Glossary: W

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

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W abbrev

The completed model is the fiber product of the normalized section-cone Proj with the completed polynomial blowup over the projective plane. · OAI.KummerSourceModel.W

W def

The least positive Ramsey interval size. Statements about its Ramsey property require a finiteness proof; no such proof is built into this definition. · OAI.QuantitativeVanDerWaerden.W

w def

Field of a controller is w(T) bits. Pad cell entries even further, to match a controller as a whole, for one consistent block radix. · OAI.Fifths.Single.w

w def

The parameters, with M the least power of two strictly above BΔ. · OAI.RationalHitting.Parameters.w

w def

The nonnegative weights used by both global and exceptional-set bounds. · OAI.TalagrandDiscreteConvexity.w

w def

Distinguished subtree vector, with the common suffix coefficient. · OAI.TreeEdit.BinaryLower.FiniteVectors.w

Walk inductive

Directed walks retain the chosen generator pair, including parallel edges. · OAI.EilenbergGanea.LevelTransport.Walk

Walk inductive

Multigraph walks remember the traversed port, not just endpoint vertices. · OAI.TorsionFreeZeroDivisors.PortSubdivision.PortGraph.Walk

walk def

The integer propagation table also defines a normalized finite law. · OAI.ContingencyTables.IntegerLaw.walk

walk def

The actual link walk traced by successive corners, without assuming its image is simple. This is why reduced closed walks, rather than only cycles, are needed. · OAI.Release075.CyclicLink.walk

walkBall def

A finite over-approximation of vertices reached in at most r edges. · OAI.CoarseAssembly.walkBall

WalkCellCover structure

Literal voltage data for a regular cover of the indexed cell target. offset is the position of a side's initial vertex relative to its cell lift. These are local edge equations, not a chain-vanishing assumption. · OAI.HyperbolicObstruction.PictureSwitch.WalkCellCover

WalkCellTarget structure

Indexed oriented steps in polygonal cells, with selected cells permitted to have arbitrary closed template walks. In the cone target these are precisely top-cone walks; all fan triangles and decoration disks are frozen. Template vertices … · OAI.HyperbolicObstruction.PictureSwitch.WalkCellTarget

walkCharge def

Sum an integer edge charge along an actual graph walk. · OAI.CoarseAssembly.walkCharge

walkData def

For each actual path edge, store its complete base relation, then the start-to-tail mask, then the end-to-head mask. Every bit is computed. · OAI.MaxCutGames.Foundations.PCP.PoweringRowData.walkData

walkData def

For each actual path edge, store its complete base relation, then the start-to-tail mask, then the end-to-head mask. Every bit is computed. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringRowData.walkData

walkLaw def

Each transition uses a fresh independent finite experiment. · OAI.ContingencyTables.FirstSuccess.walkLaw

walkLength def

Sum of the given positive real edge lengths along a graph walk. · OAI.BoundedTreewidthL1.walkLength

walkLength def

The sum of edge lengths along a finite graph walk. · OAI.PlanarL1.walkLength

walkLength def

The ordinary sum of edge lengths of a finite graph walk. · OAI.PlanarL1.walkLength

WalkOpinionRow structure

One finite row for each constraint checked along a recorded walk. · OAI.MaxCutGames.Foundations.PCP.PoweringOpinionTables.WalkOpinionRow

WalkOpinionRow structure

One finite row for each constraint checked along a recorded walk. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringOpinionTables.WalkOpinionRow

walkPath def

Follow the barycentric unit edges of a graph walk. · OAI.CoarseAssembly.Rips.walkPath

walkRows def

The only target vertices inspected are the actual edge endpoints. · OAI.MaxCutGames.Foundations.PCP.PoweringOpinionTables.walkRows

walkRows def

The only target vertices inspected are the actual edge endpoints. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringOpinionTables.walkRows

walkSet def

Inputs of the first r path queries, in temporal order forgotten. · OAI.CubeShuffle.walkSet

walkTarget def

Literal target for the actual top occurrences and their subdivided macro edges. No simple-cycle or normal-picture assertion is included in this data. In particular two DISTINCT top polygons using a common indexed cone side are recognized … · OAI.HyperbolicObstruction.ConeSkeleton.walkTarget

walkWeight def

A frozen oriented face contributes its cellular coefficient once, at its zero-th POSITION, not once per side. Top steps contribute their individually indexed oriented elementary cone triangles. · OAI.HyperbolicObstruction.ConeSkeleton.walkWeight

wall def

Source Z_s in the max metric on R x R: the two boundary images and the images of actual switches between opposite OPEN arcs. · OAI.PlanarL1.Columns.Model.Field.wall

wallPhase def

Reflection at a whole supporting line, on the ambient phase space. · OAI.TriangularBilliards.wallPhase

wallReflection def

Reflection of a point in the supporting line of a side. · OAI.TriangularBilliards.wallReflection

walsh def

The Walsh character indexed by a coordinate set. · OAI.TwoPointCorrelations.walsh

walshBasis def

The concrete Walsh basis, indexed by coordinate subsets. · OAI.LeanBlast.GotsmanLinial.walshBasis

walshChar def

The complex-valued Walsh character associated to a subset of coordinates. · OAI.LeanBlast.GotsmanLinial.walshChar

walshCoefficientLinear def

Unnormalized Fourier coefficient as a linear functional. · OAI.LeanBlast.GotsmanLinial.walshCoefficientLinear

