Glossary: R

Definitions in this project whose names start with R. 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

R structure

Record of numeric registers (time separate from rest). · OAI.ExactDerandomization.Base.R

R def

Universal ordered parameter choices. The two free widths are chosen strictly smaller than every displayed upper bound, not adapted to a graph. · OAI.PlanarL1.Parameters.R

R def

The component entry is generic-normal Read; other generic profiles and unsorted literal expected sorts use exactly typed top. · OAI.PureTypeSystem.ObservationValues.R

r def

Actual rational-coordinate vector, including its constant first entry. · OAI.EuclideanRamsey.NineRational.r

r abbrev

A fixed moment exponent sufficient for the density recurrence. · OAI.Problem337.RawDensityParameters.r

racks def

Push a fixed (top-first) string onto each selected stack. · OAI.Fifths.SM.racks

rademacher def

Each mark coordinate is a genuine independent fair Rademacher variable. · OAI.DilutedSpinGlass.MarkSelector.rademacher

Radial def

Radiality of a real function on the Euclidean plane. · OAI.SharpPlanar.Radial

radial def

The radial unit vector (at nonzero points) and the tangential polar direction. · OAI.Bernoulli.Axisymmetric.radial

RadialBusemann def

Radial comparison for any two constant-speed geodesics with common start. · OAI.HyperbolicObstruction.RadialBusemann

RadialCapBound def

Both directional cap estimates, using the original opposite probability. · OAI.Falconer.RadialCapBound

radialCoefficient def

The nonzero radial coefficient in the finite Fischer decomposition. · OAI.PettyProjection.Spherical.radialCoefficient

RadialCompactnessSequence structure

All analytic preconditions of the published local compactness theorem. No projected Euler equation or affine approximation is part of this data. · OAI.RegularPointCoordinates.RadialCompactnessSequence

radialCompactTest def

The standard radial plateau, viewed as a compact continuous test. · OAI.RegularPointCoordinates.radialCompactTest

radialContact def

The actual point of contact, retaining its unit-sphere subtype proof. · OAI.Tingley.radialContact

radialCoordinate def

Radial coordinate recovered directly from the diagonal entries. · OAI.PositiveEinsteinFour.Bianchi.radialCoordinate

radialCrossing def

The ordered crossing invariant from equation crossing-invariant. Here a, b, and c are the values of the metric on (v,v), (w,w), and (v,w) respectively, and K is the Gauss curvature. · OAI.ClosedSurfaceR4.RadialMetricAlgebra.radialCrossing

radialCut def

A continuous cutoff equal to one on the inner half ball. · OAI.SharpLiebThirring.CubeFlags.radialCut

radialCutoff def

A conjugation-invariant smooth cutoff, zero near coordinate hyperplanes and one near the unit quaternion locus. · OAI.EilenbergGanea.radialCutoff

radialCutoff def

A radial auxiliary cutoff; its lack of smoothness is harmless because only Lipschitz regularity is needed for the flow and barriers. · OAI.SharpLiebThirring.ODEProof.radialCutoff

radialCutoff def

A comparison density, equal to one on the competitor disk. It adds no spacing or local-count hypothesis to the density-only LP argument. · OAI.TriangularUniversal.radialCutoff

radialDerivative def

The radial derivative formula on 0<r<1. Its removable value at r=0 is not asserted by this definition. · OAI.SymmetricMahler.radialDerivative

radialDirection def

Radial direction, assigned zero on the diagonal. The application uses separated supports, so this arbitrary diagonal value has no effect. · OAI.Falconer.radialDirection

radialDirichletKernel def

This is the regular form of ∫ t in r..R, t⁻¹¹ * ∫ s in 0..t, s¹¹*f s. · OAI.DefocusingNLS.radialDirichletKernel

radialDisk def

Radial straight-line deformation, in the full disk. · OAI.GroupRingTopology.Attachment.radialDisk

radialExtension def

The actual radial extension, with a separate zero branch. · OAI.Tingley.radialExtension

radialFitError def

The dimensionless least-squares error of the actual radial derivative. · OAI.RegularPointCoordinates.radialFitError

RadialHamiltonianChart structure

Data produced by the local Hamiltonian theorem, retaining the actual coordinate map and the common interval of agreement with radial motion. · OAI.PositiveEinsteinFour.Geometry.RadialHamiltonianChart

radialHeat def

Radial transition density of three-dimensional Brownian motion. · OAI.FKCRT.SimpleWalk.radialHeat

radialHomotopy def

The punctured attachment deformation-retracts to the original space. · OAI.GroupRingTopology.Attachment.radialHomotopy

RadialIndex abbrev

The very same finite connected radial cells, at an arbitrary prescribed width; the secondary construction uses width g, not the main width D. · OAI.PlanarL1.Columns.Model.BandTrees.RadialIndex

RadialInnerData structure

Only numerical parameter conditions; no analytic conclusion is part of the data. · OAI.DefocusingNLS.RadialInnerData

RadialInterior def

All column representatives, not only the chosen one, pass the radial interior test. The quantification is over zero-distance geometric points. · OAI.PlanarL1.Columns.Model.BandTrees.RadialInterior

radialIsometryEquiv def

The metric equivalence is built only after isometry and surjectivity are proved. · OAI.Tingley.radialIsometryEquiv

radialKernel def

The radial projection law, with the pin as its kernel parameter. · OAI.Falconer.radialKernel

radialKernelDensity def

A single jointly measurable choice of the radial angular densities. · OAI.Falconer.radialKernelDensity

radialLinearIsometryEquiv def

Mazur–Ulam applied to the derived whole-space isometry fixing zero. · OAI.Tingley.radialLinearIsometryEquiv

radialLogCoefficient def

The scalar coefficient multiplying the path vector in the derivative. · OAI.LipschitzHilbert.radialLogCoefficient

radialLogSlope def

The radial size of the logarithmic-energy gradient. · OAI.LipschitzHilbert.radialLogSlope

radialMap def

The actual M(r)=F(r) on the real diameter, taking its real part for a real-valued function on all of ℝ. The estimates use only 0≤r<1. · OAI.SymmetricMahler.radialMap

radialMassLaplacian def

The mass density times the scalar weighted Laplace operator, in its nonsingular divergence form. · OAI.DefocusingNLS.radialMassLaplacian

RadialMatrixEstimates def

The estimates already proved for the actual Poisson radial matrices, packaged only to retain their common choices of rows. · OAI.RegularPointCoordinates.RadialMatrixEstimates

radialNormalize def

Radial normalization is kept as an actual map of Euclidean four-space. · OAI.ClosedSurfaceR4.SphericalJets.radialNormalize

radialPairing def

A comparison scalar product at two positive positions on radial segments. Its supremum as the positions decrease is the tangent scalar product. · OAI.HyperbolicObstruction.CATZero.radialPairing

radialPairingLp def

The actual radial pairing, as an element of the local L2 space. · OAI.RegularPointCoordinates.radialPairingLp

radialPowerRep def

The continuous representative of the Sobolev radial test. · OAI.RegularPointCoordinates.radialPowerRep

radialPowerTest def

The actual radial Sobolev test obtained by clipping an inward distance. · OAI.RegularPointCoordinates.radialPowerTest

radialPowerWeight def

The global derivative used before clipping at the inner radius. · OAI.RegularPointCoordinates.radialPowerWeight

radialPrimitiveDensity def

The radial form of the vertical primitive, before proving its equality with the integral in the image domain. · OAI.SymmetricMahler.radialPrimitiveDensity

radialRadius def

The radius is chosen from the proved existence theorem. · OAI.Tingley.radialRadius

radialShrink def

Radial truncation by the distance δ; its value at the zero vector is zero. · OAI.LipschitzHilbert.radialShrink

radialSphere def

Radial projection of a nonzero disk point onto its sphere. · OAI.GroupRingTopology.Attachment.radialSphere

radialSupport def

The support-plane height ν(o-X) in original graph coordinates. · OAI.AffineBernstein.radialSupport

radialTemplate def

Read only the indexed radial port, not its numerical letter assignment, back into the subdivided macro template. · OAI.HyperbolicObstruction.ConeSkeleton.radialTemplate

radialTransposedTest def

The transposed radial derivative includes the exact Jacobian. · OAI.DefocusingNLS.radialTransposedTest

radialUnit def

Unit radial vector in the background Cartesian metric. · OAI.HarmonicCounterexample.Cartesian.radialUnit

radialValue def

Actual distance field of an old chart. Outside its component we choose zero only to make a whole-band scalar function; radial sublevels retain the source's +infinity convention and never include those columns. · OAI.PlanarL1.Columns.Model.Point.radialValue

radialWeightedSchwartzTest def

Absorb the twelve-dimensional radial Jacobian into a Schwartz test. · OAI.DefocusingNLS.radialWeightedSchwartzTest

radianFourierKernel def

The radian-normalized Fourier transform of a cutoff is still Schwartz. · OAI.DefocusingNLS.radianFourierKernel

radical_comapFun def

The underlying function of Proj ℬ ⟶ Proj 𝒜 on the level of points. · OAI.AlgebraicGeometry.ProjectiveSpectrum.radical_comapFun

radical_comapStructureSheaf def

The underlying ring hom of Proj ℬ ⟶ Proj 𝒜 on the level of structure sheaves. · OAI.AlgebraicGeometry.Proj.radical_comapStructureSheaf

radical_comapStructureSheafFun def

The underlying function of Proj ℬ ⟶ Proj 𝒜 on the level of structure sheaves. · OAI.AlgebraicGeometry.Proj.radical_comapStructureSheafFun

radical_sheafedSpaceMap def

The underlying map of Proj ℬ ⟶ Proj 𝒜 on the level of sheafed spaces. · OAI.AlgebraicGeometry.Proj.radical_sheafedSpaceMap

radicalBasis def

Basis of the source field by its radical monomials. · OAI.KummerLines.radicalBasis

radicalCharacter def

The action character of a radical monomial. · OAI.KummerLines.radicalCharacter

radicalComap def

The underlying continuous function of Proj ℬ ⟶ Proj 𝒜 on the level of points. · OAI.AlgebraicGeometry.ProjectiveSpectrum.radicalComap

radicalMap def

Functoriality of Proj. · OAI.AlgebraicGeometry.Proj.radicalMap

radicalQuotientForm def

The bilinear form induced on the quotient by its radical. · OAI.SidorenkoCounterexample.radicalQuotientForm

radius def

The coarse consecutive range, written in increasing order for moment summation. · OAI.KServer.Pilot.radius

radius def

The number of shared trajectory tapes needed by a single instruction. · OAI.MaxCutGames.Foundations.PCP.PoweringFieldPlan.Instruction.radius

radius def

The physical radius is the actual radius of the chosen replacement. · OAI.RegularPointCoordinates.ActualProjectionConfiguration.radius

radius def

The number of shared trajectory tapes needed by a single instruction. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringFieldPlan.Instruction.radius

RadiusDensity structure

An ordinary nonnegative probability density supported on a radius interval. · OAI.DiskKernels.RadiusDensity

RadiusDensity structure

An ordinary nonnegative probability density supported on a radius interval. · OAI.SignedDisk.DiskKernels.RadiusDensity

radiusMatrix def

Radius matrix in projected constant tangent frames. At a unit `e` and an orthonormal tangent frame this is exactly `∇²_S h + h Id`; its first derivatives are the round covariant derivatives in a normal frame. · OAI.AffineBernstein.radiusMatrix

radiusMultiply def

Radius multiplication between actual homogeneous parts. · OAI.HarmonicCounterexample.Angular.radiusMultiply

radiusPolynomial def

Radius squared as an ambient polynomial. · OAI.PettyProjection.Spherical.radiusPolynomial

radiusScaledColumns def

Coordinate rescaling uses the same Sobolev representatives. · OAI.RegularPointCoordinates.radiusScaledColumns

radiusSphereArea def

Euclidean sphere area at radius r, transported from the volume-induced unit sphere measure with its positive tangential Jacobian. · OAI.Mahler.radiusSphereArea

radiusSphereFlux def

Flux on the radius-r sphere itself. For nonpositive radii we use zero; this branch is irrelevant to the right-sided small-sphere limit. · OAI.Mahler.radiusSphereFlux

radiusSq def

The squared Euclidean radius, in real coordinates with standard Lebesgue measure. · OAI.MahlerStokes.radiusSq

radiusSq def

The square of the distance to the origin, in Cartesian coordinates. · OAI.StationaryNS.radiusSq

radix def

Work within radix, one chosen polynomial in n. cs: space coefficient, k,e time coefficient/exponent. · OAI.ExactDerandomization.Base.R.radix

radix def

Most-significant-word first, with leading zero words retained in the fixed-length domain. Mathlib's ofDigits uses the opposite list order. · OAI.MaxCutGames.Reduction.CanonicalAddress.radix

radix def

Most-significant-word first, with leading zero words retained in the fixed-length domain. Mathlib's ofDigits uses the opposite list order. · OAI.UniqueGamesTheorem.Reduction.CanonicalAddress.radix

radonHeatProbability def

Heat evaluation as a probability measure, constructed from the actual semigroup on compact continuous tests. · OAI.RegularPointCoordinates.NoncollapsedRCD.radonHeatProbability

radonNikodymIsometry def

The actual scalar L² coordinate inclusion over a dominating measure. · OAI.KadisonSimilarity.radonNikodymIsometry

raise def

Mesh 2^(M-l) in multiples. · OAI.FourierLLogL.Dyadic.raise

raising def

Off-diagonal raising generator, in the source order +, vacant, -. · OAI.CriticalHoneycomb.Transfer.FluxWard.raising

ramanujanSum def

The standard Ramanujan sum at any nonzero modulus, with its usual unit-residue restriction. · OAI.Problem337.RamanujanProduct.ramanujanSum

ramifiedIdealPrimes def

All prime ideals dividing the rational integer three. · OAI.CubicFirstMoment.ramifiedIdealPrimes

ramp def

A globally smooth, fixed ramp, constant on neighborhoods of both endpoints. · OAI.HarmonicCounterexample.Pulses.ramp

ramp def

The heavy profile is zero up to radius 3 and one from radius 4. · OAI.KServer.HeavyCenters.ramp

ramp def

The heavy profile is zero up to radius 9 and one from radius 10. · OAI.KServer.HeavyEdit.ramp

ramp def

Exact increasing clipped ramp on [1/8,1/4]. · OAI.KServer.ShortRoster.ramp

ramp def

The clipped ramp on [9,10]. · OAI.UniformKServer.HeavyEditRamp.ramp

ramp def

The clipped ramp on [3,4]. · OAI.UniformKServer.HeavyPilot.ramp

rampPiece def

Rational polynomial on a quarter interval; no real oracle is used in its data. · OAI.ExactQuantumFactoring.OrderTrial.rampPiece

rampSq def

A positive-part square. This supplies a globally C¹ rational spline primitive. · OAI.ExactQuantumFactoring.OrderTrial.rampSq

rampSum def

A ramp can change sign only once along an arithmetic progression. · OAI.ExactQuantumFactoring.OrderTrial.rampSum

RamseyProperty def

The red-cycle/blue-clique property at order N. · OAI.CycleClique.Construction.RamseyProperty

RamseyProperty def

Every graph has an s-clique or an independent set of size t. · OAI.SharpLogRamsey.RamseyProperty

randomCoefficientAverage def

Expectation over external disorder and independent Gaussian coordinates. · OAI.InvariantIsing.randomCoefficientAverage

randomizedAcceptance def

A finite randomized strategy, with the common coin including both private coins. · OAI.MaxCutGames.Reduction.IncidenceProbability.randomizedAcceptance

randomizedAcceptance def

A finite randomized strategy, with the common coin including both private coins. · OAI.UniqueGamesTheorem.Reduction.IncidenceProbability.randomizedAcceptance

randomizedAgreement def

Only randomized entries count; an unchanged entry contributes zero, even when its old label already matches the target. · OAI.MaxCutGames.Inverse.RowErasure.SliceFamily.randomizedAgreement

randomizedAgreement def

Only randomized entries count; an unchanged entry contributes zero, even when its old label already matches the target. · OAI.UniqueGamesTheorem.Inverse.RowErasure.SliceFamily.randomizedAgreement

randomizedRecoveryFamily def

The finite unitary family before the final coordinate-row choice. · OAI.KadisonKastler.randomizedRecoveryFamily

randomizedRecoveryLift def

The three independent choices used in scalar recovery, as unitary sequences. · OAI.KadisonKastler.randomizedRecoveryLift

RandomMachine structure

A uniform randomized Post-Turing machine: a single finite transition table. · OAI.MatchingFPRAS.RandomMachine

randomWalkConvolution def

The root law after the first increment. · OAI.KadisonKastler.randomWalkConvolution

randomWalkScalarBoundaryMean def

The original action of the countable group on its random-walk boundary has the exact scalar mean needed by the stability argument. · OAI.KadisonKastler.randomWalkScalarBoundaryMean

range def

Images of actual conservative maps are conservative subobjects. · OAI.PartitionConsistency.ConservativeMonoids.Subobject.range

range abbrev

Stage images in the actual colimit, used only to control finite families and to apply the proved realization exhaustion theorem. · OAI.Thomason.Construction.InfiniteExtension.range

rangeAttachment def

Identify an original executed cell with the literal ranges inside its terminal monoid. Surjectivity here is proved from the presentation. · OAI.PartitionConsistency.ConservativeMonoids.Cell.rangeAttachment

rangeCalls def

The actual number of permutation range calls for all four families. · OAI.LowSpaceSubsetSum.SeparatorExpectation.RoundedParameters.rangeCalls

rangeEligible def

Rejection predicate for the clog₂ D fresh low bits. · OAI.LowSpaceSubsetSum.rangeEligible

rangeEmbedding def

Embed the finite range, rather than the entire ambient label type. · OAI.MetricEntropyDuality.FixedLabels.rangeEmbedding

rangeEquiv def

The image of the actual column projector is the concrete two-sided path corner. · OAI.LittleFinitistic.CornerProjection.rangeEquiv

rangeEquiv def

A conservative embedding identifies its domain with its actual range. · OAI.PartitionConsistency.ConservativeMonoids.Hom.rangeEquiv

rangeEquiv def

Collapse is a membership isomorphism onto its transitive range. · OAI.PartitionConsistency.Mostowski.rangeEquiv

rangeEquivOfKerEq def

The range identification induced by a common kernel. · OAI.MaxCutGames.Inverse.KMSKernelOrbits.rangeEquivOfKerEq

rangeEquivOfKerEq def

The range identification induced by a common kernel. · OAI.UniqueGamesTheorem.Inverse.KMSKernelOrbits.rangeEquivOfKerEq

RangeFiber def

The actual fiber of injective maps over a specified range subspace. · OAI.MaxCutGames.Inverse.KMSBasisComparison.RangeFiber

RangeFiber def

The actual fiber of injective maps over a specified range subspace. · OAI.UniqueGamesTheorem.Inverse.KMSBasisComparison.RangeFiber

rangeFiberEquiv def

A range fiber is the set of linear equivalences onto that range. · OAI.MaxCutGames.Inverse.KMSBasisComparison.rangeFiberEquiv

rangeFiberEquiv def

A range fiber is the set of linear equivalences onto that range. · OAI.UniqueGamesTheorem.Inverse.KMSBasisComparison.rangeFiberEquiv

rangeFiberEquivAut def

After choosing one basis of a subspace, its ordered-basis fiber is exactly the domain's automorphism group. · OAI.MaxCutGames.Inverse.KMSBasisComparison.rangeFiberEquivAut

rangeFiberEquivAut def

After choosing one basis of a subspace, its ordered-basis fiber is exactly the domain's automorphism group. · OAI.UniqueGamesTheorem.Inverse.KMSBasisComparison.rangeFiberEquivAut

rangeIntoClosure def

The range map, with its codomain restricted to the closed range. · OAI.KadisonSimilarity.rangeIntoClosure

rangeLink def

Link two ranges whose displayed partial isometries have the same source. · OAI.Generator.Tracial.PartialIsometry.rangeLink

rangeProjection def

The projection onto the closed range of x, expressed via ker(x*). · OAI.Generator.Tracial.rangeProjection

rangeSeed def

Tabulate the supplied finite random-word tuple once. The extra reserve counts both tabulation storage and its construction stack. · OAI.LowSpaceSubsetSum.SamplerVectorCells.rangeSeed

rangeSupport def

The orthogonal projection onto the closed operator range. · OAI.KadisonSimilarity.rangeSupport

rangeSupport def

The actual orthogonal projection onto the closed range of an operator. · OAI.Nonhyperlinear.Diffuseness.rangeSupport

rangeTransport def

Changing coordinates induces a genuine linear equivalence on ranges. · OAI.MaxCutGames.Appendix.CoordinateTransport.rangeTransport

rangeTransport def

Changing coordinates induces a genuine linear equivalence on ranges. · OAI.UniqueGamesTheorem.Appendix.CoordinateTransport.rangeTransport

rangeVertex def

The actual Grassmann vertex represented by an injective map. · OAI.MaxCutGames.Inverse.KMSBasisComparison.rangeVertex

rangeVertex def

The actual Grassmann vertex represented by an injective map. · OAI.UniqueGamesTheorem.Inverse.KMSBasisComparison.rangeVertex

rank def

zero-based descending rank, padded by zeros · OAI.ExactDerandomization.Analysis.rank

rank def

A leaf-elimination certificate, rooted at p. Ranks strictly rise towards the unique common line and the root; the exceptional rank subdivides its edge. This is a formula on the actual seven Fano vectors, not a graph hypothesis. · OAI.HyperbolicObstruction.MacroGraph.rank

rank def

The source's numerical rank array, before deletion of zero pieces. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.ComplementaryPatch.rank

rank def

The true post-filtering, post-drift ranks and the filtered old ranks on an arbitrary finite experiment with increasing observations. · OAI.KServer.PosteriorRanks.rank

rank def

Exact rank descends to mutual-restriction classes. · OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.rank

rank def

The literal names used for the ordinal ranks and order type. · OAI.PartitionConsistency.ProductForcing.CohenSelection.rank

rank def

Preorder rank of a path. On paths of length at most d, this is the usual zero-based root-first rank in the complete tree. · OAI.Problem310.TreePreorderRank.rank

RankArray def

Finite numerical choices for the local replacement. In a global family every vertex uses an independent occurrence of this bounded array. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.ComplementaryPatch.RankArray

RankBad def

Concrete bad event: an included collection of distinct trials has more unsigned coincidences than its information budget permits. · OAI.HyperbolicObstruction.Coincidence.RankBad

rankCell def

Rank inside a column, sorted by an arbitrary column-injective row assignment. · OAI.CubeShuffle.Specht.rankCell

rankComponent def

The rank-level component uses actual image dimension. · OAI.MaxCutGames.Inverse.KMSAnalytic.rankComponent

rankComponent def

The rank-level component uses actual image dimension. · OAI.UniqueGamesTheorem.Inverse.KMSAnalytic.rankComponent

rankData def

The actual rank measure, extended by zero outside the range where the balanced schedule is used. The small values do not enter the eventual bounds. · OAI.Problem335.LowerParameters.rankData

rankDifference def

d(x,y) in the manuscript, used only for comparable endpoints. · OAI.KLInvariance.rankDifference

rankDimension def

The source dimension for the configured bidegree rank measure. · OAI.Problem335.LowerParameters.rankDimension

ranked def

A direct realization of the displayed program, one literal passage for each recorded adjacent crossing. It does not choose an arbitrary permutation word after forgetting the no-recrossing property. · OAI.IntegralCharacterVarieties.DirectedSort.Program.ranked

RankedBagTree structure

This records ordinary bag axioms, not lift solvability or any WL promise. · OAI.ParityWL.RankedBagTree

rankedBitMemoCells def

Sixteen scalar/frame slots suffice in addition to eight slots per key bit and the supplied address-construction workspace; the larger fixed reserve also covers the result record and the empty-trie case. · OAI.EditApproximation.rankedBitMemoCells

RankedChain inductive

A finite ranked gallery. The constructor stores a verified vertex, not a geometric bundle hypothesis; lengths and unused ports remain explicit. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.RankedChain

rankedQuery def

Total extension of the declared tape decoder, for the query loop's natural index register. Out-of-range values are never visited by that loop. · OAI.MaxCutGames.Foundations.Hastad.SourceLoopOrder.rankedQuery

rankedQuery def

Total extension of the declared tape decoder, for the query loop's natural index register. Out-of-range values are never visited by that loop. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceLoopOrder.rankedQuery

rankedUncachedCalls def

Number of requests in the same adaptive program without cache hits. It is a dominating execution tree, not the size of the full table. · OAI.EditApproximation.rankedUncachedCalls

rankFailure def

Once the rank is zero, later draws are declared successful for the purpose of counting failures. Before then, success means a strict rank decrease. · OAI.LeanBlast.SecondKahnKalai.rankFailure

rankFiberEquivFixedKernel def

The kernel condition already forces the rank condition. · OAI.MaxCutGames.Inverse.KMSKernelFiberCard.rankFiberEquivFixedKernel

rankFiberEquivFixedKernel def

The kernel condition already forces the rank condition. · OAI.UniqueGamesTheorem.Inverse.KMSKernelFiberCard.rankFiberEquivFixedKernel

rankFilter def

The unselected multiplier in factored form, after multiplying numerator and denominator by 2^(2*r). Selected characters have multiplier one. · OAI.MaxCutGames.Appendix.Level.rankFilter

rankFilter def

The unselected multiplier in factored form, after multiplying numerator and denominator by 2^(2*r). Selected characters have multiplier one. · OAI.UniqueGamesTheorem.Appendix.Level.rankFilter

rankFlip def

Switch the two vertices at every strictly internal rank, leaving all other ranks fixed. The diagram involution supplies the intrinsic switch. · OAI.KLInvariance.Dihedral.rankFlip

rankFor def

Computable rank exceeding any prescribed rational MaxCutGames.Gadget.Harmonic.harmonic level. · OAI.MaxCutGames.Gadget.Parameters.rankFor

rankFor def

Computable rank exceeding any prescribed rational harmonic level. · OAI.UniqueGamesTheorem.Gadget.Parameters.rankFor

rankFrequencyImageEquiv def

Exact rank frequencies are in bijection with a choice of actual image and a surjective map onto it. · OAI.MaxCutGames.Inverse.KMSAnalyticHybridEnergy.rankFrequencyImageEquiv

rankFrequencyImageEquiv def

Exact rank frequencies are in bijection with a choice of actual image and a surjective map onto it. · OAI.UniqueGamesTheorem.Inverse.KMSAnalyticHybridEnergy.rankFrequencyImageEquiv

rankGapEventEquiv def

The actual event of seeing specified source slots at specified output ranks is in bijection with independent choices of the retained slots in each gap. · OAI.TraceReconstruction.rankGapEventEquiv

RankImage abbrev

Actual image subspaces of the required dimension. · OAI.MaxCutGames.Inverse.KMSAnalyticHybridEnergy.RankImage

RankImage abbrev

Actual image subspaces of the required dimension. · OAI.UniqueGamesTheorem.Inverse.KMSAnalyticHybridEnergy.RankImage

rankKernel def

The exact conditional rank kernel, including the fixed count normalization, before any coarse-gap estimates or restrictions. · OAI.TraceReconstruction.rankKernel

RankLossAt def

A simultaneous rank-loss bound for relation quotients with unit lifts. · OAI.OneWayLiveness.RankLossAt

RankModel structure

The properties of the ordinal rank which are used by the global assembly. · OAI.InfiniteMatroidCounterexample.RankModel

rankMomentScale def

The common path-count and occupation-mean factor in the two trace moments. · OAI.Problem335.LowerParameters.rankMomentScale

rankOn def

Coincidence rank on an actual subset of field positions. · OAI.HyperbolicObstruction.Coincidence.rankOn

rankOne def

The orthogonal projection onto a normalized one-qubit vector. · OAI.QAC.rankOne

RankOneClaim def

The rank-one comparison for the symmetric-cutoff matrix field. · OAI.SharpLiebThirring.RankOneClaim

rankOneEquivChartNeighbors def

Difference from a fixed matrix is an exact bijection between rank-one maps and adjacent matrices. Subtraction also makes this construction valid without invoking characteristic-two simplification. · OAI.MaxCutGames.Inverse.MatrixChart.rankOneEquivChartNeighbors

rankOneEquivChartNeighbors def

Difference from a fixed matrix is an exact bijection between rank-one maps and adjacent matrices. Subtraction also makes this construction valid without invoking characteristic-two simplification. · OAI.UniqueGamesTheorem.Inverse.MatrixChart.rankOneEquivChartNeighbors

rankOneUpdate def

The actual rank-one update of an ordered basis map. · OAI.MaxCutGames.Inverse.KMSBasisComparison.rankOneUpdate

rankOneUpdate def

The actual rank-one update of an ordered basis map. · OAI.UniqueGamesTheorem.Inverse.KMSBasisComparison.rankOneUpdate

rankOneUpdateNeighbor def

The updated image is a genuine Grassmann neighbor, with the full ambient neighbor predicate rather than its chart restriction. · OAI.MaxCutGames.Inverse.KMSBasisComparison.rankOneUpdateNeighbor

rankOneUpdateNeighbor def

The updated image is a genuine Grassmann neighbor, with the full ambient neighbor predicate rather than its chart restriction. · OAI.UniqueGamesTheorem.Inverse.KMSBasisComparison.rankOneUpdateNeighbor

RankPairChoices abbrev

A column and an ordered list of candidate lit pairs have at most this many descriptions. Repeated or invalid descriptions only enlarge the bound. · OAI.TwoPointCorrelations.RankPairChoices

rankProjection def

Projection onto trace characters whose index has bounded rank on C. · OAI.MaxCutGames.Fourier.MatrixEnergy.rankProjection

rankProjection def

Exact rank projection, defined directly by trace characters. · OAI.MaxCutGames.Inverse.KMSBasisInvariant.rankProjection

rankProjection def

Projection onto trace characters whose index has bounded rank on C. · OAI.UniqueGamesTheorem.Fourier.MatrixEnergy.rankProjection

rankProjection def

Exact rank projection, defined directly by trace characters. · OAI.UniqueGamesTheorem.Inverse.KMSBasisInvariant.rankProjection

ranks def

This concrete process discharges the signed all-step filtering premise. · OAI.KServer.FiniteExperiment.ranks

RankSamplingStage structure

A genuine stage with a bounded-rank certificate for the same arrays. The size cutoff is arbitrary and fixed before the alphabet is selected. · OAI.HyperbolicObstruction.IncidenceTemplate.RankSamplingStage

RankShape def

Finite, exhaustive intersection-rank indexing set. Zero blocks and repeated flag terms are retained at this stage, rather than excluded by genericity. · OAI.IntegralCharacterVarieties.TwoFlagBand.RankShape

RankSpace def

The finite Grassmannian at a rank chosen before orientation. · OAI.MaxCutGames.Gadget.Orientation.RankSpace

RankSpace def

The finite Grassmannian at a rank chosen before orientation. · OAI.UniqueGamesTheorem.Gadget.Orientation.RankSpace

rankSpaceEquiv def

A fixed coordinate equivalence transports the complete rank-indexed Grassmannian bijectively, so it preserves its uniform distribution. · OAI.MaxCutGames.Gadget.FreshGenericity.rankSpaceEquiv

rankSpaceEquiv def

A fixed coordinate equivalence transports the complete rank-indexed Grassmannian bijectively, so it preserves its uniform distribution. · OAI.UniqueGamesTheorem.Gadget.FreshGenericity.rankSpaceEquiv

rankVector def

All ranks are taken under the same true observed history. · OAI.KServer.AdaptiveMinimization.rankVector

RapidMachineHaltingUndecidable def

Finite transition tables with a distinguished terminal state. · OAI.ForcedComputation.RapidMachineHaltingUndecidable

rareWord def

All zero bits select correction; every other word selects the ordinary draw. · OAI.ContingencyTables.ProfilePrograms.rareWord

RatExpr structure

Rational arithmetic as fixed finite syntax over nonnegative registers. No canonical-fraction reduction or unit-cost large-integer operation is required. · OAI.ExactQuantumFactoring.RatExpr

RatExprPoly def

Polynomial literal numerator and denominator syntax; no bound on dynamic values is assumed. · OAI.ExactQuantumFactoring.RatExprPoly

RatHeightPoly def

Bit-length bounds for exact normalized rational constants appearing in the emitted circuit. These are not bounds on their exponentially large values. · OAI.ExactQuantumFactoring.RatHeightPoly

ratio def

A numerical tag probability. · OAI.FourierLLogL.Compress.ratio

ratio def

The normalized projection-body volume. · OAI.ProjectionCounterexample.ratio

ratioInvolution def

The fractional-linear involution exchanges the two exceptional points with themselves, making it a permutation of the entire finite field. · OAI.Ostmann.ratioInvolution

rationalArc def

A closed rational neighborhood of radius δ / q. · OAI.Problem337.ThreePrimeAnalysis.rationalArc

rationalBasis def

The rational cotangent basis induced from an affine étale coordinate chart agrees with the differentials of those very same coordinates. · OAI.CanonicalCoordinates.rationalBasis

rationalBasis def

An integer basis of a full lattice is a rational basis of the field. · OAI.DukePrimeDegree.FullLattice.rationalBasis

rationalBlockRealSpan def

The real span of the rational block centralizer in lattice coordinates. · OAI.DukePrimeDegree.rationalBlockRealSpan

rationalBlockStabilizer def

The actual rational lattice stabilizer intersected with the block centralizer. Both conditions refer to the given lattice and partition. · OAI.DukePrimeDegree.rationalBlockStabilizer

RationalComplexCore structure

A dense rational subspace on which all complex-linearity tests remain countable, since multiplication by i preserves the subspace. · OAI.KadisonKastler.RationalComplexCore

rationalConjugationQuotient def

The centralizer quotient acts faithfully on the rationalization. · OAI.PolycyclicRecognition.rationalConjugationQuotient

RationalConvexExpr inductive

A countable syntax for rational convex combinations. Clamping the rational parameter avoids proof fields in the syntax. · OAI.KadisonKastler.RationalConvexExpr

rationalDiagonalActionAt def

Evaluation of the action at a fixed vector is rational linear. · OAI.DukePrimeDegree.rationalDiagonalActionAt

rationalDiagonalEval def

Evaluation in any coordinate is faithful on the actual stabilizer. · OAI.DukePrimeDegree.rationalDiagonalEval

rationalDiagonalStabilizer def

Real diagonal coordinate functions preserving the rational space. · OAI.DukePrimeDegree.rationalDiagonalStabilizer

rationalEncode def

Full signed canonical rational output. No gcd, comparison or sign oracle occurs in this fixed finite binary-stack program. · OAI.RationalHitting.StackLanguage.rationalEncode

rationalEncodingBits def

Signed numerator plus positive denominator encoding length. · OAI.EditApproximation.rationalEncodingBits

RationalEntry structure

The two binary fields are a numerator and a positive denominator. · OAI.CurrentKS.RationalEntry

RationalError structure

The rational number numerator / denominator lies strictly between 0 and 1/2. No normalization to lowest terms is needed for the gap predicates. · OAI.MaxCutGames.Foundations.Target.RationalError

RationalError structure

The rational number numerator / denominator lies strictly between 0 and 1/2. No normalization to lowest terms is needed for the gap predicates. · OAI.UniqueGamesTheorem.Foundations.Target.RationalError

RationalFixedScaleLaw def

The joint limit for rational fixed-scale thresholds. · OAI.JointDickman.RationalFixedScaleLaw

rationalHomologyCongr def

The singular-homology functor turns a homeomorphism into a rational linear equivalence in every degree. · OAI.PositiveEinsteinFour.Topology.rationalHomologyCongr

rationalHomologyZeroEquiv def

Augmentation identifies the singular zeroth homology with its coefficient field on a path-connected space. · OAI.PositiveEinsteinFour.Topology.rationalHomologyZeroEquiv

rationalizeNuclei def

Rational nuclear data produced from real centers and integral charges. · OAI.ContinuumCoulomb.rationalizeNuclei

rationalizeUnitNuclei def

In the unit-charge case the output uses the unit-charge input structure. · OAI.ContinuumCoulomb.rationalizeUnitNuclei

rationalLatticePoint def

A rational pair represented by one positive natural denominator. · OAI.GammaFPP.rationalLatticePoint

RationalLaw structure

A finite probability law with exactly represented rational masses. · OAI.ContingencyTables.ResidualMixture.RationalLaw

rationalLogApproximationWithWork def

Three operations form the logarithm parameter, and one final multiplication doubles the partial sum. · OAI.EditApproximation.rationalLogApproximationWithWork

rationalLogPartialWithWork def

The direct finite logarithm summation, with every power, division and addition counted. · OAI.EditApproximation.rationalLogPartialWithWork

rationalLowParameter def

The floor of the rational cutoff, with natural truncation at zero. · OAI.LowSpaceSubsetSum.rationalLowParameter

rationalMatrix def

Coefficientwise extension from the integral to the rational group ring. · OAI.GroupRingDeterminant.rationalMatrix

rationalMatrixCast def

Entrywise rational-to-real scalar extension for matrices. · OAI.DukePrimeDegree.rationalMatrixCast

rationalMaximal def

A countable measurable envelope, sufficiently rich for every actual disk. · OAI.DiskMaximalTools.rationalMaximal

rationalMaximal def

A countable measurable envelope, sufficiently rich for every actual disk. · OAI.SignedDisk.DiskMaximalTools.rationalMaximal

rationalMean def

A concrete rational mean; no representative is chosen nonconstructively. · OAI.GammaFPP.rationalMean

rationalOptimizerIterate def

Exact rational conditional-gradient iterates. · OAI.EditApproximation.rationalOptimizerIterate

rationalPairCast def

The rational embedding uses the standard product (maximum) metric. · OAI.GammaFPP.rationalPairCast

rationalPowerWithWork def

Costed repeated multiplication, counting one rational multiplication at each successor. · OAI.EditApproximation.rationalPowerWithWork

RationalPrefix abbrev

A rational probability vector on an initial, nonempty list of indices. · OAI.Tingley.RationalPrefix

rationalProgressionPrimes def

Rational primes in one residue class, with the literal real cutoff. · OAI.CubicFirstMoment.rationalProgressionPrimes

RationalQuartetData structure

The fixed data above a bottom quartet; its four leaf products remain free. · OAI.Ostmann.RationalQuartetData

rationalResult def

The phase result as a function of just the rational query values. · OAI.ContingencyTables.CompanionPhaseFinishProgram.rationalResult

rationalRoundedScaledCharge def

Rational rounding used by the charge-output algorithm. · OAI.ContinuumCoulomb.rationalRoundedScaledCharge

rationalRoundProgram def

Ties round upward, exactly as the manuscript's rational charge function. · OAI.ContinuumCoulomb.ChargePrograms.rationalRoundProgram

rationalRowMap def

Rational linear combinations of the rows of a real lattice matrix. · OAI.DukePrimeDegree.rationalRowMap

rationalStorage def

One word each for the tag and two child pointers, plus the value bits. Counting bits individually is an upper bound on logarithmic-word storage. · OAI.EditApproximation.BinaryMemo.rationalStorage

rationalSupplyBase def

The prime product indexed by the argument used in the rational supply. · OAI.Problem337.rationalSupplyBase

RationalTraceIncrements structure

This is the positive-increment and interval-support assertion used in the pathwise extension. Local applications restrict all measures to one compact set, making the terminal measure finite. · OAI.SLEExactGauge.RationalTraceIncrements

rationalTreeAmplitude def

The manuscript's diagram value, permitting arbitrary conjugation choices at the leaves. · OAI.Ostmann.rationalTreeAmplitude

rationalTreeArgumentEquiv def

With the other root data fixed, exposing the total leaf product gives a uniform argument. · OAI.Ostmann.rationalTreeArgumentEquiv

rationalValue def

The rational denominator is parsed and reduced by the actual gcd-based machine routine, so noncanonical input fractions are handled correctly. · OAI.ContinuumCoulomb.SourcePrograms.rationalValue

rationalVectorCast def

Coordinatewise inclusion of rational vectors into real vectors. · OAI.MatrixMultiplication.AuxiliarySeparation.rationalVectorCast

rationalWedge def

A wedge of two rational differentials. · OAI.CanonicalCoordinates.rationalWedge

RationalWeights structure

Normalized rational occurrence weights; zero weights are also permitted. · OAI.UniqueGamesTheorem.Explicit.RationalWeights

rationalWindow def

A rational arc written without requiring its displayed fraction to be reduced. · OAI.Problem337.MinorArc.rationalWindow

RatioOrder structure

Both functionals are actual dual vectors; the root order is not a label-carrying addition to the interval isomorphism. · OAI.KLInvariance.TitsSpace.RatioOrder

ratioStep def

The normalized scalar recurrence used for the exterior jet enclosure. Unlike unnormalized coefficient arithmetic it cancels the largest jet scale. · OAI.SepticProfile.ExteriorJet.ratioStep

rAux def

Source R_{aux,i}: the auxiliary is the first wire, physical site i is at position i+1. All other wires are spectators. · OAI.CriticalHoneycomb.Transfer.PlanarNormalization.rAux

Raw abbrev

The unchanging request register dimensions depend only on the original n. · OAI.ExactQuantumFactoring.UniversalSplit.Raw

Raw structure

A literal raw-letter stack program, with its finite control part. · OAI.FiniteAlgebraicEnvelopes.MachineRealization.CodeMachine.Raw

Raw inductive

Unreduced strings, including a compulsory terminal old factor. · OAI.PartitionConsistency.ProbeAlgebra.Raw

raw def

The finite data used after accepting an unskipped band. · OAI.EditApproximation.UnskippedBand.raw

raw def

Literal raw singleton; in particular no program is evaluated on it. · OAI.FiniteAlgebraicEnvelopes.Compiler.Surface.raw

raw def

The polynomial matrix expression, before imposing any inverses or incidence conditions. The sign retains both orientations of every transport. · OAI.IntegralCharacterVarieties.MatrixExpression.Term.raw

raw def

An explicit smooth radial bump. It is one on the half-unit disk, positive in the unit disk and zero outside it. · OAI.TriangularBilliards.SpatialSmoothing.raw

rawAcceptNet def

Exact ordinary trial decoder, including the sampler's full workspace test, continued-fraction reconstruction, arithmetic validation, and all three retained coin tests. Its inputs are actual retained raw bits. · OAI.ExactQuantumFactoring.OrderTrial.rawAcceptNet

rawAugmentationRank def

The augmentation rank on the free abelian group of idempotents, before imposing isomorphism and direct-sum relations. · OAI.TorsionFreeBass.rawAugmentationRank

rawBackground def

Eight disjoint background allocations in a fixed order. This uses all outside indices, regardless of which ones belong to the fixed solution. · OAI.LowSpaceSubsetSum.RepresentationSetup.rawBackground

rawBianchiProjection def

The polynomial formula for the actual orthogonal Bianchi compression. · OAI.PositiveEinsteinFour.Bianchi.rawBianchiProjection

rawBlockFiberEquiv def

The set-level filter is exactly the corresponding fiber of the sampled balanced coloring. · OAI.LowSpaceSubsetSum.RepresentationSetup.rawBlockFiberEquiv

