidences. · OAI.CliqueFreeIndependence.WeightedGraph.Splitting.Survives
Survives structure: Retain every surviving step, without changing any surviving frozen face. Unlike the stronger Retains, this permits cancellation of entire frozen faces. Protected outside faces will be retained separately at that boundary. · OAI.HyperbolicObstruction.PictureSwitch.WalkPicture.Survives
surviving def: The surviving through relation U relative to e. · OAI.TwoWayComplementation.Diagram.surviving
survivingConstituentEquiv def: Surviving constituent slots are exactly a surviving atom and one of its original prime positions. · OAI.Ostmann.survivingConstituentEquiv
survivingSwitchProbability def: Iterate the actual independent pairs of increment samples, killing a path at its first exit from the prescribed next set. · OAI.PolycyclicRecognition.survivingSwitchProbability
survivingWeights def: The image contains distinct weights rather than repeated subset occurrences. · OAI.LowSpaceSubsetSum.survivingWeights
survivors def: The source comparison independent set after sacrificing the fair mask. · OAI.MatroidProphet.CandidateSpecification.survivors
survivorTrace def: A binary trace suffices on surviving vertices: at a fixed predecessor, copying and retaining are mutually exclusive. · OAI.Ostmann.survivorTrace
Swap inductive: A single exchange of independent neighboring occurrences. · OAI.EilenbergGanea.TraceWords.Swap
swap def: Exchange two quoted word fields, without a unit-cost tuple operation. · OAI.ExactQuantumFactoring.BitStackProgram.Procedure.swap
swap def: Swap the order of two successive face expansions. · OAI.SharpLiebThirring.CubeFlags.Flag.swap
swap def: Interchange the ordinary polynomial parameter and the Laurent parameter. · OAI.StableCoordinate.ActionWeightDerivative.swap
swap23Character def: The reflected character exchanges the second and third tensor legs. · OAI.MatrixMultiplication.AuxiliarySeparation.Character.swap23Character
swapAlgHom def: The parameter interchange respects an arbitrary coefficient algebra. · OAI.StableCoordinate.ActionWeightDerivative.swapAlgHom
swapAt def: SWAP is three named CNOT gates, not an extra primitive. · OAI.ExactQuantumFactoring.swapAt
swapAutomorphism def: The elementary Hurwitz automorphism fixes every other free generator. · OAI.IntegralCharacterVarieties.SurfaceSurgery.Hurwitz.swapAutomorphism
swapCoarsePair def: Swapping the two maps preserves all six bounds. · OAI.PolycyclicRecognition.swapCoarsePair
swapCylinderEquiv def: Braiding identifies the two orders of the simplicial cylinder. · OAI.Thomason.Construction.swapCylinderEquiv
swapDeletion def: Swap the order of two vertex deletions, accounting for the index shift. · OAI.Thomason.Construction.SimplexComplex.swapDeletion
swapEdgeEquiv def: The actual undirected edge swap is a bijection, not just a sign reversal. · OAI.GammaFPP.swapEdgeEquiv
swapHom def: The genuine diagram automorphism of an arbitrary rank-two Coxeter system, including the infinite one. · OAI.KLInvariance.Dihedral.swapHom
swapIncidence def: Switch the two incidence-graph parts without altering any incidence. · OAI.Release075.MarkedLineData.swapIncidence
swapInflationEquiv def: Swapping product factors identifies the two actual inflation constructions. · OAI.LittleFinitistic.ProductModules.swapInflationEquiv
swapPairs def: Exchange the current and future pairs, leaving all old coordinates fixed. This makes the common summation domain explicit. · OAI.ContingencyTables.IntegerWeightedTransport.swapPairs
swapReflect def: The flip and interchange relating the two physical maps. · OAI.RuledApproximation.swapReflect
swapScalar def: The involutive scalar interchanging the two corners. · OAI.FiniteAlgebraicEnvelopes.Steinberg.CornerPair.swapScalar
swatch def: layer singleton builders · OAI.PowerSaving.Slant.Reel.swatch
switch def: Crosswise re-pairing of the two specified outgoing sides. All original face cycles (hence all original corners and their maps) are unchanged. · OAI.HyperbolicObstruction.PictureSwitch.PictureMap.switch
switch def: Conjugating by the transposition of equally labelled ports. · OAI.TorsionFreeZeroDivisors.ConditionedMatchings.Matching.switch
switch def: Transpose two equally oriented ports. Their reverse ports are not moved. · OAI.TorsionFreeZeroDivisors.PortSubdivision.PortGraph.switch
switchCount def: Switches in a Boolean labeling along a finite collection of oriented edges. · OAI.Problem335.SwitchRestoration.switchCount
SwitchData structure: A finite original cell-weighted pair law and its measurable switch. The carrier is not normalized and no sigma-algebra on the retained subtype is silently substituted for the ambient original pair law. · OAI.SmoothObstruction.SwitchData