WalshDegreeLE def

A concrete finite representation of a real multilinear polynomial of degree at most t, restricted to the Boolean cube. · OAI.TwoPointCorrelations.WalshDegreeLE

Wares structure

constants chosen once for the analytic part of the algorithm. · OAI.ExactDerandomization.Stage.Wares

warm def

At least subexponential mass. · OAI.HadwigerCounterexample.warm

wasserstein2 def

The formalization proves one-third Hölder stability of Brenier maps from the uniform measure on a compact convex body with nonempty interior in $\mathbb R^d$, for every $d\ge2$. For targets supported in one fixed nonempty compact set, the … · OAI.Problem358.wasserstein2

wave def

A piece effect for a target chunk s, intermediary chunk c and source chunk j. · OAI.PowerSaving.Slant.wave

wave def

The oscillator as an actual equivalence class in L²([0,1]; ℝ²). · OAI.ThompsonNonamenability.Analytic.wave

waveDeriv def

The proposed derivative, independently defined from its concrete representative. · OAI.ThompsonNonamenability.Analytic.waveDeriv

wDirection def

The image-side outward unit direction. · OAI.Tingley.wDirection

weakAmbient def

Pointwise weak values in their ambient vector space. · OAI.ReflexiveFixedPoints.WeakSelfMap.weakAmbient

WeakBalanced structure

The same balances, allowing temporary zero edges during cancellation. · OAI.KaplanskyCounterexample.Planar.WeakBalanced

WeakCarrier abbrev

The weak carrier is distinct from the norm subtype K. · OAI.ReflexiveFixedPoints.WeakCarrier

weakCarrierPoint def

Underlying original-norm point of a weak carrier element. No weak-to-norm continuity assertion is made. · OAI.ReflexiveFixedPoints.weakCarrierPoint

WeakColoring def

Exactly the manuscript weak measurable condition, using normalized Haar angular measure. A function is understood up to a plane-null set; no regularity is imposed on a proper coloring. · OAI.EuclideanFiveColor.WeakColoring

weakDualOperator def

Transport a bounded linear operator to the weak dual. · OAI.BoundedHochschild.KadisonRingrose.weakDualOperator

weaken def

Enlarge the permitted support and total-order bound. · OAI.Problem335.FactorialExpansion.weaken

weakFunctionalClosure def

Weak-star closure of a family of bounded functionals. · OAI.BoundedHochschild.KadisonRingrose.weakFunctionalClosure

WeakGradient def

Distributional gradient, tested in every constant direction. · OAI.Bernoulli.WeakGradient

weakGradient def

A representative of the minimal weak upper gradient (zero off its domain). · OAI.WeakHessian.weakGradient

weakHessian def

The weak Hessian, with ∇_(∇g)∇g = ∇(Γ(g)/2). · OAI.WeakHessian.weakHessian

weakify def

Coordinatewise algebraic change from norm values to weak values. · OAI.ReflexiveFixedPoints.WeakSelfMap.weakify

weakIntegral def

Hilbert-space weak invariant integral of bounded, boundedly supported functions, with no ordinary real-variable measurability hypothesis. · OAI.PlaneFiveColor.InvariantMean.weakIntegral

weakIsometry def

The weak action associated to a linear isometry. · OAI.BoundedHochschild.KadisonRingrose.weakIsometry

weakLaplacianDomain def

The existence domain, with no bound on the size of its Laplacian. · OAI.RegularPointCoordinates.weakLaplacianDomain

weakLaplacianOperator def

Uniqueness in L² makes the weak Laplacian a linear operator. No continuity on the whole Sobolev domain is claimed. · OAI.RegularPointCoordinates.weakLaplacianOperator

WeaklyClosed def

Actual weak closedness in the ambient weak topology. · OAI.ReflexiveFixedPoints.WeaklyClosed

WeaklyCompact def

Actual weak compactness in the ambient weak topology. · OAI.ReflexiveFixedPoints.WeaklyCompact

WeaklyIntegrable def

Integrability of the scalar matrix coefficients of an operator field. · OAI.KadisonSimilarity.WeaklyIntegrable

WeaklyNull def

Weak convergence to zero, expressed by all continuous real linear tests. · OAI.BoundedTreePotentials.WeaklyNull

WeaklyNull def

Weak nullity in its exact scalar-functional form, over the real field. · OAI.DiamondDistortion.Midpoint.WeaklyNull

WeaklyNull def

Weak convergence to zero, stated directly against continuous real linear functionals. · OAI.IndependentProducts.WeaklyNull

WeaklyTendsto def

Scalar-test characterization of weak convergence over the real field. · OAI.TriangularBilliards.Analytic.WeaklyTendsto

weakMatrixReconstruction def

Reconstruct a matrix of weak-dual values in the original algebra. · OAI.BoundedHochschild.KadisonRingrose.weakMatrixReconstruction

weakMean def

Hilbert-space weak mean; it exists by the proved scalar mean and Riesz, without any real-variable measurability assumption on V. · OAI.PlaneFiveColor.InvariantMean.weakMean