RawBlockPencil structure

The finite blocks after splitting the support of the first positive pencil component. · OAI.Paper256.RawBlockPencil

rawBoxMeasure def

Only the two finite boxes contribute an exponential factor; the numeric walk, rejection and reserved-word schedules are polynomial. · OAI.ContingencyTables.DensePrograms.rawBoxMeasure

rawCoefficient def

The actual punctured Cartesian tensor; J may switch on round gaps. · OAI.HarmonicCounterexample.Cartesian.rawCoefficient

rawCost def

The recursive cost makes both the suffix scale and the old gate's cost available simultaneously. · OAI.PartitionConsistency.ProbeAlgebra.Word.rawCost

rawCostCoefficient def

Quaternion coefficient of the total derivative of (7-K)L. · OAI.PositiveEinsteinFour.Bianchi.rawCostCoefficient

rawCountStep def

Only finite materialized tables are passed between consecutive iterations. · OAI.Superstring.Computable.rawCountStep

rawCovectorJet def

Forget continuity only after taking the actual bundled derivative. · OAI.Mahler.rawCovectorJet

rawCovectorVariation def

The uncurried derivative of a covector field, before alternation. · OAI.Mahler.rawCovectorVariation

RawDensitySupply def

The remaining arithmetic supply interface for the main upper bound. The residue distribution and shifted divisor moments are explicit fields of DensityResidueFamily; this definition does not assert their existence. · OAI.Problem337.RawDensitySupply

rawDifferential def

Tensor the actual arrow with O; no differential is assumed. · OAI.LittleFinitistic.OrdinarySimulation.VertexTensor.rawDifferential

rawEdge def

The original oriented edges really are arrows in the symmetrified quiver. · OAI.Release075.BlockPresentation.rawEdge

rawEntries def

Mathematical list of valid entries emitted by this task. The executable collector keeps only the bounded deduplicated array. · OAI.LowSpaceSubsetSum.RepresentationParentData.rawEntries

rawEntries def

All fifty coordinates before imposing derivative trace and Bianchi. · OAI.PositiveEinsteinFour.HighestHessian.rawEntries

rawExponent def

Raw exponent conditions of (exponent-set), using real indices so the numerical argument does not depend on integrality. d / 3 replaces its integer floor; for integral j the bounds are equivalent. · OAI.Nagata.ExponentEnclosure.rawExponent

rawFrame def

A genuine polynomial determinant frame exists by unique factorization. · OAI.StableCoordinate.NormalizedDeterminantLift.rawFrame

rawFrontFin def

The derivative slot is numbered zero; all old slots follow in order. · OAI.Mahler.rawFrontFin

rawH def

Original independent variable order: h,u,v,w. · OAI.AbhyankarSathaye.CommutingDerivations.ParameterBridge.rawH

rawHeatKernel def

The unmodified heat operator, set to zero only at nonpositive times. · OAI.ForcedComputation.PlaneScalarMild.rawHeatKernel

rawHeight def

Recursive heights on literal irreducibles; the same formula is harmless on unreduced strings. · OAI.PartitionConsistency.ProbeAlgebra.Word.rawHeight

rawIntegral def

Integration from zero to a fixed real time is bounded on the space of continuous periodic functions, even when that time is outside one period. · OAI.ClosedSurfaceR4.PeriodicIntegralOperator.rawIntegral

rawInterval def

The source interval before splitting labels into lattice slots. · OAI.LipschitzHilbert.SourceLabeledLattice.rawInterval

rawJetFamily def

All raw jets have the same lattice and fixed envelope constant. · OAI.LipschitzHilbert.SourceLatticePacket.rawJetFamily

rawLeaf def

Table 1's actual leaf domains and rounded prescribed counts. · OAI.LowSpaceSubsetSum.RepresentationSetup.rawLeaf

rawLetters def

Candidate extraction followed by a literal check recognizes only full, clean raw sectors. Dirty tags or absent boundaries cannot silently pass. · OAI.FiniteAlgebraicEnvelopes.Compiler.Word.rawLetters

rawLineWeight def

The literal real line mass, without changing any conditioning. · OAI.CoordinateSweeps.Grid.rawLineWeight

rawMatrixCurvature def

The actual trace-free anticommutator, applied independently in each derivative direction. · OAI.PositiveEinsteinFour.Bianchi.rawMatrixCurvature

rawMoveDerivative def

Move a distinguished derivative slot past the left factor without changing the order within either factor. · OAI.Mahler.rawMoveDerivative

rawNorm def

Frobenius norm of an arbitrary matrix of real four-vectors. · OAI.PositiveEinsteinFour.Bianchi.rawNorm

RawObservationFibre abbrev

H fixes gamma and the outside incidence pairs. K is a deterministic function of gamma, so including K explicitly in H gives precisely this same event. · OAI.MaxCutGames.Soundness.ConditionalIncidences.RawObservationFibre

RawObservationFibre abbrev

H fixes gamma and the outside incidence pairs. K is a deterministic function of gamma, so including K explicitly in H gives precisely this same event. · OAI.UniqueGamesTheorem.Soundness.ConditionalIncidences.RawObservationFibre

rawOperatorFamily def

The full substitution operator restricted to a fixed bidegree source. Keeping the ambient polynomial codomain avoids unnecessary degree transports. · OAI.Problem335.rawOperatorFamily

RawOperatorField abbrev

The discrete topology is used only to package arbitrary bounded families; all measurable assertions use the original measure space. · OAI.KadisonKastler.RawOperatorField

rawOutput def

Literal coordinatewise output of countably many signed segments: first sum all rows with a common start/end, then use the absolutely summable path map. · OAI.DiamondDistortion.StartModel.rawOutput

rawPartitionBlock def

The exact labeled block supplied by the sampled partition. · OAI.LowSpaceSubsetSum.RepresentationSetup.rawPartitionBlock

RawPath structure

Raw infinite path data, retaining the actual rational pieces, declared weights and metadata. Nonzero pieces are included here to define the prefix endpoint without arbitrary defaults. Type-I membership is not needed for the combinatorial … · OAI.SeparableQuotient.PathCoding.RawPath

rawPinchingMass def

The raw full mass before conjugation by the patch's mean. · OAI.RegularPointCoordinates.rawPinchingMass

RawPosition abbrev

Indexed positions, NOT generator labels. Two fields carrying an equal numerical letter remain distinct positions. · OAI.HyperbolicObstruction.ConeSkeleton.RawPosition

rawPower def

The raw ordered product of k bilinear factors. · OAI.Mahler.rawPower

rawPrefix def

Adjoin the actual directional derivative as the first raw slot. · OAI.Mahler.rawPrefix

rawProduct def

Tensor product of scalar multilinear forms before alternatization. · OAI.Mahler.rawProduct

rawProgram def

Every step is compiled from actual binary add/multiply/divide routines; the repetition count is polynomial in the unary input size. · OAI.ContinuumCoulomb.AmplificationProgram.rawProgram

rawPurifiedSecretBitError def

The operational bit error written directly in terms of the output entries. · OAI.ZeroKey.BitProtocol.rawPurifiedSecretBitError

rawRepresentationModel abbrev

Repetition labels are retained: each start node admits a countable sequence of signed coefficient arrays over all its possible endpoints. Its mass is the ℓ¹ sum of the ℓ¹ norms of these arrays, before any cancellation. · OAI.DiamondDistortion.StartModel.rawRepresentationModel

RawRowEncoding abbrev

The unrestricted row record consists of its actual support and its labels. · OAI.TalagrandThreshold.Selector.RawRowEncoding

rawRowLookup def

Reading one point avoids any dependence on the chosen subtype proof. · OAI.TalagrandThreshold.Selector.rawRowLookup

rawRun def

The input is read from the genuine initList configuration. Header, every row, final guard, and reversal are all actual transitions of this one program. · OAI.MaxCutGames.Foundations.Complexity.FinalCNFMachine.Program.rawRun

rawRun def

The input is read from the genuine initList configuration. Header, every row, final guard, and reversal are all actual transitions of this one program. · OAI.UniqueGamesTheorem.Foundations.Complexity.FinalCNFMachine.Program.rawRun

rawSelectorTreeValue def

Arbitrary coordinate multipliers on the actual raw packets. · OAI.LipschitzHilbert.SourceLabeledLattice.rawSelectorTreeValue

rawSelectProgram def

All arithmetic is on extra tapes. The old-dart branch literally contains the shared selector program, with only static tape, label and state transport. · OAI.MaxCutGames.Foundations.Complexity.MachineRegularInternalRow.rawSelectProgram

rawSelectProgram def

All arithmetic is on extra tapes. The old-dart branch literally contains the shared selector program, with only static tape, label and state transport. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRegularInternalRow.rawSelectProgram

RawSlot abbrev

None is ζ₀; the left summand gives two coefficients per full position; the right summand gives one coefficient per single position. · OAI.MaxCutGames.Soundness.ConditionalIncidences.RawSlot

RawSlot abbrev

None is ζ₀; the left summand gives two coefficients per full position; the right summand gives one coefficient per single position. · OAI.UniqueGamesTheorem.Soundness.ConditionalIncidences.RawSlot

rawSourceList def

The entire finite random-tape population, including repeated visible contexts and all equation multiplicities, in a fixed explicit order. · OAI.MaxCutGames.Foundations.Hastad.SourceOccurrences.rawSourceList

rawSourceList def

The entire finite random-tape population, including repeated visible contexts and all equation multiplicities, in a fixed explicit order. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceOccurrences.rawSourceList

RawStep inductive

Literal clean-anchor slides, oriented to the leftmost reachable head gap. · OAI.FiniteAlgebraicEnvelopes.Compiler.Surface.RawStep

rawTangentCurve def

A polynomial curve on the nonzero isotropic cone realizing a horizontal vector. · OAI.ArnoldCounterexample.Quadric.rawTangentCurve

rawTop def

The source's fixed c_top in the raw wrapper, used only by the markers. · OAI.PureTypeSystem.Labelled.RelationalCode.rawTop

rawToPosition def

Reindexing uses the original edge/decoration field, never its label value. · OAI.HyperbolicObstruction.ConeSkeleton.rawToPosition

RawTuple abbrev

The conditional coefficient sample, with full pairs and singleton values kept at their occurrence-local positions. · OAI.MaxCutGames.Decoder.PaddedLaw.RawTuple

RawTuple abbrev

The conditional coefficient sample, with full pairs and singleton values kept at their occurrence-local positions. · OAI.UniqueGamesTheorem.Decoder.PaddedLaw.RawTuple

rawVisits def

Raw occurrences are a separate type and retain all multiplicities. Task branching is decided by U, the distinct entries, rather than by T. · OAI.LowSpaceSubsetSum.rawVisits

rawWeight def

A real outer vector tensored with an arbitrary matrix-valued four-vector. · OAI.PositiveEinsteinFour.HighestHessian.rawWeight

ray def

An infinite nearest-neighbor ray, used to certify supercriticality at one. · OAI.CriticalZ3.ray

RayBoundaryComparison abbrev

Volume data for the actual source and target ray actions. · OAI.PolycyclicRecognition.CartanWordModel.RayBoundaryComparison

RayBoundaryModel structure

The actual integrated ray quotient, with its descended ambient action. · OAI.PolycyclicRecognition.CartanWordModel.RayBoundaryModel

rayCondition def

The measurable condition whose planar cost is bounded by the ray-length estimate. · OAI.DiskGeometry.rayCondition

rayCondition def

The measurable condition whose planar cost is bounded by the ray-length estimate. · OAI.SignedDisk.DiskGeometry.rayCondition

rayFlux def

Signed intersections with the positive horizontal ray; dual vertex (a,b) represents the point (a-1/2,b-1/2). · OAI.PlanarBoundary.rayFlux

rayMarkedSection def

The actual nine-mark multiplier section for ray lifts τ^ξᵢ. · OAI.Nagata.W22.rayMarkedSection

rayPoint def

The covering-space point tau^s, expressed through the genuine real logarithm. · OAI.Nagata.Workers.W05.rayPoint

raySetoid def

Finite synchronous distance, equivalent to asymptoticity for based rays. · OAI.CannonRelease.raySetoid

RCap def

uniformly bounded, in roots · OAI.ExactDerandomization.Est.RCap

reach def

budget reached by group i. Difference indices m denote the paired evaluation at m+1 and m. · OAI.ExactDerandomization.Est.Stair.reach

reach def

Reachability in the actual signed graph, not equality of word images. · OAI.HyperbolicObstruction.SignedWalks.reach

Reachable inductive

These are precisely the extensions generated from the double-dual model by finitely many torsion-free divisor quotients. · OAI.SourceConeMorphism.ExtensionState.Reachable

Reachable inductive

Reachability includes every binary choice, not merely an almost-sure set. The index counts requests already served. · OAI.UniformKServer.Reachable

reachable def

The coefficients that can be reached from one input form a two-sided ideal; unitality of the inclusion into B is deliberately not required. · OAI.FiniteAlgebraicEnvelopes.Enveloping.reachable

Reaches def

The nonnegative-time particle observation used in the manuscripts. · OAI.ForcedComputation.Reaches

Reaches def

The right divisibility relation on a right monoid-set. · OAI.PartitionConsistency.MergerAlgebra.RightFreeAction.Reaches

reachesCompleted def

The actual refinement path stops at the first dictionary that fits. · OAI.LowSpaceSubsetSum.reachesCompleted

read def

All internal tournament keys use the concrete list-mask evaluator. · OAI.LowSpaceSubsetSum.RepresentationKeys.read

readAnchor def

Tags following the copy phase; unused full-tag fields remain literally clean, as required by the validator/simulator convention. · OAI.FiniteAlgebraicEnvelopes.Compiler.Simulator.readAnchor

readBit def

Move exactly one bit, leaving the suffix and the old output accumulator in place. Empty input is a no-op, but valid parser calls always supply the bit. · OAI.Superstring.BitCode.Block.readBit

readCoord def

none is the literal input zero, not a physical workspace coordinate. · OAI.ExactFourier.readCoord

readCoordinates def

Read the physical saved digit positions in reverse coordinate order. · OAI.MaxCutGames.Reduction.OccurrenceTupleOrder.readCoordinates

readCoordinates def

Read the physical saved digit positions in reverse coordinate order. · OAI.UniqueGamesTheorem.Reduction.OccurrenceTupleOrder.readCoordinates

readEncodedSlots def

Read one zero-delimited unary Boolean field for each fixed register slot. The second pop occurs exactly when the value is one. · OAI.MaxCutGames.Foundations.Complexity.PoweringMachineRow.readEncodedSlots

readEncodedSlots def

Read one zero-delimited unary Boolean field for each fixed register slot. The second pop occurs exactly when the value is one. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringMachineRow.readEncodedSlots

readerStateEquiv def

Merely rearranges finite registers to match the affine-emitter state. No tape content or unbounded integer is moved into the internal state. · OAI.MaxCutGames.Foundations.Complexity.FinalCNFMachine.readerStateEquiv

readerStateEquiv def

Merely rearranges finite registers to match the affine-emitter state. No tape content or unbounded integer is moved into the internal state. · OAI.UniqueGamesTheorem.Foundations.Complexity.FinalCNFMachine.readerStateEquiv

readGap def

A path that reads a word one letter at a time, always returning to empty. This supplies the gap case of ordered windows without adding non-substring words. · OAI.Superstring.Hierarchical.Path.readGap

readInput 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.readInput

readKeys def

Finite output trace, used only to state and prove the loop's behavior. · OAI.LowSpaceSubsetSum.BinaryTrie.readKeys

ReadOnly def

A syntactic read-only rule suffices; no hypothesis about the state prepared by the algorithm or its unknown factors occurs here. · OAI.ExactQuantumFactoring.ReadOnly

readoutProgram def

The readout is a clean reversible Boolean network, not a classical hidden postprocessor. The preparation's complete history is retained. · OAI.ExactQuantumFactoring.readoutProgram

readRelation def

Fixed finite program fragment; it can be inserted before any continuation. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.ReadRelation.readRelation

readRelation def

Fixed finite program fragment; it can be inserted before any continuation. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.ReadRelation.readRelation

readRowInTime def

Read both unary fields using actual copy loops, then execute the checked fixed predicate parser. Only the two field destinations must initially be empty; the cursor suffix and every unrelated tape are retained exactly. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Lookup.readRowInTime

readRowInTime def

Read both unary fields using actual copy loops, then execute the checked fixed predicate parser. Only the two field destinations must initially be empty; the cursor suffix and every unrelated tape are retained exactly. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Lookup.readRowInTime

Reads def

f read only first p entries for the first q output entries · OAI.PowerSaving.Slant.Reads

readSigns def

The head defaults make the finite program total on malformed tapes too; the correctness theorem below supplies the actual unary-field representation. · OAI.MaxCutGames.Foundations.Hastad.SourceSignLoad.readSigns

readSigns def

The head defaults make the finite program total on malformed tapes too; the correctness theorem below supplies the actual unary-field representation. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceSignLoad.readSigns

readSlot def

Read one encoded Boolean into a fixed slot of the finite relation register. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.ReadRelation.readSlot

readSlot def

Read one encoded Boolean into a fixed slot of the finite relation register. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.ReadRelation.readSlot

readSpin def

N=0 is an irrelevant harmless extension; for every positive N this reads exactly the spin at the uniformly sampled original site. · OAI.DilutedSpinGlass.ConcreteReservoir.readSpin

readUnarySlots def

Read fixed register slots from the actual unary 0/1 word encoding. Each slot consumes one symbol for false and two symbols for true. · OAI.MaxCutGames.Foundations.Complexity.FinalCNFMachine.readUnarySlots

readUnarySlots def

Read fixed register slots from the actual unary 0/1 word encoding. Each slot consumes one symbol for false and two symbols for true. · OAI.UniqueGamesTheorem.Foundations.Complexity.FinalCNFMachine.readUnarySlots

readVertex def

The inverse reading of the vertices of categorification. · OAI.Thomason.Construction.NormalizedSimplexIdentities.readVertex

readWord def

Finite letters on a chain, extended only to make the integer indexing convenient. The fallback is never used at an edge of the pattern. · OAI.TorsionFreeZeroDivisors.StageExpectation.readWord

Ready def

Necessary future checks at each phase. Already-scanned cells need not be constrained: backwards propagation recovers exactly the checks made. · OAI.FiniteAlgebraicEnvelopes.Compiler.Validator.Ready

Ready structure

Conditions on real input/work stacks; arbitrary output suffixes remain permitted. The two query/scan tapes are empty at a row boundary. · OAI.MaxCutGames.Explicit.MachineProductField.Ready

Ready structure

The input-dependent data needed for one whole row, held invariant by all physical field executions. · OAI.MaxCutGames.Explicit.MachineProductRow.Ready

Ready structure

The complete representation needed at one odometer leaf. Remaining-digit stacks, output history, and retained unused headers are arbitrary. · OAI.MaxCutGames.Foundations.Hastad.SourceTupleBody.Ready

Ready structure

Guard-to-guard tape contract. Only scan and indexScan may contain residues. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.LoopState.Ready

Ready def

Numerical conditions shared by every bounded pattern, with the threshold chosen before its carrier types, lengths, offsets, words or bin profile. · OAI.TorsionFreeZeroDivisors.PatternUniform.Ready

Ready structure

Conditions on real input/work stacks; arbitrary output suffixes remain permitted. The two query/scan tapes are empty at a row boundary. · OAI.UniqueGamesTheorem.Explicit.MachineProductField.Ready

Ready structure

The input-dependent data needed for one whole row, held invariant by all physical field executions. · OAI.UniqueGamesTheorem.Explicit.MachineProductRow.Ready

Ready structure

The complete representation needed at one odometer leaf. Remaining-digit stacks, output history, and retained unused headers are arbitrary. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceTupleBody.Ready

Ready structure

Guard-to-guard tape contract. Only scan and indexScan may contain residues. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.LoopState.Ready

realArcFourierCoefficient def

The real Fourier coefficient, including the centered zero frequency. · OAI.KadisonSimilarity.realArcFourierCoefficient

realArray def

Real finitely supported vectors in the usual ℓp space. · OAI.SeparableQuotient.LpFinite.realArray

realCartanInComplex def

The original real Cartan algebra inside its complexification. · OAI.PolycyclicRecognition.realCartanInComplex

realCartanRotation def

Removing rotational weights uses actual automorphisms of the original real Lie algebra, with inverse given by the negative parameter. · OAI.PolycyclicRecognition.realCartanRotation

realCartesianHarmonic def

Literal real polynomial equivalence: the split-coordinate real form is exactly the Cartesian harmonic space, not merely an abstract dimension copy. · OAI.HarmonicCounterexample.Control.realCartesianHarmonic

RealCharacterZeroExpansion structure

A genuine zero sequence, indexed with multiplicities. The Hadamard identity specifies its sum; no downstream prime-mass bound is included. · OAI.Ostmann.RealCharacterZeroExpansion

realCirclePathLift def

Canonical lift, used only to make the path increment independent of the choice of real branch. · OAI.CircleDomainRigidity.FiniteTransfer.realCirclePathLift

realComplexPart def

The canonical projection from real-linear maps to complex-linear maps. · OAI.Kaplansky.realComplexPart

realConjugatePolynomial def

Conjugation in split coordinates of a real Cartesian polynomial: it also interchanges z and bar-z, rather than merely conjugating their coefficients. · OAI.HarmonicCounterexample.ComplexAngular.realConjugatePolynomial

realCost def

Real-valued three-particle Coulomb cost on an arbitrary metric space. · OAI.Problem356.CoulombEstimates.realCost

realDensity def

Ordinary natural density over 2 ≤ n ≤ X, normalized by the real endpoint. · OAI.JointDickman.realDensity

realDensity def

The paper's real-endpoint ordinary density. The natural floor is harmless for the positive endpoints in the limit, and includes exactly 2 ≤ n ≤ X. · OAI.JointDickman.realDensity

realDistance def

Real graph distance, on the same literal original map. · OAI.FKCRT.Inventory.CoreTree.realDistance

RealEigenfunction structure

Classical representative of a real weak second-order eigenfunction. · OAI.SharpLiebThirring.SobolevProof.RealEigenfunction

realEmbeddingLinear def

The ordered real embedding, regarded as an integer-linear map. · OAI.DukePrimeDegree.realEmbeddingLinear

realEmbeddingMatrix def

Rows are basis vectors and columns are ordered real embeddings. · OAI.DukePrimeDegree.realEmbeddingMatrix

realEncode def

Literal messages at leaves, with no spin terminal transition. · OAI.DilutedSpinGlass.KernelTower.realEncode

realField def

Projection of a complex physical displacement to an actual real trial field. The map w will be real or imaginary part. · OAI.Elasticity.realField

realFormCLM def

Real-part projection on continuous forms, as a continuous linear map. · OAI.Mahler.realFormCLM

realFunctionalComplexification def

Complexification of a real functional on the self-adjoint part. · OAI.WeakPureInfiniteness.realFunctionalComplexification

realFunctionHom def

The real function algebra mapped into the ambient finite matrix algebra. · OAI.Kaplansky.realFunctionHom

realGram def

Real Gram pairing on a uniformly sampled finite space. · OAI.MUB6.realGram

RealGraphReduction structure

Polynomial-time graph reduction with a real independence-ratio threshold. · OAI.LargeIndependentSets.RealGraphReduction

realHarmonicTupleGenerator def

Real generators on the actual real harmonic spaces, not formal matrix copies with an assumed complexification. · OAI.HarmonicCounterexample.Control.realHarmonicTupleGenerator

realImagEquiv def

The fixed real form of a conjugation really complexifies to the entire complex space; this is an actual linear equivalence, not a dimension assumption. · OAI.HarmonicCounterexample.RealForm.realImagEquiv

realImaginaryDerivation def

The infinitesimal compact action as a real derivation. · OAI.PolycyclicRecognition.realImaginaryDerivation

realImaginaryDerivationLinear def

The imaginary derivation depends linearly on the real Cartan parameter. · OAI.PolycyclicRecognition.realImaginaryDerivationLinear

realImaginaryOperator def

The real imaginary semisimple operator, obtained from the actual complex weight spaces by real descent. · OAI.PolycyclicRecognition.realImaginaryOperator

realInner def

Real polarization; its bilinearity follows from the trace identities above. · OAI.TraceConeClassification.ExtendedTrace.realInner

RealizableForestPaths def

This range includes the union over all finite graph realizations. · OAI.TwoPointCorrelations.RealizableForestPaths

Realization structure

Starts and drift vectors of an actual local path. · OAI.QuantitativeVanDerWaerden.AnchoredPatterns.Realization

realization def

The proved one-step realization uses the actual endomap of the finite family. · OAI.LittleFinitistic.FiniteSelectionFamily.realization

realization def

The one fixed selecting bimodule realizes H on every finite-dimensional Y. This is an isomorphism in the full B-module derived category. · OAI.LittleFinitistic.SelectedRealization.realization

realization def

The source convention places the two summands in shifts b and b+3. · OAI.LittleFinitistic.SelectedShiftedRealization.realization

realization def

The canonical cell base-change is identified with the given apex by its universal property. · OAI.PartitionConsistency.ConservativeMonoids.Subobject.Square.realization

realization def

A positive augmentation-preserving chain map gives a strict functor. · OAI.Thomason.Construction.DirectedComplex.Hom.realization

realization def

Steiner's positive-table construction as a strict omega-category. · OAI.Thomason.Construction.DirectedComplex.realization

realizationBody def

The source's seven structural branches; evaluation tag 7 is excluded. · OAI.PureTypeSystem.Labelled.DoubleCode.realizationBody

realizationComponents def

The map to discrete edge components descends through actual paths. · OAI.Thomason.Construction.RealizationPrism.realizationComponents

realizationComponentsEquiv def

Actual path components of geometric realization are exactly the simplicial edge components, without a Kan or model-category hypothesis. · OAI.Thomason.Construction.RealizationPrism.realizationComponentsEquiv

realizationCone def

The actual global continuous cone map, with no KQ assumption. · OAI.Thomason.Construction.ConePrism.realizationCone

realizationCubeProductHomeomorph def

Iterated cylinder coordinates give an actual product homeomorphism when the right simplicial factor is a finite cube. · OAI.Thomason.Construction.realizationCubeProductHomeomorph

realizationCubeProductIso def

The canonical cube product comparison as an explicit isomorphism. · OAI.Thomason.Construction.realizationCubeProductIso

realizationDeformations def

Maps whose actual realizations admit a stationary deformation retraction. · OAI.Thomason.Construction.realizationDeformations

realizationFiniteProductHomeomorph def

The homeomorphism for finite parameter spaces retains both actual realized projections, by construction from the canonical comparison. · OAI.Thomason.Construction.realizationFiniteProductHomeomorph

realizationFunctor def

Strict realization of positive chain maps is a functor. · OAI.Thomason.Construction.DirectedComplex.realizationFunctor

realizationHomEquiv def

On the free globes, taking strict realization is a full and faithful bijection of positive chain maps with actual strict functors. · OAI.Thomason.Construction.DirectedGlobeComplex.realizationHomEquiv

realizationHomotopyEquiv def

Simplicial homotopy inverses give actual topological homotopy inverses. · OAI.Thomason.Construction.realizationHomotopyEquiv

realizationIso def

The two positive-table realizations are isomorphic as strict omega-categories. · OAI.Thomason.Construction.DirectedComplex.realizationIso

realizationNested def

The concrete two summands for one application of the actual shifted tensor. · OAI.LittleFinitistic.SelectedShiftedRealization.realizationNested

realizationProductComparison def

The canonical comparison uses the two actual realized projections. · OAI.Thomason.Construction.realizationProductComparison

realizationProductRightComparison def

Regard the actual product comparison as a map of colimit diagrams. · OAI.Thomason.Construction.realizationProductRightComparison

realizationProductRightMap def

The right-factor map on the product of actual realizations. · OAI.Thomason.Construction.realizationProductRightMap

realizationProductSubcomplexHomeomorph def

Any established product comparison restricts to an actual product homeomorphism on every subcomplex of its right factor. · OAI.Thomason.Construction.realizationProductSubcomplexHomeomorph

realizations def

Concrete realization count for one fixed immersed labeling. · OAI.TorsionFreeZeroDivisors.StageRealization.realizations

realizationSimplexProductHomeomorph def

Product realization with a simplex, with the actual two projections. · OAI.Thomason.Construction.realizationSimplexProductHomeomorph

realizationTime def

The usual affine coordinate on the realization of the simplicial interval. · OAI.Thomason.Construction.PrismBarycentric.realizationTime

realizationUnitHomotopy def

A continuous homotopy between realized simplicial maps gives an actual simplicial homotopy after the realization/singular unit. Its evaluation is exact; no weak-equivalence assertion about that unit is used. · OAI.Thomason.Construction.realizationUnitHomotopy

realizationWeakEquivalences def

Kan–Quillen weak equivalences as weak homotopy equivalences of geometric realizations. No model structure is presupposed by this definition. · OAI.Thomason.Construction.realizationWeakEquivalences

realizationWhiteheadEquiv def

A weak map between geometric realizations has a constructed continuous homotopy inverse. The proof uses the actual mapping-path fibration and the proved singular boundary lifts, without a Kan–Quillen model structure. · OAI.Thomason.Construction.realizationWhiteheadEquiv

Realize def

Satisfaction in an arbitrary relational structure, not necessarily externally well founded or transitive. In particular, E need not be Lean's membership. · OAI.PartitionConsistency.Formula.Realize

realize def

Expand a depth assignment into the actual leveled prescribed tree. Outside admissible assignments this is still total; no such assignment is used in the geometry or analytic theorems below. · OAI.DilutedSpinGlass.ReducedTopology.realize

realize def

Every germ follows its own port, retaining all repeated facet uses. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.PortAssembly.realize

realizeBoundaryAttachment def

Realization carries a simultaneous simplicial boundary attachment to the corresponding simultaneous topological attachment. · OAI.Thomason.Construction.RealizationPrism.realizeBoundaryAttachment

Realized def

A word is realized by a finite genus-zero relator ribbon. The dart and vertex sets range over ordinary finite types; the coefficient group has arbitrary universe. · OAI.KervaireRibbon.Realized

Realized def

The fixed-pattern event before any scale or bins are chosen. · OAI.TorsionFreeZeroDivisors.ActualPatternEvent.Realized

Realized def

Literal whole-path comparisons, before cutting at the marked vertices. · OAI.TorsionFreeZeroDivisors.OriginalPattern.Realized

realizeDart def

Pick an actual occurrence realizing each directed edge; this is injective because both endpoint vertices of an occurrence are fixed. · OAI.TorsionFreeZeroDivisors.MapEuler.realizeDart

RealizedBand structure

Entire finite band, including all split branch ports. The exposed ports are the complement of the internal sewing, not chosen subsets. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.RealizedBand

realizedCellComplexOfMono def

The topological skeletal presentation with the original realized boundary arrows. · OAI.Thomason.Construction.RealizationPrism.realizedCellComplexOfMono

realizedChart def

Map a barycentric chart to the actual geometric realization of the prism. · OAI.Thomason.Construction.PrismBarycentric.realizedChart

realizedCoproductData def

Actual realization of a coproduct inherits the componentwise deformations. · OAI.Thomason.Construction.realizedCoproductData

realizedCylinderComparison def

The continuous comparison from the ordinary cylinder to the realized Gray cylinder. · OAI.Thomason.Construction.RightGrayInput.realizedCylinderComparison

realizedDeformation def

The deformation is stationary on the entire realized horn, so it can be glued after arbitrary horn attachments. · OAI.Thomason.Construction.HornBarycentric.realizedDeformation

realizedFiberAlongInterval def

Attach the actual realized base edge to the initial point fibre. · OAI.Thomason.Construction.realizedFiberAlongInterval

realizedHomotopy def

A strict Gray homotopy induces an actual continuous homotopy of realizations. · OAI.Thomason.Construction.RightGrayInput.realizedHomotopy

realizedHomotopyEquivOfSimplicial def

The specified simplicial homotopy inverses realize to actual continuous homotopy inverses, with those same forward and inverse maps. · OAI.Thomason.Construction.realizedHomotopyEquivOfSimplicial

realizedHornData def

The constructed strong deformation data for a realized horn inclusion. · OAI.Thomason.Construction.realizedHornData

realizedInternalPathComparison def

Realizing the evaluation cylinder and currying gives the genuine compact-open path of every realized internal path. · OAI.Thomason.Construction.realizedInternalPathComparison

realizedIntervalPath def

The path realized by a base interval, with the exact image vertices. · OAI.Thomason.Construction.realizedIntervalPath

realizedMappingPathComparison def

Realization of the actual simplicial pullback maps to the actual continuous mapping-path pullback, without asserting preservation of pullbacks. · OAI.Thomason.Construction.realizedMappingPathComparison

realizedPointFiberComparison def

The canonical comparison attaches the constant path at the specified vertex. No preservation of pullbacks by realization is assumed. · OAI.Thomason.Construction.realizedPointFiberComparison

realizedPointFiberHomeomorph def

Realization commutes homeomorphically with a fibre over a simplicial vertex, using the closed subcomplex embedding. · OAI.Thomason.Construction.realizedPointFiberHomeomorph

realizedPointIsTerminal def

The realization of the zero simplex is an actual terminal topological space. · OAI.Thomason.Construction.realizedPointIsTerminal

realizedRetraction def

The actual continuous retraction from a realized simplex to its horn. · OAI.Thomason.Construction.HornBarycentric.realizedRetraction

realizedSequenceData def

A countable simplicial sequence of realized deformations has a realized stationary deformation on its actual colimit inclusion. · OAI.Thomason.Construction.realizedSequenceData

realizedSigns def

Realized patterns on any specified parameter subset. · OAI.QuantitativeVanDerWaerden.realizedSigns

realizedSimplexHomotopyFiberMap def

Restriction to a base simplex induces the literal continuous map of homotopy fibres, by projecting the point and applying the simplex map to its path. · OAI.Thomason.Construction.realizedSimplexHomotopyFiberMap

realizedSiteKernel def

Realizing a finite site kernel on one independent configuration. · OAI.JointDickman.realizedSiteKernel

realizedTestAcceptance def

The actual test using one uniform Fin D tape entry per noise coordinate. · OAI.MaxCutGames.Foundations.Hastad.realizedTestAcceptance

realizedTestAcceptance def

The actual test using one uniform Fin D tape entry per noise coordinate. · OAI.UniqueGamesTheorem.Foundations.Hastad.realizedTestAcceptance

realizeHomSection def

Interpret a section of the constructed core as linear maps between its original bundles, using the core's selected chart in each fiber. · OAI.Kaplansky.realizeHomSection

Realizes def

The finite-machine adapter has to prove this execution specification for any supplied partial-recursive program. It is not an extra main hypothesis. · OAI.FiniteAlgebraicEnvelopes.Compiler.Word.Realizes

realJordanInterior def

A physical Jordan disk in the real coordinates used for the energy and patching calculation. · OAI.CircleDomainRigidity.FiniteTransfer.realJordanInterior

realKnots def

The initial endpoint followed by the right endpoints of the cells. · OAI.ThompsonNonamenability.GapChain.realKnots

RealL1 abbrev

The full infinite-dimensional real ℓ₁ space. · OAI.TreeEdit.RealL1

realLinearCharacter def

A real-valued complex linear character regarded as a real character. · OAI.DukePrimeDegree.realLinearCharacter

realLocalValuation def

Regard a nonnegative valuation as an extended-real valuation. · OAI.CartierSections.realLocalValuation

realLongAverage def

The mean on the dyadic interval (X, 2X]. · OAI.JointDickman.PublishedInputs.realLongAverage

realMatrix def

Interpret a rational matrix as a real matrix. · OAI.MUB6.Certificate.realMatrix

realMatrixOperator def

A finite real matrix, acting with its Hilbert-space operator norm. · OAI.KadisonSimilarity.realMatrixOperator

realMeanSeminorm def

Construct, rather than assume, a real seminorm from rational data. · OAI.GammaFPP.realMeanSeminorm

RealNormed structure

A local bundle used only to recurse over the finite tree height. · OAI.DiamondDistortion.StartModel.RealNormed

RealPacketFamily structure

A finite literal real smooth packet list. Repeated frequencies are permitted; no orthogonality is part of this bookkeeping structure. · OAI.ThreeTorus.RealPacketFamily

realPacketPair def

A pair of opposite angular modes with real coefficient. · OAI.ThreeTorus.realPacketPair

realPhaseHom def

Additive Cartan parameters act through the compact rotation group. · OAI.PolycyclicRecognition.realPhaseHom

realPhaseVector def

The genuine rotations by real Cartan parameters lie in the compact subgroup preserving the real form. · OAI.PolycyclicRecognition.realPhaseVector

realPositiveConeNullIdeal def

The closed ideal attached to the real weak-star cone in Lemma 7.17(ii). · OAI.WeakPureInfiniteness.realPositiveConeNullIdeal

realRetainedWeight def

The literal real lift of a retained coefficient weight. The smooth cutoff is taken before the final passage integer, as in the source. · OAI.ThreeTorus.realRetainedWeight

realRotationMap def

The real rotation, obtained by taking real coordinates of the complex weight rotation. · OAI.PolycyclicRecognition.realRotationMap

realRotationTorus def

Those compact diagonal automorphisms which preserve the original real Lie algebra, expressed by commutation with conjugation. · OAI.PolycyclicRecognition.realRotationTorus

realSelfAdjointEntryCLM def

Real entry evaluation on the actual symmetric-matrix normed space. · OAI.SharpLiebThirring.ActionProof.realSelfAdjointEntryCLM

RealSeparated def

Functional Hausdorffness is convenient for iterated attachments: separating functions, unlike chosen open neighborhoods, extend across every new disk. · OAI.GroupRingTopology.Attachment.RealSeparated

RealSeqBounded def

A real sequence has a uniform absolute-value bound. · OAI.Tingley.RealSeqBounded

realShadowCartanHom def

The Cartan factor retains its original Lie bracket. · OAI.PolycyclicRecognition.realShadowCartanHom

realShadowKernel def

The same weight kernel, now inside the modified Lie algebra. · OAI.PolycyclicRecognition.realShadowKernel

realShadowKernelEquiv def

On the nilpotent ideal the modified and original brackets agree. · OAI.PolycyclicRecognition.realShadowKernelEquiv

realShortAverage def

A real short average, with the manuscript's open-left endpoint convention. · OAI.JointDickman.PublishedInputs.realShortAverage

RealShortIntervalInput def

The mean-square comparison needed from Matomäki and Radziwiłł, Multiplicative functions in short intervals, Ann. of Math. 183 (2016), Theorem 1, DOI 10.4007/annals.2016.183.3.6, arXiv:1501.04585v4. · OAI.JointDickman.PublishedInputs.RealShortIntervalInput

realSign def

Archived complete binary-sign support, reused without a new-result claim. The real-valued sign character, not a replacement for the permutation. · OAI.CoordinateSweeps.realSign

realSmooth def

The real smooth coefficients are embedded pointwise, not abstractly. · OAI.ElasticityPhysicalSolvability.realSmooth

realSourceGroundEnergy def

The finite source bottom, now known to be a real number. · OAI.ContinuumCoulomb.realSourceGroundEnergy

realTorusAutomorphism def

Actual real Lie automorphisms, indexed by the compact torus. · OAI.PolycyclicRecognition.realTorusAutomorphism

realTorusLinearAction def

The underlying linear representation of the compact group. · OAI.PolycyclicRecognition.realTorusLinearAction

realTorusOperator def

The compact torus action restricted to the original real form. · OAI.PolycyclicRecognition.realTorusOperator

realTorusUnitAction def

The compact rotations as units of the continuous endomorphism algebra. · OAI.PolycyclicRecognition.realTorusUnitAction

realTrace def

The real part of the trace, viewed as a real-linear functional. · OAI.KadisonKastler.realTrace

realTracialHilbertSequence def

Real algebra vectors give the actual fixed orthonormal sequence in the centered tracial Hilbert space. · OAI.KadisonSimilarity.realTracialHilbertSequence

RealTracialSequence structure

A real orthonormal sequence expressed entirely in the tracial algebra. Its Hilbert realization is derived below. · OAI.KadisonSimilarity.RealTracialSequence

realTracialSequenceOfHaar def

The algebraic real orthonormal sequence required by the tracial matrix tower is constructed from the Haar unitary. · OAI.KadisonSimilarity.realTracialSequenceOfHaar

realULim def

The chosen real limit along an ultrafilter. Use tendsto_realULim to establish that this value is a limit for a bounded sequence. · OAI.Tingley.realULim

RealValuedMeasurableContinuum def

The probability measure is defined on the full powerset of the continuum. The last clause quantifies over every smaller index set (represented as a subset of ℝ); ENNReal tsum is the supremum of finite subsums. · OAI.SeparableQuotient.RealValuedMeasurableContinuum

realVectorState def

Real part of the ambient vector state, as a real-linear functional. · OAI.KadisonKastler.realVectorState

rearm def

Routes the packed choice back to the walk head (outer i replaced by the packed rep index). · OAI.ExactDerandomization.Reed.Board.rearm

reassociateTorus def

Reassociation retains the same translation, linear and stable coordinates. · OAI.FiniteAlgebraicEnvelopes.LimitAction.reassociateTorus

rebase def

Simplifying two bases via coefficient equivalence. · OAI.GeneralMahler.rebase

rebase def

Change the common initial corner by a displayed equivalence. · OAI.Generator.Tracial.EquivalentPartition.rebase

rebase def

Equal supporting diagrams can be identified without changing any name code. This explicit version has a definitionally unchanged underlying name. · OAI.PartitionConsistency.ProductForcing.LocalName.rebase

rebase def

A patch can be installed after other patches which have not changed its closed domain. Its actual activity and all clipping data are retained. · OAI.PlanarL1.Columns.Model.SetPatch.rebase

rebase def

Rebase a complete chain at a time strictly between its collisions. The integer relabeling is forced by the designated initial flight. · OAI.TriangularBilliards.FlightChain.rebase

rebasedRadialOperator def

The covector coefficient of a radial fit after changing its input frame from M to N. · OAI.RegularPointCoordinates.rebasedRadialOperator

rebaseGlued def

Changing the root inside its component changes only the proof that a representative belongs to the component. · OAI.PartitionConsistency.LocalDiagramSystem.System.rebaseGlued

rebaseMap def

The actual unique lift between based local-geodesic covers. · OAI.HyperbolicObstruction.LocalGeodesic.rebaseMap

rebaseUnit def

Equal-rank fibers may be based on different original facets. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.rebaseUnit

receiveSecond def

The second graph remainder completes the final lookup in the fixed table. The square return ports are ordered second, first, as in stepLookup. · OAI.MaxCutGames.Foundations.PCP.ExpanderRowControl.receiveSecond

receiveSecond def

The second graph remainder completes the final lookup in the fixed table. The square return ports are ordered second, first, as in stepLookup. · OAI.UniqueGamesTheorem.Foundations.PCP.ExpanderRowControl.receiveSecond

recenteredReferenceData abbrev

Literal recentering of original physical reference data. · OAI.LipschitzHilbert.PacketLatticeData.recenteredReferenceData

recenteredReferenceData abbrev

Literal recentering of original physical reference data. · OAI.LipschitzHilbert.SourceLabeledLattice.recenteredReferenceData

recenteredReferenceSelection def

Original packet indices are selected through the exact reference-label map. · OAI.LipschitzHilbert.PacketLatticeData.recenteredReferenceSelection

recenteredReferenceSelection def

Original packet indices are selected through the exact reference-label map. · OAI.LipschitzHilbert.SourceLabeledLattice.recenteredReferenceSelection

recentNames def

The deterministic, diameter-independent list at a node. · OAI.PlanarL1.Columns.Model.Point.RadialTree.recentNames

reciprocal def

The reciprocal is the pointwise reciprocal, not an abstract inverse assumption. · OAI.ElasticityCoeffBounds.reciprocal

reciprocalChart def

Reciprocal projective chart of the Gauss covector. In the chart where coordinate i dominates the normal, its image lies in a fixed cube. · OAI.AffineBernstein.reciprocalChart

reciprocalFirst def

The FIRST reciprocal degree moment of the concrete symmetric-group irreducible family. Specht.degree is the actual complex dimension. · OAI.Thorp.Current.reciprocalFirst