switchedSet def: Successive independent fair transpositions, listed in reverse time order. · OAI.CubeShuffle.switchedSet
switchEndpoint def: The last state after the specified sequence of switches. · OAI.PolycyclicRecognition.switchEndpoint
switches def: Internal positions where adjacent bits differ. A word has n + 1 letters. · OAI.Problem335.WordSwitches.switches
switchFun def: Switch the first-coordinate pairs, without rotating. · OAI.CubeShuffle.switchFun
switchHeight def: Physical outward height of a switch in a butterfly. · OAI.CubeShuffle.switchHeight
SwitchIndex def: One index for each independent coin of the recursive butterfly. · OAI.CubeShuffle.SwitchIndex
switchingTreeDataOfAnchor def: Assemble the proved conclusions of the actual finite-stage recursion. · OAI.ReflexiveFixedPoints.switchingTreeDataOfAnchor
switchLeaves def: The actual leaf assignment after every leaf at or to the right of the cutoff has switched. · OAI.ReflexiveFixedPoints.TreeEvaluation.switchLeaves
SwitchParameters abbrev: Two independent samples per switch, in their prescribed order. · OAI.PolycyclicRecognition.SwitchParameters
switchRoot def: One fully evaluated actual root state of the switching experiment. · OAI.ReflexiveFixedPoints.TreeEvaluation.switchRoot
SwitchSamples def: An explicit product of independent fair coins, of a recursively specified length. · OAI.CubeShuffle.SwitchSamples
SwitchSuccessful def: A parameter is successful precisely when every intermediate state belongs to the next retained set. · OAI.PolycyclicRecognition.SwitchSuccessful
switchThresholdProduct def: The product of the successive transition thresholds, in the same recursive order as the independent sampling. · OAI.PolycyclicRecognition.switchThresholdProduct
switchUsers def: The two distinct INPUT labels whose paths traverse a given switch. This records encounters without any independence or random-route assumption. · OAI.CubeShuffle.switchUsers
switchVector def: The normalized difference associated to a single actual leaf switch. · OAI.ReflexiveFixedPoints.TreeEvaluation.switchVector
symAlgMap def: The algebra map induced by a complex-linear map. · OAI.Problem346.symAlgMap
Symbol abbrev: A symbol retains both its leaf tag and its cyclic state. · OAI.TreeEdit.BinaryLower.FiniteTree.Symbol
symExponent def: The exponent vector of a multiset of basis indices. · OAI.MetricEntropyDuality.symExponent
symFiberCard def: Number of ordered tuples representing the given multiset. · OAI.MetricEntropyDuality.symFiberCard
symm def: Reverse a comparison using its specified coarse inverse. · OAI.PolycyclicRecognition.MetricQuasiEquivalence.symm
symm def: Reverse the two-sided gauge without changing any coordinate basis. · OAI.SmoothObstruction.ManifoldCoordinates.BilateralGauge.symm
symMatrixCoordinates def: A symmetric matrix is exactly a function on unordered pairs with repetition. · OAI.SidorenkoCounterexample.symMatrixCoordinates
Symmetric abbrev: A symmetric edge interaction; nonedge values are irrelevant. · OAI.BoundedTreewidthL1.TreeMarkov.Symmetric
Symmetric def: A name is symmetric when its literal set-name stabilizer belongs to the specified subgroup filter. This does not mention its interpretation. · OAI.PartitionConsistency.SetForcingNames.ConditionAction.Name.Symmetric
symmetricAction def: The algebra action is induced by the linear reflection, not postulated. · OAI.KLInvariance.ReflectionFrobenius.symmetricAction
SymmetricAlphabet abbrev: The alphabet of rank n+1, including each generator and its formal inverse. This is an arbitrarily large cofinal set of the literal source alphabets. · OAI.HyperbolicObstruction.IncidenceTemplate.SymmetricAlphabet
symmetricArray def: A symmetric derivative array from its ten matrix entries. · OAI.PositiveEinsteinFour.HighestHessian.symmetricArray
SymmetricForm abbrev: Free literal coefficients on multisets of basis indices. · OAI.MetricEntropyDuality.SymmetricForm
symmetricGain def: The associated symmetric bilinear gain. · OAI.BoltzmannNonuniqueness.symmetricGain
symmetricInverse def: Extend inversion by fixing singular forms. Unlike the total matrix inverse, this is a permutation of the entire symmetric-matrix probability space. · OAI.SidorenkoCounterexample.symmetricInverse
symmetricMultilinearSubmodule def: The submodule of multilinear maps invariant under permutations of their arguments. · OAI.Problem346.symmetricMultilinearSubmodule
SymmetricSystem structure: A set symmetric system with a named presentation of the conditions. Every piece of data is algebraic: no forcing axiom or satisfaction assertion is included in this structure. The equivalence also ensures that the poset is a set in the … · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem
symmetricUnit def: The symmetric elementary off-diagonal coefficient. · OAI.PositiveEinsteinFour.HighestHessian.symmetricUnit