WeakMTW def

Weak MTW only at interior velocities and only for orthogonal test vectors. · OAI.WeakMTWTransport.WeakMTW

WeakNull def

Weak convergence against the entire continuous dual. · OAI.ForestSpace.WeakNull

weakOperatorBall def

The norm ball of linear maps, expressed by closed pointwise conditions. The conjugate scalar appears because the chosen predual identification is conjugate linear. · OAI.BoundedHochschild.KadisonRingrose.weakOperatorBall

WeakOperatorConverges def

Weak operator convergence, in the only form needed for scalar recovery. It records all vector pairings of the actual operators. · OAI.KadisonKastler.WeakOperatorConverges

weakOperatorImage def

A linear map viewed as a pointwise weak-star continuous-dual valued function. No continuity of the original map for the weak-star topology is required. · OAI.BoundedHochschild.KadisonRingrose.weakOperatorImage

weakOperatorImageLinear def

The weak-star realization is real linear. · OAI.BoundedHochschild.KadisonRingrose.weakOperatorImageLinear

weakOperatorIntegral def

The weak integral of a uniformly bounded, weakly integrable field. · OAI.KadisonSimilarity.weakOperatorIntegral

WeakOperatorUnitBall abbrev

The operator unit ball, with the topology induced by the weak-operator topology. · OAI.KadisonSimilarity.WeakOperatorUnitBall

weakPositiveOrderInterval def

The positive order interval below a fixed positive functional, represented in the weak dual of the algebra. · OAI.BoundedHochschild.KadisonRingrose.weakPositiveOrderInterval

weakPredualMap def

The normal-functional embedding with both weak topologies. · OAI.BoundedHochschild.KadisonRingrose.weakPredualMap

weakPredualRealMap def

The weak predual embedding is real-linear. · OAI.BoundedHochschild.KadisonRingrose.weakPredualRealMap

weakPreserving def

A single attaching map preserves weak maps under pushout. · OAI.Thomason.Construction.CobaseChangeCellularClosure.weakPreserving

weakRightAverage def

Right averaging acts on the ambient space of all weak-dual-valued functions, so its continuity also applies to closures of operator families. · OAI.BoundedHochschild.KadisonRingrose.weakRightAverage

weakRightAverageOrbit def

Pointwise weak-star closure of the finite averaging orbit. · OAI.BoundedHochschild.KadisonRingrose.weakRightAverageOrbit

WeakSelfMap abbrev

Actual selfmaps of K, with pointwise weak topology on their values. · OAI.ReflexiveFixedPoints.WeakSelfMap

weakTailConvexFamily def

Actual selfmaps whose ambient representatives belong to the algebraic convex hull of the resolvent tail. The ambient function space is used for convexity, and the weak selfmap space is used for topology. · OAI.ReflexiveFixedPoints.weakTailConvexFamily

weakUnitaryPullback def

Unitary pullback is weak-star continuous. · OAI.BoundedHochschild.KadisonRingrose.weakUnitaryPullback

WeakUnitBallDense def

A family of contractions whose coefficientwise limits cover the whole unit ball. No metrizability or choice of a sequence is required. · OAI.KadisonKastler.WeakUnitBallDense

weave def

One parity of output, sum of odd-numbered positions. · OAI.FourierLLogL.Dyadic.Procedure.Paths.Run.Streak.weave

wedge def

The exterior product uses the shuffle convention, with scalar multiplication after the tensor-valued alternating product. · OAI.Mahler.wedge

wedgeCoordinates def

The vector v₀u′−u₀v′ in the rank-one and operator-norm identities. · OAI.Paper256.wedgeCoordinates

wedgePower def

Iterated shuffle exterior power of an actual alternating two-form. · OAI.Mahler.wedgePower

WedgePowerSlots def

Slot types retain the order of each two-form factor. · OAI.Mahler.WedgePowerSlots

wedgePowerVariation def

The exact variation of an iterated shuffle power, without suppressing any terms by a formal commutativity or integration-by-parts assumption. · OAI.Mahler.wedgePowerVariation

wedgeSection def

The wedge of sections, followed by the sheafification map. · OAI.ActualExterior.wedgeSection

weight def

The stripe fraction is exactly one quarter in the supplied manuscript. · OAI.BoltzmannNonuniqueness.AxialBackground.weight

weight def

Weights are normalized counts of incident top-dimensional faces. · OAI.CoboundaryExpanders.Complex.weight

weight def

Source equation (weights), with ambient pure dimension s. · OAI.CoboundaryExpanders.Complex.weight

weight def

Length counts centers; the port half-edges contribute no factors. · OAI.CriticalHoneycomb.weight

weight def

The homogeneous subspace, defined by actual tensors rather than formal signs. · OAI.DerivedBar.SquareZero.weight

weight def

Actual syntax length, sharing-sensitive compiler cost and constant bit size. · OAI.ExactQuantumFactoring.NatExpr.weight

weight def

Absolute mass in the common top grid · OAI.FourierLLogL.Dyadic.Procedure.Scene.weight

weight def

Literal independent product weights, before any requestPrefix is revealed. · OAI.KServer.FiniteExperiment.weight

weight def

A finite lifetime d is alive at ages 0,...,d. · OAI.KServer.FiniteSurvival.weight