ReciprocalTerminalPowerBound def

The terminal estimate needed at one fixed differencing order. · OAI.Problem337.ReciprocalTerminalPowerBound

recital def

Enough to test against dual relations. · OAI.HadwigerCounterexample.PredictionClass.recital

recode def

Reinterpret equal input/output words, with no additional computation. · OAI.RationalHitting.StackLanguage.Procedure.recode

Recognizes def

A single recognition predicate handles both distinct words and every nontrivial splice, so the symbol-scale induction cannot drop an offset. · OAI.SmoothObstruction.WordConcatenation.Recognizes

recoil def

Reciprocal sequence, with finite truncated calls. · OAI.PowerSaving.Spectral.recoil

ReComp abbrev

Reduce repetition in local uniformity proofs. · OAI.Fifths.Comp.ReComp

reconstruct def

Affine reconstruction and its coordinate readout. · OAI.IntegralCharacterVarieties.ExactSequenceChart.MatrixSplitData.reconstruct

reconstruct def

Reconstruct from a projective base point and a normalized chart fiber. · OAI.Nagata.W16.AssociatedBundle.reconstruct

reconstructedGame def

The reconstructed outer question law is the actual pushforward of the inner question law under the separate local reconstruction functions. · OAI.MaxCutGames.Soundness.RepeatedGameBounds.Simulation.reconstructedGame

reconstructedGame def

The reconstructed outer question law is the actual pushforward of the inner question law under the separate local reconstruction functions. · OAI.UniqueGamesTheorem.Soundness.RepeatedGameBounds.Simulation.reconstructedGame

reconstructedOperator def

The scalar kernel gives a bounded operator at every good point, and zero on the exceptional null set. · OAI.KadisonKastler.ScalarOperatorKernel.reconstructedOperator

reconstructedPivot def

The positive integer retained by a transferred frequency, when its integrality and range indicators hold. · OAI.Ostmann.reconstructedPivot

reconstructedRepresentative def

A bounded measurable representative reconstructed from the scalar kernel. · OAI.KadisonKastler.ScalarOperatorKernel.reconstructedRepresentative

reconstructedRule def

A concrete secretary rule from a hidden rule and a precommitted decoder. The seed type is finite, but its law is arbitrary and need not be uniform. · OAI.MatroidProphet.SecretaryReplay.reconstructedRule

reconstructFirst def

The first reconstruction receives only fixed outside occurrence/name data, the fixed raw coefficients, and the inside occurrence draw. · OAI.MaxCutGames.Soundness.ZeroInformation.reconstructFirst

reconstructFirst def

The first reconstruction receives only fixed outside occurrence/name data, the fixed raw coefficients, and the inside occurrence draw. · OAI.UniqueGamesTheorem.Soundness.ZeroInformation.reconstructFirst

reconstructionSuccess def

Independent masks generate independent complete traces; no retained positions, boundaries or other latent variables are revealed to A. · OAI.TraceReconstruction.reconstructionSuccess

reconstructSecond def

The second reconstruction receives no inside occurrence: only inside names. · OAI.MaxCutGames.Soundness.ZeroInformation.reconstructSecond

reconstructSecond def

The second reconstruction receives no inside occurrence: only inside names. · OAI.UniqueGamesTheorem.Soundness.ZeroInformation.reconstructSecond

recontext def

Context conversion retains every nonvariable core and all sort choices. · OAI.PureTypeSystem.TypingTree.recontext

record def

The actual record appended by this adaptive step. · OAI.ReflexiveFixedPoints.CandidateContext.ParentReturn.record

recordCount def

Count record delimiters, skipping both the escape bit and payload bit. · OAI.RationalHitting.StackLanguage.recordCount

recordedWalk def

The full recorded finite-word path therefore retains the proved joint-history distribution, rather than merely its terminal marginal. · OAI.ContingencyTables.CompanionArithmeticStep.recordedWalk

recordedWalk def

The same finite step word records the newly sampled state, so the history law is physically realized by the concatenated step words. · OAI.ContingencyTables.CompanionBooleanPhase.recordedWalk

recordedWalk def

The full recorded finite-word path therefore retains the proved joint-history distribution, rather than merely its terminal marginal. · OAI.ContingencyTables.RootedArithmeticStep.recordedWalk

recordedWalk def

The same finite step word records the newly sampled state, so the history law is physically realized by the concatenated step words. · OAI.ContingencyTables.RootedBooleanPhase.recordedWalk

recordFirst def

The distinguished component is selected from the supplied record, not by factoring the requested integer a second time. · OAI.ExactQuantumFactoring.recordFirst

recordPhi def

The closed statistics are evaluated over the supplied exponent record, not by enumerating residues or invoking a factorization oracle. · OAI.ExactQuantumFactoring.recordPhi

recordPrimes def

Expand ONLY the supplied exponent record. No factoring operation is hidden in the list used by the retrospective controller. · OAI.ExactQuantumFactoring.recordPrimes

Records def

The finite set encodes precisely those test coordinates whose pivot has mark one. · OAI.MatroidProphet.MarkedPivots.Records

Records def

The entire recording criterion is the exit window at the occurrence start. · OAI.Superstring.Hierarchical.Layer.Records

records def

The full candidate group retains every record, including discarded ones. · OAI.ReflexiveFixedPoints.CandidateContext.ParentReturn.records

recordTapes def

Full caller frame after the clause fields have been consumed. · OAI.MaxCutGames.Foundations.PCP.RawInitialMachineReadClause.recordTapes

recordTapes def

Full caller frame after the clause fields have been consumed. · OAI.UniqueGamesTheorem.Foundations.PCP.RawInitialMachineReadClause.recordTapes

recordTypeAt def

A record's terminal offspring and its increase over the previous running maximum. This definition uses only the first t offspring, even for streams which have no later records. · OAI.FKCRT.TreeCoding.recordTypeAt

recordWords def

Words: tag, name, occurrence, followed by three split coefficient pairs. Padding is zero. A single record uses only the first coefficient pair. · OAI.MaxCutGames.Reduction.CanonicalEncoding.recordWords

recordWords def

Words: tag, name, occurrence, followed by three split coefficient pairs. Padding is zero. A single record uses only the first coefficient pair. · OAI.UniqueGamesTheorem.Reduction.CanonicalEncoding.recordWords

recover def

Restriction to all residual occurrences is an invertible matching, not a choice of matching with the right cardinality. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.CyclicInterface.recover

recoveredLabels def

Search only the polynomially many Euclidean convergents. Rational labels not satisfying the exact bin rule are rejected, not rounded into a candidate. · OAI.ExactQuantumFactoring.OrderTrial.recoveredLabels

recoverLabel def

The operational decoder either returns the unique exact bin label or rejects. · OAI.ExactQuantumFactoring.OrderTrial.recoverLabel

Recovers def

Each path of a retained tree recovers a path of the original tree after the already collected coordinates are added. · OAI.LeanBlast.SecondKahnKalai.ProbabilityTree.Recovers

recoveryBackwardRow def

The backward row is formed from the adjoint polynomial. · OAI.KadisonKastler.recoveryBackwardRow

recoveryCompression def

The compressed square used by the scalar recovery argument. · OAI.KadisonKastler.recoveryCompression

recoveryContraction def

A concrete contraction obtained from the adjoint of the clipped polar approximation inside the projection corner. · OAI.KadisonKastler.recoveryContraction

recoveryCornerContraction def

The original-factor contraction obtained from the recovery operator. · OAI.KadisonKastler.recoveryCornerContraction

recoveryCutoff def

The cutoff whose scaled trace loss is exactly one tenth. · OAI.KadisonKastler.recoveryCutoff

recoveryForwardRow def

The actual forward row of the normalized polynomial. · OAI.KadisonKastler.recoveryForwardRow

recoveryObservable def

The expression averaged over the finite random choices in the scalar recovery argument. · OAI.KadisonKastler.recoveryObservable

recoverySequenceBackwardRows def

Backward polynomial rows at the same sequence coordinate. · OAI.KadisonKastler.recoverySequenceBackwardRows

recoverySequenceContractions def

The corner entries are actual elements of the original algebra. · OAI.KadisonKastler.recoverySequenceContractions

recoverySequenceForwardRows def

Forward polynomial rows for the unitary at a sequence coordinate. · OAI.KadisonKastler.recoverySequenceForwardRows

rect def

The same finite pure rectangle as the tensor grid, given by its literal integer-coordinate inequalities for the geometric continuation proof. · OAI.CriticalHoneycomb.FiniteContinuation.rect

Rectangle structure

A closed orthogonal rectangle; long and short are full side lengths. · OAI.LipschitzHilbert.Rectangle

rectangle def

Rectangle associated with a recurrent representative. · OAI.TwoWayComplementation.Diagram.rectangle

RectangleFrame structure

Disjoint source/target points and a separate intermediate hub. · OAI.OneWayLiveness.RectangleFrame

rectangleLineMass def

Integration over the coordinate rectangle; this is an explicit upper bound, not a definition of the original ball restriction. · OAI.BoltzmannNonuniqueness.GeometricSlice.rectangleLineMass

rectangleMaximum def

The finite-rectangle maximal function appearing in the needed specialization. · OAI.LipschitzHilbert.rectangleMaximum

RectangleRouche def

Ahlfors's Rouché corollary, specialized to the one family of rectangles used for the spectral count. No spectral conclusion is included in this input. · OAI.DefocusingNLS.RectangleRouche

RectanglesSurvive def

Nondegenerate rectangular words survive in the finite group. · OAI.Release075.RectanglesSurvive

rectangleSwap def

Transfer mass from the diagonal corners to the off-diagonal corners. · OAI.Problem358.FiniteCoupling.rectangleSwap

RectangularCompleteBound def

A uniform square-matrix bound for an operator-valued map between two spaces. · OAI.KadisonSimilarity.RectangularCompleteBound

rectangularCoordinateChart def

A coordinate permutation followed by currying. · OAI.PolycyclicRecognition.rectangularCoordinateChart

rectangularCorner def

A rectangular matrix as an upper-right corner of a square matrix. · OAI.KadisonKastler.rectangularCorner

rectangularDiagonalStarHom def

The diagonal of the triangular representation is a star representation. · OAI.KadisonSimilarity.rectangularDiagonalStarHom

rectangularMatrixOperator def

Finite rectangular operator matrices act between the two Hilbert sums. · OAI.KadisonSimilarity.rectangularMatrixOperator

rectangularRepresentation def

The bounded unital triangular homomorphism of a rectangular derivation. · OAI.KadisonSimilarity.rectangularRepresentation

RectangularSum abbrev

The Hilbert direct sum used in the rectangular triangular representation. · OAI.KadisonSimilarity.RectangularSum

rectangularTriangular def

The diagonal plus one upper-right block. · OAI.KadisonSimilarity.rectangularTriangular

rectBracket def

Lower block of the bracket of two off-diagonal skew matrices. · OAI.KervaireIncidenceCurrent.rectBracket

RectBridge def

A literal bottom-to-top bridge in width W+1 and height h+1, with terminal position summed. No endpoint or length information is assumed. · OAI.CriticalHoneycomb.FiniteContinuation.RectBridge

rectCoordBox def

A coordinate box, expressed in the Euclidean-space model used by the challenge. · OAI.Problem358.rectCoordBox

rectifyingMap def

The relation null-homotopy extends p over the first cone. · OAI.LittleFinitistic.RectificationTotal.rectifyingMap

recurrence def

The recurrent relation obtained by closing through traversals. · OAI.TwoWayComplementation.Diagram.recurrence

recursiveConvexSet def

The closed convex hull of the one actual recursively constructed sequence. · OAI.Tingley.recursiveConvexSet

recursiveFamily def

The next-level predicate is the disjunction of the literal row clauses. · OAI.Paper320.recursiveFamily

recursiveLeafLaw def

The actual mixed leaf oracle at arbitrary exact metadata. The one latent law governs the whole map X; no independent-anchor law is substituted. · OAI.TraceReconstruction.recursiveLeafLaw

recursivePrefix def

Ordinary finite-prefix recursion; no admissibility proof enters the data. · OAI.Tingley.recursivePrefix

recursiveSupportFactor def

Leaf bins and all intermediate arithmetic/range gates are real. They remain real under every conjugation in the frequency tree. · OAI.Ostmann.recursiveSupportFactor

recursiveTransferWeight def

The node indicator contains precisely the arithmetic support used for history rigidity. Other original range/bin cutoffs are carried by cutoff. · OAI.Ostmann.recursiveTransferWeight

redirectStmt def

Structural redirection changes only the destinations of goto and halt. · OAI.MaxCutGames.Foundations.Complexity.FinalCNFCleanup.redirectStmt

redirectStmt def

Structural redirection changes only the destinations of goto and halt. · OAI.UniqueGamesTheorem.Foundations.Complexity.FinalCNFCleanup.redirectStmt

Reduce def

A literal equation in either direction, used without any grade increase. · OAI.FiniteAlgebraicEnvelopes.Compiler.Word.Reduce

reduce def

The canonical distinct-site reduction; none denotes zero. · OAI.KadisonKastler.MatrixSiteWord.reduce

reduce def

Delete the prescribed caps of an idempotent. Its retained left endpoints are used as the smaller index set, avoiding an arbitrary numbering. · OAI.OneWayLiveness.Pairing.reduce

reduceBlock def

A single block's intercept correction and tagged retained record. · OAI.MaxCutGames.Decoder.TableKeys.reduceBlock

reduceBlock def

A single block's intercept correction and tagged retained record. · OAI.UniqueGamesTheorem.Decoder.TableKeys.reduceBlock

Reduced def

A walk has no immediate backtracking when it has no closed subwalk of length two. This formulation is hereditary under taking subwalks. · OAI.SimpleGraph.Walk.Reduced

Reduced def

Both kinds of shortening forbidden in a least marked finite certificate. · OAI.TorsionFreeZeroDivisors.FinitePicture.Data.Reduced

ReducedBagTree structure

Bags on a finite tree. Occurrences form the subtree above their unique highest occurrence top; the interval axiom is precisely subtree convexity. The intersections on nonroot edges have size strictly below the bag bound; this is the … · OAI.ParityWL.ReducedBagTree

ReducedCohomologyInput def

The needed Hilbert-space specialization of Corollary 1.9. Continuity and absence of invariant vectors are hypotheses of the published theorem. · OAI.PolycyclicRecognition.ReducedCohomologyInput

reducedCreator def

Insert the centered first letter and then use the reduced-word inclusion. · OAI.KadisonSimilarity.reducedCreator

ReducedCutDecomposition structure

Standard finite tree-decomposition cuts: the incident outer sides and the bag partition the graph, and each cut is separated by the intersection of the incident bags. The strict intersection bound is the reduced form. No parity or … · OAI.ParityWL.ReducedCutDecomposition

reducedDensity def

The finite-rank reduced density; independent of any chosen tensor expansion. · OAI.NuclearUltrapower.reducedDensity

reducedFrequency def

After cancelling the gcd, the numerator is an invertible residue class. · OAI.Problem337.reducedFrequency

ReducedGallery def

A genuinely fresh ranked gallery whose literal passage trace never repeats an ordered pair of occurrence labels. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.ReducedGallery

reducedHaarWord def

A reduced word with an initial Haar power. · OAI.KadisonKastler.reducedHaarWord

reducedInclusion def

The inclusion after reduction, viewed over the actual factor ring. · OAI.KLInvariance.EdgeImage.reducedInclusion

reducedIntegrand def

The explicit upper expression obtained after the quartic projection and quadratic axial reductions. It retains the two actual total derivative energies and the inner product of the two norm gradients. · OAI.PositiveEinsteinFour.reducedIntegrand

ReducedSet def

The reduction uses every integral member of U. · OAI.RationalHitting.LaurentReduction.ReducedSet

ReducedTopology inductive

Finite rooted topology with unary vertices suppressed. Children have fixed distinct positions, hence leaves remain labeled throughout realization. · OAI.DilutedSpinGlass.ReducedTopology

ReducedWalk def

Internal reduction of an edge walk; no condition is imposed at the closing seam. · OAI.Release075.ReducedWalk

ReducedWordCompressionModel structure

The vacuum/creation coordinates for multiplication by the second factor. The projection singles out words beginning in that factor, and before is its orthogonal complement. No operator norm estimate is part of this data. · OAI.KadisonSimilarity.ReducedWordCompressionModel

ReducedWordCreationModel structure

Grouping ordinary reduced words by their first factor. Both the tail space and the A-leading space are actual closed subspaces of the GNS space. · OAI.KadisonSimilarity.ReducedWordCreationModel

ReducedWordGenerated def

Being generated in the tracial Hilbert space is a density hypothesis, not an operator conclusion. · OAI.KadisonSimilarity.ReducedWordGenerated

reducedWordVectors def

The actual alternating vectors whose span is dense in a reduced product. · OAI.KadisonSimilarity.reducedWordVectors

reduceSteps def

Reduction of each left move, preserving its signed base charge. · OAI.CoarseAssembly.Congruence.reduceSteps

ReducesToEmpty inductive

Literal finite surgery trace, including the empty terminal picture. This records no geometric realization or implicit essentiality hypothesis. · OAI.TorsionFreeZeroDivisors.SphericalPicture.ReducesToEmpty

reduceSubspace def

The image of a subspace intersected with Sᗮ in the quotient. · OAI.SidorenkoCounterexample.reduceSubspace

reduceTree def

Explicit good-node pruning and normalization. The depth is preserved in the result type. Every retained label is the actual fragment of its original label. · OAI.LeanBlast.SecondKahnKalai.reduceTree

reducingExtension def

Extension by zero of the operator under study. · OAI.KadisonSimilarity.reducingExtension

reducingRestriction def

Restriction to a closed subspace invariant under the given star algebra. · OAI.KadisonKastler.reducingRestriction

reduction def

Satisfiability equivalence plus an actual raw-bit machine; no external verifier, parsing-time assertion, or recoding hypothesis is an input. · OAI.MaxCutGames.BinaryInputReduction.reduction

reduction def

Choose the parity error only after the matrix tuple length is fixed, then compose the actual raw-input, matrix-test, and final graph machines. · OAI.MaxCutGames.FromMatrixGap.reduction

reduction def

Satisfiability equivalence plus an actual raw-bit machine; no external verifier, parsing-time assertion, or recoding hypothesis is an input. · OAI.UniqueGamesTheorem.BinaryInputReduction.reduction

reduction def

Choose the parity error only after the matrix tuple length is fixed, then compose the actual raw-input, matrix-test, and final graph machines. · OAI.UniqueGamesTheorem.FromMatrixGap.reduction

ReductionAt inductive

The alternatives furnished by the spherical positive-height reduction and the nonspherical harmonic-minimum reduction. The split case expresses the invisible direct factor, including its normalization property. · OAI.HarmonicArtin.ParabolicIntersections.ReductionAt

reductionKernel def

The copy of an isotropic space inside its orthogonal complement. · OAI.SidorenkoCounterexample.reductionKernel

ReductionStatement def

Literal main objective, with arbitrary universes, arbitrary metric spaces, repeated initial positions, and one policy and additive constant for all finite inputs. Real constants are finite; an embedding witnesses at least k+1 points. The … · OAI.KServer.ReductionStatement

ReductionStep def

Reduction of traces, not a choice of ordered representative. · OAI.EilenbergGanea.TraceWords.ReductionStep

reductionWord def

The literal binary output of the compressed-computation graph reduction. The separate cost obligation is a genuine ComputesWithin statement. · OAI.WLIdentification.Machine.reductionWord

Reel structure

Array of strokes; slots read for t < beats. · OAI.PowerSaving.Slant.Reel

rees def

The extended Rees algebra with integer-indexed, increasing convention. · OAI.StableCoordinate.FilteredRees.rees

reesDerivation def

Restriction of the shifted Laurent derivation to the Rees algebra. · OAI.StableCoordinate.FilteredRees.reesDerivation

reesMap def

The homogeneous polynomial map into the extended Rees algebra. · OAI.StableCoordinate.GradedPresentationMap.reesMap

reference def

One feasible reference bijection is a permitted choice of identifications of the remaining input and output sites (02-preliminaries:5). · OAI.CoordinateSweeps.Grid.Holes.reference

referenceChain def

The exact target row is the source Metropolis chain, slowed by its dyadic neighbor-code padding. The stationary distribution is unchanged. · OAI.ContingencyTables.CompanionBooleanPhase.referenceChain

referenceChain def

The exact target row is the source Metropolis chain, slowed by its dyadic neighbor-code padding. The stationary distribution is unchanged. · OAI.ContingencyTables.RootedBooleanPhase.referenceChain

referenceComponent def

The component top inside the same common reference geometry. · OAI.LipschitzHilbert.SourceLabeledLattice.referenceComponent

referenceDivergence def

A proposed fresh split at depth d along a reference path. Every child containing any reference through the chosen prefix is forbidden. · OAI.DilutedSpinGlass.PrescribedTree.referenceDivergence

referenceFamily def

The reference lattice derives both dimensionless bounds from its cone. No bound on the slopes of the original packet family is used. · OAI.LipschitzHilbert.PacketLatticeData.referenceFamily

referenceFamily def

Only labels in the fixed short reference cone remain. All spatial items in those labels retain the original lattice and profile. · OAI.LipschitzHilbert.SourceLabeledLattice.referenceFamily

referenceFibreEquiv def

Actual retained completion fibre to the fixed-reference dense block. · OAI.ContingencyTables.SmallGraphProfiles.referenceFibreEquiv

referenceHistoryExpectation def

The actual signed history sum with a retained test. Each fresh branch carries its probability kernels, and every further extension transports ALL retained variables with oldSample. Old-leaf choices introduce no new variable. · OAI.DilutedSpinGlass.PrescribedTree.referenceHistoryExpectation

referenceHistoryTotal def

The signed sum over the actual sequential old-or-fresh choices. Every fresh choice changes the entire current tree with grow, and transports all previous references. No constant normalization is built into this recursion. · OAI.DilutedSpinGlass.PrescribedTree.referenceHistoryTotal

referenceHorizontalDilation def

The same horizontal scaling in the reference packet convention (y,x). · OAI.LipschitzHilbert.referenceHorizontalDilation

referenceInput def

The empty input fixes the spatial normalization of the repeated body. · OAI.ForcedComputation.BalancedPlanar.referenceInput

referenceItems def

Retain the common scale/label geometry while supplying actual packet sites and profiles from a subcollection or disjoint union. · OAI.LipschitzHilbert.SourceLabeledLattice.referenceItems

referenceLinear def

Exact middle-time complementary matching. No determinant or angle nondegeneracy is assumed: it follows from the splitting identities. · OAI.SmoothObstruction.FiniteGreen.Splitting.referenceLinear

referenceMaskedPacket def

The actual top mask multiplies the original reference-line packet. · OAI.LipschitzHilbert.SlopeBox.referenceMaskedPacket

referencePartial def

Partial endpoint matrix written on a single placement set using the reference sweep; every variable is a literal conditional path probability. · OAI.CoordinateSweeps.Grid.Holes.referencePartial

ReferencePlan inductive

A finite color prescription. The deepest-reference witness is only positional data; exists_deepest_reference supplies it for every new leaf. · OAI.DilutedSpinGlass.PrescribedTree.ReferencePlan

referencePlanNormalizer def

The deterministic product of one-step reference factors. · OAI.DilutedSpinGlass.PrescribedTree.referencePlanNormalizer

referencePlanRefs def

The actual final set of leaf positions in a finite reference prescription. · OAI.DilutedSpinGlass.PrescribedTree.referencePlanRefs

referencePoint def

The point at horizontal coordinate w.1 on the reference line through z. · OAI.LipschitzHilbert.SlopeBox.referencePoint

referenceSelector def

The local family carries the original selector as a function of the physical direction. Only its label type changes. · OAI.LipschitzHilbert.PacketLatticeData.referenceSelector

refine def

Every edge is replaced by countably many internally disjoint two-edge paths. All old vertices remain, and each pair (edge,index) is a new midpoint. · OAI.ForestSpace.Diamond.Network.refine

refine def

Refinement is performed on the common initial corner and transported. · OAI.Generator.Tracial.EquivalentPartition.refine

refineBlocks def

All blocks are replaced by their chosen finite subpartitions. · OAI.Generator.Tracial.CornerPartition.refineBlocks

refinedAtomicBand def

This named-root recoloring is obtained from the exact common-atom gallery, not existentially chosen from its rank and endpoint signature. · OAI.IntegralCharacterVarieties.TwoFlagBand.RankShape.refinedAtomicBand

refinedAtomicGraft def

Complete occurrence assembly, with every untouched component of the literal band double a DISTINCT fresh strip disk. No new incidence is guessed. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.PortAssembly.refinedAtomicGraft

refinedBoundaryEnumeration def

Actual marked inverse-band boundary enumeration. Short-root injection is proved from the atomic row/column topology; exclusion of principal children is derived from the author's proper-seam and maximal-rank hypotheses. No cyclic coverage, … · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.refinedBoundaryEnumeration

refinedCutDiagram def

Literal inverse-band occurrence diagram with a distinct lower-rank disk for each uncontacted strip. ALL circles are taken from the actual boundary successor; none is supplied as a cyclic-template coverage assumption. The full solution lift … · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.refinedCutDiagram

refinedCutHandleValues def

The original handles are retained except for the last handle of the selected parent. New strip facets are disks. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.refinedCutHandleValues

refinedCutOldSideValues def

Side values of the cut diagram expressed in the original seam ranks. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.refinedCutOldSideValues

refinedCutVertexFrames def

The original and atomic frames satisfy the local vertex equations before the global gauge changes. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.refinedCutVertexFrames

refinedDecoration def

Replace coarse atom colors by the actual connected strip roots on the SAME band. Contacted strips inherit old facets; untouched strips are separate disks. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.RealizedBand.refinedDecoration

refinedFacetRank def

Rank of a fresh connected strip, not a rank-only grouping of disks. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.RealizedBand.refinedFacetRank

refinedMap def

Subtracting the actual reference point makes the component assignment integrable, even when the original label measure is infinite. · OAI.PlanarL1.Columns.Model.SetSystem.refinedMap

refinedMarkedDiagram def

The marked actual inverse-band diagram, retaining the order and starting occurrence of every old boundary circle. Its mirror boundary is appended at the selected parent; every fresh strip is precisely one genus-zero disk. Exhaustion and … · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.refinedMarkedDiagram

refinedWordPreDiagram def

The old diagram with its ranks extended to the grafted facets. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.refinedWordPreDiagram

refineFamily def

Refining each chart by an independently chosen finite partition preserves its actual cube permutation and transports the tail with the original label. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Table.refineFamily

refinementFiberEntries def

Additional stabilizer blocks after refining a coarse block. · OAI.IntegralCharacterVarieties.refinementFiberEntries

refinementUpdate def

The nonzero branch of the refinement update, for a nonempty online list. An empty list returns the cone minimum directly. · OAI.EditApproximation.refinementUpdate

Refines inductive

Branches of the reference query are selected only by the values of answers actually read. The relation can therefore be composed stage by stage. · OAI.EditApproximation.BitQuery.Refines

Refines structure

A literal chart refinement: attach a finite prefix to each tail, then transport it with the original coordinate label. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Table.Refines

refl def

The machine-based definition includes the identity reduction. · OAI.ContinuumCoulomb.PolynomialManyOne.refl

reflect def

The real scalar spelling makes the coefficient budget transparent. · OAI.QAC.reflect

reflect def

Coordinate reflections preserve the entire flag subdivision. · OAI.SharpLiebThirring.CubeFlags.Flag.reflect

reflect def

Reflection in the real line spanned by the nonzero tangent e. In coordinates with e real this is complex conjugation, preserving the tangential component and reversing the normal component. · OAI.TriangularBilliards.reflect

reflect def

Reflection reverses the order of multiplication. · OAI.TwoWayComplementation.Diagram.reflect

reflectedDirection def

Unit outgoing velocity, with exactly the specular law. · OAI.TriangularBilliards.reflectedDirection

reflectedIndexEquiv def

Nonzero reflected mode lists are exactly their positive half and its negative copy. This equivalence is used BEFORE the ratio cutoff choices. · OAI.ThreeTorus.reflectedIndexEquiv

reflectedLabelFiber def

The full zero fibre, kept inside the positive finite category. · OAI.Thomason.Construction.FiniteReflectionInput.reflectedLabelFiber

reflectedOneLabelFiber def

The full one fibre, kept inside the positive finite category. · OAI.Thomason.Construction.FiniteReflectionInput.reflectedOneLabelFiber

reflectedS def

The opposite-copy contribution after unfolding through wall i. · OAI.TriangularBilliards.reflectedS

reflectedSmoothing def

A bounded global extension of the local disk mollifier. Away from vertices, at most one reflected term can contribute. This fact is proved geometrically instead of assuming a chart or an injectivity-radius bound. · OAI.TriangularBilliards.reflectedSmoothing

reflection def

The actual centered reflection obtained from the completed projection. · OAI.KadisonSimilarity.HalfTraceProjectionData.reflection

reflectionDuality def

A genuine simple-reflection integral Frobenius extension in characteristic zero, for the symmetric algebra of any representation carrying the reflection. · OAI.KLInvariance.ReflectionFrobenius.reflectionDuality

ReflectionLayer structure

A parallel layer of product-projection reflections, with disjoint supports. · OAI.QAC.ReflectionLayer

reflectionTerms def

Expansion of a disjoint reflection product. The real coefficients carry all multiplicities; no commutation of the middle projection is required. · OAI.QAC.reflectionTerms

reflectIsometry def

The reflection is a genuine real-linear Euclidean isometry. · OAI.TriangularBilliards.reflectIsometry

reflectiveInclusion def

The actual truncation inclusion, with the given reflection adjunction. · OAI.Thomason.Construction.FiniteReflectionInput.reflectiveInclusion

reflectLinearEquiv def

Fixed-length coefficient reversal is linear, unlike degree-dependent polynomial reversal. · OAI.RationalHitting.Expansion.reflectLinearEquiv

reflectZ def

Reflection in the horizontal plane. · OAI.StationaryNS.reflectZ

Reflexive def

Canonical (not merely abstract) reflexivity. · OAI.ForestSpace.Reflexive

refNoise def

The finite-time noise is monotone in the literal chart constant. · OAI.SmoothObstruction.SwitchScales.refNoise

refPt def

Reflection combined with swapping · OAI.GeneralMahler.SCal.SE.refPt

reframe def

Copy the coefficient data while enlarging bounds or changing the displayed function by pointwise equality. Unlike a type cast, this keeps all data projections transparent. · OAI.Problem335.FactorialExpansion.reframe

refreshedStrength def

The source's exact positive strength after residue refresh. · OAI.ThreeTorus.refreshedStrength

refreshResidueError def

Summed residue discrepancy, for one color. · OAI.ThreeTorus.refreshResidueError

refVector def

A vector of ordinary references; this constructor emits no graph nodes. · OAI.ExactQuantumFactoring.AIGCompiler.refVector

region def

A region is exactly the old identifier's inverse image in the full-column ambient model. No approximation of its boundary is used. · OAI.PlanarL1.Columns.Model.BState.region

region def

Border X has t free axes; Y the same mask with zero address removed. · OAI.PowerSaving.Cluster.region

region def

The closed region on which the prism cut passes through vertex j. · OAI.Thomason.Construction.PrismBarycentric.region

Register abbrev

A physical placement of a register uses distinct existing qubits. · OAI.ExactQuantumFactoring.Register

registerEmitInTime def

Timed witness retaining the exact 2*n+5-transition execution. · OAI.MaxCutGames.Foundations.Complexity.MachineRegisterEmit.registerEmitInTime

registerEmitInTime def

Timed witness retaining the exact 2*n+5-transition execution. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRegisterEmit.registerEmitInTime

registers def

A concrete polynomial TM2 parser for the entire source instance. · OAI.ContinuumCoulomb.SourcePrograms.registers

regroup def

Permutation changing the contiguous fiber chart to any given ambient chart. · OAI.CoordinateSweeps.FiberBlocks.regroup

regroup def

Finite regrouping of amplitudes. Used only for equal-mode terms; this constructor does not assert an orthogonality or spectral identity. · OAI.ThreeTorus.PassagePackets.regroup

regroupingIsometry def

The isometry sending matrix summand k to summand e (i,k). · OAI.KadisonKastler.FullMatrixCorner.regroupingIsometry

regroupPhase def

The actual Koszul regrouping phase from (V⊗V̄)^⊗l to V^⊗l⊗V̄^⊗l. It will cancel, not be dropped, under partial trace. · OAI.CoordinateSweeps.SignedTensor.regroupPhase

Regular def

Exactly the separated, interior distinct-coordinate domain of decor:average-definition. Ancestry/order restrictions may further restrict it. · OAI.DilutedSpinGlass.DepthAverage.Regular

Regular def

The ordinary forcing relation is regular: density of atomic forcing below p already implies forcing at p. This property is separately verified by the actual name recursion, not assumed for any target axiom. · OAI.PartitionConsistency.ForcingQuotient.AtomicData.Regular

regular structure

A measurable field with polynomial growth. · OAI.GeneralMahler.HMode.regular

regular def

Literal regular gap weight; zero coordinates are harmless placeholders. · OAI.KServer.HeldWeights.regular

regular def

Entrywise regular representation, flattening and fixed row-major reindexing. · OAI.RationalHitting.Generator.regular

regular def

Exact h on all regular coordinates; no clamped-log discrepancy occurs because the epoch geometry proves ai < A. · OAI.UniformKServer.EpochAlpha.regular

regular_open_coordinate_data def

The actual regular-neighborhood harmonic coordinates satisfy every hypothesis of the constructive open-map argument. · OAI.RegularPointCoordinates.regular_open_coordinate_data

regularActionIntegral def

The convolution convention uses the action at -t. · OAI.KadisonKastler.regularActionIntegral

regularAt def

Select the actual center-regular solution by its value at a positive late sphere. No arbitrary two-sided ODE solution is introduced. · OAI.HarmonicCounterexample.LinearODE.regularAt

regularAt def

Remove only the fixed-time collision set from the conull regular states. · OAI.TriangularBilliards.regularAt

regularBasisSummand def

The copy of H belonging to one scalar basis vector. · OAI.KadisonKastler.regularBasisSummand

RegularBilinearData structure

A record of the already constructed heat and harmonic objects. No chart, directional pinching, or metric-iteration conclusion is a field. · OAI.RegularPointCoordinates.RegularBilinearData

regularBochnerForm def

The full weak Bochner form retains both L² source terms. · OAI.RegularPointCoordinates.regularBochnerForm

RegularBound def

Bounds the current regular product at every actual state of the tree. · OAI.Ostmann.MovingSlotData.RegularBound

RegularBoundaryAtlas structure

A finite cover extracted from actual derivative-based charts. · OAI.MahlerStokes.RegularBoundaryAtlas

RegularBoundaryPatch structure

A bounded open portion of a genuine regular-level chart, inside U. · OAI.MahlerStokes.RegularBoundaryPatch

RegularCenterRadialData structure

Sources, normalizing frames and radial fits all use the same harmonic map. · OAI.RegularPointCoordinates.RegularCenterRadialData

regularCharacterRepresentation def

The actual unitary representation defining the dual-character action. · OAI.KadisonKastler.regularCharacterRepresentation

regularCharacterUnitary def

The multiplication unitary Q_a ξ(t) = exp(i a t) ξ(t). · OAI.KadisonKastler.regularCharacterUnitary

regularCoefficient def

Polynomial-normalized coefficient. This is constant in the Euclidean core. · OAI.HarmonicCounterexample.Cartesian.regularCoefficient

RegularCoherent def

The child regular slots contain the same compensation vector and the corresponding parent's regular slots, with their actual per-leaf allocation. · OAI.Ostmann.MovingSlotData.RegularCoherent

regularCommutantAlgebra def

The actual von Neumann algebra generated by the commutant generators. · OAI.KadisonKastler.regularCommutantAlgebra

regularCommutantGenerators def

The concrete generators on the commutant side of the regular crossed-product commutation theorem. · OAI.KadisonKastler.regularCommutantGenerators

RegularCompactCutoff structure

Qualitative cutoff data, without a fixed ratio between the two radii. · OAI.RegularPointCoordinates.RegularCompactCutoff

regularConstantHom def

The constant copy of B(H) on a Hilbert-valued L² space. · OAI.KadisonKastler.regularConstantHom

regularConstantOperator def

Pointwise action of a constant bounded operator on a Hilbert-valued L² space. · OAI.KadisonKastler.regularConstantOperator

regularConstantUnitary def

The constant copy of a unitary as an actual unitary on L². · OAI.KadisonKastler.regularConstantUnitary

regularCoreAction def

The common regular action U_s ⊗ ρ_s, without any fixed-point premise. · OAI.KadisonKastler.regularCoreAction

regularCoreAlgebra def

The von Neumann algebra generated by the regular representation. · OAI.KadisonKastler.regularCoreAlgebra

regularCoreDiagonalAlgebra def

Adjoin the full scalar diagonal to the actual regular core. · OAI.KadisonKastler.regularCoreDiagonalAlgebra

regularCoreEmbedding def

The regular embedding π(x) = gauge* · constant(x) · gauge. · OAI.KadisonKastler.regularCoreEmbedding

regularCoreGenerators def

The actual generating set of the regular core. · OAI.KadisonKastler.regularCoreGenerators

RegularCoreHilbert abbrev

The actual real-line regular Hilbert space. · OAI.KadisonKastler.RegularCoreHilbert

regularCurve def

The proved elementary geometry of the actual radial oscillating curve. · OAI.ThompsonNonamenability.Analytic.regularCurve

RegularCurveData structure

Differential and separation data at the fixed parameter scale 1 / 32. · OAI.ThompsonNonamenability.Analytic.RegularCurveData

regularDiagonalAlgebra def

Constants from P together with the complete scalar diagonal. · OAI.KadisonKastler.regularDiagonalAlgebra

regularDirichletCore def

The same core as a submodule of the actual zero-boundary space. · OAI.RegularPointCoordinates.regularDirichletCore

regularEdges def

The regular edges for the actual high-degree initial family. · OAI.TalagrandDiscreteConvexity.Covering.regularEdges

regularFamily def

Edges avoiding all prescribed initial generators. · OAI.TalagrandDiscreteConvexity.Covering.regularFamily

regularFieldOperator def

Multiplication by an essentially bounded, strongly measurable operator field, as a genuine bounded operator. · OAI.KadisonKastler.regularFieldOperator

regularFrequencyNest def

Reindexing the upper cuts produces an increasing real-indexed nest. · OAI.KadisonKastler.regularFrequencyNest

regularFrequencyProjection def

The positive-frequency cut of the actual L² Fourier transform. · OAI.KadisonKastler.regularFrequencyProjection

regularFrequencyWindow def

A bounded frequency interval, used for the finite-weight core corners. · OAI.KadisonKastler.regularFrequencyWindow

regularGauge def

Multiplication by U_t, as a unitary on the regular Hilbert space. · OAI.KadisonKastler.regularGauge

regularGaugedCoordinates def

Scalar coordinates after undoing the gauge in the packet inclusions. · OAI.KadisonKastler.regularGaugedCoordinates

regularGaugedPacket def

A scalar wave packet in the regular representation, with its gauge removed. · OAI.KadisonKastler.regularGaugedPacket

regularHarmonic def

Evaluating the actual global flow as a function on R^n. · OAI.HarmonicCounterexample.Angular.regularHarmonic

RegularHeatConstants structure

The constants retained before choosing the pinching tolerance. · OAI.RegularPointCoordinates.RegularHeatConstants

regularHilbertCoordinates def

Actual Hilbert coordinates on the regular representation. · OAI.KadisonKastler.regularHilbertCoordinates

regularIndicatorProjection def

Multiplication by the indicator of a measurable subset of the line. · OAI.KadisonKastler.regularIndicatorProjection

regularize def

The regularization descends to an additive, normalized monotone state on the actual Cuntz monoid. · OAI.Kaplansky.CuntzDimensionState.regularize

regularize def

The actual regularized output rows; no runtime proof or positivity input. · OAI.MaxCutGames.Foundations.PCP.PreprocessingRegularTables.regularize

regularize def

The actual regularized output rows; no runtime proof or positivity input. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingRegularTables.regularize

regularized def

Cutoff regularization of the constructed dimension state, as a finite real supremum. · OAI.Kaplansky.CuntzDimensionState.regularized

regularizedHessian def

The full six-dimensional regularized Hessian. · OAI.Problem356.LocalHessian.regularizedHessian

regularizedLogScalar def

Scalar logarithmic generator at a positive regularization scale. · OAI.KadisonKastler.regularizedLogScalar

regularizedModularGroup def

The regularized unitary groups used to approach the modular group. · OAI.KadisonKastler.regularizedModularGroup

regularizedModularLog def

The bounded approximation to log ((1-R)/R). · OAI.KadisonKastler.regularizedModularLog

regularizedPlusWeylScalar def

Positive regularization makes the Weyl cube root smooth even on its zero set. · OAI.PositiveEinsteinFour.Geometry.regularizedPlusWeylScalar

regularizedPrimitive def

The regularized primitive, including beta = 0 without a special convention. · OAI.SharpLiebThirring.regularizedPrimitive

regularizedSlowKernel def

The manuscript's regularized Laplace representation. · OAI.DefocusingNLS.regularizedSlowKernel

regularizedSobolevGram def

In the heat construction, w is the cutoff heat-kernel weight and a is the positive scalar regularization lambda * t. · OAI.RegularPointCoordinates.regularizedSobolevGram

regularizedSobolevGramEquiv def

The actual inverse exists in finite dimension; it need not be added as a hypothesis in the later heat-matrix estimates. · OAI.RegularPointCoordinates.regularizedSobolevGramEquiv

regularizedStateHessian def

The regularized state-Jacobian, written in component form suitable for the parameter/state stationarity equation. · OAI.Problem356.CoulombCalculus.regularizedStateHessian

regularizedZeta def

The real regularized zeta function, filled in at its pole by the positive class-number-formula residue. · OAI.DukePrimeDegree.regularizedZeta

RegularJacobi structure

Precisely the local ODE and regularity bounds used by the discretization. All constants can be chosen on a compact interval of a smooth Jacobi curve. · OAI.PositiveEinsteinFour.Jacobi.RegularJacobi

regularLevelChartedSpace def

The actual real atlas induced by implicit-function charts on a regular level. · OAI.CircleDomainRigidity.regularLevelChartedSpace

regularLocalFrame def

The translated-cutoff analysis map is bounded with its exact frame constant. · OAI.KadisonKastler.regularLocalFrame

regularLocalFrameValue def

The continuous analysis map of the translated scalar cutoff. · OAI.KadisonKastler.regularLocalFrameValue

regularLocalizationAverage def

The unnormalized Haar average of localized copies of an operator. · OAI.KadisonKastler.regularLocalizationAverage

regularLocalizationAverageLinear def

Linearity in the operator being localized and averaged. · OAI.KadisonKastler.regularLocalizationAverageLinear

regularMatrix def

The left regular permutation representation over any coefficient semiring. · OAI.Release075.regularMatrix

regularNetRadius def

The net is substantially finer than the pinching mean radius, so that all active energy centers satisfy the reverse mass-covering condition. · OAI.RegularPointCoordinates.regularNetRadius

regularNormalizedIntervalAverage def

The interval-length-normalized localization is a contraction on operators. · OAI.KadisonKastler.regularNormalizedIntervalAverage

regularPacketCoefficient def

An operator coefficient between two wave packets. · OAI.KadisonKastler.regularPacketCoefficient

regularPacketConvolution def

The finite packet average has a factored, algebra-valued realization. · OAI.KadisonKastler.regularPacketConvolution

regularPacketOperator def

A rank-one scalar packet matrix with an arbitrary bounded vector coefficient. · OAI.KadisonKastler.regularPacketOperator

regularPacketProjection def

Projection onto finitely many whole vector-valued scalar packets. · OAI.KadisonKastler.regularPacketProjection

regularPairProjectionSize def

The actual normalized least-squares residual of the two replacements. · OAI.RegularPointCoordinates.regularPairProjectionSize

regularPatchRegion def

The region is small enough that all pair comparison scales follow the common heat comparison index. · OAI.RegularPointCoordinates.regularPatchRegion