symmetricValues def: The actual symmetric interpretation of ground set-name codes. Membership in this domain is witnessed by a ground HS name, not by a required target axiom. · OAI.PartitionConsistency.SetForcingNames.ConditionAction.symmetricValues
symmetrize def: Averaging over the cyclic group already gives identical coordinate marginals. · OAI.Problem356.Cyclic.symmetrize
symmetrizedProfile def: The Section 5 symmetrized value with mean singleton-leg exponent t. · OAI.MatrixMultiplication.AuxiliarySeparation.Character.symmetrizedProfile
symmetryExpectation def: The trace-orthogonal projection onto the span of a centered symmetry. · OAI.KadisonSimilarity.symmetryExpectation
symmetryUnit def: An involution, regarded as a unit with itself as inverse. · OAI.KadisonSimilarity.symmetryUnit
symMonomialMultilinear def: The product of degree-one generators, with its multilinearity bundled. · OAI.Problem346.symMonomialMultilinear
symMonomialMultilinearRaw def: Multiplication of degree-one generators as a multilinear map. · OAI.Problem346.symMonomialMultilinearRaw
symplectic def: The ambient symplectic bilinear form, with the factor 2 normalization. · OAI.ArnoldCounterexample.Field.symplectic
symplectic def: Exactly the normalization d(sum x dy-y dx). · OAI.ArnoldCounterexample.Forms.symplectic
symPowCast def: Transport a homogeneous element across an equality of degrees. · OAI.Problem346.symPowCast
symPowDualEquiv def: Symmetric powers commute with algebraic duality in finite dimension. · OAI.Problem346.symPowDualEquiv
symPowDualMap def: The canonical pairing from the symmetric power of the dual space. · OAI.Problem346.symPowDualMap
symPowEquiv def: A linear equivalence induces an equivalence on each symmetric power. · OAI.Problem346.symPowEquiv
symPowLift def: A chosen linear descent of a symmetric multilinear map. · OAI.Problem346.symPowLift
symPowLiftLinear def: Descent from symmetric multilinear maps is linear in the map being descended. · OAI.Problem346.symPowLiftLinear
symPowMap def: Symmetric powers act covariantly on linear maps. · OAI.Problem346.symPowMap
symPowMul def: Graded multiplication in the symmetric algebra, as a bilinear map. · OAI.Problem346.symPowMul
symPowMulCast def: Multiplication followed by transport to a specified total degree. · OAI.Problem346.symPowMulCast
symPowOneEquiv def: The degree-one symmetric power is canonically the original vector space. · OAI.Problem346.symPowOneEquiv
symPowOneIntro def: The canonical inclusion of a vector into the degree-one symmetric power. · OAI.Problem346.symPowOneIntro
symPowPairing def: The normalized symmetric-power pairing induced by a bilinear pairing. · OAI.Problem346.symPowPairing
symProjector def: Orthogonal projection onto vectors invariant under ordinary slot permutations, implemented by their exact histogram blocks. · OAI.CoordinateSweeps.OrdinaryTensor.symProjector
syncBox def: A support condition checks the Gate/projection interaction. · OAI.PowerSaving.Cluster.syncBox
synthesis def: Fourier synthesis in the actual trace-character orthonormal basis. · OAI.MaxCutGames.Inverse.KMSAnalytic.synthesis
synthesis def: Synthesis using only a selected finite set of spectral coordinates. · OAI.Problem313.FiniteParseval.synthesis
synthesis def: Fourier synthesis in the actual trace-character orthonormal basis. · OAI.UniqueGamesTheorem.Inverse.KMSAnalytic.synthesis
System structure: A covariant diagram system with injective restriction and the actual support-intersection property. This is purely diagrammatic infrastructure; no counting, choice or partition principle is part of this definition. · OAI.PartitionConsistency.LocalDiagramSystem.System
system def: Side zero is the sum block I−R; side one is the difference block I+R. · OAI.AtomicTriangular.FiniteCertificate.AtomicMatrix.system
system def: A proved Coxeter system on the actual maximal plane subgroup. · OAI.KLInvariance.TitsSpace.PlaneBasis.system
system def: Actual affine matrix restrictions instantiate the generic deletion interface. · OAI.MaxCutGames.Fourier.MatrixRestrictions.system
system def: The literal Cohen/diagram symmetric system, with exactly the small-support subgroup filter. Neither ZF nor any target axiom is a structure field. · OAI.PartitionConsistency.ProductForcing.system
system def: The actual coherently nested finite closures on the well-order. · OAI.SeparableQuotient.CoherentClosures.system
system def: Appending the actual short closures changes neither original comparison nor its underlying graph lifts. Every newly added position is unpaired. · OAI.TorsionFreeZeroDivisors.SampleGraph.ClosedWords.system
system def: Actual affine matrix restrictions instantiate the generic deletion interface. · OAI.UniqueGamesTheorem.Fourier.MatrixRestrictions.system