weight def

Multiplication of an error density by a nonnegative path coefficient. Zero coefficients are retained as genuine zero-density pieces. · OAI.LocalCancellation.ErrorPiece.weight

weight def

Exact integer weight of an indexed subset. · OAI.LowSpaceSubsetSum.weight

weight def

Square-length averages avoid square-root denominators while retaining the exact isometry on the support of an arbitrary nonunital representation. · OAI.NuclearUltrapower.CuntzAveraging.weight

weight def

The alternating sum over actual strict chains stabilized by J. · OAI.PCoreCancellation.weight

weight def

Spectral weighting by a Hilbert vector, exactly |v|² μ. · OAI.PlaneFiveColor.LpMultiplier.weight

weight def

Bernoulli weight on an arbitrary finite set of coordinates. · OAI.Problem313.RestrictionInfluence.weight

weight def

The indicator weight of the derivative-variable side of a partition. · OAI.Problem335.Bidegree.weight

weight def

The actual weighted word summand, with A = alpha⁻¹, B = beta, and x = n⁻¹. The relation records internal adjacency. · OAI.Problem335.IsolatedWeight.weight

weight def

The exact local weights after division by the first-moment square. · OAI.Problem335.LocalMomentWeights.weight

weight def

Source degree minus destination degree. · OAI.Problem346.PolynomialShift.weight

weight def

The coefficient norm required to pay for expanding powers of a count. · OAI.QAC.weight

weight def

Each independent fair-bit assignment has this product weight. · OAI.QuantitativeVanDerWaerden.FairBitTail.weight

weight def

Syntactic statement bound, fixed once the finite machine is fixed. · OAI.RationalHitting.TMTime.OneToZero.weight

weight def

Multiplication of an error density by a nonnegative path coefficient. Zero coefficients are retained as genuine zero-density pieces. · OAI.SignedDisk.LocalCancellation.ErrorPiece.weight

weight def

A finite left-to-right sampling law. The kernel at time k may depend on all k preceding labels; no independence inside a group is assumed. · OAI.SmoothObstruction.SequentialLaw.weight

weight def

Expanded length, attached to undirected edges. · OAI.TorsionFreeZeroDivisors.WeightedCycles.weight

weight def

Source w(t,u), including zero on cancelling turns. · OAI.TorsionFreeZeroDivisors.WordEstimate.weight

weight def

The extended trace taking zero on an ideal and infinity outside it. · OAI.TraceConeClassification.ClosedIdeal.weight

weightCode def

The integer coefficient of every rounded weight is within the finite code range. The positivity and interval hypotheses are explicit. · OAI.MetricEntropyDuality.weightCode

weightCodeSize def

The number of possible rounded values of a weight in [0, 1]. · OAI.MetricEntropyDuality.weightCodeSize

weightComplex def

The actual complex formed by the fixed-count homogeneous bar subspaces. · OAI.DerivedBar.SquareZero.weightComplex

weightCoupling def

The coupled action on a real rank-one weight tensor. · OAI.PositiveEinsteinFour.Bianchi.weightCoupling

weightCrop def

Legal crop selecting an interval of original weights, without recoding. · OAI.SeparableQuotient.Norming.FinitePath.weightCrop

weightDecomposition def

The generalized Cartan decomposition over the complex numbers. · OAI.PolycyclicRecognition.ComplexCartan.weightDecomposition

weightDoubleCoupling def

The twice-coupled action, retaining the ordered generator products. · OAI.PositiveEinsteinFour.Bianchi.weightDoubleCoupling

weightedBilinear def

Integration is bilinear in the coordinate directions, including the off-diagonal signed measures. · OAI.RegularPointCoordinates.PositiveBilinearMeasure.weightedBilinear

weightedBinAverage def

The short average of the actual centered bin label with a real weight. · OAI.JointDickman.weightedBinAverage

weightedBinInterpolant def

The real multiplicative interpolants after multiplying by the real weight. · OAI.JointDickman.weightedBinInterpolant

WeightedBound def

A fixed finite family of localized coordinate norms. · OAI.ClosedSurfaceR4.FiniteOrderSmoothing.SmoothingAtlas.WeightedBound

weightedCharacter def

A character with values in the actual residual cyclic group algebra. · OAI.CirculantHadamard.PrimeCharacterEvaluations.weightedCharacter

WeightedCleanGallery def

The same fresh, crossing-free gallery conserves every labeled weight. Weights are instantiated below by atom occupancies; hence this records actual partitions and not only the numerical rank equation. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.WeightedCleanGallery

weightedCLM def

The actual exponentially weighted integration operator on the time interval. Its norm is at most 1/k; this supplies a uniform contraction for arbitrary bounded linear ODE coefficients, without an assumed global flow theorem. · OAI.ElasticityODE.weightedCLM

weightedCoordinateBox def

The coordinate weighted box is an actual linear image of a cube. · OAI.PolycyclicRecognition.weightedCoordinateBox

weightedCorrection def

Undo diagonal normalization and subtract the entries corresponding to invalid repeated removals. Zero scale coordinates also restore zero rows. · OAI.ContingencyTables.weightedCorrection

weightedCorrection def

Polynomial correction for a single input time b, before taking its output coefficient. Matrix multiplication is in the original order J then G. · OAI.ExactFourier.Boundary.weightedCorrection

weightedCorrelation def

Finite weighted pair correlation, with no probability-library dependency. · OAI.Problem337.weightedCorrelation

weightedCubicTrace def

The trace appearing in the varying matrix-slot coefficient. · OAI.PositiveEinsteinFour.Bianchi.weightedCubicTrace

weightedCut def

Weighted cut on unordered distinct vertex pairs, on the original weight scale. · OAI.OptimalMaxCut.Unweighted.weightedCut

weightedDegree def

The weighted number of edges containing a prescribed vertex set. · OAI.TalagrandDiscreteConvexity.Covering.weightedDegree