RegularPatchSelection structure

Extending the selected net centers outside the net makes the operator family total without changing any active patch. · OAI.RegularPointCoordinates.RegularPatchSelection

regularPatchSelection def

The selector is constructed from the already proved finite covers and positive energy floors. · OAI.RegularPointCoordinates.regularPatchSelection

RegularPinchingField structure

These are constructed from the original full Bochner measure and the actual Poisson sources. The common positive floor permits a fixed energy minimizer in each finite patch. · OAI.RegularPointCoordinates.RegularPinchingField

regularPositions def

A position is regular exactly when the pattern records a label there. · OAI.QuantitativeVanDerWaerden.RecordedSupport.regularPositions

regularPositiveFiberDomain def

The largest open positive base domain on which the literal transverse fibers are compact and contain zero in their interiors. It is defined directly from the convex body's fibers, without any estimate or derivative hypothesis. · OAI.AffineBernstein.regularPositiveFiberDomain

regularPrepare def

This is the prepared minimizer from the manuscript; no drift inequality is an input to its definition. · OAI.KServer.AdaptiveAllocation.regularPrepare

RegularPrimeSet def

The last dyadic endpoint and all later endpoints impose an automatic condition when C is nonnegative, so the tail condition is stated for all i. · OAI.JointDickman.RegularPrimeSet

RegularProjectionEnvironment structure

Actual heat data and actual small-source replacements with identical root radii. Every later coordinate estimate uses heat.coordinates. · OAI.RegularPointCoordinates.RegularProjectionEnvironment

regularRadialEnergyCoefficient def

The dimensionless energy coefficient for the same transported residual. · OAI.RegularPointCoordinates.regularRadialEnergyCoefficient

regularRadialValueCoefficient def

The dimensionless value coefficient after restriction and frame transport. · OAI.RegularPointCoordinates.regularRadialValueCoefficient

regularRightTranslation def

The concrete right-translation unitary ρ_s ξ(t)=ξ(t+s). · OAI.KadisonKastler.regularRightTranslation

RegularRunStart def

A regular segment starts at the first run or immediately after an omitted run. Empty segments are not counted. · OAI.TwoPointCorrelations.RegularRunStart

regularScalarBasis def

One fixed scalar basis; its only role is to select Hilbert coordinates. · OAI.KadisonKastler.regularScalarBasis

regularScalarMultiplier def

The actual multiplication operator associated to a bounded Borel scalar function on the real line. · OAI.KadisonKastler.regularScalarMultiplier

regularShapeDomain def

The literal single-copy depth domain: one evaluation coordinate, the branching coordinates, their strict ancestry and the regular interior cube. · OAI.DilutedSpinGlass.ReducedTopology.regularShapeDomain

regularSingletonAverage def

The unnormalized regular-depth average of the actual physical singleton covariance energy; L+1 is the physical height, not the number of RSB steps. · OAI.DilutedSpinGlass.UniversalDictionary.regularSingletonAverage

regularSingletonDepths def

The singleton depth domain of the manuscript, with no artificial normalization by its cardinality. At height L+1 its mass is at most one. · OAI.DilutedSpinGlass.UniversalDictionary.regularSingletonDepths

regularSlots def

Current regular slots, before sampling at this node. · OAI.Ostmann.MovingSlotData.regularSlots

regularSmallMassCoefficient def

All volume and cutoff losses in the local smallness estimate. · OAI.RegularPointCoordinates.regularSmallMassCoefficient

regularStep def

The final update after genuine preparation and transport. · OAI.KServer.AdaptiveAllocation.regularStep

regularTail def

Continuous primitive of the regular B/s term, including the improper endpoint. · OAI.KServer.SimplexAnalytic.regularTail

regularTensorAlgebra def

The concrete tensor algebra, specified by its constant commutant. · OAI.KadisonKastler.regularTensorAlgebra

regularTensorVector def

An elementary scalar-vector tensor, as a continuous bilinear map. · OAI.KadisonKastler.regularTensorVector

regularTranslateApply def

Right translation of actual L² classes. · OAI.KadisonKastler.regularTranslateApply

regularTransport def

Delete and insert zero coordinates following the prepared minimizing update, with a canonical simplex state only when its weight is exactly zero. · OAI.KServer.AdaptiveAllocation.regularTransport

regularUnary def

All frequency dependence of a regular unary is its current root power. · OAI.Ostmann.regularUnary

regularUnitaryField def

The gauge operator as a genuine unitary on the regular Hilbert space. · OAI.KadisonKastler.regularUnitaryField

regularUnitaryFieldApply def

Multiplication of L² vectors by a measurable unitary field. · OAI.KadisonKastler.regularUnitaryFieldApply

regularValue def

The genuine center-regular ODE solution in the actual angular polynomial space. · OAI.HarmonicCounterexample.Angular.regularValue

regulator def

The regulator equals one at either endpoint of an active interval. · OAI.SLEExactGauge.Multipoint.regulator

reindex def

Reindex only the genuine finite family, preserving its exact sum. · OAI.ClosedSurfaceR4.FiniteOrderSmoothing.SmoothPrimitiveFamily.reindex

reindex def

Reindexing does not change either defining equation. · OAI.HarmonicCounterexample.Berger.ComplexStructure.reindex

reindex def

Relabeling a finite frame preserves its initial projection and resolution. · OAI.KadisonSimilarity.MatrixUnitColumnFamily.reindex

reindex def

The normalized corner trace is preserved when a finite frame is relabeled. · OAI.KadisonSimilarity.TracialMatrixUnitColumnFamily.reindex

reindex def

The unitary permutation representation on ℓ², for arbitrary index types. · OAI.KervaireRegular.reindex

reindex def

Read the same family using another finite set of support degrees. · OAI.LittleFinitistic.FiniteSupportReindex.reindex

reindex def

Explicit changes of vertex, dart, and label coordinates. · OAI.MaxCutGames.Foundations.PCP.ConstraintGraph.reindex

reindex def

Reindexing changes the names of outcomes, retaining their uniform law. · OAI.MaxCutGames.Integration.NoiseTables.reindex

reindex def

Transport both normalized question laws and the projection predicate. · OAI.MaxCutGames.Repetition.ProjectionKernel.reindex

reindex def

Restricting to a cofinal subsequence keeps the original neighboring law, affine profiles and offsets. · OAI.PolycyclicRecognition.CartanWordModel.AffineNeighborRow.reindex

reindex def

Every increasing subsequence keeps the original limiting law. · OAI.PolycyclicRecognition.CartanWordModel.NeighborSamplingRow.reindex

reindex def

Concatenate retained wrapper syntax, never selecting new positive witnesses. · OAI.PureTypeSystem.DataWrapper.reindex

reindex def

Transport a genuine augmented contraction along a graph/order reindexing. · OAI.SimplicialChains.Contraction.reindex

reindex def

Restrict a simultaneously transported finite family to a chosen root list. This is the SAME physical passage, not another existence choice. · OAI.ThreeTorus.NeutralStep.reindex

reindex def

Re-enumerating occurrence chunks does not change their geometric partition. · OAI.TorsionFreeZeroDivisors.ChunkComparisons.Chunking.reindex

reindex def

Rename states along an equivalence, retaining all transitions and moves. · OAI.TwoWayComplementation.TwoNFA.reindex

reindex abbrev

The bounded unitary implementation of a permutation on ell2. · OAI.UltrapowerBridge.reindex

reindex def

Explicit changes of vertex, dart, and label coordinates. · OAI.UniqueGamesTheorem.Foundations.PCP.ConstraintGraph.reindex

reindex def

Reindexing changes the names of outcomes, retaining their uniform law. · OAI.UniqueGamesTheorem.Integration.NoiseTables.reindex

reindex def

Transport both normalized question laws and the projection predicate. · OAI.UniqueGamesTheorem.Repetition.ProjectionKernel.reindex

reindexAssignments def

Reindexing an assignment transports it contravariantly along the variable equivalence. · OAI.MaxCutGames.Quadratic.reindexAssignments

reindexAssignments def

Reindexing an assignment transports it contravariantly along the variable equivalence. · OAI.UniqueGamesTheorem.Quadratic.reindexAssignments

reindexConfiguration def

Reindexing preserves the actual points and changes only the tuple's index type. · OAI.Nagata.W06.TorusConfiguration.reindexConfiguration

Reindexed def

Matching old raw images and parameters under declaration reindexing. No equality is imposed on undeclared indices. · OAI.PureTypeSystem.ObservationValues.Environment.Reindexed

reindexed def

Reindex the row/column recursion without changing its arithmetic. · OAI.ContingencyTables.GreedyFeasibleTable.reindexed

reindexedBooleanDraw def

Move a largest row and column to the distinguished indices inside the sampler, then transport the output back to the fixed original indices. · OAI.ContingencyTables.GridBoundary.reindexedBooleanDraw

reindexedCycleLabelingsEquiv def

Vertex reindexing does not change cycle-constant labels. · OAI.KadisonSimilarity.reindexedCycleLabelingsEquiv

reindexedHilbertBasis def

Reindex a Hilbert basis while retaining its concrete vectors. · OAI.KadisonKastler.reindexedHilbertBasis

reindexedModuleSummandIso def

The exact block tensor after reindexing supported summands by Shape. · OAI.DerivedBar.RecursiveTotal.reindexedModuleSummandIso

reindexEquivalence def

The actual Hilbert-space equivalence determined by all finite words. · OAI.KadisonKastler.PointedMatrixSites.reindexEquivalence

reindexHiddenSeed def

The sacrificed mask and every core decision use the same decoded seed. · OAI.MatroidProphet.reindexHiddenSeed

reindexInstruction def

The program depends on the fixed degree and tape placement, never on input-dependent vertices, reverse indices, or predicate values. · OAI.MaxCutGames.Foundations.Complexity.MachineLazyRows.reindexInstruction

reindexInstruction def

The program depends on the fixed degree and tape placement, never on input-dependent vertices, reverse indices, or predicate values. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineLazyRows.reindexInstruction

ReindexIso structure

A reindexing isomorphism of diagram systems. The compatibility square uses the already-defined literal restriction maps. · OAI.PartitionConsistency.LocalDiagramSystem.ReindexIso

reindexIso def

Sectionwise exterior powers commute with coefficient-ring and open-index isomorphisms, using exactly the standard pushforward functor. · OAI.ActualExterior.reindexIso

reindexLinear def

Coordinate permutation before adding the isometric structure. · OAI.KervaireRegular.reindexLinear

reindexLogCoordinates def

The original last coordinate is restored before the coordinates are permuted. · OAI.DukePrimeDegree.reindexLogCoordinates

reindexObj def

A definitionally normalized presentation of restriction, with no second bundling of the module before taking its exterior power. · OAI.ActualExterior.reindexObj

reindexOnlineSeed def

Change only the representation of the complete initial seed. · OAI.MatroidProphet.reindexOnlineSeed

reindexParameters def

This map permutes coordinates, preserving both the cube and Lebesgue measure. · OAI.PolycyclicRecognition.reindexParameters

reindexSecretarySeed def

Reindex a secretary rule through any measurable decoding map. · OAI.MatroidProphet.reindexSecretarySeed

reindexSequence def

Reindex square-summable integer sequences by a permutation. · OAI.CoarseAssembly.Analytic.reindexSequence

reindexStrategy def

Each player only permutes that player's own questions and answers. · OAI.MaxCutGames.Foundations.Games.Game.reindexStrategy

reindexStrategy def

Each player only permutes that player's own questions and answers. · OAI.UniqueGamesTheorem.Foundations.Games.Game.reindexStrategy

reindexVector abbrev

Reindex a square-summable vector by a permutation. · OAI.UltrapowerBridge.reindexVector

rejectingImages def

Rejection produces actual nonempty scalar-image sets, including correlated buffers. This does not assert an unavailable simultaneous global assignment. · OAI.WLIdentification.ProductCircuit.rejectingImages

rejectionFailureProbability def

Rejection probability per trial. · OAI.EditApproximation.rejectionFailureProbability

rejectionPrefix def

Executable finite prefix of the rejection loop. Each failed trial is discarded before the next trial is examined. · OAI.EditApproximation.rejectionPrefix

rejectionStateCells def

The stream keeps the current word and one optional accepted value; failed words are overwritten and the failure count belongs only to the probability-space description, not to the machine state. · OAI.EditApproximation.rejectionStateCells

rejectionTransition def

A streaming rejection sampler only retains its current accepted value and the current bounded-width random word. · OAI.EditApproximation.rejectionTransition

rejectionTrial def

A trial accepts exactly the words lying in the requested range. · OAI.EditApproximation.rejectionTrial

Rejects def

Rejection here means the actual finite table has no forward accepting trace; it is stronger than merely failing a chosen simulation trace. · OAI.FiniteAlgebraicEnvelopes.Compiler.Simulator.Rejects

Rel def

Literal equations of the stabilized infinite table. Restricting a stage to its own budget removes only premature copies of the fixed absorption rules. · OAI.FiniteAlgebraicEnvelopes.Bootstrap.Quotient.Rel

Rel abbrev

The compiler's finite monoid presentation is formed from the literal rule table; neither a normalizer nor program totality enters its definition. · OAI.FiniteAlgebraicEnvelopes.Compiler.Word.Rel

Rel def

The two halves of the integer tape share one physical cell. The immutable origin flag makes crossing the fold a finite-control, zero-movement operation. · OAI.FiniteAlgebraicEnvelopes.MachineRealization.TapeFold.Rel

Rel def

The stabilizing summand records the cancellation which is unavailable in the original monoid. · OAI.Kaplansky.SemigroupDifference.Rel

Rel def

Equality modulo norm-null sequences, with unrestricted varying dimensions. · OAI.NuclearUltrapower.VariableUltra.Rel

Rel def

Equality forced by every strict cocone on the diagram. · OAI.Thomason.Construction.StrictOmegaCategory.ColimitConstruction.Rel

Rel inductive

The least composition congruence killing all cells above dimension n. · OAI.Thomason.Construction.StrictOmegaCategory.IntelligentTruncation.Rel

Rel def

The relation described by a table, without enumerating its product points. · OAI.WLIdentification.ProductMap.Rel

relabel def

A change of alphabet coordinates keeps endpoints and edge identities. · OAI.MaxCutGames.Explicit.BipartiteGame.relabel

relabel def

Relabelling columns does not change geometry. The two order conditions are explicit, so the construction never assumes invariance under an arbitrary permutation of the circular order. · OAI.PlanarL1.Columns.Model.relabel

relabel def

Relabel the alphabet, retaining the actual graph dart and endpoint data. This does not change the path, partner, rotation, or exterior marker. · OAI.TorsionFreeZeroDivisors.FinitePicture.Data.relabel

relabel def

A change of alphabet coordinates keeps endpoints and edge identities. · OAI.UniqueGamesTheorem.Explicit.BipartiteGame.relabel

relabelledUnitary def

This declaration is an exact comparator-selected statement. Read the Lean statement below for its mathematical content. · OAI.CubeShuffle.Specht.relabelledUnitary

relabelRule def

Conjugate an actual online rule by a fixed label enumeration. In particular, the translated rule does not receive a future order or a future value. · OAI.MatroidProphet.Labeling.relabelRule

Relation structure

A prescribed output of finite input coordinates, allowing all countable finite-cost representations rather than only finite ones. · OAI.DiamondDistortion.InfimalCost.Relation

Relation inductive

The inverse equations are imposed in both directions over the full ring. Every incidence equation is included individually, even for repeated sides. · OAI.IntegralCharacterVarieties.FiniteMatrixPresentation.Relation

relation def

The defining polynomial of the cyclic group algebra. · OAI.CirculantHadamard.CyclicPolynomial.relation

relation def

A complete 64-by-64 Boolean relation in the shared row-major table format. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Relations.relation

relation def

The least-index graph, regularized in both the value and index names. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.LeastOccurrence.relation

relation def

Substitution into a binary formula with n additional parameters. · OAI.PartitionConsistency.SetLanguage.relation

relation def

Source observational relation, with its actual Good predicate callback. · OAI.PureTypeSystem.Labelled.ChannelCode.relation

relation def

Fixed-bin future/past observations and the two old coordinate tubes. All auxiliary frames have disappeared from this boundary relation. · OAI.SmoothObstruction.MeasurableSwitch.relation

relation def

The defining equation of the five-variable hypersurface. · OAI.StableCoordinate.MatrixChange.relation

relation def

Modes zero and one use internal intervals. Modes two and three use the earlier and later comparison, respectively. · OAI.TwoPointCorrelations.WitnessSystemData.relation

relation def

A complete 64-by-64 Boolean relation in the shared row-major table format. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Relations.relation

relationAddress def

Graph rows store the right endpoint index fastest. · OAI.MaxCutGames.Foundations.Complexity.FinalCNFPattern.relationAddress

relationAddress def

Graph rows store the right endpoint index fastest. · OAI.UniqueGamesTheorem.Foundations.Complexity.FinalCNFPattern.relationAddress

RelationalCode structure

Fixed syntax shared by both source logical channels and the A-independent probe type. The raw path is literally shared, not an isomorphism hypothesis. · OAI.PureTypeSystem.Labelled.RelationalCode

RelationalExclusion def

Relational exclusion for finite profile components. This internal proposition is proved from weak normalization, not assumed in the main theorem. · OAI.PureTypeSystem.RelationalExclusion

relationBasis def

The finite relation basis used by the manuscript exists for this exact subspace. · OAI.LittleFinitistic.RelationSequence.relationBasis

relationClass def

The canonical name for {y ∈ A | (x,y) ∈ E}. No equivalence-relation axiom is needed to construct this set. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.relationClass

relationEquiv def

The relation register is a finite tuple of bits, with no unbounded storage. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.ReadRelation.relationEquiv

relationEquiv def

Exact currying/uncurrying equivalence of the two full relation alphabets. · OAI.TwoWayComplementation.CompanionBridge.relationEquiv

relationEquiv def

The relation register is a finite tuple of bits, with no unbounded storage. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.ReadRelation.relationEquiv

relationFintype def

Executable enumeration used when assembling the finite machine state. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.ReadRelation.relationFintype

relationFintype def

Executable enumeration used when assembling the finite machine state. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.ReadRelation.relationFintype

relationGraph def

The simple bipartite graph of a relation (the same definition used by the links). · OAI.Release075.relationGraph

relationHomotopy def

The actual chain-representative relation homotopies survive the forgetful functor. · OAI.LittleFinitistic.UnderlyingRepresentatives.relationHomotopy

relationIdeal abbrev

The ideal generated by a single equation. · OAI.StableCoordinate.IteratedQuotient.relationIdeal

relationIndex def

The fixed row-major ordering of the complete predicate table. · OAI.MaxCutGames.Foundations.PCP.GenericGraphTables.relationIndex

relationIndex def

The fixed row-major ordering of the complete predicate table. · OAI.MaxCutGames.Foundations.PCP.GraphTables.relationIndex

relationIndex def

The fixed row-major ordering of the complete predicate table. · OAI.UniqueGamesTheorem.Foundations.PCP.GenericGraphTables.relationIndex

relationIndex def

The fixed row-major ordering of the complete predicate table. · OAI.UniqueGamesTheorem.Foundations.PCP.GraphTables.relationIndex

relationInTime def

The fixed unary relation reader occupies one finite TM2 statement. · OAI.MaxCutGames.Foundations.Complexity.FinalCNFMachine.Program.relationInTime

relationInTime def

The fixed unary relation reader occupies one finite TM2 statement. · OAI.UniqueGamesTheorem.Foundations.Complexity.FinalCNFMachine.Program.relationInTime

relationMap def

The actual finite relation evaluated on the chosen chain maps. · OAI.ExternalTensor.Section4.relationMap

RelationMonoid abbrev

Relations in path order, using the opposite of Mathlib's endomorphism monoid. · OAI.TwoWayComplementation.RelationMonoid

relationPolynomial def

The literal inhomogeneous relation polynomial. · OAI.LittleFinitistic.QuadraticEncoding.relationPolynomial

relationProduct def

Relation product for an input word; the empty product is the identity. · OAI.TwoWayComplementation.relationProduct

Relations structure

The Coxeter relations on the first n adjacent generators. · OAI.FiniteAlgebraicEnvelopes.Adjacent.Relations

Relations structure

Coefficients of the finite quadratic relation list, including degrees zero and one. · OAI.LittleFinitistic.QuadraticEncoding.Relations

Relations def

The exact defining relation of a single block, retaining the order L⁻¹R. · OAI.Release075.CartesianBlock.Relations

Relations def

The defining additive relation for one actual strict composition. · OAI.Thomason.Construction.StrictOmegaCategory.Abelianization.Relations

relations def

The two usual relations: isomorphic objects have the same class, and short exact sequences are additive. · OAI.CoherentK0.relations

relations def

The balancing relations for the right A-action d·a = d*f(a). · OAI.DerivedBar.Balanced.relations

relations def

Noncommutative balancing relations for an arbitrary (D,D)-bimodule X. · OAI.DerivedBar.BimoduleTensor.relations

relations def

The generators of the kernel, exactly d_y=y⊗1−1⊗yᵒᵖ. · OAI.FiniteAlgebraicEnvelopes.Enveloping.relations

relations def

Balancing relations for a right A-module and a left A-module, over k. · OAI.LittleFinitistic.BalancedTensor.relations

relations def

Ordinary atomic set forcing, on actual sets of Kuratowski pairs. The first component is membership and the second is equality. No truth values or atomic laws are supplied as parameters. · OAI.PartitionConsistency.SetForcingNames.AtomicForcing.relations

relationSpace def

Exactly the span of a'_h s - s' a_h and all homogenized p_j. · OAI.LittleFinitistic.QuadraticEncoding.relationSpace

relationStateEquiv def

Only finite register coordinates change; all tapes remain fixed. · OAI.MaxCutGames.Foundations.Complexity.MachineLazyRows.relationStateEquiv

relationStateEquiv def

Only finite register coordinates change; all tapes remain fixed. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineLazyRows.relationStateEquiv

relative def

The parent's scalar relative reference, and later the side-total reference, use exactly the manuscript's two-sided factor test, including zero crossings. · OAI.KServer.CoarseProcess.relative

relativeActive def

Active colors at their last recorded incidence. · OAI.HarmonicArtin.ParabolicIntersections.relativeActive

relativeAlgebra def

The algebra structure is exactly the specified map c ↦ F+1. · OAI.AbhyankarSathaye.CommutingDerivations.relativeAlgebra

relativeArtinHom def

The canonical homomorphism into the Artin group of the relative diagram. · OAI.HarmonicArtin.ParabolicIntersections.relativeArtinHom

relativeB def

The relative map at the covariant end of a span. · OAI.Thomason.Construction.SimplicialSpan.relativeB

relativeBackward def

Evaluation at the displayed X,Y,T, with the explicit relative coefficient map. · OAI.AbhyankarSathaye.CommutingDerivations.relativeBackward

relativeBoundaryExtension def

A concrete continuous extension preserving the prescribed map on a subcomplex and its image under p. · OAI.Thomason.Construction.RealizationPrism.relativeBoundaryExtension

relativeChainsHomeomorphIso def

A genuine homeomorphism of pairs acts invertibly on relative singular chains. · OAI.Kaplansky.relativeChainsHomeomorphIso

relativeConjugationAverage def

Equal-weight conjugation by a finite family from the source algebra. · OAI.KadisonKastler.relativeConjugationAverage

relativeCore def

The core of A in U, pulled back to a subgroup of A. · OAI.FiniteLattice.CommonCore.relativeCore

relativeCoverChainSequence def

The relative gluing sequence formed using the actual singular chains of A. Its right term compares directly with C(X)/C(A). · OAI.Kaplansky.relativeCoverChainSequence

relativeDensity def

Relative density, with value zero when the conditioning set is empty. · OAI.MaxCutGames.Inverse.KMS.relativeDensity

relativeDensity def

Relative density, with value zero when the conditioning set is empty. · OAI.UniqueGamesTheorem.Inverse.KMS.relativeDensity

relativeExitScale def

Relative radii, starting below one half as required by the pathwise argument. · OAI.LipschitzHilbert.relativeExitScale

RelativeGauge structure

A genuine extended gauge of a relative symmetric convex body. The lower bound is only on its supporting subspace, so cylinders are retained. · OAI.PettyProjection.RelativeGauge

relativeGeneratorLocus def

The entire relative-generator locus in the original unitary group. · OAI.RelativeGeneration.relativeGeneratorLocus

RelativeHaarBlock structure

A nonempty Haar block with centered base separators and a base prefix. · OAI.KadisonKastler.RelativeHaarBlock

relativeHaarKernel def

The centered part of a containing algebra for the finite base projection. · OAI.KadisonKastler.relativeHaarKernel

relativeHomologyBoundary def

The boundary homomorphism of an actual injective topological map. · OAI.Kaplansky.relativeHomologyBoundary

RelativeInput def

Threshold-removal uses a single head for all deviations at this radius. · OAI.DiamondDistortion.PathApplications.RelativeInput

relativeInvisible def

Absence of recorded incidence, restricted to the group currently acting. · OAI.HarmonicArtin.ParabolicIntersections.relativeInvisible

RelativeLocalizationContract def

A relative algebra equivalence over ℂ[c,c⁻¹], with c = F + 1, and its coordinate equations. · OAI.AbhyankarSathaye.CommutingDerivations.RelativeLocalizationContract

relativeMatrix def

The six transitions from a fixed reference basis, with that basis omitted. · OAI.MUB6.relativeMatrix

relativeMayerVietorisBoundary def

The connecting map on the actual full relative homology groups. · OAI.Kaplansky.relativeMayerVietorisBoundary

relativeMayerVietorisSumSequence def

The exact relative Mayer–Vietoris sequence at the direct sum, with the standard signed intersection map and actual pair inclusions. · OAI.Kaplansky.relativeMayerVietorisSumSequence

relativeOpenCoverGlue def

Compatible local linear normalizations glue to a genuine global normalization when the preceding overlap group vanishes. · OAI.Kaplansky.relativeOpenCoverGlue

relativePairProjection def

The common initial projection of a two-column row. · OAI.WeakPureInfiniteness.relativePairProjection

RelativePartBoundary abbrev

One side of a selected INTERIOR fragment. The complement includes the protected outside, even when its arbitrary partition label agrees. · OAI.HyperbolicObstruction.PictureSwitch.RelativePartBoundary

relativePositiveSign def

A closed positive null component, isolated from all higher levels. · OAI.HarmonicArtin.ParabolicIntersections.relativePositiveSign

RelativePrismData structure

The poset data used by the relative prism construction. · OAI.Thomason.Construction.RelativePrismData

relativeRowProduct def

The actual product cocycle, with the corner unit as its zeroth term. · OAI.WeakPureInfiniteness.relativeRowProduct

relativeSingularChains def

Relative chains are the actual cokernel of the subspace chain map. · OAI.Kaplansky.relativeSingularChains

relativeStabilizer def

Stabilizer of the transported family, restricted to a standard subgroup. For a family of homotopy classes, take V to be the product of the classes. · OAI.HarmonicArtin.ParabolicIntersections.relativeStabilizer

relativeUnit def

This relative comparison retains both the original target and the entire replacement of its prescribed subcomplex. · OAI.Thomason.Construction.InfiniteExtension.relativeUnit

relativeUnitaryExtension def

Extend a relative unitary by the identity on the complementary corner. · OAI.WeakPureInfiniteness.relativeUnitaryExtension

relativeUnitaryOrbit def

The orbit under the unitary group of the given subalgebra map. · OAI.KadisonKastler.relativeUnitaryOrbit

relativeValue def

The affine value relative to the constant-slope branch t. · OAI.Problem326.FixedMinimum.relativeValue

relativeVector def

Relative tail products, algebraically equal to sums of outside-head jumps. · OAI.IndependentProducts.FiniteHead.relativeVector

Relator inductive

The literal source letter relators: all included fans, all included top occurrences (with all seven fan slots), and every included square/six-letter row, even rows that do not occur in any master match. · OAI.HyperbolicObstruction.LetterPresentation.Relator

Relators def

Relators of the actual voltage-cover edge presentation, not a quotient of base-cell coefficients. Every deck coordinate remains an independent index. · OAI.HyperbolicObstruction.VoltageWords.Relators

relators def

The usual finite type-A Coxeter presentation. · OAI.FiniteAlgebraicEnvelopes.Adjacent.relators

relaxedEncoding def

This encoding preserves all four paths, not just their pairing word. · OAI.Problem335.MomentPairing.relaxedEncoding

RelaxedPaths abbrev

Four endpoint-fixed paths respecting a prescribed relaxed pairing in every layer. · OAI.Problem335.Pairings.RelaxedPaths

relaxedPathsEquivAssignments def

The explicit product decomposition used in the second trace count. · OAI.Problem335.Pairings.relaxedPathsEquivAssignments

relaxedWordWeight def

The normalized pairing-word summand in the second-moment reduction. · OAI.Problem335.MomentPairing.relaxedWordWeight

relCWComplex def

Attaching disks is a classical relative CW complex, not merely an abstract cell-complex object in a category. · OAI.GroupRingTopology.Attachment.relCWComplex

Reloc structure

helper for Nat ranged permutations · OAI.PowerSaving.Grove.Reloc

relocateVal def

Move the physical input after the branch bit to the front, and place the readout immediately after the true input. The retained history is permuted, never erased or initialized with input-dependent advice. · OAI.ExactQuantumFactoring.relocateVal

relStar def

Reflexive transitive closure, including the empty path. · OAI.TwoWayComplementation.relStar

remainder def

The difference between coefficient blocks at distance m. · OAI.CirculantHadamard.DyadicRemainder.remainder

remainder def

The relative complement is the actual difference of the operators. · OAI.Generator.Tracial.AlgebraProjection.remainder

remainder def

The fractional part of Q * (p i / q), with denominator q. · OAI.MaxCutGames.Explicit.Rounding.remainder

remainder def

Bytes remaining after the selected row's actual tail field. · OAI.MaxCutGames.Foundations.Complexity.MachineCloudSelect.remainder

remainder def

The Boolean remainder after deleting the union of bad rows. · OAI.MaxCutGames.Fourier.Deletion.remainder

remainder def

Remove a specified amount of mass from one vertex. · OAI.Thomason.Construction.PrismBarycentric.remainder

remainder def

The fractional part of Q * (p i / q), with denominator q. · OAI.UniqueGamesTheorem.Explicit.Rounding.remainder

remainder def

Bytes remaining after the selected row's actual tail field. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineCloudSelect.remainder

remainder def

The Boolean remainder after deleting the union of bad rows. · OAI.UniqueGamesTheorem.Fourier.Deletion.remainder

remainder def

The coisometry filling the complement of the first binary range. · OAI.WeakPureInfiniteness.BufferAbsorptionCorner.remainder

remainderCollisionCount def

Ordered residue-collision pairs, expressed with natural remainders so no typeclass proof is part of the collision function's data. · OAI.Problem337.CompositeSpectral.remainderCollisionCount

remainderOrder def

Executable merge-sort of occurrence indices, rather than their records. · OAI.MaxCutGames.Explicit.Rounding.remainderOrder

remainderOrder def

Executable merge-sort of occurrence indices, rather than their records. · OAI.UniqueGamesTheorem.Explicit.Rounding.remainderOrder

remaining def

Macroscopic bridge parameters expressed directly in successive durations. · OAI.FKCRT.SimpleWalk.remaining

remainingBackwardTube def

The tube followed by the backward diagonal orbit, with every remaining matrix entry bounded by exp (-ρ) through time N. · OAI.DukePrimeDegree.EliminationChain.remainingBackwardTube

remainingBackwardTubeInformation def

Negative log mass of an actual finite-time diagonal tube in the intrinsic remaining-subgroup leaf. · OAI.DukePrimeDegree.EliminationChain.remainingBackwardTubeInformation

remainingEvents def

The actual unprocessed suffix of the finite table's stored row indices. · OAI.MaxCutGames.Foundations.Complexity.FinalCNFMachine.Program.remainingEvents

remainingEvents def

The actual unprocessed suffix of the finite table's stored row indices. · OAI.UniqueGamesTheorem.Foundations.Complexity.FinalCNFMachine.Program.remainingEvents

remainingIndicator def

The unchanged factors in the one-cut-at-a-time expansion of a cell. · OAI.Problem358.FiniteCellVariation.remainingIndicator

remainingPathParameter def

The remaining part of an interval after time s, parametrized again by the unit interval. · OAI.Thomason.Construction.remainingPathParameter

remainingRows def

Completed rows at guard remainder r are precisely vertices r,...,n-1. This is a semantic description of the output tape, not a machine instruction. · OAI.MaxCutGames.Foundations.Complexity.PoweringMachineLoop.remainingRows

remainingRows def

Completed rows at guard remainder r are precisely vertices r,...,n-1. This is a semantic description of the output tape, not a machine instruction. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringMachineLoop.remainingRows

RemainingStabilityPublishedInputs structure

Existence of expected matrix subfactors in the absorbed type III₁ setting. · OAI.KadisonKastler.RemainingStabilityPublishedInputs

remainingSuccessor def

Remove the two old circle factors. Unlike an arbitrary factor, this one will be proved to fix every occurrence on either selected circle. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.remainingSuccessor

RemezPolynomialInput def

The precise published input used to keep mass away from a polynomial's zero set. The constant depends only on the dimension and degree. · OAI.PolycyclicRecognition.RemezPolynomialInput

remoteNatDoubling def

The left copy starts beyond the prefix, which belongs to the right copy. · OAI.KadisonKastler.remoteNatDoubling

RemoteProductBlock structure

Finite data obtained by moving a block to unused sites of the same types. Complements witness the product formula on every finite stage. · OAI.KadisonKastler.RemoteProductBlock

remove def

The zero-chain backtracking bigon is first pinched off by a legal vertex switch and then deleted as a whole component. Its other edge contacts need not initially meet one another. · OAI.HyperbolicObstruction.PictureSwitch.PictureMap.Bigon.remove

removeAux def

Cap two vacant auxiliary wires at the output boundary. Boundary K is implemented by one adjacent tile before this cap. · OAI.CriticalHoneycomb.Transfer.PlanarNormalization.removeAux

removeCaps def

Tighten inverse end caps of one literal word boundary. The matching is spliced, and its genuine genus zero remains valid even for an empty interior. · OAI.TorsionFreeZeroDivisors.WordBlocks.Matching.removeCaps

removeConjugator def

Remove exactly the actual conjugating word, not a replacement reduced representative. Each removal is the proved cap splice on the same boundary. · OAI.TorsionFreeZeroDivisors.WordBlocks.Matching.removeConjugator

Removed def

A lower face is obtained by deleting one direction and crossing it. · OAI.CubicalChains.Removed

removed def

The Boolean part removed from the original function. · OAI.MaxCutGames.Fourier.Deletion.removed

removed def

The Boolean part removed from the original function. · OAI.UniqueGamesTheorem.Fourier.Deletion.removed

removedCoverSequenceMap def

The inclusion of the removed subspace, with its induced two-set cover. · OAI.Kaplansky.removedCoverSequenceMap

removedField def

Conditional field induced by an actual fixed spin. · OAI.BufferedIsing.DSS.removedField

removeDipole def

Delete the isolated dipole; every unremoved face and dart retains its original index, and every surviving side pairing is unchanged. · OAI.HyperbolicObstruction.PictureSwitch.PictureMap.removeDipole

removedRestrictionChainsIso def

The relative chain group in a chart uses its actual removed subspace. · OAI.Kaplansky.removedRestrictionChainsIso

rename def

Capture-avoiding renaming; in this relational language all terms are variables. · OAI.PartitionConsistency.Formula.rename

rename def

Thinning/reindexing transports every stored witness and preserves the whole tree shape. Binder annotations are transported too. · OAI.PureTypeSystem.TypingTree.rename

rename def

Change the names of registers; loops and instructions are unchanged. · OAI.Superstring.BitCode.Block.rename

renewalRegular def

The coefficients of the regular factor are uniformly bounded by four. · OAI.TotientAsymptotic.renewalRegular

Renormalization structure

The renormalizations of Definition 2.1, with the right derivative at zero. · OAI.BoltzmannNonuniqueness.Renormalization

rent def

Contextual cost of a coupling with tag-sum output and random error ø of alphabet E. · OAI.FourierLLogL.Compress.rent

rEntry def

The unswapped source R matrix, in the vacant/occupied encoding. · OAI.CriticalHoneycomb.Transfer.PlanarNormalization.rEntry

renyiIntegral def

The positive of minus the N-Rényi entropy; singular mass contributes zero. · OAI.WeakHessian.renyiIntegral

reorderBoundary def

Only the ordered boundary enumeration is changed. Every side, corner, seam, genus and facet remains the old one. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.reorderBoundary

reorderCircleBases def

Evaluate those same paths over an arbitrary coefficient ring. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.reorderCircleBases

reordered def

The reordered tower, inside the same T. Its initial submonoid is S whenever S contains the original initial stage. · OAI.PartitionConsistency.ConservativeMonoids.CellTower.reordered

reorderGenerators def

This producer merely transfers the identical named matrix coordinates; the noncommutative re- framing is performed before this transfer. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.reorderGenerators

reorderPathWords def

The choices depend only on the finite permutation, not on a solution or on a coefficient field. They are based path words. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.reorderPathWords

reorderPunctures def

Puncture scalars and genus are not changed by the re-basing. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.reorderPunctures

reorient def

Reorientation preserves the actual weak topologies, not only cellular chains. · OAI.EilenbergGanea.ArtinCubicalTopology.reorient

rePair def

Change the pairing on two edges, creating pairs (a,b) and (reverse a, reverse b). Formulated by conjugation, so involutivity and absence of fixed points are inherited rather than assumed for the result. · OAI.Release075.PermCycles.rePair

repairedViews def

The final pair stores the source and receiver respectively. · OAI.ContingencyTables.ProfilePrograms.repairedViews

repairMatrix def

The crossing receiver (i',j) in the manuscript. · OAI.ContingencyTables.repairMatrix

reparamEquiv def

Reparameterizing a group does not alter the Hom multiplicity. · OAI.Representation.IntertwiningMap.reparamEquiv

reparametrize def

Use the original quotient/kernel probability cell as the probe space. Measurable reparametrization keeps every prefix, coefficient and endpoint, so the independent probe law can be chosen directly on that cell. · OAI.PolycyclicRecognition.CartanWordModel.BoxProbeChains.reparametrize

reparametrize def

Reparametrize every path by the same endpoint-preserving interval map. · OAI.Thomason.Construction.PathSpace.reparametrize

reparametrizeHomotopy def

Straight-line interpolation is continuous in both the path and its parameter. · OAI.Thomason.Construction.PathSpace.reparametrizeHomotopy

repeat def

Repeat a returning routine until it returns true, removing the halt marker on false. Restarting takes one local stack step and does not touch data. · OAI.ParityWL.Complexity.StackMachine.repeat

repeatAccept def

The executable finite OR of complete trial outcomes. · OAI.LowSpaceSubsetSum.repeatAccept

repeated def

common even relation on four scores. · OAI.HadwigerCounterexample.repeated

repeated def

The complete finite reindexing into underlying chain and occurrence number. · OAI.TorsionFreeZeroDivisors.ChunkComparisons.repeated

repeatedBlockMatrix def

The usual unital inclusion M_α → M_(α×β), sending a matrix to repeated diagonal blocks (equivalently, tensoring by the identity matrix). · OAI.KadisonSimilarity.repeatedBlockMatrix

repeatedDihedralInclusion def

The repeated-word column at the actual normalized projection endpoint. · OAI.KadisonSimilarity.TracialCornerGeometry.repeatedDihedralInclusion

RepeatedLetterWords structure

Local tracial-word data for one fixed positive word level. The fields are operator identities, word isometries, and the tensor-coordinate formula for the exceptional projection. There is no approximation or propagation bound among the … · OAI.KadisonSimilarity.RepeatedLetterWords

repeatedLineMap def

The actual bundle map obtained by tensoring each scalar block with its line identity, expressed in the finite-sum core fibers. · OAI.Kaplansky.repeatedLineMap

repeatedMultiplicityInclusion def

Insertion of a normalized repeated centered letter, with its exact norm. · OAI.KadisonSimilarity.repeatedMultiplicityInclusion

repeatedMultiplicityWord def

The repeated centered word s ⊗ ⋯ ⊗ s. · OAI.KadisonSimilarity.repeatedMultiplicityWord

repeatedPairs def

Ordered distinct pairs in one assigned bin. · OAI.EuclideanFiveColor.PhaseCollision.repeatedPairs

repeatedSuccess def

The repeated-success expression used by both real transitions, without an exponentially expanded balanced-integer numerator syntax. · OAI.ExactQuantumFactoring.RatExpr.repeatedSuccess

repeatedThroughCode def

Every repeated multiset through a is either aa followed by a remainder, or abb followed by a remainder. · OAI.KadisonKastler.repeatedThroughCode