WeightedDomination def

A weighted change of variables that only needs an upper change-of-variables bound. · OAI.Problem358.TriangularDomination.WeightedDomination

weightedEuler def

The Euler derivation associated with natural-number weights. · OAI.Problem346.PolynomialWeights.weightedEuler

weightedEval def

The actual finite polynomial evaluated on homogeneous base and fiber coordinates. · OAI.Nagata.W16.AssociatedBundle.weightedEval

weightedEval def

Evaluation with each variable tagged by its integral weight. · OAI.StableCoordinate.LaurentWeightObstruction.weightedEval

weightedFourierFlag def

The weighted Fourier flag, shifted by one so that stage zero is bottom. · OAI.LeanBlast.GotsmanLinial.weightedFourierFlag

weightedFourierSpace def

The weighted degree-k subspace of the counting Hilbert space. · OAI.LeanBlast.GotsmanLinial.weightedFourierSpace

weightedGame def

Interpret a concrete predicate and a normalized finite question law as a game with the common foundation interface. · OAI.MaxCutGames.Soundness.RepeatedGameBounds.Simulation.weightedGame

weightedGame def

Interpret a concrete predicate and a normalized finite question law as a game with the common foundation interface. · OAI.UniqueGamesTheorem.Soundness.RepeatedGameBounds.Simulation.weightedGame

weightedGaussianCoefficient def

The normalized H^k coefficients, represented as an ℓ² sequence. · OAI.DefocusingNLS.weightedGaussianCoefficient

weightedGaussianLaw def

The corresponding probability law in the Hilbert model. · OAI.DefocusingNLS.weightedGaussianLaw

weightedGaussianVector def

Measurable Hilbert-valued sum of the weighted Gaussian coordinates. · OAI.DefocusingNLS.weightedGaussianVector

weightedIndex def

One trial returns an index or requests another independent word. All integers are processed in binary; no weight is expanded into copies. · OAI.ContingencyTables.ExactIntegerPrograms.weightedIndex

weightedIntegral def

Finite linear model of the actual occurrence integral on expressions vanishing on singleton and empty cells. Multiplicities are in m. · OAI.WeakPinned.Variance.weightedIntegral

weightedKernel def

Exponential conjugation changes only the time kernel. · OAI.ForcedComputation.WeaklySingular.weightedKernel

weightedMatrix def

The rank-one scalar matrix f fᵀ tensored with a matrix. · OAI.Release075.weightedMatrix

weightedMatrixHistory def

A complete-matrix history with a general terminal observable. The full ordered old-erase/fresh-grow history remains literal. · OAI.DilutedSpinGlass.PrescribedTree.weightedMatrixHistory

weightedMean def

A finite weighted mean, as a linear statistic. · OAI.JointDickman.weightedMean

weightedMean def

The weighted intercept normalization functional. · OAI.Problem358.CellLaplacian.weightedMean

weightedOperatorIntegral def

Integrating an integrably weighted, uniformly bounded field on vectors. · OAI.KadisonKastler.weightedOperatorIntegral

weightedPathDisplacement def

The same ancestor displacements weighted by their actual root path mass. · OAI.ReflexiveFixedPoints.TreeEvaluation.weightedPathDisplacement

WeightedPlanarScalarCoarea def

Federer (1969), §3.2.12, the nonnegative weighted coarea formula specialized to a scalar C¹ function on a planar open set. This is the identity used by eq:weighted-level-bound in §1 of the Koebe manuscript. · OAI.CircleDomainRigidity.WeightedPlanarScalarCoarea

weightedProduct def

Tensor density on an n-fold product probability space. · OAI.ThreeTorus.weightedProduct

weightedProductForm def

The actual weighted product equation, including zero coefficients and empty products. Its degree is proved by homogeneity and nonzeroness. · OAI.Nagata.W15.weightedProductForm

weightedProfile def

The scalar majorant after exponential time weighting. · OAI.ForcedComputation.WeaklySingular.weightedProfile

weightedProjection def

Coefficients are mapped by φ, the coordinate f is kept as a variable, and all other character values become coefficients. · OAI.CirculantHadamard.PrimeCharacterEvaluations.weightedProjection

weightedProjection def

Actual orthogonal projection onto weighted Fourier layer k. · OAI.LeanBlast.GotsmanLinial.weightedProjection

weightedProjectionMatrix def

The projection represented in the point-mass orthonormal basis. · OAI.LeanBlast.GotsmanLinial.weightedProjectionMatrix

weightedQuadratic def

Polarization is applied before any symmetry is imposed on the weight tensor. · OAI.PositiveEinsteinFour.HighestHessian.weightedQuadratic

WeightedReducedGallery def

A fresh, directed/no-recrossing family-conserving gallery. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.WeightedReducedGallery

WeightedScalarCoareaIdentity def

Precisely the nonnegative weighted scalar coarea formula. · OAI.CircleDomainRigidity.WeightedScalarCoareaIdentity

weightedSchwartzKernel def

Applying any real Bessel weight preserves the Schwartz localization kernel. · OAI.DefocusingNLS.weightedSchwartzKernel

weightedShift def

This includes both shifts: S has coefficient labels, while Theta has only scalar phase labels. · OAI.KervaireMatrix.weightedShift

weightedShiftProfile def

The actual weighted progression-correlation profile. · OAI.TwoPointCorrelations.weightedShiftProfile

weightedSum def

Finite complex weighted sum of a real arithmetic function. · OAI.Problem337.Vaughan.weightedSum

weightedSupplyPrimeTriples def

The logarithmically weighted ordered prime triple count, with exactly the same triples as the rational-divisor supply reduction. · OAI.Problem337.weightedSupplyPrimeTriples

WeightedTree structure

Positive edge weights, normalized separately over each parent's children. · OAI.ReflexiveFixedPoints.WeightedTree

weightedTrial def

A rational choice with integer weights, after a single binary rejection trial. The sum of these masses is totalWeight / 2^k. · OAI.ContingencyTables.ExactIntegerChoice.weightedTrial

weightedVector def

Encode the weighted coefficient norm in a Euclidean space. · OAI.Problem358.WeightedEndpointEnergy.weightedVector

weightedVector def

Literal packet spectral calculus, evaluated in the current coordinates. · OAI.ThreeTorus.ColoredState.weightedVector

WeightedWitness structure

The supported, symmetric dual certificate of the actual parity-lift system. · OAI.ParityWL.WeightedWitness

weightedWordGroupNorm def

A norm below the assigned costs makes the weighted seminorm a norm. · OAI.PolycyclicRecognition.weightedWordGroupNorm

weightedWordNorm def

Weighted word norm, allowing every finite product expression. · OAI.PolycyclicRecognition.weightedWordNorm

weightedZonotope def

The sum of all weighted intervals from the fixed vector list. · OAI.PolycyclicRecognition.weightedZonotope

weightImaginaryAction def

The infinitesimal imaginary action depends only on the actual weight quotient; this is the derivative used in the rotational modification. · OAI.PolycyclicRecognition.ComplexTriangularAdjoint.weightImaginaryAction

weightMap def

Multiplication by t^a, written as its identical Laurent monomial. · OAI.RationalHitting.LaurentReduction.weightMap

weightOperator def

Multiplication by an exponential scalar weight on the time interval. · OAI.ForcedComputation.WeaklySingular.weightOperator

weightOperator def

A scalar on each generalized weight block. · OAI.PolycyclicRecognition.ComplexCartan.weightOperator

weightOperatorLinear def

Linear dependence of a block diagonal operator on its coefficients. · OAI.PolycyclicRecognition.ComplexCartan.weightOperatorLinear

WeightParameters structure

The numerical parameter estimates needed by the circuit upper rank bound. · OAI.Problem335.WeightParameters

weightPlusPart def

The explicit polynomial (7-K)(a ⊗ Z)/12, where K = Σ I_outer,s ⊗ (I_inner,s + Σ_s). · OAI.PositiveEinsteinFour.Bianchi.weightPlusPart

weightRotationCharacter def

The rotational character on the image of the full complex adjoint weight map, viewed as a real additive vector group. · OAI.PolycyclicRecognition.ComplexTriangularAdjoint.weightRotationCharacter

Weights abbrev

Undirected edge weights; the domain contains only actual edges. · OAI.GammaFPP.LoopGraph.Weights

Weights structure

Integer weights on the occurrences, with fan slots retained at every level. · OAI.HyperbolicObstruction.IncidenceTemplate.Occurrences.Shape.Weights

weights def

One fixed countable full-support law suffices for all finitely many colors/requirements and their signed rational polarizations. · OAI.DilutedSpinGlass.UniversalDictionary.weights

weights def

Weights of the actual finite radius outcome. · OAI.KServer.FiniteRadius.weights

weights def

Incoming-window lengths listed in the tree's edge preorder. · OAI.Problem310.TreeSpanBridge.weights

weightSamplingInterior def

Strict halfspaces give its interior sampling region. · OAI.PolycyclicRecognition.weightSamplingInterior

weightSamplingPolytope def

Halfspaces tangent to the unit ball, one per oriented generator ray. · OAI.PolycyclicRecognition.weightSamplingPolytope

weightsCode def

Fractions share the total integer weight before canonical reduction. · OAI.ContingencyTables.ExactIntegerPrograms.weightsCode

weightSize def

Unit-scale derivatives of the fixed partition coefficient. · OAI.ClosedSurfaceR4.PhasePartitions.CompactPhasePartition.weightSize

weightTensor def

The unsymmetrized weight derivative tensor dm ⊗ ∇T. · OAI.PositiveEinsteinFour.HighestHessian.weightTensor

weightVanishingIdeal def

The ideal of series whose coefficients vanish up to the specified weight. · OAI.CartierSections.weightVanishingIdeal