RepeatedThroughLabel abbrev

Multisets containing a prescribed label and at least one repetition. · OAI.KadisonKastler.RepeatedThroughLabel

repeatedWordInclusion def

The concrete column at a fixed word degree, endpoint, and repeated letter. · OAI.KadisonSimilarity.repeatedWordInclusion

repeatedWordRealization def

The concrete repeated-multiplicity insertion, together with raw word factorization and level actions, gives the propagation realization. Complete prefix isometry and uniform boundedness are both proved by this constructor. · OAI.KadisonSimilarity.TracialCreationFormulas.repeatedWordRealization

repeatStrategy def

Playing a fixed base strategy separately in each coordinate is one permitted repeated strategy, though repeated strategies are more general. · OAI.MaxCutGames.Foundations.Games.Game.repeatStrategy

repeatStrategy def

Playing a fixed base strategy separately in each coordinate is one permitted repeated strategy, though repeated strategies are more general. · OAI.UniqueGamesTheorem.Foundations.Games.Game.repeatStrategy

repeatTuple def

Repeat the same row tuple a times. · OAI.TalagrandDiscreteConvexity.repeatTuple

repeatWord def

Periodic repetition is used solely as a finite-admissibility witness; it is not the ergodic measure ultimately selected. · OAI.SmoothObstruction.PhaseDynamics.repeatWord

repetition def

Parallel repetition retains arbitrary dependence on each full local tuple. · OAI.MaxCutGames.Foundations.Games.Game.repetition

repetition def

Parallel repetition retains arbitrary dependence on each full local tuple. · OAI.UniqueGamesTheorem.Foundations.Games.Game.repetition

repetitionCount def

The repetition count is computed once from the gap, target error, and noise denominator, independently of the input formula. · OAI.MaxCutGames.Foundations.Hastad.SourceGap.repetitionCount

repetitionCount def

The repetition count is computed once from the gap, target error, and noise denominator, independently of the input formula. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceGap.repetitionCount

repetitionLength def

Exact rational search, independent of the input formula. · OAI.MaxCutGames.Integration.SourceParameters.repetitionLength

repetitionLength def

Exact rational search, independent of the input formula. · OAI.UniqueGamesTheorem.Integration.SourceParameters.repetitionLength

replace def

Replace just one placed register, keeping every other physical wire. · OAI.ExactQuantumFactoring.Register.replace

replace def

Delete centers at distance at most 100r, then insert the heavy request. · OAI.KServer.HeavyCenters.replace

replaceComparisons def

Number of cached comparisons made by a leaf replacement. · OAI.LowSpaceSubsetSum.Tournament.replaceComparisons

replacement def

The instance of Replacement for an entirely arbitrary formula p(params,x,y): if it is single-valued and total on A, its image is a set. As with Separation, the parameters are all universally closed. · OAI.PartitionConsistency.SetLanguage.replacement

replacement def

A positive middle boundary suffices to make the replacement a strict globular cell, with every lower boundary equation retained. · OAI.Thomason.Construction.DirectedComplex.Cell.replacement

replacementBound def

An orbit-closed bound for one Replacement problem. Instead of a uniform name-rank cutoff, take the whole set-group orbit of the ground-chosen local witnesses. This set is invariant and is still small. No symmetric maximum principle, global … · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.replacementBound

replacementGraph def

The equality constraints and inherited original constraints on the actual rotation. · OAI.MaxCutGames.Foundations.PCP.DegreeReplacement.replacementGraph

replacementGraph def

The equality constraints and inherited original constraints on the actual rotation. · OAI.UniqueGamesTheorem.Foundations.PCP.DegreeReplacement.replacementGraph

replacementMass def

0/1 are top/bottom of width 0, 2/3 of width 1, 4 is the tag. In particular equal numerical widths do not identify labels. · OAI.FiniteFactor.CouplingParameters.replacementMass

replacementStep def

Internal ports rotate inside a cloud; the extra port reverses an old dart. · OAI.MaxCutGames.Foundations.PCP.DegreeReplacement.replacementStep

replacementStep def

Internal ports rotate inside a cloud; the extra port reverses an old dart. · OAI.UniqueGamesTheorem.Foundations.PCP.DegreeReplacement.replacementStep

replacementTable def

Replace the upper part of X by A and retain one chosen r-boundary of X. · OAI.Thomason.Construction.DirectedComplex.Cell.replacementTable

replacementWeights def

Retain supported draws and replace all others by an independent retained-demand draw. · OAI.UniformSparsestCut.FiniteWeights.replacementWeights

replaceNew def

Fix the old subposet and send every new element to a fixed value. · OAI.Thomason.Construction.SieveRetraction.replaceNew

replaceNoise def

Replacement of exactly one current interpolation noise. · OAI.PlanarL1.Columns.Model.Point.RadialTree.replaceNoise

replaceOn def

One independent replacement value per matrix entry, not per row fiber. · OAI.MaxCutGames.Inverse.RowErasure.replaceOn

replaceOn def

One independent replacement value per matrix entry, not per row fiber. · OAI.UniqueGamesTheorem.Inverse.RowErasure.replaceOn

replaceRecord def

Replace the front variable-width record, consuming the replacement in 5. · OAI.RationalHitting.StackLanguage.replaceRecord

Replaces def

A single admissible replacement, with both endpoints in the stipulated set of alignments. Its right factor is the same as that of the path indices. · OAI.PartitionConsistency.DiagramOrder.Replaces

replayEquationInput def

The reference is fixed public advice. The only remaining argument contains the equation player's own clean equations; it contains no selected positions. · OAI.UniqueGamesTheorem.Clean.LocalReplay.replayEquationInput

replayMatrix def

Jam the arbitrary dirty memory in an identity summand. The lower-left block is the designated rectangular map extended by zero on dirty scratch. · OAI.ExactFourier.replayMatrix

replayOrder def

Move the dirty gate memory into a literal identity summand. · OAI.ExactFourier.replayOrder

replayStep def

Proposed transcript Z has controller at each digit. Scan local block i from block data B. Continue applying the visits even on a failed test. Normalize initial B (no-op on semantic arguments). · OAI.Fifths.Tour.replayStep

replayVariableInput def

The only remaining argument contains the variable player's own clean names; it contains no clean equation or right-hand side. · OAI.UniqueGamesTheorem.Clean.LocalReplay.replayVariableInput

replica def

A part is shared over one copy-symbol pair. · OAI.ExactDerandomization.Est.replica

Representable def

Order-isomorphism with the entire congruence lattice of a nonempty finite algebra. · OAI.FiniteCongruence.Representable

Representation structure

An arbitrary vector-space representation of the presented quadratic algebra. · OAI.LittleFinitistic.QuadraticEncoding.Representation

representation def

The true enveloping-ring action on the algebra. · OAI.FiniteAlgebraicEnvelopes.Enveloping.representation

representation def

The star representation on each member of the Zorn family. · OAI.KadisonSimilarity.CyclicDecompositionData.representation

representation def

The triangular algebra representation associated to a bounded derivation. · OAI.KadisonSimilarity.Derivation.representation

representation def

The representation used in the rank-gap argument. Its coefficient field is a commutative Hahn field; all inversion takes place inside the division subring, never by adjoining arbitrary matrix-field scalars. · OAI.RationalHitting.QuantumLift.representation

representation def

The signs genuinely compose; no group-action relation is assumed. · OAI.RowColumn.Signed.representation

representationBufferCapacity def

The mutable tie buffer chosen by the manuscript. · OAI.LowSpaceSubsetSum.representationBufferCapacity

representationComparison def

Conversion of the source presentation commutes naturally with the tensor. · OAI.Thomason.ConstructedInterval.representationComparison

RepresentationCompletelyBoundedBy def

Complete bound for a Hilbert-space algebra representation. · OAI.KadisonKastler.RepresentationCompletelyBoundedBy

representationControlledCochain def

Scalar coefficient extraction, with its support obligation proved. · OAI.PolycyclicRecognition.representationControlledCochain

representationCost def

The start cost of a signed segment representation. · OAI.BoundedTreePotentials.Segments.representationCost

representationEquivalence def

The two presentations describe equivalent categories of strict omega-categories. · OAI.Thomason.Construction.representationEquivalence

representationModel abbrev

At each node the root-start coefficients form ℓ¹ over the possible endpoints; the children form ℓ² of their own representation spaces. The outer ℓ¹ pair is exactly q_s=m_s+(Σ q_c²)^(1/2). · OAI.DiamondDistortion.StartModel.representationModel

representationOutput def

The literal output coefficient of a countable segment array. Equivalently this is the double sum over t≼v with t≼s≼v; Segment carries exactly that pair. · OAI.BoundedTreePotentials.Segments.representationOutput

representationOutputMap def

The linear output of literal arrays, with absolute convergence proved for every finite-cost array. · OAI.BoundedTreePotentials.Segments.representationOutputMap

RepresentationPairData structure

Runtime fields for one low-filtered child-pair stream. · OAI.LowSpaceSubsetSum.RepresentationPairData

representationPairing def

The absolutely convergent countable representation pairing on finitely supported tests. · OAI.BoundedTreePotentials.Segments.representationPairing

RepresentationParentData structure

Data available to a parent task: input, signs, leaf domains, moduli, residues, capacity, and the current partial top mask. No solution is stored. · OAI.LowSpaceSubsetSum.RepresentationParentData

RepresentationScheduleData structure

Primitive data retained before the residue loop. There is no designated solution or selected successful residue in the runtime state. · OAI.LowSpaceSubsetSum.RepresentationScheduleData

representationSequence def

The bounded coordinate representation, before taking its quotient. · OAI.NuclearUltrapower.VariableUltra.representationSequence

RepresentationSetup structure

A fixed YES instance and the three pre-order mixer locations. · OAI.LowSpaceSubsetSum.RepresentationSetup

representationSupport def

The nondegenerate support of a possibly nonunital representation. · OAI.NuclearUltrapower.representationSupport

representationVonNeumannAlgebra def

The bicommutant σ(A)″ of the represented C-star algebra. · OAI.KadisonSimilarity.representationVonNeumannAlgebra

representative def

A representative with exactly the quotient norm bound. · OAI.KadisonKastler.WeakOperatorField.representative

representative def

An actual bounded representative of each homotopy-category object. · OAI.LittleFinitistic.BoundedHomotopy.representative

representative def

A representative is used only to define operations; all evaluations below show that the resulting quotient cells are independent of it. · OAI.Thomason.Construction.StrictOmegaCategory.ColimitConstruction.Cell.representative

representativeBandWithWork def

Full stored-data representative construction, including the Cartesian product and all subsequent numerical work. · OAI.EditApproximation.BinaryFraction.representativeBandWithWork

representativeBlockProfile def

Smaller masks, complementary masks, and outside indices fill each block. · OAI.LowSpaceSubsetSum.representativeBlockProfile

representativeCategoryEquiv def

The canonical three-category domain is exactly the actual mixer. · OAI.LowSpaceSubsetSum.representativeCategoryEquiv

representativeCompleteBitDegree def

The representative tail used to construct the actual online draw range. · OAI.EditApproximation.representativeCompleteBitDegree

representativeFromActionsWithWork def

The actions have been materialized once and are reused for the wide minimum and all prefix queries. · OAI.EditApproximation.BinaryFraction.representativeFromActionsWithWork

representativeMap def

A bounded chain-map representative of a homotopy-category morphism. · OAI.LittleFinitistic.BoundedHomotopy.representativeMap

representativeMask def

One concrete mask for every distinct integer weight at the selected size. · OAI.LowSpaceSubsetSum.representativeMask

represented def

Elements with a genuine decorated-tree witness for both their algebraic value and their actual cube action form a subgroup. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Diagram.represented

representedCore def

Every finite-cost literal representation defines an element of the full incomparable dual through the core/full-dual equivalence. · OAI.BoundedTreePotentials.Segments.representedCore

representedCorner def

Compression of a von Neumann algebra to the range of one of its projections, represented on that range. · OAI.KadisonKastler.representedCorner

representedCornerAlgebra def

The algebra of the concrete corner; its identity extends to the given orthogonal projection in the ambient algebra. · OAI.KadisonKastler.representedCornerAlgebra

representedDoubleMatrix def

The four-by-four dilation acts on two successive two-coordinate sums. · OAI.KadisonSimilarity.representedDoubleMatrix

representedGraphIso def

A common erased shape gives an isomorphism of the represented graphs. · OAI.TwoPointCorrelations.representedGraphIso

representedInnerDerivation def

The represented commutator, with its boundedness proved from multiplication. · OAI.KadisonKastler.representedInnerDerivation

RepresentedLocalChart structure

Only the concrete local chart data needed for the analytic atlas. · OAI.PolycyclicRecognition.RepresentedLocalChart

representedMatrix def

A C-star matrix acts entrywise on actual finite Hilbert sums. · OAI.KadisonSimilarity.representedMatrix

representedMatrixHom def

The concrete matrix action is a unital star representation. · OAI.KadisonSimilarity.representedMatrixHom

representedMatrixUnitary def

A matrix unitary acts isometrically through the actual left representation. · OAI.KadisonSimilarity.representedMatrixUnitary

representedSubcorner def

Both corners act on the same pH; only the membership predicate changes. · OAI.Generator.Tracial.representedSubcorner

Represents def

A normalized real root representing a genuine Coxeter reflection. · OAI.KLInvariance.TitsSpace.Represents

Represents def

A retained entry has a subset witness within its parent's allowed domain. · OAI.LowSpaceSubsetSum.Represents

Represents def

Eventual representation is explicitly insensitive to all entries in a finite initial segment. · OAI.PartitionConsistency.ProductForcing.ChainCode.Represents

Represents def

Exact existence on members, no witness on nonmembers, and uniqueness of the whole witness on every input. No coordinates are hidden outside w. · OAI.SingleFold.Represents

Represents structure

Finite array semantics: no periodic/oracle tests are hidden in the code. · OAI.Superstring.ThreadArray.Represents

Represents def

The representation invariant retains all capped paths, not just paths with positive probability under a particular table. · OAI.UniformKServer.Represents

representTraceFamily def

Taking trace coordinates preserves linear dependence on the family index. · OAI.MaxCutGames.Quadratic.representTraceFamily

representTraceFamily def

Taking trace coordinates preserves linear dependence on the family index. · OAI.UniqueGamesTheorem.Quadratic.representTraceFamily

reprogram def

Transport only the proof-facing program index, leaving the executed result intact. · OAI.LowSpaceSubsetSum.WordSeedMeasured.reprogram

Request structure

A call changes only the requested slot, by the indicated multiple. · OAI.Fifths.Circuit.Request

Request abbrev

All the dense sets needed for the external induction on standard formulas. Because each formula has finitely many free variables, the requests are countable whenever the set of name codes is countable. · OAI.PartitionConsistency.ForcingQuotient.AtomicData.Request

Request abbrev

The two actual operations to execute in a round. · OAI.PartitionConsistency.MonoidConstruction.Request

Request structure

The primitive-layer requester uses an actual child vertex at this record. · OAI.Superstring.Hierarchical.HostedGroup.Request

request def

Reassociate the two rational parameters without re-encoding them. · OAI.ContingencyTables.CountingBranchProgram.request

RequestCell def

Forgetting heights takes each executed request to precisely one of the cells above, not to an object merely satisfying the same axioms. · OAI.PartitionConsistency.ConservativeMonoids.RequestCell

requestHistory def

True request history alone, for computing the metric posterior independently of the partition tape. · OAI.KServer.FiniteExperiment.requestHistory

requestIso def

Identification of the next stage with the actual operation's normal forms. · OAI.PartitionConsistency.MonoidConstruction.Round.requestIso

requestPrefix def

A finite observation sufficient for exactly the request requestPrefix. · OAI.KServer.FiniteExperiment.requestPrefix

reRoot def

Changing the bookkeeping root does not change a geometric point. · OAI.PlanarL1.Columns.Model.Point.reRoot

resample def

Only the numerical tuple products change. Orientations, padding, and labels remain fixed. · OAI.TwoPointCorrelations.LabeledPrimeWord.resample

ResampledProhibitedEvent def

The numerical event retains every arithmetic assumption used to get the last-step coefficient. The reciprocal sum may subsequently enlarge to coincident numerical assignments. · OAI.TwoPointCorrelations.ResampledProhibitedEvent

ResampledWitnessEvent def

Conditions on an actual numerical assignment, before enlarging its event into the finite union of symbolic relation systems. · OAI.TwoPointCorrelations.ResampledWitnessEvent

resamplingFragment def

The retained target fragment; no quotienting of the label indices occurs. · OAI.LeanBlast.SecondKahnKalai.resamplingFragment

resamplingTransition def

The finite transition matrix for the actual two-stage cluster resampling. · OAI.HyperedgeComparison.resamplingTransition

resamplingZ def

The actual normalizing density Z(Y). · OAI.LeanBlast.SecondKahnKalai.resamplingZ

rescaleCompactProfile def

The operator on compact profiles induced by a change of chart. · OAI.PolycyclicRecognition.rescaleCompactProfile

rescaledContinuousProfile def

The actual rescaled profile as a continuous function of the chart point. · OAI.PolycyclicRecognition.rescaledContinuousProfile

rescaledGraphAction def

The common graph action implementing x ↦ r*x*s. · OAI.KadisonKastler.rescaledGraphAction

rescaledGraphMap def

The conjugation in the second coordinate makes the graph map complex-linear rather than conjugate-linear. · OAI.KadisonKastler.rescaledGraphMap

rescaledGraphOperatorMap def

The graph formula is linear on all ambient operators. · OAI.KadisonKastler.rescaledGraphOperatorMap

rescaledGraphProjection def

Orthogonal projection onto the closed graph. · OAI.KadisonKastler.rescaledGraphProjection

rescaledGraphSignalMap def

The actual graph signal, viewed as a real-linear map for sign sums. · OAI.KadisonKastler.rescaledGraphSignalMap

rescaledGraphSubspace def

The closed linear rescaled graph. · OAI.KadisonKastler.rescaledGraphSubspace

rescaledGraphVector def

Graph coordinates after identifying the conjugate Hilbert space by J. · OAI.KadisonKastler.rescaledGraphVector

rescaledLieData def

The coordinate law and coordinate generators belonging to one rescaling. · OAI.PolycyclicRecognition.rescaledLieData

rescaledLog def

The rescaling epsilon^(-2m) tau(epsilon z), followed by log. · OAI.Mahler.rescaledLog

rescaledLogEnergy def

The exact logarithm in the rescaled mass argument. · OAI.MahlerRescaling.rescaledLogEnergy

rescaledMapProfile def

The map profile on an exponential chart, centered at x with suffix k. · OAI.PolycyclicRecognition.rescaledMapProfile

rescaleField def

The usual reciprocal-Lipschitz normalization of a vector field. · OAI.LipschitzHilbert.rescaleField

RescalingNetFamily structure

Every radius and accuracy has a fixed finite label type across the whole sequence. The point at infinity is not introduced at this stage. · OAI.RegularPointCoordinates.RescalingNetFamily

reserve def

The reserve is used for |m| < 1 and 0 ≤ I < H m. · OAI.SoftChannel204.reserve

reservoirRate def

The old-site Poisson intensity, written so that its deficit from αN is manifestly nonnegative. The exponent is p-1. · OAI.DilutedSpinGlass.reservoirRate

reset def

Reset conserves the digit's unary budget, then enters the next coordinate. · OAI.MaxCutGames.Foundations.Complexity.MachineTupleOdometer.reset

reset def

Reset only the specified member to zero, the first color index. · OAI.TalagrandThreshold.Selector.reset

reset def

Reset conserves the digit's unary budget, then enters the next coordinate. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineTupleOdometer.reset

residual def

The full right residual of the rational inverse candidate. · OAI.AtomicTriangular.FiniteCertificate.AtomicMatrix.residual

residual def

The residual of the implicit velocity equation at the proposed velocity u. · OAI.BoltzmannNonuniqueness.AxialFlow.residual

residual def

A reference completion identifies the free input and output sites. · OAI.CoordinateSweeps.Grid.Holes.residual

residual def

The remaining permutation relative to those identifications. · OAI.CoordinateSweeps.Grid.Holes.residual

residual def

The same residual formula as Rule II on a fixed star universe. · OAI.KServer.OutputDynamics.residual

residual def

Numerator after substituting z=(Z-Hs)/R, u=P(s)/Q in the exact normalized source ODE. All arithmetic in certificates is integral. · OAI.SepticProfile.IntegerPolynomial.residual

residual def

Actual one-step Taylor residuals. The affine matching and terminal constraints really have zero residual. · OAI.SmoothObstruction.ArrayRemainder.residual

residual def

The actual normal residual in the ambient inner-product space. · OAI.ThompsonNonamenability.Analytic.RegularCurveData.residual

residualBound def

Upper bound F₀ = 5.1 × 10⁻⁷ for the stationary residual. · OAI.StationaryNS.residualBound

residualColumnBound def

The residual column optimum after deleting the sampled endpoints. · OAI.JointDickman.residualColumnBound

residualCutoff def

The lower edge used for the residual integer in the comparison lemma. · OAI.TotientAsymptotic.residualCutoff

residualFamily def

Evaluation rows with the rank-one interpolation correction. · OAI.CriticalHoneycomb.FiniteFunctional.StripAsymptotic.residualFamily

residualFamily def

An injective tuple contributes a residual generator exactly when all of its ordered pair indices were avoided. · OAI.TalagrandDiscreteConvexity.Covering.residualFamily

residualFraction def

The off-path fraction of the complete child denominator. · OAI.LipschitzHilbert.residualFraction

residualGroupAutomorphism def

The actual group automorphism induced by the Cartan residual derivation. · OAI.PolycyclicRecognition.CartanWordModel.residualGroupAutomorphism

residualInput def

The finite frontier's residual parent input, before any current map is chosen. · OAI.ReflexiveFixedPoints.Prediction.Prefix.residualInput

residualInput def

The common point receives all unassigned mass in each parent group. · OAI.ReflexiveFixedPoints.Prediction.residualInput

residualLaw def

A residual law exists whenever the ordinary branch is pointwise dominated by the target law. No additional existence assumption is used. · OAI.ContingencyTables.ResidualMixture.residualLaw

residualLogCorrection def

The logarithmic correction from the off-path denominator. · OAI.LipschitzHilbert.residualLogCorrection

residualMass def

The residual point mass used on the rare correction branch. · OAI.ContingencyTables.ResidualMixture.residualMass

residualNumerator def

The integer numerator in equation eq:residual-integers of 200-01. Here Q = 2^R, T = 2^D, and a x is the tabulated dyadic numerator. · OAI.ContingencyTables.ResidualMixture.residualNumerator

residualP def

The literal normalized source polynomial residual, expanded about z=1. · OAI.SepticProfile.ParametricJets.residualP

residualP def

The literal normalized source polynomial residual, expanded about z=1. · OAI.SepticProfile.SourceJets.residualP

ResidualPath def

A directed residual path with k arcs and no repeated vertices. · OAI.ContingencyTables.PublishedEdmondsKarp.ResidualPath

residualPoint def

The parent input predicted from earlier candidates and one common terminal point. · OAI.ReflexiveFixedPoints.Prediction.residualPoint

residualPolynomial def

The literal residual polynomial of two finite operator/amplitude series. · OAI.ElasticityBoundaryMVTower.residualPolynomial

residualProduct def

Descendant-only factor after the first exit. · OAI.IndependentProducts.residualProduct

residualRelation def

The residual relation after setting a=0; variables are d,b,c,u. · OAI.StableCoordinate.DeterminantPrimeA.residualRelation

ResidualVectorCache abbrev

Candidate matrices, ordinary numerators, denominator/branch size, and table count. · OAI.ContingencyTables.ProfilePrograms.ResidualVectorCache

residualWeight def

Exact FK polynomial of the graph with C contracted and R remaining. Using connected-component counts avoids any relabelling or suppressed-edge convention for the residual multigraph. · OAI.FKCRT.residualWeight

residueAmplitude def

Equation (order-P), before squaring the norm. · OAI.ExactQuantumFactoring.OrderTrial.residueAmplitude

residueCenteredKernel def

Center a kernel in both of its residue coordinates. · OAI.JointDickman.residueCenteredKernel

residueCenteredTransition def

Subtract the uniform residue average within each base cell. · OAI.JointDickman.residueCenteredTransition

residueCharacterFourier def

The finite Fourier transform of the actual residue multiplicative character. · OAI.CubicFirstMoment.residueCharacterFourier

ResidueCharacterInduces def

Agreement of characters on ideals prime to the original modulus. · OAI.CubicFirstMoment.ResidueCharacterInduces

residueCircuitInputs def

Inputs to an actual Boolean circuit, with each input depending on one specified residue coordinate. Equality with a fixed residue is a special case. · OAI.TwoPointCorrelations.residueCircuitInputs

residueCoefficientMap def

Restrict an actual coefficient map to the manuscript's finite residue interval. · OAI.Nagata.Workers.W10.residueCoefficientMap

residueColor def

The two contiguous residue blocks used by the explicit drawing. · OAI.Paper170.residueColor

residueComponent def

The multiplicative pieces of a function restricted to a residue class. · OAI.TwoPointCorrelations.residueComponent

residueCount def

Number of offsets 0 ≤ j < N whose translated residue equals r. · OAI.TwoPointCorrelations.residueCount

ResidueData structure

The weaker, directly consumable arithmetic data: only actual ceiling residues are bounded. In particular a non-strict residue bound is sufficient. · OAI.Problem337.GeometricDensity.ResidueData

residueData_of_two_lists def

Combine the deterministic and sampled lists according to the level cutoff. The index set is kept fixed, so repeated sampled products retain their full multiplicity. The same choice also changes the auxiliary factor from C to 1. The … · OAI.Problem337.GeometricDensity.residueData_of_two_lists

residueEntry def

Extend finite sample data by zero outside their canonical natural indices. · OAI.Problem337.residueEntry

residueEval def

Evaluation at the point extends to its actual local ring, because the allowed denominators have nonzero values in the coefficient field. · OAI.Nagata.AffineMultiplicity.residueEval

residueFiber def

Starting labels producing a prescribed diagonal-sum residue. · OAI.Paper170.residueFiber

residueFourier def

The positive-sign discrete Fourier transform, convenient for residue products. · OAI.Problem337.residueFourier

residueFourierAverage def

The ordinary normalized exponential sum, defined also at modulus zero. All residue conclusions below explicitly require a positive modulus. · OAI.Problem337.residueFourierAverage

residueFourierChar def

Canonical primitive additive character modulo any nonzero Eisenstein integer. · OAI.CubicFirstMoment.residueFourierChar

residueFrequencies def

The finite Fourier frequency window used at every scale. · OAI.Problem337.RandomProducts.residueFrequencies

residueGaussianNumerator def

A literal finite integer sum, used to prove dyadicity independently of the bounded-piece evaluator. This sum is NOT asserted to be polynomial-time. · OAI.ExactQuantumFactoring.OrderTrial.residueGaussianNumerator

residueIndices def

Natural indices for a finite sample space. Distinct samples remain distinct indices even when their integer-valued data agree. · OAI.Problem337.residueIndices

residueIntegral def

I(u) in equation (majorant-integral), without a restriction on u needed for its analytic bounds. · OAI.ExactQuantumFactoring.OrderTrial.residueIntegral

residueLedger def

No right-hand work is counted after dictionary overflow. The control count includes even the unused fuel suffix of the original run loop. · OAI.LowSpaceSubsetSum.CompressionData.residueLedger

residueLoop def

Tail-recursive, constant-state enumeration of a consecutive residue range. · OAI.LowSpaceSubsetSum.CompressionSetup.residueLoop

residueMass def

The probability mass function obtained by reduction modulo p. · OAI.Ostmann.residueMass

ResidueOffsets structure

The offsets may depend on the layout, guesses, and subblock partition; they are fixed before the common prime tuple and its target residues. · OAI.LowSpaceSubsetSum.RepresentationSetup.ResidueOffsets

ResidueOneAssociated def

x differs from y by multiplication by a unit whose residue is one. · OAI.CirculantHadamard.LocalComparison.ResidueOneAssociated

residueOrbit def

The nonnegative exponent orbit. In a finite group this is the manuscript's H; using its submonoid presentation makes the m^u p^v representatives explicit. · OAI.ExactQuantumFactoring.Primality.residueOrbit

residueOrder def

Residues are ordered 0,...,p-1, independently of their field operations. · OAI.KaplanskyCounterexample.Prescription.residueOrder

residuePass def

One explicitly selected residue pass, using concrete streams and tries. · OAI.LowSpaceSubsetSum.CompressionSetup.residuePass

residuePeriod def

The actual common period is built from the cleared formula denominators and the two retained frequency moduli. · OAI.Ostmann.MovingFormulaNode.residuePeriod

residuePermutation def

Reduction modulo two, acting by affine permutations on its eight points. · OAI.NuclearUltrapower.Elementary.residuePermutation

residuePermutation def

Reduction modulo two, acting by affine permutations on its eight points. · OAI.NuclearUltrapower.residuePermutation

ResiduePositiveWord def

Positivity expressed in the one common set of residue coordinates. · OAI.TwoPointCorrelations.ResiduePositiveWord

residuePrefix def

A prefix sum on any residue class, without a coprimality assumption. · OAI.TwoPointCorrelations.residuePrefix

residueProbability def

The residue law of a uniformly selected origin from an interval. · OAI.TwoPointCorrelations.residueProbability

residueQueries def

The query counter of the actually enabled two joins. The right join is skipped on overflow, and all joins are skipped when a compact-list test fails. · OAI.LowSpaceSubsetSum.CompressionData.residueQueries

residueQuotientEquiv def

The radical quotient is unchanged by a square-zero kernel. · OAI.LittleFinitistic.PDSquareZero.residueQuotientEquiv

residueReduction def

Reduction between actual Eisenstein residue rings. · OAI.CubicFirstMoment.residueReduction

residueRingEquiv def

The right-module quotient used by tensor detection is the underlying vector space of the usual quotient ring A/rad A. · OAI.LittleFinitistic.PDTensor.residueRingEquiv

residueSamplePairs def

The same labelled samples, in the paired-block interface. · OAI.Problem337.RandomProducts.residueSamplePairs

residueSubsetProduct def

The deterministic subset list on the same Boolean-word index type as the sampled lists. This changes indices only, never the list entries or multiplicities. · OAI.Problem337.RandomProducts.residueSubsetProduct

residueTensorEquiv def

The actual noncommutative tensor/quotient identification in cor:pd-detection. · OAI.LittleFinitistic.PDTensor.residueTensorEquiv

residueTorIso def

Actual residue Tor is computed by tensoring the free D-bar with D/rad D. This proves the needed change-of-rings comparison on the actual models. · OAI.LittleFinitistic.PDResidueAssociativity.residueTorIso

residueTotalVariation def

Half the ℓ¹ difference from the uniform residue law. · OAI.TwoPointCorrelations.residueTotalVariation

residueUsable def

The usable-slot test is just range plus Euclidean gcd, not an order oracle. · OAI.ExactQuantumFactoring.residueUsable

residueWordEquiv def

Reindex Boolean product words by the two-element finite type used in the auxiliary-denominator construction. No product values are identified. · OAI.Problem337.RandomProducts.residueWordEquiv

resizeInTime def

The resize certificate is built from actual primitive and table executions; its caller supplies no phase trace or computational hypothesis. · OAI.MaxCutGames.Foundations.Complexity.MachineExpanderFamily.resizeInTime

resizeInTime def

The resize certificate is built from actual primitive and table executions; its caller supplies no phase trace or computational hypothesis. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineExpanderFamily.resizeInTime

resizeTable def

Change only a proof-equal size index, without selecting any new numbering. · OAI.MaxCutGames.Foundations.PCP.PreprocessingRegularTables.resizeTable

resizeTable def

Change only a proof-equal size index, without selecting any new numbering. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingRegularTables.resizeTable

resizeWord def

Ordinary zero extension/truncation, without copying a word circuit once per bit. · OAI.ExactQuantumFactoring.BitArithmetic.resizeWord

Resolution structure

The bivalent parts of a finite graph can be suppressed. In addition to its boundary matching we retain extension of every antisymmetric edge weight. · OAI.ArtinCAT0.WireSystem.Resolution

resolution def

A projective resolution, including its augmentation quasi-isomorphism. · OAI.LittleFinitistic.PDMinimal.resolution

resolution def

The actual fixed resolution supplies every generic native/bar premise. · OAI.LittleFinitistic.SelectedBarComparison.resolution

resolution def

The genuine absolute bar projective resolution of each actual inflated module. It can have infinitely many nonzero degrees. · OAI.LittleFinitistic.SelectedBarResolution.resolution

resolution def

A genuine minimal projective resolution of each actual constructed A-module. · OAI.LittleFinitistic.SelectedMinimalResolution.resolution

resolutionAugmentation def

The actual augmentation of the bundled complex in its last degree. · OAI.LittleFinitistic.OrdinarySimulation.ChainAlgebra.resolutionAugmentation

resolutionDifferential def

Actual left-arrow differentials, with no transport of module types. · OAI.LittleFinitistic.OrdinarySimulation.ChainAlgebra.resolutionDifferential

resolutionLift def

Full faithfulness of precomposition by the localization descends that map. · OAI.DerivedBar.resolutionLift

resolutionLiftUp def

A candidate map into the resolved model, before descending the localization. · OAI.DerivedBar.resolutionLiftUp

resolutionMap def

The differential is actual left multiplication by the source arrow. · OAI.LittleFinitistic.OrdinarySimulation.ChainAlgebra.resolutionMap

resolutionModel def

The actual Q_T augmentation with all source resolution obligations proved. · OAI.LittleFinitistic.OrdinarySimulation.TensorResolution.resolutionModel

resolutionModel def

The actual Q_X resolution with no assumed augmentation, projectivity, or support theorem. · OAI.LittleFinitistic.OrdinarySimulation.XResolution.resolutionModel

resolveColumnReference def

A reference resolves either to a perfect label or to a fresh imperfect class. · OAI.TwoPointCorrelations.resolveColumnReference

Resolved inductive

After fixing all occupied wire sets, a physical honeycomb history is a single planar word or zero, not a linear superposition of connectivities. · OAI.CriticalHoneycomb.Transfer.PlanarNormalization.Resolved

resolvedBimoduleComparison def

The source's resolved T tensor O comparison in the native derived category of genuine B-bimodules. Its left operand includes the actual bounded right-BC-projective resolution and augmentation to T. · OAI.LittleFinitistic.OrdinarySimulation.resolvedBimoduleComparison

resolvedMap def

A map out of a nonsingular source has a specified map into the resolution after passing to its nondegenerate face poset. · OAI.Thomason.Construction.FaceSimplexResolution.resolvedMap

resolvedSquareComparison def

The source's native derived square with both tensor functors tied to their certified full-support models. · OAI.LittleFinitistic.OrdinarySimulation.resolvedSquareComparison

resolvedTensor def

Tensor of X with the verified absolute bar projective resolution. This construction applies the genuine balanced functor to every differential. · OAI.DerivedBar.BimoduleTensor.resolvedTensor

resolvedTensorIso def

The categorical derived-resolution model and the explicit balanced bar complex agree. · OAI.DerivedBar.BimoduleTensor.resolvedTensorIso

resolvedWeights def

Physical tile with the positive center-count factor removed. The forbidden double-return connectivity still has coefficient exactly zero. · OAI.CriticalHoneycomb.Transfer.PlanarNormalization.resolvedWeights

resolvent def

The inverse of id-sX, for a smooth uniformly small parameter s. · OAI.ArnoldCounterexample.Midpoint.resolvent

resolventAnchorPoint def

The anchor is constructed from actual resolvent tails. No idempotent, return law, or tail-limit convex-hull statement is an extra premise. · OAI.ReflexiveFixedPoints.resolventAnchorPoint

resolventAverage def

A velocity average of the damped free-transport resolvent. · OAI.BoltzmannNonuniqueness.Published.resolventAverage

resolventDifference def

The scalar difference whose positive-matrix functional calculus is integrated. · OAI.SharpLiebThirring.ScalarProof.resolventDifference

resolventNormalization def

The normalization constant for the resolvent representation. · OAI.SharpLiebThirring.ScalarProof.resolventNormalization

resonanceSum def

The ordinary fourth-moment quantity in the manuscript. · OAI.LipschitzHilbert.PhaseCombinatorics.resonanceSum

Respects def

A coloring is constant on precisely the components specified by a setoid. · OAI.BufferedIsing.Respects

Respects def

Literal oriented-edge and vertex labels into any target one-skeleton. The vertex label of a dart is the initial vertex of its oriented edge label. No filling or normal-form conclusion is included in this compatibility. · OAI.HyperbolicObstruction.PictureSwitch.PictureMap.Respects

Respects def

The interval is exactly covered by partition cells contained in it. · OAI.ThompsonNonamenability.BasicPartition.Respects

RespectsSector def

Matrix entries cannot change the read-only input. · OAI.ExactQuantumFactoring.RespectsSector

response def

Responses through a constant finite state space, with time-dependent laws. · OAI.BoundedTreewidthL1.ReservationCoupling.response

Responses abbrev

A pair of response kernels, each indexed only by its own complete input. · OAI.MaxCutGames.Clean.StochasticBound.Responses

Responses abbrev

A pair of response kernels, each indexed only by its own complete input. · OAI.UniqueGamesTheorem.Clean.StochasticBound.Responses

responseSign def

The best-response sign, with the manuscript's convention at zero. · OAI.JointDickman.responseSign

ResponseWitness structure

The part of a normalized affine witness needed by the left sampler. · OAI.MaxCutGames.Decoder.VisiblePolicies.ResponseWitness

ResponseWitness structure

The part of a normalized affine witness needed by the left sampler. · OAI.UniqueGamesTheorem.Decoder.VisiblePolicies.ResponseWitness

rest def

remainder of scalar profile. bstar uses A profile. · OAI.GeneralMahler.Segment.rest

restore def

Restore the prescribed boundary values. · OAI.TriangularSurface.FiniteModel.restore

restoredData def

Restore the full affine intercept after labeling the orbit body. · OAI.MaxCutGames.Decoder.TableKeysRestoration.restoredData

restoredData def

Restore the full affine intercept after labeling the orbit body. · OAI.UniqueGamesTheorem.Decoder.TableKeysRestoration.restoredData

restrict def

Restriction of a line bundle to any open subscheme. · OAI.ActualSheafTensor.LineTrivialization.restrict

restrict def

Restrict inputs to a central corner and compress the output. · OAI.BoundedHochschild.CentralCorner.restrict

restrict def

A weight vector restricted to a vertex set. · OAI.CliqueFreeIndependence.WeightedGraph.restrict

restrict def

A label-preserving permutation has genuine induced local permutations. · OAI.CoordinateSweeps.FiberBlocks.restrict

restrict def

Forget the parameter delimiter as an input/output letter. Work symbols and transitions are not changed. · OAI.FiniteAlgebraicEnvelopes.MachineRealization.UniversalMachine.restrict

restrict def

A union of whole polygonal components can be kept, discarding the others. This is a literal restriction of both permutations. · OAI.HyperbolicObstruction.PictureSwitch.PictureMap.restrict

restrict def

Restrict a ribbon to a union of its connected components. · OAI.KaplanskyCounterexample.Planar.Ribbon.restrict

restrict def

Discard components not containing the exception. · OAI.KaplanskyCounterexample.Planar.WeakBalanced.restrict

restrict def

Restrict any output labeling to the original vertices. · OAI.MaxCutGames.Explicit.Subdivision.restrict

restrict def

The values of a full Boolean tape on the legal coordinates. · OAI.MaxCutGames.Foundations.PCP.UniformRestriction.restrict

restrict def

Restrict a real observable to the actual affine quotient-map domain. · OAI.MaxCutGames.Fourier.MatrixRestrictions.restrict

restrict def

Restricting a lift preserves its prescribed logical characters. · OAI.MaxCutGames.Gadget.Lift.restrict

restrict def

Restriction to an invariant subspace, with the inherited Hilbert norm. · OAI.NuclearUltrapower.FiniteUnitaryRep.restrict

restrict def

Restriction to a set of label coordinates. · OAI.PartitionConsistency.CohenForcing.Condition.restrict

restrict def

Retaining the same representative gives the restriction/inclusion map. · OAI.PartitionConsistency.ProductForcing.LocalQuotient.restrict

restrict def

Subcounts with the same axes, allowing repeated monomial indices to collapse. · OAI.QAC.Count.restrict

restrict def

Restriction keeps the same energy representative and the same measure. · OAI.RegularPointCoordinates.LocalHarmonicEnergy.restrict

restrict def

Restriction of an actual bilateral gauge keeps exactly the same bases, not a recomputed QR path or a newly chosen stationary witness. · OAI.SmoothObstruction.ManifoldCoordinates.BilateralGauge.restrict

restrict def

Restriction to any full subposet containing the old part preserves the right-adjoint retraction. Downward closure of the subposet is not needed. · OAI.Thomason.Construction.SieveRetraction.restrict

restrict def

The genuine partition obtained by restriction and affine normalization. · OAI.ThompsonNonamenability.BasicPartition.restrict

restrict def

The actual symmetric subgraph on a specified set of original darts. · OAI.TorsionFreeZeroDivisors.ImageComplexity.restrict

restrict def

Restriction needs only injectivity, not a bijection onto the whole image. In particular, inactive marked vertices are never carried into a stage. · OAI.TorsionFreeZeroDivisors.RealizationTransport.restrict

restrict def

Restriction to the real forced coordinates, with only equality transport of the child-height index. · OAI.TraceReconstruction.OccupiedParityTree.Forced.restrict

restrict def

Restrict any output labeling to the original vertices. · OAI.UniqueGamesTheorem.Explicit.Subdivision.restrict

restrict def

The values of a full Boolean tape on the legal coordinates. · OAI.UniqueGamesTheorem.Foundations.PCP.UniformRestriction.restrict

restrict def

Restrict a real observable to the actual affine quotient-map domain. · OAI.UniqueGamesTheorem.Fourier.MatrixRestrictions.restrict

restrict def

Restricting a lift preserves its prescribed logical characters. · OAI.UniqueGamesTheorem.Gadget.Lift.restrict

restrictAssignmentEquiv def

Split all assignments into their values on an embedded coordinate set and on its complement. · OAI.Problem335.IndependentCoordinates.restrictAssignmentEquiv

restrictBagCut def

Literal restriction of a bag assignment to a smaller bag. · OAI.BoundedTreewidthL1.ParticleSchedule.restrictBagCut

restrictCells def

Affinely normalized cells. Membership is exactly membership after insertion. · OAI.ThompsonNonamenability.BasicPartition.restrictCells

restrictCircuit def

Restrict a circuit to the subfield generated by its occurring coefficients. · OAI.Problem335.restrictCircuit

restrictCube def

Restriction of a function to a coordinate subset, with exterior values fixed. · OAI.Problem313.restrictCube

restrictDependentAssignmentEquiv def

Dependent-coordinate version of restriction to an embedded coordinate set. · OAI.Problem335.IndependentCoordinates.restrictDependentAssignmentEquiv

restrictDerivation def

Restrict an sheaf derivation along an open immersion. · OAI.ActualCotangent.restrictDerivation

restrictedAlternate def

Alternate synthesis profiles are restricted along the same item map. · OAI.LipschitzHilbert.SourceLabeledLattice.restrictedAlternate

RestrictedCompletion def

Ordered completions in W of Q to an independent spanning tuple for L. · OAI.MaxCutGames.Inverse.KMSBasisComparisonPseudorandom.RestrictedCompletion

RestrictedCompletion def

Ordered completions in W of Q to an independent spanning tuple for L. · OAI.UniqueGamesTheorem.Inverse.KMSBasisComparisonPseudorandom.RestrictedCompletion

restrictedCompletionEquiv def

All nonempty restricted-completion fibers are equivalent. · OAI.MaxCutGames.Inverse.KMSBasisComparisonPseudorandom.restrictedCompletionEquiv

restrictedCompletionEquiv def

All nonempty restricted-completion fibers are equivalent. · OAI.UniqueGamesTheorem.Inverse.KMSBasisComparisonPseudorandom.restrictedCompletionEquiv

restrictedConjugate def

The convex conjugate with the source variable restricted to K. · OAI.Problem358.restrictedConjugate

restrictEdges def

Edges remaining after a vertex deletion. · OAI.AsymptoticallyMinimalLittlewood.Nibble.restrictEdges

restrictedIdeal def

Restriction of the ideal inclusion. · OAI.ActualSheafTensor.restrictedIdeal

restrictedIdentity def

Identity restricted to the retained points. · OAI.TwoWayComplementation.restrictedIdentity

restrictedLiftDensity def

The denominator counts all remaining tuples drawn uniformly from W. Dependent completions are rejected, without conditioning away their mass. · OAI.MaxCutGames.Inverse.KMSBasisComparisonPseudorandom.restrictedLiftDensity

restrictedLiftDensity def

The denominator counts all remaining tuples drawn uniformly from W. Dependent completions are rejected, without conditioning away their mass. · OAI.UniqueGamesTheorem.Inverse.KMSBasisComparisonPseudorandom.restrictedLiftDensity

restrictedLine def

The restricted ideal line atlas is induced by the kernel restriction isomorphism. · OAI.ActualCartier.restrictedLine

restrictedMixture def

The probability obtained by retaining a positive-measure collection of components and renormalizing. · OAI.DukePrimeDegree.restrictedMixture

restrictedOldIso def

Restricting the ambient poset retains a canonically isomorphic old part. · OAI.Thomason.Construction.restrictedOldIso

restrictedOperatorAction def

The restricted automorphisms form a group action. · OAI.KadisonKastler.NonsingularAction.restrictedOperatorAction

restrictedOperatorBimoduleMean def

The restricted mean retains the two module identities. · OAI.KadisonKastler.ScalarBoundaryMean.restrictedOperatorBimoduleMean

restrictedOperatorEquiv def

The boundary automorphism restricted to the M-valued field algebra. · OAI.KadisonKastler.NonsingularAction.restrictedOperatorEquiv

restrictedOperatorMean def

Restrict the scalar-coefficient operator mean to M-valued fields. · OAI.KadisonKastler.ScalarBoundaryMean.restrictedOperatorMean

restrictedOperatorMeanValue def

The restricted mean value, with its range proof separated from linearity. · OAI.KadisonKastler.ScalarBoundaryMean.restrictedOperatorMeanValue

restrictedOracle def

A failed domain test supplies the same zero word to the adaptive continuation. · OAI.EditApproximation.BitQuery.restrictedOracle

restrictedPacketEquiv def

The restricted packet type is precisely the selected finite set. · OAI.LipschitzHilbert.SourceLabeledLattice.restrictedPacketEquiv

restrictedPairChainMap def

Inclusion of restricted pairs (U,U∩A) into (V,V∩A). · OAI.Kaplansky.restrictedPairChainMap

restrictedPairSubsetIso def

Nested restrictions give the same actual pair, with ambient subspace topology and the same removed set. · OAI.Kaplansky.restrictedPairSubsetIso

restrictedPairToAmbientChainMap def

Inclusion of the actual chart pair into the whole pair. · OAI.Kaplansky.restrictedPairToAmbientChainMap

restrictedPolynomialFunctions def

Restricted polynomial functions on the punctured affine cone of G. The source construction lands in this genuine image submodule. · OAI.Nagata.Workers.W24.restrictedPolynomialFunctions

restrictedPositiveTrace def

Restrict the actual positive functional to a closed star subalgebra. · OAI.KadisonSimilarity.restrictedPositiveTrace

restrictedShiftIso def

Align the finite resolution's degree n with the source's degree a+n. · OAI.LittleFinitistic.OrdinarySimulation.restrictedShiftIso

restrictedTestFamily def

Restrict coefficient tests to an injected coordinate set, without imposing any condition on the omitted coefficients. · OAI.BoundedTreePotentials.restrictedTestFamily

restrictedTracialGNSInclusion def

The inclusion of the generated algebra induces its concrete GNS inclusion. · OAI.KadisonSimilarity.restrictedTracialGNSInclusion

restrictedYonedaIso def

Removing the harmless universe lift identifies the restricted Yoneda functor of the Kan-extension theorem with the actual Street nerve. · OAI.Thomason.Construction.NormalizedSimplexIdentities.restrictedYonedaIso

restrictFamily def

Restrict the source of a linear operator family to a chosen homogeneous subspace, leaving the codomain ambient. This avoids unnecessary transports between equivalent descriptions of the target homogeneous space. · OAI.Problem335.RankMeasure.restrictFamily

restrictHom def

Restriction to a subspace preserved by the whole star algebra is a genuine star homomorphism (both the subspace and its orthogonal complement are reducing). · OAI.RowColumn.FiniteStarAlgebra.restrictHom

restrictHomologyIso def

Restriction of scalars preserves the actual homology of a D-complex. · OAI.LittleFinitistic.PDBarQuotient.restrictHomologyIso

restrictInner def

Restrict an inner automorphism of the containing algebra to its ideal. · OAI.WeakPureInfiniteness.RightIdealRange.restrictInner

restriction def

Ordinary Cartier restriction on coherent sheaves. · OAI.ActualCartier.restriction

restriction def

The three-sector construction is an actual polynomial restriction degeneration in the existing tensor API, with leading order one. · OAI.MatrixMultiplication.AuxiliarySeparation.Sector.restriction

restriction def

Restriction of a continuous homotopy to a realized simplicial subobject. · OAI.Thomason.Construction.RealizationHomotopyRectification.restriction

restriction def

The ordinary restriction to the two sieves. · OAI.Thomason.Construction.SieveMap.restriction

restriction_freeIso def

The coherent restriction used by the Cartier resolution sends the finite free sheaf to the same finite free sheaf on the divisor. · OAI.ActualCartier.restriction_freeIso

restriction_powerIso def

Restricting each ambient ideal power gives the corresponding conormal tensor power on the divisor. · OAI.ActualCartier.restriction_powerIso

restriction_supported_iso def

Ordinary restriction of a closed pushforward is the original coherent sheaf on the divisor. · OAI.ActualCartier.restriction_supported_iso

restriction_tensorLine_iso def

Cartier restriction sends an ideal twist to the conormal twist by preservation of cokernels and centrality of multiplication. · OAI.ActualCartier.restriction_tensorLine_iso

restrictionCokernelIso def

Ordinary restriction is the cokernel of the ideal action. · OAI.ActualCartier.restrictionCokernelIso

restrictionDegeneration_approximation def

The existing polynomial-approximation API applies to a polynomial restriction as soon as the source tensor has a rank bound. · OAI.MatrixMultiplication.AuxiliarySeparation.restrictionDegeneration_approximation

RestrictionGlobal def

Rational restriction globality of a Boolean function. · OAI.MaxCutGames.Fourier.Deletion.RestrictionGlobal

RestrictionGlobal def

Rational restriction globality of a Boolean function. · OAI.UniqueGamesTheorem.Fourier.Deletion.RestrictionGlobal

restrictionGraph def

Restriction is a bounded application graph, not a change of interpretation of an old function name. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.restrictionGraph

restrictionIso def

On the actual supported degrees, mapping and extension have the same original terms. · OAI.LittleFinitistic.FunctorExtension.restrictionIso

restrictionIso def

Every parameter is recovered, since the parameter inclusion is full and faithful. This statement assumes no global preservation by the extension. · OAI.Thomason.Construction.StrictOmegaCategory.GlobePasting.restrictionIso

restrictionProductEquiv def

The values inside and outside a finite set reconstruct the full tuple. · OAI.MaxCutGames.Soundness.PartnerFullSampling.restrictionProductEquiv

restrictionProductEquiv def

The values inside and outside a finite set reconstruct the full tuple. · OAI.UniqueGamesTheorem.Soundness.PartnerFullSampling.restrictionProductEquiv

restrictionPullbackIso def

The descended ordinary restriction is the geometric pullback. This spells out its universal property, rather than using the desired identification as an extra hypothesis. · OAI.ActualCartier.restrictionPullbackIso

RestrictionSystem structure

A family of admissible restrictions and their containing rows. Rows may overlap, and there is no requirement that different restrictions share the same ancillary data. · OAI.MaxCutGames.Fourier.Deletion.RestrictionSystem

RestrictionSystem structure

A family of admissible restrictions and their containing rows. Rows may overlap, and there is no requirement that different restrictions share the same ancillary data. · OAI.UniqueGamesTheorem.Fourier.Deletion.RestrictionSystem

restrictionUnitIsColimit def

This is the cokernel fork in all sheaves of modules after forgetting coherence. · OAI.ActualCartier.restrictionUnitIsColimit

restrictIso def

Restriction of an actual ambient orthogonal map to a stable subspace. · OAI.SubsphereCurrent.Mixture.restrictIso

restrictKappa def

Completeness and Page uniqueness persist when the exceptional strip is narrowed. The prime-progression estimate and its actual zero are unchanged. · OAI.Ostmann.PublishedProgressionInput.restrictKappa

restrictLength def

Remove the unused long sources without changing either source budget. · OAI.LipschitzHilbert.LineErrorSources.restrictLength

restrictOverHom def

Restrict a slice-site module morphism to a smaller open. · OAI.ActualCartier.restrictOverHom

restrictPairing def

Restricting an integral perfect pairing along a Frobenius trace is again integral and perfect; no localization or fraction-field argument is used. · OAI.KLInvariance.FrobeniusDuality.restrictPairing

restrictScalarsFullyFaithful def

Restriction along a quotient is fully faithful on modules. · OAI.CoherentGlobal.restrictScalarsFullyFaithful

restrictSource def

Restrict both parameter charts to the same additional open neighborhood. · OAI.Problem356.LocalImplicit.AnalyticStationaryCharts.restrictSource

restrictSquare def

Open pushforward comparison viewed as a square of functors. · OAI.CoherentBaseChange.restrictSquare

restrictStage def

View the same finite matrix stage inside any algebra containing it. · OAI.Generator.FiniteMatrixStage.restrictStage

restrictStarRepresentation def

Restriction of a star representation to a closed invariant subspace. · OAI.KadisonSimilarity.restrictStarRepresentation

restrictSubgroup def

Restriction changes only the action, not the Hilbert carrier. · OAI.NuclearUltrapower.restrictSubgroup

restrictTime def

Restrict only the time interval of a constructed local flow. · OAI.PositiveEinsteinFour.Geometry.KineticLocalFlow.restrictTime

restrictToStable def

Restrict to a stable finite full subposet. It need not be a sieve in E. · OAI.Thomason.Construction.SieveRetraction.restrictToStable

restrictTupleColor def

Restricting a joint color retains an entire replacement vector at once. · OAI.WLIdentification.restrictTupleColor

restrictWord def

Restriction of a full stage string to this prescribed side. · OAI.TorsionFreeZeroDivisors.StageSideEmbedding.restrictWord

result def

Descending multiplication of factors 33 through 0. · OAI.DefocusingNLS.ProfileCertificate.result

result def

Change the mathematical output only when its actual encoded word agrees. · OAI.ExactQuantumFactoring.BitStackProgram.Procedure.result

resultCells def

The projected return value has no reader or array fields. · OAI.LowSpaceSubsetSum.PreparedCollectCells.resultCells

resultKey def

full query data key function · OAI.HadwigerCounterexample.Template.resultKey

resultLabel def

The two explicit control outcomes on a complete encoded table. · OAI.MaxCutGames.Foundations.Complexity.MachineLookup.resultLabel

resultLabel def

The two explicit control outcomes on a complete encoded table. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineLookup.resultLabel

resultOutput def

Successful lookup prefixes a complete unary word; failure writes nothing. · OAI.MaxCutGames.Foundations.Complexity.MachineLookup.resultOutput

resultOutput def

Successful lookup prefixes a complete unary word; failure writes nothing. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineLookup.resultOutput

resultTapes def

Exact header-stage endpoint. The occurrence counter is retained for the body loop; generated output is accumulated in reverse. · OAI.MaxCutGames.Explicit.MachineClone100Headers.resultTapes

resultTapes def

Exact final tapes, including the lookup's retained scan suffixes. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Body.resultTapes

resultTapes def

Exact header-stage endpoint. The occurrence counter is retained for the body loop; generated output is accumulated in reverse. · OAI.UniqueGamesTheorem.Explicit.MachineClone100Headers.resultTapes

resultTapes def

Exact final tapes, including the lookup's retained scan suffixes. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Body.resultTapes

retain def

The global increasing retained subsequence. · OAI.ReflexiveFixedPoints.OrderedGroupConcat.retain

Retained def

The actual surviving test is just the two descending-path tests and the absence of a low separating tip, after the literal earlier tests. · OAI.PlanarL1.Columns.Model.PairGeometry.Retained

retained def

Original entry payloads and their array slots remain shared throughout all filtering, sorting and recursion. · OAI.LowSpaceSubsetSum.PreparedCheckerCells.retained

retained def

Retained candidate words, input indices, permutation and scalar fields. · OAI.LowSpaceSubsetSum.SampledCompressionCells.retained

retained def

The weight-zero part in the original coordinate spaces. · OAI.MatrixMultiplication.AuxiliarySeparation.Sector.retained

retained def

Continue all old lifetime trials using fresh independent keep bits; compulsory cutoffs apply on the same age increment. The new entry's ramp value is supplied separately so duplicates of physical centers are harmless. · OAI.UniformKServer.ShortRoster.retained

retained def

Retain a specified family of the original fibers, restricted to D. This never alters a source or creates a new distance coincidence. · OAI.WeakPinned.Core.Fibers.retained

retainedAmplitude def

The literal new amplitude at a retained angular mode. · OAI.ThreeTorus.retainedAmplitude

RetainedBimodule def

An ordinary bimodule packaged as a module over its actual tensor algebra. · OAI.LittleFinitistic.RetainedBimodule

retainedBudget def

The retained budget uses the original global candidate label. · OAI.ReflexiveFixedPoints.OrderedTree.retainedBudget

retainedCandidate def

Transport the retained embedding to the actual next level of the infinite tree. · OAI.ReflexiveFixedPoints.AnchoredCandidateContext.retainedCandidate

retainedCandidate def

Retained children keep their positions in the complete candidate sequence. · OAI.ReflexiveFixedPoints.CandidateContext.retainedCandidate

retainedCircleWeight def

The exact circle weights of one retained passage mode. · OAI.ThreeTorus.retainedCircleWeight

retainedEquiv def

Actual retained-hook subgroup chart inside the original hole stabilizer. · OAI.CoordinateSweeps.Grid.Holes.retainedEquiv

retainedFavorable def

All component computations read the same retained, verified occurrence log. Only primes dividing the requested cofactor contribute to the comparison. · OAI.ExactQuantumFactoring.NodeMachine.retainedFavorable

retainedFiberEquiv def

Grouping the fiber coordinates recovers m copies of every stage bundle. · OAI.Kaplansky.retainedFiberEquiv

retainedGraphRow def

The erased pivot is represented only by its total integer. Its outgoing row has already been removed by Cauchy--Schwarz. · OAI.Ostmann.retainedGraphRow

retainedGridEdge def

The three center tests for a fixed pair of angular caps. · OAI.Falconer.retainedGridEdge

retainedGroupedPhase def

The phase left after the outgoing pivot rows have been removed. · OAI.Ostmann.retainedGroupedPhase

retainedGroupingEquiv def

Every retained line occurrence is accounted for exactly once. · OAI.Kaplansky.retainedGroupingEquiv

retainedHyperplaneMultiplicity def

Count all copies of L_(i+1) in m copies of E_0⊕...⊕E_R. The stages are enumerated in descending order. · OAI.Kaplansky.retainedHyperplaneMultiplicity

retainedInputs def

Four shared coordinate vectors and four sampled arrays remain resident. · OAI.LowSpaceSubsetSum.PreparedIncidenceCells.retainedInputs

retainedLaw def

The actual normalized child law, after removing all zero-mass targets. · OAI.LeanBlast.SecondKahnKalai.retainedLaw

retainedLiftEntries def

Free blocks in the identified-quotient lift itself. · OAI.IntegralCharacterVarieties.retainedLiftEntries

retainedMask def

The lookup is a deterministic analysis choice on distinct retained weights. · OAI.LowSpaceSubsetSum.RepresentationSetup.retainedMask

retainedOrder def

The literal order network already available at any retained stage, before adding its reversible phase-flag wrapper. · OAI.ExactQuantumFactoring.NodeMachine.retainedOrder

retainedOrderPhi def

The numerator of the exact order-trial success probability, with no recovery of an exponent record and no factorization call at runtime. · OAI.ExactQuantumFactoring.NodeMachine.retainedOrderPhi

retainedPrimeProduct def

Prime product of one retained bit vector. · OAI.JointDickman.retainedPrimeProduct

RetainedRealCircleModel structure

A circle-domain normal limit with the original finite scalar list retained on the actual finite-transfer exhaustion. Sobolev continuation of its traces is a separate conclusion proved from these data. · OAI.CircleDomainRigidity.RetainedRealCircleModel

retainedSpectralCell def

A retained cell has a coordinate at least two lattice units from zero. · OAI.WeakPureInfiniteness.retainedSpectralCell

retainedTotient def

The totient network reads only the actual retained rectangular table. · OAI.ExactQuantumFactoring.NodeMachine.retainedTotient

retainedTrivialMultiplicity def

The exact rank of the retained trivial block, in descending stage order. · OAI.Kaplansky.retainedTrivialMultiplicity

retainedTwistedSourceUnitary def

Exact slot regrouping as an actual unitary bundle isomorphism. · OAI.Kaplansky.retainedTwistedSourceUnitary

retainFun def

Retaining the prescribed ordered subsequence inside each parent group. · OAI.ReflexiveFixedPoints.OrderedGroupConcat.retainFun

Retains structure

Retention maps leave every retained face and its individual side indices unchanged. They may discard whole faces; switches only change side pairings. · OAI.HyperbolicObstruction.PictureSwitch.CellPicture.Retains

Retains structure

Retain every individual surviving step and EVERY frozen face. Only free cone steps may disappear; on frozen faces the original cyclic indexing is retained exactly, not just up to a word or a chain class. · OAI.HyperbolicObstruction.PictureSwitch.WalkPicture.Retains

retarget def

Retarget only the root; all core data and all children are unchanged. · OAI.PureTypeSystem.TypingTree.retarget

retention def

The second rounding uses EXACTLY J+2, not the transition completion's slightly finer W+t+2 grid. One surplus fair retention bit is harmless. · OAI.ExactQuantumFactoring.Quarter.retention

retention def

Uniform start in S, followed by a uniform Grassmann neighbor. · OAI.MaxCutGames.Inverse.KMS.retention

retention def

Uniform start in S, followed by a uniform Grassmann neighbor. · OAI.UniqueGamesTheorem.Inverse.KMS.retention

retentionBits def

One clock-independent number of bits works for all three modes, on every valid label, including d=1,j=0. · OAI.ExactQuantumFactoring.OrderTrial.retentionBits

retentionNet def

Compile the dyadic threshold to a finite Boolean network. All arithmetic, including signed rational division and floor, is reduced to bounded word gates. · OAI.ExactQuantumFactoring.retentionNet

retires def

Only an old selected covering record can trigger this event. The selector may depend on the entire old roster and its persistent radii, but not on the fresh lifetime trials. · OAI.UniformKServer.ActualRoster.retires

Retrace structure

mount at offset and effect only on zone. · OAI.PowerSaving.Slant.Retrace

retract def

Retraction onto an affine plane along the transverse direction. · OAI.Problem358.TransverseSurface.retract

retractCocone def

Retractions form a cocone to the initial stage. · OAI.Thomason.Construction.TopologicalDeformation.Sequential.retractCocone

retractCommon def

The common projection remembers the entire shared k-globe. · OAI.Thomason.Construction.StrictOmegaCategory.PastingGlobe.retractCommon

retraction def

Literal continuous retraction of the prism to bottom and sides. · OAI.HyperbolicObstruction.CubicalHomotopyExtension.retraction

retraction def

The face-chain inclusion retracts on precisely the chains meeting the old faces, with the order adjunction. · OAI.Thomason.Construction.ChainSieve.retraction

retraction def

The actual continuous retraction to the geometric horn. · OAI.Thomason.Construction.HornBarycentric.retraction

retraction def

The compatible stage retractions descend to the colimit. · OAI.Thomason.Construction.TopologicalDeformation.Sequential.retraction

retractLeft def

Retract the left half, collapsing the right half to the shared target face. · OAI.Thomason.Construction.StrictOmegaCategory.PastingGlobe.retractLeft

retractMap def

The retraction obtained by traversing the given arrow retract. · OAI.Thomason.Construction.TopologicalDeformation.Data.retractMap

retractPolicy def

Ambient extension by a retraction. The extension is deliberately arbitrary on requests outside the finite submetric, while agreeing exactly on all its histories. It is one policy, not a collection indexed by horizons. · OAI.KServer.retractPolicy

retractRight def

Retract the right half, collapsing the left half to the shared source face. · OAI.Thomason.Construction.StrictOmegaCategory.PastingGlobe.retractRight

retryIntegerLaw def

none records that all supplied independent trials rejected; it is not a fallback table. This is the finite prefix law of exact rejection. · OAI.ContingencyTables.ExactIntegerChoice.retryIntegerLaw

returnCode def

Strict first return: the initial path entry is omitted. · OAI.ContingencyTables.CompanionTraceProgram.returnCode

Returns def

The returned string begins at position zero and is delimited by its first blank. Extra blanks may lie outside the finite represented portion. · OAI.FiniteAlgebraicEnvelopes.Compiler.Simulator.Returns

returnSphere def

The genuine return map on the unit sphere. · OAI.Tingley.returnSphere

returnWord def

The search consumes a fixed cap worth of disjoint words even when the first return occurs before the last word. · OAI.ContingencyTables.CompanionTraceWords.returnWord

returnWord def

The search consumes a fixed cap worth of disjoint words even when the first return occurs before the last word. · OAI.ContingencyTables.RootedTraceWords.returnWord

returnWordDefault def

A concrete total draw: a missing capped return keeps the input. · OAI.ContingencyTables.CompanionCappedReturn.returnWordDefault

revealBitWeight def

Probability of a prescribed observation-mask membership bit. · OAI.MatroidProphet.Secretary.revealBitWeight

revealedProductWeight def

mask i = false reveals the first question, and true the second. · OAI.MaxCutGames.Foundations.Repetition.revealedProductWeight

revealedProductWeight def

mask i = false reveals the first question, and true the second. · OAI.UniqueGamesTheorem.Foundations.Repetition.revealedProductWeight

revealFallback def

The default row still preserves the displayed endpoint if that endpoint has zero probability in the base question law. · OAI.MaxCutGames.Foundations.Repetition.revealFallback

revealFallback def

The default row still preserves the displayed endpoint if that endpoint has zero probability in the base question law. · OAI.UniqueGamesTheorem.Foundations.Repetition.revealFallback

revealLaw def

Independently reveal either endpoint with probability one half. · OAI.MaxCutGames.Foundations.Repetition.revealLaw

revealLaw def

Independently reveal either endpoint with probability one half. · OAI.UniqueGamesTheorem.Foundations.Repetition.revealLaw

reverse def

Reversing a listed gate stays inside the same twenty gate names. · OAI.ExactQuantumFactoring.Gate.reverse

reverse def

Formal dart reversal. Loop edges and parallel indexed edges remain distinct. · OAI.HyperbolicObstruction.IntegralIncidence.reverse

reverse def

Both endpoints of each literal strip run are exposed. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.LayeredTrail.reverse

reverse def

Reversal changes every seam endpoint, and nothing in the occurrence table. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.reverse

reverse def

left = true retains the first giant; false retains the second. · OAI.Ostmann.GiantRows.reverse

reverse def

Reverse the column word and all its zero-cost switches. · OAI.PlanarL1.Columns.Model.Path.reverse

reverse def

Time reversal keeps the original gauge and negates its rates; it is not an independently selected certificate. · OAI.SmoothObstruction.ManifoldCoordinates.BilateralGauge.reverse

reverse def

Reverse a path continuously. · OAI.Thomason.Construction.PathSpace.reverse

reverseAppend def

The stack update performed by one record append. · OAI.MaxCutGames.Foundations.Hastad.SourceLoopOrder.reverseAppend

reverseAppend def

The stack update performed by one record append. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceLoopOrder.reverseAppend

reverseAtomSource def

None is the newly reconstructed pivot. Every other atom reads one particular copied or retained atom from the parent assignment. · OAI.Ostmann.reverseAtomSource

reverseBits def

The physical input word reversal is an explicit wire permutation. · OAI.ExactQuantumFactoring.reverseBits

reverseCopyLabelMap def

The chosen child reads its H copy and the common Y. The erased pivot is represented by the coordinate that the arithmetic node reconstructs. · OAI.Ostmann.reverseCopyLabelMap

reversed def

The opposite orientation is the pullback of the original orientation through the concrete single-axis reflection. · OAI.PositiveEinsteinFour.Geometry.FrameOrientation.reversed

reverseDart def

Reversal preserves the recorded walk and changes its direction flag. · OAI.MaxCutGames.Foundations.PCP.PoweringTest.reverseDart

reverseDart def

Reversal preserves the recorded walk and changes its direction flag. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringTest.reverseDart

reversedComputableInPolyTime def

Reversed transducer output in polynomial 2 * X + 1 time. · OAI.MaxCutGames.Reduction.MachineTransducer.reversedComputableInPolyTime

reversedComputableInPolyTime def

Reversed transducer output in polynomial 2 * X + 1 time. · OAI.UniqueGamesTheorem.Reduction.MachineTransducer.reversedComputableInPolyTime

reverseFunction def

Original darts use the original reversal; dummy darts stay fixed. · OAI.MaxCutGames.Foundations.PCP.VertexPadding.reverseFunction

reverseFunction def

Original darts use the original reversal; dummy darts stay fixed. · OAI.UniqueGamesTheorem.Foundations.PCP.VertexPadding.reverseFunction

reverseIndex def

Reversal identifies de Bruijn indices in a finite context with fixed levels. Outside that context it is the identity. · OAI.PureTypeSystem.Expr.reverseIndex

reverseInTime def

The actual driver's reversal loop enters its first cleanup label. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Finish.reverseInTime

reverseInTime def

The actual driver's reversal loop enters its first cleanup label. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Finish.reverseInTime

reverseMap def

The machine's fixed-degree numerical map for an old reverse-row index. · OAI.MaxCutGames.Foundations.PCP.PreprocessingLazyWords.reverseMap

reverseMap def

The machine's fixed-degree numerical map for an old reverse-row index. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingLazyWords.reverseMap

reverseMove def

Reverse one stack onto another, preserving all symbols. · OAI.ExactQuantumFactoring.BitStackProgram.reverseMove

reverseNorthRotation def

Preparation uses the direction y-x; the inverse phase uses x-y. · OAI.Falconer.reverseNorthRotation

reversePorts def

Reversal records each arrival port, not the original outgoing port. · OAI.MaxCutGames.Foundations.PCP.PoweringReach.reversePorts

reversePorts def

Reversal records each arrival port, not the original outgoing port. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringReach.reversePorts

reverseSlots def

Reverse only digit positions. All six fixed work/control roles stay put. · OAI.MaxCutGames.Foundations.Hastad.SourceRankPhase.reverseSlots

reverseSlots def

Reverse only digit positions. All six fixed work/control roles stay put. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceRankPhase.reverseSlots

reverseWord def

Reversing a word reverses its order and its signs. · OAI.TwoPointCorrelations.reverseWord

reverseWordEquiv def

Reversal is a permutation of the finite word alphabet. · OAI.DukePrimeDegree.reverseWordEquiv

reverseWordEquiv def

Reverse the digit positions before using the little-endian arithmetic encoding from mathlib. Both directions are the same executable operation. · OAI.MaxCutGames.Foundations.PCP.PoweringEnumeration.reverseWordEquiv

reverseWordEquiv def

Reverse the digit positions before using the little-endian arithmetic encoding from mathlib. Both directions are the same executable operation. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringEnumeration.reverseWordEquiv

rewardCost def

Backward expected reward for an arbitrary time-dependent finite Markov law. · OAI.SignedDisk.SignedRuns.rewardCost

rewardCost def

Backward expected reward for an arbitrary time-dependent finite Markov law. · OAI.SignedRuns.rewardCost

rewind def

Rewind only the public head. Does not need a position counter. · OAI.Fifths.SM.rewind

rewire def

Wire routing is free: it refers to values already physically present. · OAI.ExactQuantumFactoring.BooleanNetwork.rewire

rewire def

A finite output permutation/quotation involves no Boolean node. · OAI.ExactQuantumFactoring.NetworkEmission.rewire

rewire def

Rewire WHOLE ports of the same ordered arity, never individual children. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.PortWiring.rewire

rewireFacets def

Carry a cycle-constant facet assignment through the actual whole-port surgery. The two invariances are finite combinatorial checks, not a solution, coverage, or chart hypothesis. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.PortWiring.rewireFacets

RF abbrev

pointwise tower of derivative functions on S · OAI.GeneralMahler.Jet.RF

rho def

The fixed primitive root used in the p-direction. · OAI.CirculantHadamard.CyclotomicRings.rho

rho def

The continuous delay-equation construction of the Dickman function. · OAI.Erdos970.NumberTheoryLean.Dickman.rho

rho def

The actual unital tensor embedding from the algebra construction, not supplied as a hypothesis. · OAI.FiniteAlgebraicEnvelopes.Bootstrap.Source.rho

rho def

Exactly the disjoint-segment norm, written as the square root of its squared testing supremum. · OAI.ForestSpace.rho

rho def

The oriented companion spectral moment. · OAI.PositiveEinsteinFour.Bianchi.Shape.rho

rho def

The relative density used in the mixed rank measure. · OAI.Problem335.BalancedSchedule.rho

rho def

Distance to the cut, expressed using the centered representative. · OAI.QuantitativeVanDerWaerden.rho

rho def

The first positive parameter at which α enters the closure of β. The irrelevant case α > β is set to zero. · OAI.SeparableQuotient.CoherentClosures.rho

rho def

A radius with slack in both the separation and curvature estimates. · OAI.ThompsonNonamenability.Analytic.RegularCurveData.rho

rho def

The chosen radial mollifier of unit mass. · OAI.TriangularBilliards.SpatialSmoothing.rho

rhoAway def

The omitted root is replaced by one; all remaining roots lie in the actual away ring. · OAI.CirculantHadamard.PrimeCharacterEvaluations.rhoAway

rhoB def

The chosen primitive prime root in the literal global coefficient ring. · OAI.CirculantHadamard.PrimeCharacterEvaluations.rhoB

rhsCorrection def

Cancellation of the two copies of g j₀ leaves this explicit RHS. · OAI.MaxCutGames.Foundations.Hastad.FoldedEquation.rhsCorrection

rhsCorrection def

Cancellation of the two copies of g j₀ leaves this explicit RHS. · OAI.UniqueGamesTheorem.Foundations.Hastad.FoldedEquation.rhsCorrection

Ribbon structure

A bipartite rotation system. Each edge is oriented from black to white; rotations at its two endpoints are permutations of the whole edge set. · OAI.KaplanskyCounterexample.Planar.Ribbon

ribbon def

A nonempty labeled filling has its connected boundary component as an actual Euler-characteristic-one triangle ribbon. No geometric embedding is postulated. · OAI.Release075.TriangleFilling.ribbon

RibbonHistory structure

Data at an actual attained ribbon front, before any restart. · OAI.SLEExactGauge.RibbonHistory

ribbonPathStep def

The path-space step event is deterministic once the chart and integer front are fixed, although the actual Loewner history is random. · OAI.SLEExactGauge.ribbonPathStep

ribbonStepCompact def

A padded compact patch in coordinates relative to the current front. · OAI.SLEExactGauge.RibbonCoordinates.ribbonStepCompact

ribbonStepPatch def

The length-four patch used for one step and its two-sided waypoint routes. · OAI.SLEExactGauge.RibbonCoordinates.ribbonStepPatch

RicardXuCompressionInput def

Section 1 and Lemma 2.3 of Ricard--Xu, restricted to one creation letter and its preceding-word space in a scalar tracial reduced product. · OAI.KadisonSimilarity.RicardXuCompressionInput

RicardXuCreationInput def

Published input specialized to the two actual tracial factors. · OAI.KadisonSimilarity.RicardXuCreationInput

RicardXuWordDecompositionInput def

Published input: the alternating centered tensor model for a normalized faithful tracial reduced product. Its only output is the word coordinates. · OAI.KadisonSimilarity.RicardXuWordDecompositionInput

ricci def

Ricci tensor, with the convention in which the round sphere has positive Ricci. · OAI.HarmonicCounterexample.ricci

ricci def

Pullback of the actual chart Ricci tensor. · OAI.PositiveEinsteinFour.Geometry.OrthonormalFrame.ricci

ricciGradient def

This is the uncompressed right side after applying Ricci commutation and divergence freedom, with Ricci normalized to 3g. Its coefficients are 6 + 3, -12, and the full bundle action -(1+t_s) I_s Σ_s. · OAI.PositiveEinsteinFour.Bianchi.ricciGradient

ricciMatrix def

Contract the output and first curvature-input indices. · OAI.PositiveEinsteinFour.Geometry.ricciMatrix

ricciTensorCoordinates def

The Ricci field in one fixed chart, retaining both covariant slots. · OAI.PositiveEinsteinFour.Geometry.ricciTensorCoordinates

richEnvelope def

The exact exponent appearing in the rich-progression union bound. · OAI.QuantitativeVanDerWaerden.Parameters.richEnvelope

ridge def

Use only diagonal of a picture as one vertex. Y conditioned nominally at rate 1, also pin at the diagonal for the mix. · OAI.ExactDerandomization.Est.ridge

riemannianExp def

Evaluation at time one of the geodesic determined by initial data. · OAI.WeakMTWTransport.riemannianExp

Rig structure

Gear and ready input full role tape. · OAI.PowerSaving.RAM.Rig

Right abbrev

false is p_e, and true is r_e. · OAI.MaxCutGames.Explicit.Subdivision.Right

Right abbrev

false is p_e, and true is r_e. · OAI.UniqueGamesTheorem.Explicit.Subdivision.Right

right def

Right multiplication, with its order correctly reversed. · OAI.FiniteAlgebraicEnvelopes.Enveloping.right

right def

Actual right multiplication, obtained from the tracial conjugation. · OAI.KadisonSimilarity.TracialCornerGeometry.right

right def

Diagonal right translations on the finite multiple of the regular Hilbert space. · OAI.KervaireMatrix.right

right def

The upper endpoint is excluded. · OAI.QuantitativeVanDerWaerden.UniformMesh.right

right def

The right-component extraction is left extraction after the graph sum swap. · OAI.Release075.SumWalk.right

right def

Normalize a right-globe face to its canonical representative. · OAI.Thomason.Construction.PastingGlobeBasis.right

rightAction def

Multiplication X ⊗_k D → X respects the arbitrary exterior D-action. · OAI.DerivedBar.BimoduleTensor.rightAction

rightAction def

The right coefficient action is induced from the second balanced-tensor factor. · OAI.LittleFinitistic.BalancedTensor.rightAction

rightActionEquiv def

The one-sided right tensor action equals the restriction of the genuine two-sided action. · OAI.LittleFinitistic.TwoSidedTensor.rightActionEquiv

rightAppend def

Append a fixed path on the right. · OAI.Thomason.Construction.PathSpace.rightAppend

rightAppendEquiv def

Concatenating a fixed path on the right is a homotopy equivalence. · OAI.Thomason.Construction.PathSpace.rightAppendEquiv

RightAverageInvariant def

Invariance under every finite normalized right average. · OAI.BoundedHochschild.KadisonRingrose.RightAverageInvariant

rightAverageOrbit def

All finite normalized elementary right averages of one map. · OAI.BoundedHochschild.KadisonRingrose.rightAverageOrbit

rightAxis def

Every unit axis reflection has an explicit right-quaternion lift. · OAI.PositiveEinsteinFour.QuaternionFrame.QuaternionLift.rightAxis

RightBase def

D with the actual right A action through f. · OAI.LittleFinitistic.PDBarQuotient.RightBase

rightBaseInTime def

Right query-block base using the globally retained left-block size. · OAI.MaxCutGames.Foundations.Hastad.SourceBasePhase.rightBaseInTime

rightBaseInTime def

Right query-block base using the globally retained left-block size. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceBasePhase.rightBaseInTime

rightBasisEquiv def

A k-basis of the left factor gives an actual right-module direct-sum decomposition. · OAI.ExternalTensor.rightBasisEquiv

rightBracket def

The outer pair of the right-associated composite. · OAI.Thomason.Construction.StrictOmegaCategory.TripleGlobe.rightBracket

RightBranch def

Parametrization of the right translated copy of C(a,h). · OAI.MatrixMultiplication.AuxiliarySeparation.Sector.RightBranch

rightCancellation def

Cancelling a path and its reverse on the right, uniformly in the preceding path. · OAI.Thomason.Construction.PathSpace.rightCancellation

rightColumnRepresentation def

Left multiplication by the full algebra on a right-supported Hilbert column. No standard-form representation theorem is used here. · OAI.KadisonSimilarity.TracialCornerGeometry.rightColumnRepresentation

rightCongr def

Transport a first-factor vector-space equivalence as an actual right-module equivalence. · OAI.LittleFinitistic.TensorProjection.rightCongr

rightContactStripSite def

Right-path internal sites use an origin chosen so their far endpoint is17. · OAI.ContinuumCoulomb.rightContactStripSite

rightCosetRepresentative def

The representative selected for the right coset containing g. · OAI.PolycyclicRecognition.rightCosetRepresentative

rightCylinder abbrev

The directed chain tensor K ⊗ M[1], with the interval on the right. · OAI.Thomason.Construction.DirectedComplex.rightCylinder

rightCylinderAbelianizationGlobeIso def

Tensoring the natural chain presentations with the right interval and realizing gives the exact globe restriction of the chain model. · OAI.Thomason.Construction.DirectedGlobeComplex.rightCylinderAbelianizationGlobeIso

rightCylinderFunctor def

The fixed right interval tensor, with its signed differential. · OAI.Thomason.Construction.DirectedComplex.rightCylinderFunctor

rightDegreeIso def

Right-degree transport keeps the actual balanced tensor and the full left action. · OAI.LittleFinitistic.BimoduleTotal.rightDegreeIso

rightDerivativeSlotForm def

The uncontracted right-quaternion curvature on the two derivative slots. · OAI.PositiveEinsteinFour.HighestHessian.rightDerivativeSlotForm

RightDivOrder def

Right divisibility in a free right monoid-set, used for the quotient letters, not for words in the new monoid. · OAI.PartitionConsistency.MergerAlgebra.RightDivOrder

rightEndpoint def

The endpoint chain maps are natural in the directed complex. · OAI.Thomason.Construction.DirectedComplex.rightEndpoint

rightEquiv def

Its right A action is precisely restriction through the canonical projection. · OAI.LittleFinitistic.PDResidueProjection.rightEquiv

rightEquiv def

Recover the exact original right action after packaging the ordinary bimodule. · OAI.LittleFinitistic.RetainedBimodule.rightEquiv

rightEquiv def

Recover exactly the original right action of the constructed ordinary X. · OAI.LittleFinitistic.SelectedBarBimodule.rightEquiv

RightFreeAction class

A right action in which each orbit map is injective. This is the property of the prescribed quotient of copies used in the merger normal forms. It does not posit the existence of any monoid extension. · OAI.PartitionConsistency.MergerAlgebra.RightFreeAction

rightGlobeCylinder def

The concrete signed right-cylinder functor on all strict globe maps. · OAI.Thomason.Construction.DirectedGlobeComplex.rightGlobeCylinder

rightGlobeEndpoint def

The two endpoint inclusions in the concrete globe cylinders. · OAI.Thomason.Construction.DirectedGlobeComplex.rightGlobeEndpoint

rightGlobeProjection def

The natural collapse returns the actual strict globe. · OAI.Thomason.Construction.DirectedGlobeComplex.rightGlobeProjection

RightGrayInput structure

The exact published input about tensoring with the oriented interval. The natural isomorphism compares to the explicitly constructed chain tensor; its endpoint and projection equations fix the right-tensor convention. · OAI.Thomason.Construction.RightGrayInput

rightIdeal def

The actual projective right ideal eA, presented as the fixed points of left multiplication. · OAI.LittleFinitistic.IdempotentTensor.rightIdeal

RightIdealRange def

A formulation of right-ideal range useful for nonunital embeddings. · OAI.WeakPureInfiniteness.RightIdealRange

rightIndex def

Index of a coordinate in the complementary block. · OAI.Problem358.FinitePrefix.rightIndex

rightInverseAverage def

Average a function over the inverse of a sampled right increment. · OAI.PolycyclicRecognition.rightInverseAverage

rightLayer structure

One-dimensional with right/left densities r=r1 or q0. Base p/a fixed. · OAI.GeneralMahler.rightLayer

rightLayer def

The layer immediately right of an internal vertex. · OAI.Problem335.BalancedSchedule.rightLayer

rightLocalizationSeed def

The lifted right localization is obtained by the actual half swap. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Generators.rightLocalizationSeed

rightMap def

The second rational-function copy, in the independent right variables. · OAI.EuclideanRamsey.IndependentTensor.rightMap

rightMap def

The same tensor map as a map of right B-modules. · OAI.ExternalTensor.rightMap

rightMap def

Tensor a k-linear map with the identity while retaining the actual right B-action. · OAI.LittleFinitistic.TensorProjection.rightMap

rightMap def

Shift the third-leg weights by one, giving weight zero to the middle interval and weight one to the two outer intervals. · OAI.MatrixMultiplication.AuxiliarySeparation.Sector.rightMap

rightMap def

The remaining part of the right surjection is its literal restriction. · OAI.MaxCutGames.Inverse.KMSAnalyticCompatibleSplitting.CompatibleSplitting.rightMap

rightMap def

The remaining part of the right surjection is its literal restriction. · OAI.UniqueGamesTheorem.Inverse.KMSAnalyticCompatibleSplitting.CompatibleSplitting.rightMap

RightMinimal def

An epimorphism is right minimal if every endomorphism over it is surjective. · OAI.LittleFinitistic.PDCovers.RightMinimal

RightModule abbrev

Right B-modules are left modules over the opposite ring. · OAI.LittleFinitistic.ProjectiveComplexes.RightModule

rightMul def

The bimodule right action, bilinear over the field. · OAI.DerivedBar.BimoduleTensor.rightMul

rightNestedWord def

A right-nested bracket in a nonempty ordered list of letters. · OAI.PolycyclicRecognition.rightNestedWord

rightOuterSlice def

The positive outer cell of the sharpness map, with all ties assigned centrally. · OAI.Problem358.SharpSlice.rightOuterSlice

rightPair def

The inclusion of the second and third globe. · OAI.Thomason.Construction.StrictOmegaCategory.TripleGlobe.rightPair

rightPolicy def

The right kernel consumes its own complete visible observation only. · OAI.MaxCutGames.Decoder.VisiblePolicies.rightPolicy

rightPolicy def

The right kernel consumes its own complete visible observation only. · OAI.UniqueGamesTheorem.Decoder.VisiblePolicies.rightPolicy

rightPower def

Whisker an endomorphism past n initial leaves of a right comb. · OAI.FiniteAlgebraicEnvelopes.BinaryGroupCategory.rightPower

rightProjection def

The interval collapse is a natural positive chain map. · OAI.Thomason.Construction.DirectedComplex.rightProjection

rightQuaternion def

Right quaternion multiplication, commuting with the fixed left frame. · OAI.PositiveEinsteinFour.HighestHessian.rightQuaternion

rightQuaternionCoefficients def

The coefficients in the orthonormal basis (L_a R_b)/2 of real four-by-four matrices, with the scalar quaternion in position zero. · OAI.PositiveEinsteinFour.HighestHessian.rightQuaternionCoefficients

rightRadical def

The actual right radical quotient of the regular module. · OAI.LittleFinitistic.PDTensor.rightRadical

rightRegular def

The right regular representation R_t ξ(x)=ξ(xt). · OAI.Dixmier.rightRegular

rightRegular def

Right translations, needed to prove that the identity vector is separating. · OAI.UltrapowerBridge.rightRegular

rightRepSequence def

Coordinate second factors, before passing to either quotient. · OAI.NuclearUltrapower.rightRepSequence

rightResidueEquiv def

The detector's canonical residue is the right D-residue inflated along A → D. · OAI.LittleFinitistic.SelectedPDDetection.rightResidueEquiv

rightRestriction abbrev

The actual outside right action obtained by restricting the enveloping action. · OAI.LittleFinitistic.TwoSidedTensor.rightRestriction

rightSector def

The four orthogonal sectors of the two real quaternion slots. · OAI.PositiveEinsteinFour.HighestHessian.rightSector

rightSquare def

The twice-right-adjoint comparison of the open base-change square. · OAI.CoherentBaseChange.rightSquare

rightSquareIso def

Commutativity of the square gives an isomorphism of composite pushforwards, without base-change assumptions about quasicoherent sheaves. · OAI.CoherentBaseChange.rightSquareIso

rightStart def

Beginning of the right sector on both the second and third legs. · OAI.MatrixMultiplication.AuxiliarySeparation.Sector.rightStart

rightSubgroupAction def

Right translations are a left action of the opposite subgroup. · OAI.DukePrimeDegree.rightSubgroupAction

rightSwap def

Factor swap, with the external right action on the source. · OAI.ExternalTensor.rightSwap

rightSweep def

The full bounding box swept when a rectangle is moved horizontally to a point on its right. endpoint is the right edge of the translated rectangle. · OAI.ForcedComputation.PlanarRouting.rightSweep