WellFormed structure

The hypotheses supplied by the actual all-substrings input construction. · OAI.Superstring.Counts.Data.WellFormed

wellorderableFormula def

A set is wellorderable exactly when it embeds in some ordinal. Both quantified witnesses are internal sets; the injection is an internal graph. · OAI.PartitionConsistency.SetLanguage.wellorderableFormula

WellOrdered def

A chosen well-order on a type of requests; nothing is quotiented or omitted. · OAI.PartitionConsistency.MonoidConstruction.WellOrdered

westRun def

Leave from the original top endpoint through a new row to the WEST wall. · OAI.CriticalHoneycomb.FiniteContinuation.westRun

weyl def

Weyl word for an involutive unit b=b⁻¹. · OAI.FiniteAlgebraicEnvelopes.Steinberg.weyl

weyl def

The Weyl monomial D^a S^b. · OAI.RationalHitting.QuantumMatrices.weyl

weylCompressionEnergy def

The true Bianchi compression energy, before any spectral coordinates. · OAI.PositiveEinsteinFour.Geometry.weylCompressionEnergy

WeylCovariantIdentities structure

The precise algebraic and covariant identities used from the classical Einstein/Weyl calculus, specialized to the plus block and Ricci 3g. All fields refer to the metric's actual tensors. · OAI.PositiveEinsteinFour.Geometry.WeylCovariantIdentities

WeylEnergyIdentities structure

Disintegration of the normalized frame measure for the metric energy calculation. · OAI.PositiveEinsteinFour.Geometry.WeylEnergyIdentities

wheel def

A planar two-vertex wheel: opposite rotations and matching white bivalent vertices. Empty and singleton rotations are included. · OAI.KaplanskyCounterexample.Planar.Ribbon.wheel

whiskerSimplicialHomotopy def

Pre- and postcompose an actual simplicial homotopy. · OAI.Thomason.Construction.whiskerSimplicialHomotopy

whitenCoefficient def

The exact whitening coefficient, filling the density's zero set with a flat vector. · OAI.ThreeTorus.whitenCoefficient

WhitneySurfaceInput def

Whitney, The Singularities of a Smooth n-Manifold in (2n-1)-Space, Ann. Math. 45 (1944), Theorem 8, p.274, specialized to compact surfaces: existence of a C1 immersion into Euclidean three-space. · OAI.ClosedSurfaceR4.PublishedInputs.WhitneySurfaceInput

whole def

Every cell of a simplex in the cosieve belongs to its full one fibre. · OAI.Thomason.Construction.CosieveSimplex.whole

whole def

Every cell of a simplex in the sieve belongs to its full zero fibre. · OAI.Thomason.Construction.SieveSimplex.whole

wholeFutureProbabilityKernel def

Normalize where the full-future mass is positive finite, using a fixed point mass on the remaining measurable exceptional set. · OAI.DukePrimeDegree.EliminationChain.wholeFutureProbabilityKernel

wholeMapTest def

Equation (6.10), allowing the chosen index and sign to depend on the row. · OAI.MaxCutGames.Soundness.ExpectationComparison.wholeMapTest

wholeMapTest def

Equation (6.10), allowing the chosen index and sign to depend on the row. · OAI.UniqueGamesTheorem.Soundness.ExpectationComparison.wholeMapTest

wholePerm def

The same local cyclic orders are used at every occurrence of a symbol. · OAI.OneWayLiveness.CellNetwork.wholePerm

wholeSausageMeasure def

The one-mesh charge is a measure, so disjoint candidates can be summed. · OAI.SLEExactGauge.wholeSausageMeasure

wholeTip def

Restoring a reflection at every outer leaf extends the tip assignment. · OAI.OneWayLiveness.CellNetwork.ColorData.wholeTip

WholeUnbalancedPaper def

Eventual odd-degree counterexamples and infinitely many counterexample ranks, without any requirement that the parts have equal size. · OAI.RyserOdd.Specification.WholeUnbalancedPaper

wideActionMinimum def

Real form of the literal local wide-action minimum; the first member is duplicated as fallback, exactly as in the rational predictor. · OAI.EditApproximation.wideActionMinimum

wideK abbrev

Absolute bounds when ears are sentinels. · OAI.ExactDerandomization.Buf.Run.wideK

Widen structure

One or several appends before a jump. · OAI.FourierLLogL.Dyadic.Procedure.Capture.Widen

widen def

Increase the allowed degree bound by one. · OAI.MatrixMultiplication.AuxiliarySeparation.DeterminantBounds.widen

widen def

The same finite path is permitted in a larger band. · OAI.PlanarL1.Columns.Model.Path.widen

widen def

Inclusion of intrinsic components into a larger band. No equality of the intrinsic metrics is asserted. · OAI.PlanarL1.Columns.Model.Point.widen

widePrimePool def

A prime pool in a fixed-width multiplicative interval. · OAI.Problem337.widePrimePool

width def

Bits per symbol (at least one). · OAI.Fifths.TM.width

width def

A single common width, chosen effectively only from finite alphabets. · OAI.FiniteAlgebraicEnvelopes.Bootstrap.Source.width

widths def

Widths in the explicit order supplied by a finite cell enumeration. · OAI.ContingencyTables.LargePairEnumeration.widths