RightTensor def

A tensor retaining the right action of the middle bimodule. · OAI.LittleFinitistic.RightTensor

rightTensor def

The right retained branch in the original coordinate spaces. · OAI.MatrixMultiplication.AuxiliarySeparation.Sector.rightTensor

rightTensorSlice def

Compression in the first leg, with values in operators on the second. · OAI.KadisonKastler.rightTensorSlice

rightTerm abbrev

Actual right projectives in the source resolution. · OAI.LittleFinitistic.OrdinarySimulation.ChainAlgebra.rightTerm

rightTotal def

Its right-module restriction, retaining the original bimodule construction. · OAI.LittleFinitistic.InducedRectification.rightTotal

rightTransfer def

Right multiplication by v* is the actual unitary from eL²r onto eL²(vv*); its inverse is right multiplication by v. · OAI.KadisonSimilarity.TracialCornerGeometry.rightTransfer

rightTwist def

Simultaneous right-quaternion action on the two derivative slots. · OAI.PositiveEinsteinFour.HighestHessian.rightTwist

rightUnitHomotopy def

The right unit law is a homotopy continuous in the entire path. · OAI.Thomason.Construction.PathSpace.rightUnitHomotopy

rightUnitIso def

The analogous canonical right unit isomorphism. · OAI.ActualSheafTensor.rightUnitIso

rightVectorEnergy def

Only the three imaginary right-quaternion coefficients enter the opposite-block action. · OAI.PositiveEinsteinFour.HighestHessian.rightVectorEnergy

rightVectorGram def

Their ordinary three-dimensional coefficient Gram form. · OAI.PositiveEinsteinFour.HighestHessian.rightVectorGram

rightVectorLinear def

Right evaluation is linear in the algebra element, despite the two conjugations. · OAI.KadisonKastler.rightVectorLinear

rightVectorOperator def

Extension of right multiplication, whenever its algebra-vector bound holds. · OAI.KadisonKastler.rightVectorOperator

rightY def

Embed the right second-leg interval. · OAI.MatrixMultiplication.AuxiliarySeparation.Sector.rightY

rightZ def

Embed the right third-leg interval. · OAI.MatrixMultiplication.AuxiliarySeparation.Sector.rightZ

rigidityCoefficient def

The source coefficient q = c D(b) - b D(c). · OAI.StableCoordinate.rigidityCoefficient

rigidityPolynomialWeight def

The weights of the source polynomial subalgebra k[a,d,u]: index 0 is a, index 1 is d, and index 2 is u. · OAI.StableCoordinate.rigidityPolynomialWeight

Ring abbrev

The actual integer algebra generated by s. · OAI.CirculantHadamard.CyclotomicAdjoin.Ring

Ring abbrev

The actual ring ℤ_(2). · OAI.CirculantHadamard.IntegerTwoLocal.Ring

ringEquivOfSpecIso def

Full faithfulness of Spec recovers the ring map from every affine scheme isomorphism. · OAI.StableCoordinate.AffineGeometry.ringEquivOfSpecIso

ringInitial def

The literal input convention, transferred to a finite line segment. · OAI.WLIdentification.Machine.ringInitial

ripple def

actValue convenience: kernel axes matrix times input · OAI.PowerSaving.Binary.ripple

ripsBoundaryRepresentative def

The constructed boundary element, before passage to the fixed ample module and before descent of the entire assembly homomorphism. · OAI.CoarseAssembly.Rips.ripsBoundaryRepresentative

ripsCharge def

The charge assigned to an oriented Rips edge, extended to all vertex pairs by choosing shortest graph walks. · OAI.CoarseAssembly.ripsCharge

ripsGraph def

The graph of the edges of the Rips complex of a connected graph at scale r. Connectedness is supplied to results, because graph distance is zero between disconnected vertices in Mathlib's natural-valued convention. · OAI.CoarseAssembly.ripsGraph

rise def

Toeplitz entries with lower triangular convention. · OAI.PowerSaving.Spectral.rise

riseCount def

Number of transitions from false to true. · OAI.Problem335.WordRuns.riseCount

RL 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.RL

rm def

The real Riemannian curvature with all indices lowered using g. · OAI.KahlerUniformization.SmoothKahlerMetric.rm

RokhlinRestrictionFormula def

The restriction case of the published entropy integral formula. · OAI.DukePrimeDegree.RokhlinRestrictionFormula

roleCode def

The finite role data is compiled into finite control, independent of table size. · OAI.ParityWL.Complexity.roleCode

roleT def

a_t read on full ambient profile using allowed tags with partner d. · OAI.HadwigerCounterexample.roleT

roleWeight def

Three roles of an active odd child in the manuscript parity expansion: constant, unsigned motif, and signed motif. · OAI.TraceReconstruction.roleWeight

Room def

Counter program transformations, correctness with initial certified zeros available FROM threshold l onwards. l is quantified: we can request a program with the scratch moved out to make room. d bounds all counters throughout … · OAI.ExactDerandomization.Counter.Room

roost abbrev

Ballot compilation including explicit reset to desired output exponent. (Called only on successful gate, fallback zero separately). · OAI.ExactDerandomization.Arena.Amber.roost

Root def

Roots are precisely the positions not constrained to earlier positions. · OAI.TorsionFreeZeroDivisors.BlockConstraints.ParentMap.Root

root def

The least actual vertex index in the reached component. · OAI.ContingencyTables.FiniteGraphRoots.root

root def

Original component root; this is distinct from a forest component index. · OAI.DiamondDistortion.PathModel.root

root def

The quenched logarithmic root energy in the actual dependent mark row. · OAI.DilutedSpinGlass.HeterogeneousMarks.root

root def

A fixed anchor and genuine paths in the finite literal core graph. · OAI.HyperbolicObstruction.MasterGraph.root

root def

The new roots which merge the original copies. · OAI.PartitionConsistency.DiagramForcing.OverlapPrescription.root

root abbrev

Positive-sign, canonical roots. Need only at positive order. · OAI.PowerSaving.root

root def

At depth zero the sole cutoff output is already the root output. · OAI.ReflexiveFixedPoints.TreeFrontier.root

root def

Canonical positive/negative root selected from the proved scalar splitting, not an assumed decomposition. · OAI.ScalarCancellation.root

root def

Canonical positive/negative root selected from the proved scalar splitting, not an assumed decomposition. · OAI.SignedDisk.ScalarCancellation.root

root def

The optional first-stage root split by prescribed side. · OAI.TorsionFreeZeroDivisors.SidePins.root

rootAction def

action on the five chosen radicals, with values in the roots of unity. · OAI.KummerLines.rootAction

rootAction def

The actual adjoint action on a Cartan generalized root space. · OAI.PolycyclicRecognition.rootAction

rootArray def

The recursive independent-root product and the finite indexed root array are the same labeled data. No exchangeability is needed here. · OAI.DilutedSpinGlass.rootArray

rootCheck def

Check a candidate root against the input integer, with its static exponent. · OAI.ExactQuantumFactoring.BitArithmetic.rootCheck

rootChildMeanSum def

The actual root child energies, not an independent paired-copy law. · OAI.DilutedSpinGlass.HeterogeneousMarks.rootChildMeanSum

RootClosed def

Root closure is the exact property satisfied by a union of root paths. It says every predecessor of a protected vertex is protected. · OAI.HyperbolicObstruction.PlanarRecursion.RootClosed

rootCoordinates def

Fixed linear coordinates put a compatible norm on the root space. · OAI.PolycyclicRecognition.rootCoordinates

rootCount def

This is the integral root part occurring in the source's exact forest completion, before multiplying by the softening strength. · OAI.ContingencyTables.ForestPenaltyProgram.rootCount

Rooted def

A code represents a particular root name from some index onward. · OAI.PartitionConsistency.ProductForcing.ChainCode.Rooted

rootedCanonicalEquiv def

Actual root-isomorphism classes with actual spanning trees, not labelled objects divided by an unproved automorphism factor. · OAI.FKCRT.rootedCanonicalEquiv

RootedCuts structure

A finite rooted tree is indexed in parent-before-child order. The cuts below are the vertex sides of its edges. Validity is recorded separately. · OAI.ParityWL.RootedCuts

rootedRep def

Prescribe the representative of one root cycle without disturbing others. · OAI.TorsionFreeZeroDivisors.CycleLists.rootedRep

RootedTest structure

Sources may have loops during root gluing. A loop contributes zero on every simple target; it is never silently deleted. · OAI.ParityWL.RootedTest

rootedTreePath def

The unique actual tree path; uniqueness is proved, not assumed by a surrogate tree structure. The distance-minimizing library witness fixes it. · OAI.HyperbolicObstruction.PlanarRecursion.rootedTreePath

rootedWordEquiv def

Exact finite rooted-map/spanning-tree to identified-word correspondence. · OAI.FKCRT.rootedWordEquiv

RootEntropyCoefficients structure

Root entropy coefficients as in EKL (2006), Proposition 3.1, with zero diagonal. · OAI.DukePrimeDegree.RootEntropyCoefficients

rootEntropyForm def

The finite positive-root expression in the entropy formula. · OAI.DukePrimeDegree.rootEntropyForm

rootEquiv def

The Kummer field carries every projective symmetry of the six line arrangement that is supplied by an rational-function automorphism. · OAI.SourceSymmetry.rootEquiv

rootEval def

Explicit finite weighted evaluation on terminal paths, for the population born at the current root. Every branching has its true capacity ratio. · OAI.IntrinsicMarks.rootEval

rootEval def

Explicit finite weighted evaluation on terminal paths, for the population born at the current root. Every branching has its true capacity ratio. · OAI.SignedDisk.IntrinsicMarks.rootEval

rootEvents def

Forbidden primes of an actual integer point. · OAI.JointDickman.rootEvents

rootExternalTreeScore def

A fresh independent insertion, retaining every original mark in z. The arguments old and new are intentionally distinct. · OAI.DilutedSpinGlass.HeterogeneousMarks.rootExternalTreeScore

rootFacet def

Every contacted strip inherits precisely its old ordered-child facet; every other strip is a separate new disk, even when ranks/colors coincide. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.RealizedBand.rootFacet

rootFlat def

A literal bounded readout for every clock count, with a harmless zero case. · OAI.ExactQuantumFactoring.PhysicalTree.rootFlat

rootInIdeal def

Inclusion of a nonzero generalized root space into the Cartan ideal. · OAI.PolycyclicRecognition.CartanWordModel.rootInIdeal

rootIntervalDirection def

Extend a fixed finite word by zero directions for the unused indices. · OAI.PolycyclicRecognition.rootIntervalDirection

rootIntervalSupport def

The closed interval image carries the complete sampling law. · OAI.PolycyclicRecognition.rootIntervalSupport

RootInvariant def

Every assigned root occurs as a minimal occurrence, and the root directions of the whole state form a clique. These assertions simultaneously express support in a simplex and permanence of old assignments. · OAI.EilenbergGanea.TraceWords.RootInvariant

RootInversions def

The actual positive ambient roots inverted by w. · OAI.KLInvariance.TitsSpace.RootInversions

rootMap def

Restriction of a lifted projective line-arrangement symmetry to the Kummer field. · OAI.SourceSymmetry.rootMap

rootMarkerObservable def

A matrix multioverlap at the actual heterogeneous physical root, retaining the literal old-test transport and root-dependent alphabet. · OAI.DilutedSpinGlass.UniversalDictionary.rootMarkerObservable

rootMatrixConjugation def

Conjugation by a root unipotent matrix. · OAI.DukePrimeDegree.rootMatrixConjugation

rootMatrixObservable def

A matrix multioverlap at the actual heterogeneous physical root, retaining the literal old-test transport and root-dependent alphabet. · OAI.DilutedSpinGlass.UniversalDictionary.rootMatrixObservable

rootNext def

The fixed choice of rule at the leftmost position, then table order. · OAI.FiniteAlgebraicEnvelopes.Bootstrap.rootNext

rootOffsetCone def

The source polar cone attached to one member of the finite family. · OAI.PolycyclicRecognition.rootOffsetCone

rootOffsetConeIndex def

One ray, and the two-ray cones independent of that ray. This is precisely the finite family needed for a single neighboring root step. · OAI.PolycyclicRecognition.rootOffsetConeIndex

rootP def

Separate Euclidean root arrays for the joined (p=2) models. · OAI.DiamondDistortion.RecursiveModel.rootP

RootPath abbrev

A finite independent block sequence, with no finiteness assumption on a block. · OAI.DilutedSpinGlass.RootPath

rootPath def

The root occurs in every path vector, including the root's own path. · OAI.BoundedTreePotentials.rootPath

rootPoly def

Exact roots at the first r positive integers. · OAI.QAC.rootPoly

rootPredicate def

A tail predicate with the values at all its possible roots fixed. · OAI.CoordinateSweeps.Coverage.rootPredicate

rootPrice def

A common root price sums the two exact coordinate defects for every member of one fixed finite palette; it does not depend on the active label. · OAI.CommonPrices.rootPrice

rootPrice def

A common root price sums the two exact coordinate defects for every member of one fixed finite palette; it does not depend on the active label. · OAI.SignedDisk.CommonPrices.rootPrice

RootProbes def

The part of probe saturation used for simultaneous ordinal choice. This is a purely algebraic statement about the actual monoid, and will be instantiated by its executed saturation, not assumed at the endpoint. · OAI.PartitionConsistency.ProductForcing.RootProbes

rootProductInformation def

Rectangular information in the product of the actual normalized root leaves, at independently prescribed real logarithmic scales. · OAI.DukePrimeDegree.rootProductInformation

rootProposals def

Root searches are clocked by the ORIGINAL input length n. · OAI.ExactQuantumFactoring.rootProposals

rootQuotientEquiv def

Exact coefficient-ring identification used for the manuscript's C=A/(alpha). This is an isomorphism of the actual quotient, not a replacement invariant. · OAI.KLInvariance.EdgeAlgebra.rootQuotientEquiv

rootQuotientPolynomial def

Polynomial coordinates on the actual symmetric-algebra quotient. · OAI.KLInvariance.EdgeAlgebra.rootQuotientPolynomial

rootResidualLinear def

Dependence of the residual operator on the Cartan parameter is linear. · OAI.PolycyclicRecognition.rootResidualLinear

rootResult def

Actual root readout from earlier retained wires, not a factor-table lookup. · OAI.ExactQuantumFactoring.PhysicalTree.rootResult

rootRing def

A finite list of prime directions, keeping i in the coefficient ring. · OAI.CirculantHadamard.CyclotomicRings.rootRing

rootScan def

Each pair records an occurrence's direction and its assigned minimal ancestral direction. Signs do not change the occurrence dependency relation. · OAI.EilenbergGanea.TraceWords.rootScan

rootsCharacter def

A finite product of the actual primary-coordinate characters. · OAI.CirculantHadamard.PrimeCharacterEvaluations.rootsCharacter

rootSchedule def

One assigned branching coordinate and all its later child coordinates. · OAI.DilutedSpinGlass.ReducedTopology.rootSchedule

rootSwap def

Three symmetries, generating the projective S₄ of the quadrangle. · OAI.SourceSymmetry.rootSwap

RootSystem def

The cone form of exact realization. Unlike checking only old right factors, this form persists under arbitrary conservative embeddings. · OAI.PartitionConsistency.DiagramForcing.OverlapPrescription.RootSystem

rootTranslationAction def

The real root group acts on the lattice space by right translation. · OAI.DukePrimeDegree.rootTranslationAction

rootVector def

The vector root comprises the two actual signed scalar surviving roots. · OAI.PricePieces.rootVector

rootVector def

The vector root comprises the two actual signed scalar surviving roots. · OAI.SignedDisk.PricePieces.rootVector

rootWidth abbrev

Width large enough for every candidate's e-th power, e≤n. · OAI.ExactQuantumFactoring.BitArithmetic.rootWidth

rootWord def

Root words, retaining their exact common length in the type. · OAI.TreeEdit.BinaryLower.FiniteTree.rootWord

roseLoop def

The subdivided x-circle in the exact 103-cell order graph. · OAI.EilenbergGanea.SeedConnection.roseLoop

Rot def

Discrete additive notation for the actual multiplicative algebraic rotation group. · OAI.PlaneFiveColor.Spectral.Rot

rotate def

(x₁,…,x_d) ↦ (x₂,…,x_d,x₁). · OAI.CubeShuffle.rotate

rotate def

An actual smooth orbit of orthogonal complex structures under any skew infinitesimal rotation; both orientation classes remain available. · OAI.HarmonicCounterexample.Berger.ComplexStructure.rotate

rotate def

The cyclic rotation of the three particles. · OAI.Problem356.Cyclic.rotate

rotate def

Conjugation by a local layer preserves the exact count support. · OAI.QAC.Count.rotate

rotatedEnergy def

The quadratic energy in spectral coordinates, equal to σᵀ Uᵀ diag(eig) U σ / 2. · OAI.InvariantIsing.rotatedEnergy

rotatedLine def

A line in the direction chosen by a unit complex multiplier. · OAI.CircleDomainRigidity.rotatedLine

rotatedPressure def

Equation (1.1), with an optional deterministic magnetic field. · OAI.InvariantIsing.rotatedPressure

rotateEdge def

On a non-simple-labelled edge this is actual left multiplication by s. Simple-labelled edges are fixed merely to make the extension a total involution; all path uses below exclude them before applying this map. · OAI.KLInvariance.AdaptedDyer.rotateEdge

rotateHead def

Advance across the first edge while retaining its reverse port. · OAI.MaxCutGames.Foundations.PCP.PoweringWalks.rotateHead

rotateHead def

Advance across the first edge while retaining its reverse port. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringWalks.rotateHead

rotatePrefix def

Rotate the first N+1 coordinates, leaving the infinite tail fixed. · OAI.CoarseAssembly.Analytic.rotatePrefix

rotateRowCell def

Move low row letters to the band and high row letters to B_h. · OAI.Saxl.rotateRowCell

Rotation abbrev

Every incoming-to-outgoing cyclic order at a vertex is permitted. · OAI.OneWayLiveness.FunctionalEdges.Rotation

rotation def

Two simultaneous identical planar rotations. · OAI.ArnoldCounterexample.Quadric.rotation

rotation def

One reverse-index lookup and the fixed mixed-radix decoding. · OAI.MaxCutGames.Foundations.PCP.PortTables.rotation

rotation def

Exponentiating an actual skew-adjoint operator produces rotations. · OAI.PettyProjection.Spherical.rotation

rotation def

Rotation is an actual Lie automorphism, inverted by negating the Cartan parameter, with no semisimplicity assumption on the original Cartan action. · OAI.PolycyclicRecognition.ComplexCartan.rotation

rotation def

Speed-two rotations on the self-dual matrix indices. · OAI.PositiveEinsteinFour.TraceHessian.rotation

rotation def

The rotation taking u to v, on the non-antipodal unit-pair domain. · OAI.ThompsonNonamenability.Analytic.rotation

rotation def

Insert an unpaired marker immediately before p. · OAI.TorsionFreeZeroDivisors.MarkerInsertion.rotation

rotation def

One reverse-index lookup and the fixed mixed-radix decoding. · OAI.UniqueGamesTheorem.Foundations.PCP.PortTables.rotation

RotationalCR def

Covariant first-derivative data, stated as actual generator graphs. The smooth seam producer below supplies these graphs for billiard fields. · OAI.TriangularBilliards.Analysis.RotationalCR

RotationalModification abbrev

The concrete graph subgroup that removes the compact rotations. · OAI.PolycyclicRecognition.ComplexWeightFactorModel.RotationalModification

rotationalModificationHomeomorph def

The modification preserves the original underlying topology. · OAI.PolycyclicRecognition.ComplexWeightFactorModel.rotationalModificationHomeomorph

rotationalTwist def

The inverse-character graph, with the multiplication inherited from the semidirect extension. The character is invariant under the compact action. · OAI.PolycyclicRecognition.rotationalTwist

rotationalTwist_properCocompact def

The same compact subgroup covers the inverse-character graph, so no metric estimate or triangulability assertion is hidden in this arrow. · OAI.PolycyclicRecognition.rotationalTwist_properCocompact

rotationalTwistCharacter def

An invariant character is still a homomorphism on the modified group. · OAI.PolycyclicRecognition.rotationalTwistCharacter

rotationalTwistChartedSpace def

The atlas comes from the original group through the graph projection. · OAI.PolycyclicRecognition.rotationalTwistChartedSpace

rotationalTwistHomeomorph def

Removing a rotation changes multiplication, while preserving the underlying topological space. · OAI.PolycyclicRecognition.rotationalTwistHomeomorph

rotationalTwistLift def

The inverse-character graph is multiplicative on this subgroup. · OAI.PolycyclicRecognition.rotationalTwistLift

rotationCharacter def

The actual quotient character followed by its compact rotational part. · OAI.PolycyclicRecognition.ComplexWeightFactorModel.rotationCharacter

rotationCLM def

The genuine infinitesimal Hopf operator as a bounded operator on V_l. · OAI.HarmonicCounterexample.Angular.rotationCLM

rotationDerivative def

Intrinsic derivative along the genuine coordinate-plane rotation. · OAI.ArnoldCounterexample.CriticalBound.rotationDerivative

rotationGenerator def

The infinitesimal generator of rotation in the oriented `u,v` plane. · OAI.AffineBernstein.rotationGenerator

rotationGenerator def

The ambient vector field generating the prescribed rotations. · OAI.ArnoldCounterexample.Quadric.rotationGenerator

rotationHamiltonian def

Minus alpha(Z), exactly the sign convention i_X omega = dH. · OAI.ArnoldCounterexample.Quadric.rotationHamiltonian

rotationIndex def

Reverse precisely the indices strictly after the pivot. · OAI.CycleClique.Construction.rotationIndex

rotationLinear def

The ambient real-linear rotation, not merely its quotient action. · OAI.ArnoldCounterexample.Quadric.rotationLinear

rotationMeeting def

Charge at a meeting after independently rotating the selected cycles. A later slot is the only possible closer, except within a single cycle where either user can be its terminal. · OAI.CubeShuffle.rotationMeeting

rotationMeeting def

Charge at a meeting after independently rotating the selected cycles. A later slot is the only possible closer, except within a single cycle where either user can be its terminal. · OAI.CubeShuffle.RoutingNetwork.rotationMeeting

rotationPolynomial def

The operator polynomial used in the rank-two rotation formula. · OAI.ThompsonNonamenability.Analytic.rotationPolynomial

rotationTorus def

Bracket-preserving unit coefficient vectors that fix the Cartan algebra. · OAI.PolycyclicRecognition.ComplexCartan.rotationTorus

rotationValue def

The fixed anchor and its deformation by an arbitrary ambient unitary. · OAI.WeakPureInfiniteness.BufferAbsorptionCorner.rotationValue

rotationWeightCharacter def

The full vector-valued character survives the modification. · OAI.PolycyclicRecognition.ComplexWeightFactorModel.rotationWeightCharacter

rotDeriv def

Derivative along an infinitesimal linear rotation. · OAI.PettyProjection.Spherical.rotDeriv

rotGen def

Elementary skew matrix rotating the (a,b) coordinate plane. · OAI.ArnoldCounterexample.CriticalBound.rotGen

roughConvolutionTail def

The part indexed by divisors greater than the chosen truncation. · OAI.JointDickman.roughConvolutionTail

roughFourierPolynomial def

The actual polynomial in (q:rough-fourier). · OAI.TwoPointCorrelations.roughFourierPolynomial

roughModulusBadBlock def

Bad moduli on one block use one common divisor list. This deliberate fixed-base choice is needed for the off-diagonal double count. · OAI.Problem337.roughModulusBadBlock

roughPart def

The part of an integer supported on primes at least R, retaining their full multiplicities. The definition at zero is harmlessly the empty product. · OAI.Problem337.roughPart

roughSquarefreeWeight def

Restriction to integers with no prime factor from E. · OAI.JointDickman.roughSquarefreeWeight

round def

Round index 0 on array columns too, so all calls can parse their local functions in uniform space regardless of type. · OAI.Fifths.Tour.round

roundCoeff def

Upward coefficient rounding by a specified integer denominator. · OAI.ExactQuantumFactoring.OrderTrial.roundCoeff

roundCounts def

Increment the first r floors, in the original occurrence order. · OAI.MaxCutGames.Reduction.WeightRounding.roundCounts

roundCounts def

Increment the first r floors, in the original occurrence order. · OAI.UniqueGamesTheorem.Reduction.WeightRounding.roundCounts

roundDerivative def

The round covariant derivative, using tangential projection of the ambient derivative. · OAI.AffineBernstein.roundDerivative

roundDivergence def

Ambient expression for the round divergence of the restriction to the unit sphere. · OAI.AffineBernstein.roundDivergence

roundedIncidentLine def

Choose the grid slope first, then the grid intercept nearest the exact line through the point. Every constructed line remains incident. · OAI.Falconer.roundedIncidentLine

roundedNumerator def

Integer numerator retained by upward rounding at an integer argument. · OAI.ExactQuantumFactoring.OrderTrial.roundedNumerator

RoundedParameters structure

Numerical consequences of the prescribed rounded sample sizes. · OAI.LowSpaceSubsetSum.SeparatorExpectation.RoundedParameters

roundedRow def

Downward rounding at every label other than the reserved allowed label. The reserved label receives the remainder, exactly as in Section 02. · OAI.UniformKServer.roundedRow

roundedStripCandidates def

The bounded candidate intercept list for one strip, retaining its original slope label even if other labels have the same rounded slope. · OAI.Falconer.roundedStripCandidates

roundedWeights def

Lost mass becomes a loop, which is deleted by exactly the original aggregation. It is never renormalized into non-loop edges. · OAI.OptimalMaxCut.LongCode.roundedWeights

roundEigenvalue def

The round eigenvalue in degree l on S^(n-1). · OAI.HarmonicCounterexample.Angular.roundEigenvalue

roundHessianContraction def

Intrinsic round Hessian contraction, formed from ambient derivatives and orthogonal projection. It is independent of the extension off the sphere. · OAI.AffineBernstein.roundHessianContraction

roundingDenominator def

ceil(4m/γ) computed from the exact numerator and denominator of γ. · OAI.MaxCutGames.Reduction.WeightedSource.RationalSource.roundingDenominator

roundingDenominator def

ceil(4m/γ) computed from the exact numerator and denominator of γ. · OAI.UniqueGamesTheorem.Reduction.WeightedSource.RationalSource.roundingDenominator

roundingStep def

The mesh used for one level of the partition compression. · OAI.MetricEntropyDuality.roundingStep

roundLabels def

Decode the actual numbered output labels by majority in each padded cloud. · OAI.MaxCutGames.Foundations.PCP.PreprocessingRegularSoundness.roundLabels

roundLabels def

Decode the actual numbered output labels by majority in each padded cloud. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingRegularSoundness.roundLabels

RoundOrdinal abbrev

The first uncountable ordinal, as a small indexing type. There is no inaccessible-cardinal or chain-length closure assumption in this index. · OAI.PartitionConsistency.MonoidConstruction.RoundOrdinal

roundPairing def

The invariant complex bilinear Fischer form in split coordinates, swapping z and bar-z in the second argument. This is the correct null-vector pairing, not the positive coefficient pairing without this swap. · OAI.HarmonicCounterexample.ComplexAngular.roundPairing

rounds def

A concrete logarithmic iteration count, including the zero input. · OAI.MaxCutGames.Foundations.PCP.AmplificationIteration.rounds

rounds def

A concrete logarithmic iteration count, including the zero input. · OAI.UniqueGamesTheorem.Foundations.PCP.AmplificationIteration.rounds

roundTensorDerivative def

The covariant derivative of an ambient two-tensor restricted to the sphere. The arguments `u,v,w` are tangent at the unit point of evaluation. The projected constant extensions of `v,w` have zero covariant derivative there. · OAI.AffineBernstein.roundTensorDerivative

roundWeight def

Downward rounding to the mesh δ. · OAI.MetricEntropyDuality.roundWeight

Route inductive

A directed walk, including its two endpoints. · OAI.ContingencyTables.FiniteDirectedBFS.Route

Route inductive

routes represent synchronous, hence restoring-program-frame execution. (They need not be cursor-restoring.) · OAI.ExactDerandomization.Relay.Grid.Route

Route inductive

Structural routes made only from tensor, associativity and exchange. The arbitrary-permutation completeness theorem below is proved, not a constructor. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Tree.Route

Routed def

All old exterior matchings with the matching parent cycle, including every separate branch occurrence. This is a subtype of finite interface data. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.CyclicInterface.Routed

routedAddress def

The terminal table coordinate consulted by the complete finite router. · OAI.Problem310.RoutedSetDensity.routedAddress

routedLeaf def

The leaf reached by the root router, as an element of the finite leaf type. · OAI.Problem310.RoutedSetDensity.routedLeaf

routeDomain def

Switches on the unique possible route from x to y. · OAI.CubeShuffle.routeDomain

routedSet def

The raw random periodic set, with both selector and terminal tables explicit. · OAI.Problem310.RoutedSetDensity.routedSet

routeFrom def

Route from a node by appending the locally selected child at each step. The choice rule is abstract: a first-success/default selector can be supplied. · OAI.Problem310.RoutingPath.routeFrom

routeRoundValue opaque

The certified value is fixed, but need never be expanded into a numeral. · OAI.ContinuumCoulomb.QuantumHistorySpatial.routeRoundValue

Routes structure

A finite choice of routes from the same base on each graph component. · OAI.TorsionFreeZeroDivisors.GraphPresentation.Routes

routes def

Contracting only the new subdivision/collar tree; no original edge is collapsed by this choice. · OAI.HyperbolicObstruction.FinitePolygonModel.Data.routes

routeValue def

Required coin values on the unique route. Values off its domain are unused. · OAI.CubeShuffle.routeValue

routine def

fields 0, fields 1, and fields 2 are the tail, reverse index, and encoded relation tapes. Every instruction reads or writes a single tape head. · OAI.MaxCutGames.Foundations.Complexity.MachineTableRows.routine

routine def

fields 0, fields 1, and fields 2 are the tail, reverse index, and encoded relation tapes. Every instruction reads or writes a single tape head. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineTableRows.routine

routingChoice def

The node-choice function induced by ordered selector tables. · OAI.Problem310.OrderedRoutePreservation.routingChoice

routingHamiltonian def

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

RoutingNetwork structure

A finite fair routing network certified by its actual test algorithms and path/meeting geometry. The fields are local routing facts, not probability or mixing assumptions. Below they are constructed for parallel butterflies. · OAI.CubeShuffle.RoutingNetwork

Row abbrev

The independent auxiliary mark row for the actually realized type list. · OAI.DilutedSpinGlass.HeterogeneousMarks.Row

Row abbrev

Compression coordinate binary vector helpers. · OAI.ExactDerandomization.Buf.Row

Row structure

An actual row includes its target subspace as part of its index. · OAI.MaxCutGames.Fourier.MatrixRestrictions.Row

Row structure

An actual row includes its target subspace as part of its index. · OAI.UniqueGamesTheorem.Fourier.MatrixRestrictions.Row

row def

A fresh deterministic lattice row. Every two new centers advance one pure-cut spacing; this construction contributes no boundary-port weight. · OAI.CriticalHoneycomb.FiniteContinuation.row

row def

Every position refers to the original arrays. A top is its independent mark, not a new independent copy of its correlated fan entries. · OAI.HyperbolicObstruction.IncidenceTemplate.Occurrences.Shape.row

row def

One explicitly addressed row, with a separately supplied complete table. · OAI.MaxCutGames.Explicit.ProductTarget.row

row def

This is a row read from the actual serialized output table. · OAI.MaxCutGames.Foundations.PCP.PreprocessingInternalRows.row

row def

The actual row vector has one coordinate for every actual column. · OAI.MetricEntropyDuality.GraphProfile.row

row def

This declaration is an exact comparator-selected statement. Read the Lean statement below for its mathematical content. · OAI.SpinAngle.TypeAngle.row

row def

Concatenation in increasing block-index order. · OAI.TreeEdit.BinaryLower.TaggedRow.row

row def

All rows are explicitly defined, even outside the construction horizon. · OAI.UniformKServer.FiniteFlow.row

row def

All rows are explicitly defined, even outside the construction horizon. · OAI.UniformKServer.RealFlow.row

row def

One explicitly addressed row, with a separately supplied complete table. · OAI.UniqueGamesTheorem.Explicit.ProductTarget.row

row def

This is a row read from the actual serialized output table. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingInternalRows.row

row0 def

Selected source equation 0 in the original pair table. · OAI.MUB6.Certificate.PairData.row0

row1 def

Selected source equation 1 in the original pair table. · OAI.MUB6.Certificate.PairData.row1

row10 def

Selected source equation 10 in the original pair table. · OAI.MUB6.Certificate.PairData.row10

row100 def

Selected source equation 128 in the original pair table. · OAI.MUB6.Certificate.PairData.row100

row101 def

Selected source equation 129 in the original pair table. · OAI.MUB6.Certificate.PairData.row101

row102 def

Selected source equation 130 in the original pair table. · OAI.MUB6.Certificate.PairData.row102

row103 def

Selected source equation 131 in the original pair table. · OAI.MUB6.Certificate.PairData.row103

row104 def

Selected source equation 133 in the original pair table. · OAI.MUB6.Certificate.PairData.row104

row105 def

Selected source equation 134 in the original pair table. · OAI.MUB6.Certificate.PairData.row105

row106 def

Selected source equation 135 in the original pair table. · OAI.MUB6.Certificate.PairData.row106

row107 def

Selected source equation 138 in the original pair table. · OAI.MUB6.Certificate.PairData.row107

row108 def

Selected source equation 139 in the original pair table. · OAI.MUB6.Certificate.PairData.row108

row109 def

Selected source equation 140 in the original pair table. · OAI.MUB6.Certificate.PairData.row109

row11 def

Selected source equation 11 in the original pair table. · OAI.MUB6.Certificate.PairData.row11

row110 def

Selected source equation 144 in the original pair table. · OAI.MUB6.Certificate.PairData.row110

row111 def

Selected source equation 145 in the original pair table. · OAI.MUB6.Certificate.PairData.row111

row112 def

Selected source equation 146 in the original pair table. · OAI.MUB6.Certificate.PairData.row112

row113 def

Selected source equation 147 in the original pair table. · OAI.MUB6.Certificate.PairData.row113

row114 def

Selected source equation 150 in the original pair table. · OAI.MUB6.Certificate.PairData.row114

row115 def

Selected source equation 151 in the original pair table. · OAI.MUB6.Certificate.PairData.row115

row116 def

Selected source equation 152 in the original pair table. · OAI.MUB6.Certificate.PairData.row116

row117 def

Selected source equation 153 in the original pair table. · OAI.MUB6.Certificate.PairData.row117

row118 def

Selected source equation 156 in the original pair table. · OAI.MUB6.Certificate.PairData.row118

row119 def

Selected source equation 157 in the original pair table. · OAI.MUB6.Certificate.PairData.row119

row12 def

Selected source equation 12 in the original pair table. · OAI.MUB6.Certificate.PairData.row12

row120 def

Selected source equation 158 in the original pair table. · OAI.MUB6.Certificate.PairData.row120

row121 def

Selected source equation 161 in the original pair table. · OAI.MUB6.Certificate.PairData.row121

row122 def

Selected source equation 162 in the original pair table. · OAI.MUB6.Certificate.PairData.row122

row123 def

Selected source equation 165 in the original pair table. · OAI.MUB6.Certificate.PairData.row123

row124 def

Selected source equation 166 in the original pair table. · OAI.MUB6.Certificate.PairData.row124

row125 def

Selected source equation 167 in the original pair table. · OAI.MUB6.Certificate.PairData.row125

row126 def

Selected source equation 168 in the original pair table. · OAI.MUB6.Certificate.PairData.row126

row127 def

Selected source equation 169 in the original pair table. · OAI.MUB6.Certificate.PairData.row127

row128 def

Selected source equation 171 in the original pair table. · OAI.MUB6.Certificate.PairData.row128

row129 def

Selected source equation 173 in the original pair table. · OAI.MUB6.Certificate.PairData.row129

row13 def

Selected source equation 13 in the original pair table. · OAI.MUB6.Certificate.PairData.row13

row130 def

Selected source equation 175 in the original pair table. · OAI.MUB6.Certificate.PairData.row130

row131 def

Selected source equation 176 in the original pair table. · OAI.MUB6.Certificate.PairData.row131

row132 def

Selected source equation 183 in the original pair table. · OAI.MUB6.Certificate.PairData.row132

row133 def

Selected source equation 184 in the original pair table. · OAI.MUB6.Certificate.PairData.row133

row134 def

Selected source equation 187 in the original pair table. · OAI.MUB6.Certificate.PairData.row134

row135 def

Selected source equation 240 in the original pair table. · OAI.MUB6.Certificate.PairData.row135

row136 def

Selected source equation 241 in the original pair table. · OAI.MUB6.Certificate.PairData.row136

row137 def

Selected source equation 242 in the original pair table. · OAI.MUB6.Certificate.PairData.row137

row138 def

Selected source equation 243 in the original pair table. · OAI.MUB6.Certificate.PairData.row138

row139 def

Selected source equation 244 in the original pair table. · OAI.MUB6.Certificate.PairData.row139

row14 def

Selected source equation 14 in the original pair table. · OAI.MUB6.Certificate.PairData.row14

row140 def

Selected source equation 245 in the original pair table. · OAI.MUB6.Certificate.PairData.row140

row141 def

Selected source equation 246 in the original pair table. · OAI.MUB6.Certificate.PairData.row141

row142 def

Selected source equation 247 in the original pair table. · OAI.MUB6.Certificate.PairData.row142

row143 def

Selected source equation 248 in the original pair table. · OAI.MUB6.Certificate.PairData.row143

row144 def

Selected source equation 249 in the original pair table. · OAI.MUB6.Certificate.PairData.row144

row145 def

Selected source equation 250 in the original pair table. · OAI.MUB6.Certificate.PairData.row145

row146 def

Selected source equation 251 in the original pair table. · OAI.MUB6.Certificate.PairData.row146

row147 def

Selected source equation 252 in the original pair table. · OAI.MUB6.Certificate.PairData.row147

row148 def

Selected source equation 253 in the original pair table. · OAI.MUB6.Certificate.PairData.row148

row149 def

Selected source equation 261 in the original pair table. · OAI.MUB6.Certificate.PairData.row149

row15 def

Selected source equation 16 in the original pair table. · OAI.MUB6.Certificate.PairData.row15

row150 def

Selected source equation 262 in the original pair table. · OAI.MUB6.Certificate.PairData.row150

row151 def

Selected source equation 263 in the original pair table. · OAI.MUB6.Certificate.PairData.row151

row152 def

Selected source equation 264 in the original pair table. · OAI.MUB6.Certificate.PairData.row152

row153 def

Selected source equation 266 in the original pair table. · OAI.MUB6.Certificate.PairData.row153

row154 def

Selected source equation 267 in the original pair table. · OAI.MUB6.Certificate.PairData.row154

row155 def

Selected source equation 268 in the original pair table. · OAI.MUB6.Certificate.PairData.row155

row156 def

Selected source equation 269 in the original pair table. · OAI.MUB6.Certificate.PairData.row156

row157 def

Selected source equation 270 in the original pair table. · OAI.MUB6.Certificate.PairData.row157

row158 def

Selected source equation 275 in the original pair table. · OAI.MUB6.Certificate.PairData.row158

row159 def

Selected source equation 276 in the original pair table. · OAI.MUB6.Certificate.PairData.row159

row16 def

Selected source equation 17 in the original pair table. · OAI.MUB6.Certificate.PairData.row16

row160 def

Selected source equation 277 in the original pair table. · OAI.MUB6.Certificate.PairData.row160

row161 def

Selected source equation 278 in the original pair table. · OAI.MUB6.Certificate.PairData.row161

row162 def

Selected source equation 285 in the original pair table. · OAI.MUB6.Certificate.PairData.row162

row163 def

Selected source equation 286 in the original pair table. · OAI.MUB6.Certificate.PairData.row163

row164 def

Selected source equation 287 in the original pair table. · OAI.MUB6.Certificate.PairData.row164

row165 def

Selected source equation 297 in the original pair table. · OAI.MUB6.Certificate.PairData.row165

row166 def

Selected source equation 298 in the original pair table. · OAI.MUB6.Certificate.PairData.row166

row167 def

Selected source equation 300 in the original pair table. · OAI.MUB6.Certificate.PairData.row167

row168 def

Selected source equation 301 in the original pair table. · OAI.MUB6.Certificate.PairData.row168

row169 def

Selected source equation 302 in the original pair table. · OAI.MUB6.Certificate.PairData.row169

row17 def

Selected source equation 18 in the original pair table. · OAI.MUB6.Certificate.PairData.row17

row170 def

Selected source equation 303 in the original pair table. · OAI.MUB6.Certificate.PairData.row170

row171 def

Selected source equation 304 in the original pair table. · OAI.MUB6.Certificate.PairData.row171

row172 def

Selected source equation 305 in the original pair table. · OAI.MUB6.Certificate.PairData.row172

row173 def

Selected source equation 306 in the original pair table. · OAI.MUB6.Certificate.PairData.row173

row174 def

Selected source equation 307 in the original pair table. · OAI.MUB6.Certificate.PairData.row174

row175 def

Selected source equation 308 in the original pair table. · OAI.MUB6.Certificate.PairData.row175

row176 def

Selected source equation 311 in the original pair table. · OAI.MUB6.Certificate.PairData.row176

row177 def

Selected source equation 312 in the original pair table. · OAI.MUB6.Certificate.PairData.row177

row178 def

Selected source equation 313 in the original pair table. · OAI.MUB6.Certificate.PairData.row178

row179 def

Selected source equation 314 in the original pair table. · OAI.MUB6.Certificate.PairData.row179

row18 def

Selected source equation 19 in the original pair table. · OAI.MUB6.Certificate.PairData.row18

row180 def

Selected source equation 315 in the original pair table. · OAI.MUB6.Certificate.PairData.row180

row181 def

Selected source equation 332 in the original pair table. · OAI.MUB6.Certificate.PairData.row181

row182 def

Selected source equation 337 in the original pair table. · OAI.MUB6.Certificate.PairData.row182

row183 def

Selected source equation 338 in the original pair table. · OAI.MUB6.Certificate.PairData.row183

row184 def

Selected source equation 339 in the original pair table. · OAI.MUB6.Certificate.PairData.row184

row185 def

Selected source equation 340 in the original pair table. · OAI.MUB6.Certificate.PairData.row185

row186 def

Selected source equation 343 in the original pair table. · OAI.MUB6.Certificate.PairData.row186

row187 def

Selected source equation 345 in the original pair table. · OAI.MUB6.Certificate.PairData.row187

row188 def

Selected source equation 346 in the original pair table. · OAI.MUB6.Certificate.PairData.row188

row189 def

Selected source equation 347 in the original pair table. · OAI.MUB6.Certificate.PairData.row189

row19 def

Selected source equation 20 in the original pair table. · OAI.MUB6.Certificate.PairData.row19

row190 def

Selected source equation 349 in the original pair table. · OAI.MUB6.Certificate.PairData.row190

row191 def

Selected source equation 350 in the original pair table. · OAI.MUB6.Certificate.PairData.row191

row192 def

Selected source equation 351 in the original pair table. · OAI.MUB6.Certificate.PairData.row192

row193 def

Selected source equation 353 in the original pair table. · OAI.MUB6.Certificate.PairData.row193

row194 def

Selected source equation 354 in the original pair table. · OAI.MUB6.Certificate.PairData.row194

row195 def

Selected source equation 355 in the original pair table. · OAI.MUB6.Certificate.PairData.row195

row196 def

Selected source equation 356 in the original pair table. · OAI.MUB6.Certificate.PairData.row196

row197 def

Selected source equation 362 in the original pair table. · OAI.MUB6.Certificate.PairData.row197

row198 def

Selected source equation 363 in the original pair table. · OAI.MUB6.Certificate.PairData.row198

row199 def

Selected source equation 364 in the original pair table. · OAI.MUB6.Certificate.PairData.row199

row2 def

Selected source equation 2 in the original pair table. · OAI.MUB6.Certificate.PairData.row2

row20 def

Selected source equation 21 in the original pair table. · OAI.MUB6.Certificate.PairData.row20

row200 def

Selected source equation 365 in the original pair table. · OAI.MUB6.Certificate.PairData.row200

row201 def

Selected source equation 368 in the original pair table. · OAI.MUB6.Certificate.PairData.row201

row202 def

Selected source equation 369 in the original pair table. · OAI.MUB6.Certificate.PairData.row202

row203 def

Selected source equation 372 in the original pair table. · OAI.MUB6.Certificate.PairData.row203

row204 def

Selected source equation 373 in the original pair table. · OAI.MUB6.Certificate.PairData.row204

row205 def

Selected source equation 374 in the original pair table. · OAI.MUB6.Certificate.PairData.row205

row206 def

Selected source equation 377 in the original pair table. · OAI.MUB6.Certificate.PairData.row206

row207 def

Selected source equation 378 in the original pair table. · OAI.MUB6.Certificate.PairData.row207

row208 def

Selected source equation 379 in the original pair table. · OAI.MUB6.Certificate.PairData.row208

row209 def

Selected source equation 381 in the original pair table. · OAI.MUB6.Certificate.PairData.row209

row21 def

Selected source equation 22 in the original pair table. · OAI.MUB6.Certificate.PairData.row21

row210 def

Selected source equation 382 in the original pair table. · OAI.MUB6.Certificate.PairData.row210

row211 def

Selected source equation 383 in the original pair table. · OAI.MUB6.Certificate.PairData.row211

row212 def

Selected source equation 384 in the original pair table. · OAI.MUB6.Certificate.PairData.row212

row213 def

Selected source equation 385 in the original pair table. · OAI.MUB6.Certificate.PairData.row213

row214 def

Selected source equation 386 in the original pair table. · OAI.MUB6.Certificate.PairData.row214

row215 def

Selected source equation 387 in the original pair table. · OAI.MUB6.Certificate.PairData.row215

row216 def

Selected source equation 388 in the original pair table. · OAI.MUB6.Certificate.PairData.row216

row217 def

Selected source equation 389 in the original pair table. · OAI.MUB6.Certificate.PairData.row217

row218 def

Selected source equation 390 in the original pair table. · OAI.MUB6.Certificate.PairData.row218

row219 def

Selected source equation 393 in the original pair table. · OAI.MUB6.Certificate.PairData.row219

row22 def

Selected source equation 23 in the original pair table. · OAI.MUB6.Certificate.PairData.row22

row220 def

Selected source equation 394 in the original pair table. · OAI.MUB6.Certificate.PairData.row220

row221 def

Selected source equation 395 in the original pair table. · OAI.MUB6.Certificate.PairData.row221

row222 def

Selected source equation 396 in the original pair table. · OAI.MUB6.Certificate.PairData.row222

row223 def

Selected source equation 397 in the original pair table. · OAI.MUB6.Certificate.PairData.row223

row224 def

Selected source equation 398 in the original pair table. · OAI.MUB6.Certificate.PairData.row224

row225 def

Selected source equation 399 in the original pair table. · OAI.MUB6.Certificate.PairData.row225

row226 def

Selected source equation 400 in the original pair table. · OAI.MUB6.Certificate.PairData.row226

row227 def

Selected source equation 401 in the original pair table. · OAI.MUB6.Certificate.PairData.row227

row228 def

Selected source equation 402 in the original pair table. · OAI.MUB6.Certificate.PairData.row228

row229 def

Selected source equation 403 in the original pair table. · OAI.MUB6.Certificate.PairData.row229

row23 def

Selected source equation 24 in the original pair table. · OAI.MUB6.Certificate.PairData.row23

row230 def

Selected source equation 404 in the original pair table. · OAI.MUB6.Certificate.PairData.row230

row231 def

Selected source equation 405 in the original pair table. · OAI.MUB6.Certificate.PairData.row231

row232 def

Selected source equation 406 in the original pair table. · OAI.MUB6.Certificate.PairData.row232

row233 def

Selected source equation 407 in the original pair table. · OAI.MUB6.Certificate.PairData.row233

row234 def

Selected source equation 408 in the original pair table. · OAI.MUB6.Certificate.PairData.row234

row235 def

Selected source equation 409 in the original pair table. · OAI.MUB6.Certificate.PairData.row235

row236 def

Selected source equation 410 in the original pair table. · OAI.MUB6.Certificate.PairData.row236

row237 def

Selected source equation 412 in the original pair table. · OAI.MUB6.Certificate.PairData.row237

row238 def

Selected source equation 419 in the original pair table. · OAI.MUB6.Certificate.PairData.row238

row239 def

Selected source equation 422 in the original pair table. · OAI.MUB6.Certificate.PairData.row239

row24 def

Selected source equation 25 in the original pair table. · OAI.MUB6.Certificate.PairData.row24

row240 def

Selected source equation 423 in the original pair table. · OAI.MUB6.Certificate.PairData.row240

row241 def

Selected source equation 425 in the original pair table. · OAI.MUB6.Certificate.PairData.row241

row242 def

Selected source equation 430 in the original pair table. · OAI.MUB6.Certificate.PairData.row242

row243 def

Selected source equation 433 in the original pair table. · OAI.MUB6.Certificate.PairData.row243

row244 def

Selected source equation 434 in the original pair table. · OAI.MUB6.Certificate.PairData.row244

row245 def

Selected source equation 436 in the original pair table. · OAI.MUB6.Certificate.PairData.row245

row246 def

Selected source equation 444 in the original pair table. · OAI.MUB6.Certificate.PairData.row246

row247 def

Selected source equation 445 in the original pair table. · OAI.MUB6.Certificate.PairData.row247

row248 def

Selected source equation 446 in the original pair table. · OAI.MUB6.Certificate.PairData.row248

row249 def

Selected source equation 447 in the original pair table. · OAI.MUB6.Certificate.PairData.row249

row25 def

Selected source equation 26 in the original pair table. · OAI.MUB6.Certificate.PairData.row25

row250 def

Selected source equation 448 in the original pair table. · OAI.MUB6.Certificate.PairData.row250

row251 def

Selected source equation 454 in the original pair table. · OAI.MUB6.Certificate.PairData.row251

row252 def

Selected source equation 455 in the original pair table. · OAI.MUB6.Certificate.PairData.row252

row253 def

Selected source equation 460 in the original pair table. · OAI.MUB6.Certificate.PairData.row253

row254 def

Selected source equation 461 in the original pair table. · OAI.MUB6.Certificate.PairData.row254

row26 def

Selected source equation 27 in the original pair table. · OAI.MUB6.Certificate.PairData.row26

row27 def

Selected source equation 28 in the original pair table. · OAI.MUB6.Certificate.PairData.row27

row28 def

Selected source equation 29 in the original pair table. · OAI.MUB6.Certificate.PairData.row28

row29 def

Selected source equation 30 in the original pair table. · OAI.MUB6.Certificate.PairData.row29

row3 def

Selected source equation 3 in the original pair table. · OAI.MUB6.Certificate.PairData.row3

row30 def

Selected source equation 31 in the original pair table. · OAI.MUB6.Certificate.PairData.row30

row31 def

Selected source equation 32 in the original pair table. · OAI.MUB6.Certificate.PairData.row31

row32 def

Selected source equation 33 in the original pair table. · OAI.MUB6.Certificate.PairData.row32

row33 def

Selected source equation 34 in the original pair table. · OAI.MUB6.Certificate.PairData.row33

row34 def

Selected source equation 35 in the original pair table. · OAI.MUB6.Certificate.PairData.row34

row35 def

Selected source equation 36 in the original pair table. · OAI.MUB6.Certificate.PairData.row35

row36 def

Selected source equation 37 in the original pair table. · OAI.MUB6.Certificate.PairData.row36

row37 def

Selected source equation 38 in the original pair table. · OAI.MUB6.Certificate.PairData.row37

row38 def

Selected source equation 39 in the original pair table. · OAI.MUB6.Certificate.PairData.row38

row39 def

Selected source equation 40 in the original pair table. · OAI.MUB6.Certificate.PairData.row39

row4 def

Selected source equation 4 in the original pair table. · OAI.MUB6.Certificate.PairData.row4

row40 def

Selected source equation 41 in the original pair table. · OAI.MUB6.Certificate.PairData.row40

row41 def

Selected source equation 42 in the original pair table. · OAI.MUB6.Certificate.PairData.row41

row42 def

Selected source equation 43 in the original pair table. · OAI.MUB6.Certificate.PairData.row42

row43 def

Selected source equation 44 in the original pair table. · OAI.MUB6.Certificate.PairData.row43

row44 def

Selected source equation 45 in the original pair table. · OAI.MUB6.Certificate.PairData.row44

row45 def

Selected source equation 46 in the original pair table. · OAI.MUB6.Certificate.PairData.row45

row46 def

Selected source equation 47 in the original pair table. · OAI.MUB6.Certificate.PairData.row46

row47 def

Selected source equation 48 in the original pair table. · OAI.MUB6.Certificate.PairData.row47

row48 def

Selected source equation 49 in the original pair table. · OAI.MUB6.Certificate.PairData.row48

row49 def

Selected source equation 50 in the original pair table. · OAI.MUB6.Certificate.PairData.row49

row5 def

Selected source equation 5 in the original pair table. · OAI.MUB6.Certificate.PairData.row5

row50 def

Selected source equation 51 in the original pair table. · OAI.MUB6.Certificate.PairData.row50

row51 def

Selected source equation 52 in the original pair table. · OAI.MUB6.Certificate.PairData.row51

row52 def

Selected source equation 53 in the original pair table. · OAI.MUB6.Certificate.PairData.row52

row53 def

Selected source equation 54 in the original pair table. · OAI.MUB6.Certificate.PairData.row53

row54 def

Selected source equation 55 in the original pair table. · OAI.MUB6.Certificate.PairData.row54

row55 def

Selected source equation 56 in the original pair table. · OAI.MUB6.Certificate.PairData.row55

row56 def

Selected source equation 57 in the original pair table. · OAI.MUB6.Certificate.PairData.row56

row57 def

Selected source equation 58 in the original pair table. · OAI.MUB6.Certificate.PairData.row57

row58 def

Selected source equation 59 in the original pair table. · OAI.MUB6.Certificate.PairData.row58

row59 def

Selected source equation 60 in the original pair table. · OAI.MUB6.Certificate.PairData.row59

row6 def

Selected source equation 6 in the original pair table. · OAI.MUB6.Certificate.PairData.row6

row60 def

Selected source equation 64 in the original pair table. · OAI.MUB6.Certificate.PairData.row60

row61 def

Selected source equation 65 in the original pair table. · OAI.MUB6.Certificate.PairData.row61

row62 def

Selected source equation 66 in the original pair table. · OAI.MUB6.Certificate.PairData.row62

row63 def

Selected source equation 67 in the original pair table. · OAI.MUB6.Certificate.PairData.row63

row64 def

Selected source equation 68 in the original pair table. · OAI.MUB6.Certificate.PairData.row64

row65 def

Selected source equation 70 in the original pair table. · OAI.MUB6.Certificate.PairData.row65

row66 def

Selected source equation 71 in the original pair table. · OAI.MUB6.Certificate.PairData.row66

row67 def

Selected source equation 72 in the original pair table. · OAI.MUB6.Certificate.PairData.row67

row68 def

Selected source equation 73 in the original pair table. · OAI.MUB6.Certificate.PairData.row68

row69 def

Selected source equation 77 in the original pair table. · OAI.MUB6.Certificate.PairData.row69

row7 def

Selected source equation 7 in the original pair table. · OAI.MUB6.Certificate.PairData.row7

row70 def

Selected source equation 78 in the original pair table. · OAI.MUB6.Certificate.PairData.row70

row71 def

Selected source equation 79 in the original pair table. · OAI.MUB6.Certificate.PairData.row71

row72 def

Selected source equation 80 in the original pair table. · OAI.MUB6.Certificate.PairData.row72

row73 def

Selected source equation 81 in the original pair table. · OAI.MUB6.Certificate.PairData.row73

row74 def

Selected source equation 82 in the original pair table. · OAI.MUB6.Certificate.PairData.row74

row75 def

Selected source equation 83 in the original pair table. · OAI.MUB6.Certificate.PairData.row75

row76 def

Selected source equation 84 in the original pair table. · OAI.MUB6.Certificate.PairData.row76

row77 def

Selected source equation 85 in the original pair table. · OAI.MUB6.Certificate.PairData.row77

row78 def

Selected source equation 87 in the original pair table. · OAI.MUB6.Certificate.PairData.row78

row79 def

Selected source equation 88 in the original pair table. · OAI.MUB6.Certificate.PairData.row79

row8 def

Selected source equation 8 in the original pair table. · OAI.MUB6.Certificate.PairData.row8

row80 def

Selected source equation 89 in the original pair table. · OAI.MUB6.Certificate.PairData.row80

row81 def

Selected source equation 90 in the original pair table. · OAI.MUB6.Certificate.PairData.row81

row82 def

Selected source equation 91 in the original pair table. · OAI.MUB6.Certificate.PairData.row82

row83 def

Selected source equation 92 in the original pair table. · OAI.MUB6.Certificate.PairData.row83

row84 def

Selected source equation 93 in the original pair table. · OAI.MUB6.Certificate.PairData.row84

row85 def

Selected source equation 94 in the original pair table. · OAI.MUB6.Certificate.PairData.row85

row86 def

Selected source equation 95 in the original pair table. · OAI.MUB6.Certificate.PairData.row86

row87 def

Selected source equation 96 in the original pair table. · OAI.MUB6.Certificate.PairData.row87

row88 def

Selected source equation 97 in the original pair table. · OAI.MUB6.Certificate.PairData.row88

row89 def

Selected source equation 98 in the original pair table. · OAI.MUB6.Certificate.PairData.row89

row9 def

Selected source equation 9 in the original pair table. · OAI.MUB6.Certificate.PairData.row9

row90 def

Selected source equation 99 in the original pair table. · OAI.MUB6.Certificate.PairData.row90

row91 def

Selected source equation 100 in the original pair table. · OAI.MUB6.Certificate.PairData.row91

row92 def

Selected source equation 101 in the original pair table. · OAI.MUB6.Certificate.PairData.row92

row93 def

Selected source equation 102 in the original pair table. · OAI.MUB6.Certificate.PairData.row93

row94 def

Selected source equation 103 in the original pair table. · OAI.MUB6.Certificate.PairData.row94

row95 def

Selected source equation 104 in the original pair table. · OAI.MUB6.Certificate.PairData.row95

row96 def

Selected source equation 105 in the original pair table. · OAI.MUB6.Certificate.PairData.row96

row97 def

Selected source equation 106 in the original pair table. · OAI.MUB6.Certificate.PairData.row97

row98 def

Selected source equation 107 in the original pair table. · OAI.MUB6.Certificate.PairData.row98

row99 def

Selected source equation 108 in the original pair table. · OAI.MUB6.Certificate.PairData.row99

rowAccepts def

The finite first-address predicate evaluator used for each stored row. · OAI.MaxCutGames.Foundations.PCP.PoweringTables.rowAccepts

rowAccepts def

The finite first-address predicate evaluator used for each stored row. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringTables.rowAccepts

rowAgree def

two abstract cross sets agree at every row coming from these frozen directions. Also apply this notion reciprocally to check columns. · OAI.HadwigerCounterexample.rowAgree

rowAnswer def

The right player's actual private table, formed from visible row advice. · OAI.MaxCutGames.Decoder.TableKeysPrivateRow.rowAnswer

rowAnswer def

The right player's actual private table, formed from visible row advice. · OAI.UniqueGamesTheorem.Decoder.TableKeysPrivateRow.rowAnswer

rowAppendInTime def

A composable actual execution certificate for a correctly encoded row. The program is fixed; row data occur only in the input-tape equalities. · OAI.MaxCutGames.Foundations.Complexity.MachineTableRows.rowAppendInTime

rowAppendInTime def

A composable actual execution certificate for a correctly encoded row. The program is fixed; row data occur only in the input-tape equalities. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineTableRows.rowAppendInTime

rowAverage def

Arbitrary matrix representations may be used: the equality is at the law level. · OAI.CoordinateSweeps.Grid.Holes.rowAverage

rowBits def

Exact literal output of the command list in field order. · OAI.MaxCutGames.Explicit.MachineProductRow.rowBits

rowBits def

Four whole rows, with the final table already in left-to-right orientation. · OAI.MaxCutGames.Explicit.MachineSubdivisionBodySpec.rowBits

rowBits def

The fully specified data block for either directed occurrence. · OAI.MaxCutGames.Foundations.PCP.PoweringRowData.rowBits

rowBits def

The bits of one actual table row, in the existing graph codec. · OAI.MaxCutGames.Foundations.PCP.PreprocessingPaddingWords.rowBits

rowBits def

Exact literal output of the command list in field order. · OAI.UniqueGamesTheorem.Explicit.MachineProductRow.rowBits

rowBits def

Four whole rows, with the final table already in left-to-right orientation. · OAI.UniqueGamesTheorem.Explicit.MachineSubdivisionBodySpec.rowBits

rowBits def

The fully specified data block for either directed occurrence. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringRowData.rowBits

rowBits def

The bits of one actual table row, in the existing graph codec. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingPaddingWords.rowBits

rowBlock def

Every powered-label pair is written in the canonical row-major order. · OAI.MaxCutGames.Foundations.Complexity.PoweringMachineRow.rowBlock

rowBlock def

Every powered-label pair is written in the canonical row-major order. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringMachineRow.rowBlock

RowBound def

Uniform row-amplification bound with the rectangular C-star matrix norm. · OAI.KadisonSimilarity.RowBound

rowBound def

A common row-size bound for all remaining vertices and ports. · OAI.MaxCutGames.Foundations.Complexity.MachinePaddingBounds.rowBound

rowBound def

A common row-size bound for all remaining vertices and ports. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachinePaddingBounds.rowBound

rowBudget def

For fixed degree and radius this is affine in the serialized graph length. · OAI.MaxCutGames.Foundations.Complexity.PoweringPlanBudget.rowBudget

rowBudget def

For fixed degree and radius this is affine in the serialized graph length. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringPlanBudget.rowBudget

rowCell def

A row pass keeps its residual margin and builds both output lists in reverse order, using only binary minimum, subtraction and list cons. · OAI.ContingencyTables.GreedyFeasibleTable.rowCell

rowCellEquiv def

Rows enumerate exactly the actual cells, including the empty diagram. · OAI.Saxl.rowCellEquiv

rowCentralizerDiagonalHom def

Diagonalizing an element of the first row commutant preserves that membership in the next norm ultrapower. · OAI.WeakPureInfiniteness.rowCentralizerDiagonalHom

RowChosen def

A selected row after the independent row reindexing. · OAI.Problem348.Construction.HostSampling.RowChosen

rowClause def

A clause uses every target in its own row and one labeled gate per outgoing edge. · OAI.Paper320.rowClause

rowColumn def

The concrete row-then-column vertex order of the contingency graph. · OAI.ContingencyTables.RootedCompletion.rowColumn

rowCommands def

Two actual affine fields followed by the 4096 stored relation entries. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.EmitRows.rowCommands

rowCommands def

Two actual affine fields followed by the 4096 stored relation entries. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.EmitRows.rowCommands

rowCosts def

Exact per-row costs at their actual accumulated output frames. · OAI.MaxCutGames.Reduction.MachineOutcomeRows.rowCosts

rowCosts def

Exact per-row costs at their actual accumulated output frames. · OAI.UniqueGamesTheorem.Reduction.MachineOutcomeRows.rowCosts

rowCount def

Number of matched source positions assigned to a specified row. · OAI.TreeEdit.BinaryLower.TaggedNonsplit.rowCount

RowEncoding def

A chosen set of positions together with labels of the prescribed profile. · OAI.TalagrandThreshold.RowEncoding

rowEncodingForget def

Forgetting the constraints retains the actual changed subset and labels. · OAI.TalagrandThreshold.Selector.rowEncodingForget

rowEnvelope def

A global-row envelope, from log T ≤ C k^(3/10) log k. · OAI.QuantitativeVanDerWaerden.Parameters.rowEnvelope

rowEquiv def

All four finite components are retained by this register permutation. · OAI.MaxCutGames.Foundations.Complexity.PoweringMasterState.rowEquiv

rowEquiv def

All four finite components are retained by this register permutation. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringMasterState.rowEquiv

rowExecution def

Every transition of an actual row run embeds with its original cost, preserving the position and both extra tapes, including its return jump. · OAI.MaxCutGames.Foundations.Complexity.MachineExpanderTable.rowExecution

rowExecution def

Every transition of an actual row run embeds with its original cost, preserving the position and both extra tapes, including its return jump. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineExpanderTable.rowExecution

rowFactorSwap def

Swap two factor positions in one nonzero row, leaving all other rows fixed. · OAI.Problem346.SecondTransfer.rowFactorSwap

RowFiber abbrev

Actual matrices with the specified observed row advice. · OAI.MaxCutGames.Decoder.AdviceFibers.RowFiber

RowFiber abbrev

Actual matrices with the specified observed row advice. · OAI.UniqueGamesTheorem.Decoder.AdviceFibers.RowFiber

rowFiberEquiv def

The exact row-fiber bijection; no rank or surjectivity premise is needed. · OAI.MaxCutGames.Decoder.AdviceFibers.rowFiberEquiv

rowFiberEquiv def

The exact row-fiber bijection; no rank or surjectivity premise is needed. · OAI.UniqueGamesTheorem.Decoder.AdviceFibers.rowFiberEquiv

rowFromDart def

One directed occurrence, without merging repeated queries or equal rows. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Table.rowFromDart

rowFromDart def

One directed occurrence, without merging repeated queries or equal rows. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Table.rowFromDart

rowHeights def

The represented heights of a genuine finite plane set. · OAI.Nagata.Workers.W10.rowHeights

rowHoles def

All holes in one actual row, with the inherited ordered trajectories. · OAI.CoordinateSweeps.Grid.Holes.rowHoles

rowIncident def

Incidence with either row role of a mode. · OAI.Problem348.ModeIncidence.rowIncident

rowIndex def

The increasing row enumeration, from raw positions to actual indices. · OAI.Problem348.OrderedHost.rowIndex

rowInTime def

Run one complete stored row, including the actual table-based head lookup. The only hypotheses describe the physical input tapes; no execution or serialization correctness of a caller-provided body is assumed. · OAI.MaxCutGames.Foundations.Complexity.FinalCNFMachine.Program.rowInTime

rowInTime def

Specialization to the standalone finite row program. · OAI.MaxCutGames.Foundations.Complexity.MachineExpanderRow.rowInTime

rowInTime def

The row subroutine's genuine execution certificate has the explicit fixed-parameter affine bound in the serialized graph length. · OAI.MaxCutGames.Foundations.Complexity.PoweringMachineRowBody.rowInTime

rowInTime def

Run one complete stored row, including the actual table-based head lookup. The only hypotheses describe the physical input tapes; no execution or serialization correctness of a caller-provided body is assumed. · OAI.UniqueGamesTheorem.Foundations.Complexity.FinalCNFMachine.Program.rowInTime

rowInTime def

Specialization to the standalone finite row program. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineExpanderRow.rowInTime

rowInTime def

The row subroutine's genuine execution certificate has the explicit fixed-parameter affine bound in the serialized graph length. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringMachineRowBody.rowInTime

rowInTimeAt def

The actual row program has a linear transition bound, with all temporary tapes empty at return. The bound holds for every valid finite input table. · OAI.MaxCutGames.Foundations.Complexity.MachineExpanderRow.rowInTimeAt

rowInTimeAt def

The actual row program has a linear transition bound, with all temporary tapes empty at return. The bound holds for every valid finite input table. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineExpanderRow.rowInTimeAt

rowKernel def

Exactly retain the old whole row with probability r, otherwise draw an independent uniform whole row. This is not independent noise on its bits. · OAI.OptimalMaxCut.TreeCode.rowKernel

rowKernel def

Coordinate product defining the alternating partial expectation. · OAI.TalagrandDiscreteConvexity.rowKernel

rowKey def

Coordinates for the row order: anchors, variables, dummy. · OAI.Problem348.AnchorFrames.rowKey

rowLaw def

A zero row is harmless and uses a fixed fallback label. · OAI.MaxCutGames.Repetition.rowLaw

rowLaw def

A zero row is harmless and uses a fixed fallback label. · OAI.UniqueGamesTheorem.Repetition.rowLaw

rowLines def

Row line chart: the other head coordinates and the fixed tail row label. · OAI.CoordinateSweeps.Grid.rowLines

rowList def

The ordered list serialized by the codec. Equal rows remain separate. · OAI.MaxCutGames.Foundations.PCP.GenericGraphTables.rowList

rowList def

The ordered list serialized by the codec. Equal rows remain separate. · OAI.MaxCutGames.Foundations.PCP.GraphTables.rowList

rowList def

The ordered list serialized by the codec. Equal rows remain separate. · OAI.UniqueGamesTheorem.Foundations.PCP.GenericGraphTables.rowList

rowList def

The ordered list serialized by the codec. Equal rows remain separate. · OAI.UniqueGamesTheorem.Foundations.PCP.GraphTables.rowList

RowMap abbrev

A row map is represented by all its binary matrix coefficients. · OAI.MaxCutGames.Inverse.RowErasureDescriptions.RowMap

RowMap abbrev

A row map is represented by all its binary matrix coefficients. · OAI.UniqueGamesTheorem.Inverse.RowErasureDescriptions.RowMap

rowMap def

The simultaneous row functionals, allowing dependent and zero rows. · OAI.MaxCutGames.Inverse.Shortcode.Slice.rowMap

rowMap def

The simultaneous row functionals, allowing dependent and zero rows. · OAI.UniqueGamesTheorem.Inverse.Shortcode.Slice.rowMap

rowMargins def

All margins in one row of the appendix, including the negative-argument tests when they are required. · OAI.DefocusingNLS.SeparatorArithmetic.rowMargins

rowMass def

The mass at each vertex, before or after determinization. · OAI.MaxCutGames.Repetition.rowMass

rowMass def

The mass at each vertex, before or after determinization. · OAI.UniqueGamesTheorem.Repetition.rowMass

rowMoment def

Zero-based row-index statistic of a partition. · OAI.Saxl.rowMoment

rowNetwork def

Combining one triangular row is just a bit permutation and masked ANDs; no arbitrary phase or exponentially large truth table is present. · OAI.ExactQuantumFactoring.Triangular.rowNetwork

rowOffset def

Offset measured in natural-word fields, including the two header fields. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Input.rowOffset

rowOffset def

Offset measured in natural-word fields, including the two header fields. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Input.rowOffset

rowOnly def

Forget only the column equations of a slice. · OAI.MaxCutGames.Inverse.RowErasureDescriptionColumnMass.rowOnly

rowOnly def

Forget only the column equations of a slice. · OAI.UniqueGamesTheorem.Inverse.RowErasureDescriptionColumnMass.rowOnly

rowOperator def

Linear noiseless observation operator, with the Euclidean output norm. · OAI.SubsphereRegression.rowOperator

rowOrder abbrev

The full raw row order, suitable for local instance installation. · OAI.Problem348.AnchorFrames.rowOrder

rowOrder abbrev

Rows: anchors first, then the minus/subtree/plus tree order, then dummy. · OAI.Problem348.OrderedHost.rowOrder

rowPairCount def

Matched positions in a specified source/target row pair. · OAI.TreeEdit.BinaryLower.TaggedNonsplit.rowPairCount

rowPerm def

A relabeling of heterogeneous marks, without identification of their alphabets. · OAI.DilutedSpinGlass.HeterogeneousMarks.rowPerm

rowPlan def

The exact fixed sequence of data-producing commands, in serialized order. · OAI.MaxCutGames.Foundations.PCP.PoweringFieldPlan.rowPlan

rowPlan def

The exact fixed sequence of data-producing commands, in serialized order. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringFieldPlan.rowPlan

rowPlanInTime def

The actual row-data plan runs within the explicit fixed-plan affine bound. · OAI.MaxCutGames.Foundations.Complexity.PoweringPlanBudget.rowPlanInTime

rowPlanInTime def

The actual row-data plan runs within the explicit fixed-plan affine bound. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringPlanBudget.rowPlanInTime

rowPointStrategies def

Transport a local strategy on genuine row maps to the complete bit-row input type. The same fixed public data may be closed over in both policies. All remaining inputs are private to the respective prover. · OAI.MaxCutGames.Clean.AnswerBridge.rowPointStrategies

rowPointStrategies def

Transport a local strategy on genuine row maps to the complete bit-row input type. The same fixed public data may be closed over in both policies. All remaining inputs are private to the respective prover. · OAI.UniqueGamesTheorem.Clean.AnswerBridge.rowPointStrategies

rowPolynomial def

A generous serialized-row bound evaluated at the actual linear vertex cap. · OAI.MaxCutGames.Foundations.Complexity.MachineRegularExecutionBounds.rowPolynomial

rowPolynomial def

Both canonical templates have exactly the same fixed token count. · OAI.MaxCutGames.Integration.AddressTupleBudget.rowPolynomial

rowPolynomial def

A generous serialized-row bound evaluated at the actual linear vertex cap. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRegularExecutionBounds.rowPolynomial

rowPolynomial def

Both canonical templates have exactly the same fixed token count. · OAI.UniqueGamesTheorem.Integration.AddressTupleBudget.rowPolynomial

rowReplace def

Replace one row, retaining the other rows. · OAI.SmoothObstruction.TriangularGauge.rowReplace

rowReplication def

Place the same row along the diagonal of a rectangular matrix. · OAI.KadisonKastler.rowReplication

rows def

The last table is the actual table for the left-to-right v → r row. · OAI.MaxCutGames.Explicit.MachineSubdivisionDynamicRows.rows

rows def

The saved fields are ordered u,p,q,r,v. All four edges are oriented left-to-right; hence the last permutation is inverted. · OAI.MaxCutGames.Explicit.MachineSubdivisionRows.rows

rows def

The row number decodes to precisely one event/slot/orientation occurrence. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Table.rows

rows def

A convenient total extension of a partial condition. This is auxiliary row data, NOT the generic rows; only values in the domain will be used to make a persistent decision. · OAI.PartitionConsistency.CohenForcing.Condition.rows

rows def

T rows of m blocks of width B. · OAI.TreeEdit.BinaryLower.TaggedExtraction.rows

rows def

The last table is the actual table for the left-to-right v → r row. · OAI.UniqueGamesTheorem.Explicit.MachineSubdivisionDynamicRows.rows

rows def

The saved fields are ordered u,p,q,r,v. All four edges are oriented left-to-right; hence the last permutation is inverted. · OAI.UniqueGamesTheorem.Explicit.MachineSubdivisionRows.rows

rows def

Apply the concrete largest-remainder list of occurrence IDs to the original list. Equal records at different indices remain separate input occurrences. · OAI.UniqueGamesTheorem.Explicit.UniformRoundingTarget.rows

rows def

The row number decodes to precisely one event/slot/orientation occurrence. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Table.rows

rowsAccepts def

Evaluate an already stored row vector by finite label lookups. The only quantified type is the bounded set of walk positions. · OAI.MaxCutGames.Foundations.PCP.PoweringOpinionTables.rowsAccepts

rowsAccepts def

Evaluate an already stored row vector by finite label lookups. The only quantified type is the bounded set of walk positions. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringOpinionTables.rowsAccepts

rowsBits def

Ordered serialized rows with successive physically stored reverse indices. · OAI.MaxCutGames.Foundations.Complexity.MachineDummyRows.rowsBits

rowsBits def

The old encoded rows, without their two header words. · OAI.MaxCutGames.Foundations.PCP.PreprocessingPaddingWords.rowsBits

rowsBits def

Ordered serialized rows with successive physically stored reverse indices. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineDummyRows.rowsBits

rowsBits def

The old encoded rows, without their two header words. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingPaddingWords.rowsBits

rowShift def

One shift after H_j combines all the prescribed mutually commuting controlled power-of-two shifts in that row. · OAI.ExactQuantumFactoring.Triangular.rowShift

rowShrink def

induced row vector transform. · OAI.HadwigerCounterexample.rowShrink

rowSizeBound def

Uniform bound on a serialized row in the actual output table. · OAI.MaxCutGames.Foundations.Complexity.MachineRegularVertexBlock.rowSizeBound

rowSizeBound def

Uniform bound on a serialized row in the actual output table. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRegularVertexBlock.rowSizeBound

RowSparse def

Variable-row sparseness on a selected prefix. · OAI.Problem348.HostAnchorRigidity.RowSparse

rowSq def

Quantitative exponent-two stability estimates. · OAI.FourRow.rowSq

rowStep def

Execute a whole row iteration from the exact guard invariant. Every run below is obtained from the concrete code at its component labels. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Body.rowStep

rowStep def

Execute a whole row iteration from the exact guard invariant. Every run below is obtained from the concrete code at its component labels. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Body.rowStep

RowStepResult structure

An execution witness, with the exact invariant needed by the next guard. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Body.RowStepResult

RowStepResult structure

An execution witness, with the exact invariant needed by the next guard. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Body.RowStepResult

rowSteps def

The actual number of transitions, retaining both concrete lookup costs. · OAI.MaxCutGames.Foundations.Complexity.MachineExpanderRow.rowSteps

rowSteps def

The actual number of transitions, retaining both concrete lookup costs. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineExpanderRow.rowSteps

rowSubgroup def

Coordinate reflections that flip whole rows and fix the homotopy time. · OAI.SharpLiebThirring.PathTopology.rowSubgroup

RowSum def

The row-sum restriction is retained explicitly; pointwise globally consistent address values alone are not used as a substitute for actual readout offsets. · OAI.WLIdentification.CycleRows.RowSum

rowSupport def

The 98 distinct array types of the exact master pattern. · OAI.HyperbolicObstruction.IncidenceTemplate.rowSupport

rowSupport def

The row maximum is over the actual nonempty finite row index set. · OAI.MetricEntropyDuality.rowSupport

rowSurcharge def

Exact increases from critical to actual array-entry costs. · OAI.HyperbolicObstruction.IncidenceTemplate.rowSurcharge

rowTime def

The exact affine polynomial of one concrete row body. · OAI.MaxCutGames.Foundations.Complexity.PoweringPolynomialBudget.rowTime

rowTime def

The exact affine polynomial of one concrete row body. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringPolynomialBudget.rowTime

rowValue def

The number emitted by the checked pure row evaluator. · OAI.MaxCutGames.Foundations.PCP.ExpanderTableEnumeration.rowValue

rowValue def

The number emitted by the checked pure row evaluator. · OAI.UniqueGamesTheorem.Foundations.PCP.ExpanderTableEnumeration.rowValue

rowWeights def

Horizontal-strip chains recording the exact ordered strip sizes. · OAI.Saxl.rowWeights

rowWindow def

A consecutive finite window of rows, concatenated in increasing order. · OAI.TreeEdit.BinaryLower.CheapTranslations.rowWindow

rowWord def

The row word corresponding to the tabloid of a tableau. · OAI.Saxl.rowWord

rowWord def

A row is itself a word of width mB. · OAI.TreeEdit.BinaryLower.BlockExtraction.rowWord

rowWords def

Exact unary game codec used by the supplied binary reduction. · OAI.OptimalMaxCut.LongCode.InstanceAdapter.rowWords

rowXs def

The first coordinates of a represented horizontal row. · OAI.Nagata.Workers.W10.rowXs

rs def

measure transformations preserving volume · OAI.GeneralMahler.Edge.rs

rTilde def

Normalized R-polynomials, obtained from the bar action rather than postulated with an inconsistent choice of descent. · OAI.KLInvariance.Hecke.rTilde

rules def

Literal selector rules, with the first member sent to the empty word. · OAI.FiniteAlgebraicEnvelopes.Bootstrap.Selectors.rules

rules def

Finite enumeration of ALL rules; no least-period restriction is imposed. · OAI.Superstring.Counts.Data.rules

Run abbrev

Auxiliary runs at any program counter. · OAI.Fifths.SM.Run

run def

Successive annulus cuts, with one finite random choice at each round. · OAI.BoundedTreewidthL1.Annulus.run

run def

The time-t permutation, with the latest physical shuffle multiplying on the left. · OAI.CubeShuffle.run

run def

`M.run input fuel` initializes `M` and iterates its one-step transition exactly `fuel` times. · OAI.CurrentKS.Machine.run

run 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.run

run def

choose i m is the prospective divisor obtained at the i-th guarded controller step from its recorded classical transition outputs. · OAI.ExactQuantumFactoring.FactorController.run

run def

Tail-recursive implementation. Every recursive call receives the updated cursor, trie and counters. A capacity violation stops immediately. · OAI.LowSpaceSubsetSum.BinaryTrie.run

run def

The four scalar counts are retained long enough to form both limits. · OAI.LowSpaceSubsetSum.ComputedParentFuels.run

run def

One branch runs at a time. The parent input, split spines and completed sibling fit in the eight-cell-per-input local allowance. · OAI.LowSpaceSubsetSum.MeasuredSortSpace.run

run def

The left array is retained while each right task regenerates its own restricted leaves and streams. An empty or overflowing left task is skipped. · OAI.LowSpaceSubsetSum.RepresentationBucketData.run

run def

The ledger completes both children, so acceptance can only save work. · OAI.LowSpaceSubsetSum.SeparatorTraversal.run

run def

The two lengths are computed once and reused as the matching limit. · OAI.LowSpaceSubsetSum.StoredEntryTerminal.run

run def

The full filter is reached only after its work charge is reserved. · OAI.LowSpaceSubsetSum.StoredNodeCells.run

run def

The capacity is supplied once; guards charge their two list-length scans. · OAI.LowSpaceSubsetSum.StoredSeparatorCheck.run

run def

A read-only random stream supplies the words in order. · OAI.LowSpaceSubsetSum.WordProgram.run

run def

Bounded execution on a finite prefix of the random tape. · OAI.MatchingFPRAS.run

run def

The actual declared-success count on a fully sampled vector. · OAI.MatroidProphet.MarkedQueryProgram.run

run def

Exactly t ordinary deterministic TM0 transitions. · OAI.RationalHitting.run

run def

A deterministic state update applied to the revealed choices. · OAI.UniqueGamesTheorem.Gadget.AdaptivePaths.run

run_fixed_rows def

Actual fixed-row execution. Physical readiness and stored header values are its only data hypotheses; no template or body-execution premise remains. · OAI.MaxCutGames.Integration.AddressTupleBody.run_fixed_rows

run_fixed_rows def

Actual fixed-row execution. Physical readiness and stored header values are its only data hypotheses; no template or body-execution premise remains. · OAI.UniqueGamesTheorem.Integration.AddressTupleBody.run_fixed_rows

runAcc def

The accumulator retains output order after one final reversal. · OAI.LowSpaceSubsetSum.MeasuredFilter.runAcc

runAcc def

The live peak is a proof-side execution annotation. The value and work projection performs exactly the tail filter. · OAI.LowSpaceSubsetSum.MeasuredFilterCells.runAcc

runB def

The check recurses on the number of grid cells. · OAI.GeneralMahler.SCal.Grid.runB

runCostAcc def

The candidate evaluator returns its Boolean answer and its actual work. Both work counters are accumulator fields, including on exhaustion. · OAI.LowSpaceSubsetSum.SamplerAttempts.runCostAcc

runCount def

True runs are counted by their first letters. · OAI.Problem335.WordRuns.runCount

runInput def

The outer confidence parameter is handled by independent repetition. · OAI.ContingencyTables.CountingAdaptiveInputProgram.runInput

runInTime def

Actual execution from the existing input codec to the complete literal halt configuration, with the sum of the three proved phase budgets. · OAI.MaxCutGames.Explicit.MachineClone100Run.runInTime

runInTime def

Actual execution from the existing input codec to the complete literal halt configuration, with the sum of the three proved phase budgets. · OAI.UniqueGamesTheorem.Explicit.MachineClone100Run.runInTime

running def

Running minimum toward a target in the circle cut at its root, index zero. The exceptional root retains its global minimum. · OAI.PlanarL1.Ordered.running

runPrefix def

No initialization or row trace is assumed: both are instantiated at this actual finite driver and this input, then composed. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Loop.runPrefix

runPrefix def

No initialization or row trace is assumed: both are instantiated at this actual finite driver and this input, then composed. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Loop.runPrefix

Runs inductive

Deterministic halting runs. f s = inl y means the transition specifies halt with that label (no more working moves). P holds also at that final configuration. · OAI.Fifths.Runs

runSector def

Workspace in its fixed surrounding anchor/cut sector. x is the cells before C in their ordinary order, y the cells following D. · OAI.FiniteAlgebraicEnvelopes.Compiler.Simulator.runSector

RunsInPolyTime def

Actual code and a clock, not an oracle or a premise of the main theorem. Ambient register types may be infinite names; certificate_any restricts them exactly to finite support. All executions start and end with the single encoded … · OAI.Superstring.BitCode.RunsInPolyTime

RunStatus abbrev

Active finite state, or stopped index and terminal finite state. · OAI.SubsphereRegression.Learner.RunStatus

runTables def

The same data computation on all validated tables; positivity is an invariant, and is erased from the runtime path. · OAI.MaxCutGames.Foundations.PCP.TableIteration.runTables

runTables def

The same data computation on all validated tables; positivity is an invariant, and is erased from the runtime path. · OAI.UniqueGamesTheorem.Foundations.PCP.TableIteration.runTables

RuntimeFavorable def

The two budgets cover the entire hypothetical execution; an early stop can only remove operations. Sampler exhaustion is not a favorable output. · OAI.LowSpaceSubsetSum.RepresentationData.RuntimeFavorable

runtimeHistory def

Convert the successful-analysis coordinates to the actual random signs. The finite partition and prime choices do not depend on the fixed solution. · OAI.LowSpaceSubsetSum.RepresentationSetup.runtimeHistory

runtimeSampler def

This is the actual input-only sampler, instantiated with the zero-target wrapper. Its definition contains no chosen solution or analysis sign change. · OAI.LowSpaceSubsetSum.RepresentationSetup.runtimeSampler

runTolerance def

A conservative positive tolerance using the complete candidate counter. Its comparison with the retained-budget tolerance is proved separately after the global labels are extracted. · OAI.ReflexiveFixedPoints.CandidateContext.runTolerance

runWithinHorizon def

A real verifier execution, enlarged to the single input-dependent clock. The machine and its output are the ones supplied in the verifier definition. · OAI.MaxCutGames.Foundations.Complexity.CookLevin.NPVerifier.runWithinHorizon

runWithinHorizon def

A real verifier execution, enlarged to the single input-dependent clock. The machine and its output are the ones supplied in the verifier definition. · OAI.UniqueGamesTheorem.Foundations.Complexity.CookLevin.NPVerifier.runWithinHorizon

russoInsert def

Reinsert one Boolean coordinate in a function on the remaining coordinates. · OAI.Problem313.russoInsert

RValue abbrev

The twenty dimensional finite atomic jet map. · OAI.AtomicTriangular.FiniteCertificate.AtomicMatrix.RValue

rValue def

R-value extracted from the genuine Hecke bar, with the classical sign and leading q-power. No recurrence is imposed in this definition. · OAI.KLInvariance.HeckeQ.rValue