Wind def

lam = walk length per unit statistical slack. · OAI.ExactDerandomization.Est.Wind

windingCoefficient def

The coefficient Im ∑ xₖ conj(yⱼ₋ₖ) of the degree-fifteen polynomial. · OAI.DefocusingNLS.SeparatorArithmetic.windingCoefficient

windingInterpolation def

Straight-line interpolation of the real lift with t ↦ n*t. · OAI.CoarseAssembly.windingInterpolation

windingPolynomial def

The full unscaled winding polynomial, with integer coefficients. · OAI.DefocusingNLS.SeparatorArithmetic.windingPolynomial

Window structure

Per tape visited positions contained in [-l,h], both nonnegative. · OAI.Fifths.TM.Window

Window structure

A common finite support window chosen from the fixed representatives alone. · OAI.LittleFinitistic.SelectedWindow.Window

window def

The complete fixed grid window. No spatial observation enters the choices. · OAI.DirectionNeighbors.window

window def

The upward and downward ramps, with the exact dwell and rest schedule. · OAI.HarmonicCounterexample.Pulses.window

window def

All labels whose slope lies in the indicated reference window. · OAI.LipschitzHilbert.SourceSlopeLabels.window

window def

Convolution of a normalized nonnegative kernel with an interval (or any finite window). · OAI.Problem337.CyclicSmoothing.window

window def

The complete fixed grid window. No spatial observation enters the choices. · OAI.SignedDisk.DirectionNeighbors.window

window def

A contiguous window of positions in a block. · OAI.TreeEdit.BinaryLower.SegmentCode.window

windowAssignmentEquiv def

A fixed bijection, independent of every bit, reveal and hypothesis. · OAI.TraceReconstruction.windowAssignmentEquiv

WindowAssignments def

All local and global orientation collections on the complete index tree, including unoccupied vertices and paths which do not fit. The type is fixed before any selected monomial or role choices. n is the local trial count, g the number … · OAI.TraceReconstruction.WindowAssignments

WindowBlocks structure

Window blocks with partition estimates, pathwise nullity, and coherent support bounds. · OAI.SeparableQuotient.ActualSpace.WindowBlocks

WindowCollection def

IDs for precisely one local and one global collection at every genuine nonleaf of the complete source window tree. · OAI.TraceReconstruction.WindowCollection

WindowData structure

Deterministic window interface shared by the finite router and its geometric/probabilistic estimates. q is the TOTAL number of children. Only the two ordering comparisons actually used by routing are required; a preorder construction can … · OAI.Problem310.WindowData

windowData def

The complete deterministic windows exist with any prescribed positive uniform threshold and gap. · OAI.Problem310.WindowConstruction.windowData

WindowFiller def

Pure filler IDs. At bottom leaf windows the two fillers have zero length; they are deterministic empty outputs, not extra information. · OAI.TraceReconstruction.WindowFiller

windowFinType def

A finite indexing of all leaves, not just the active or fitted leaves. · OAI.TraceReconstruction.windowFinType

WindowLeaf def

Complete window-tree leaf indices, with counts numbered from the bottom. The tree is fixed before any occupied subset or orientation is selected. · OAI.TraceReconstruction.WindowLeaf

windowLeafSign def

Literal top-down type recursion in the source's unconditioned surrogate. X path is the root anchor or w_j-Y_path. Dependence of different paths through the same global jitter is retained by this single full-tree map. · OAI.TraceReconstruction.windowLeafSign

windowMinusCount def

Exact number of minus leaves in a fixed realization, without a fit condition. · OAI.TraceReconstruction.windowMinusCount

windowPosteriorLaw def

Actual independent posterior orientations at exact non-leaf metadata. The common global bits in a group are drawn only once. · OAI.TraceReconstruction.windowPosteriorLaw

WindowTrial abbrev

A uniform block of r indices for each of M nondefault children. This finite trial type has exactly the exponent used in the miss formula. · OAI.Problem310.TrialAddresses.WindowTrial

WindowTrial def

Actual IDs of every local or global paired trial on the whole window tree. · OAI.TraceReconstruction.WindowTrial

Wire abbrev

A full triangular edge indexed from its up-triangle side: middle, south, west. All tensor wires are actual edges, not abstract crossings. · OAI.CriticalHoneycomb.Wire

wires def

Read-only input relabeling and arbitrary fanout are genuinely free. · OAI.ExactFourier.LinearDAG.wires

with_harmonic def

The radial and coordinate convergence data remain unchanged when an arbitrary normalized harmonic sequence replaces the harmonic member. The local energy bound follows from Jiang's published interior estimate. · OAI.RegularPointCoordinates.RadialCompactnessSequence.with_harmonic

withAffinity def

Insert a proved common affine description into the exact original row. This changes neither its weak law nor either sampled profile. · OAI.PolycyclicRecognition.CartanWordModel.NeighborSamplingRow.withAffinity

withArgs def

Preserve a program that never reads any arg tapes. · OAI.Fifths.SM.withArgs

withArrays def

Explicitly shared arrays are retained once throughout the stopped execution. · OAI.LowSpaceSubsetSum.RepresentationJoinCells.withArrays

withAuxiliary def

Simultaneously replace every ordered history, while retaining literally all old chart names and all label variables. Ordinary charts are unaltered. · OAI.PlanarL1.Columns.Model.BState.withAuxiliary

withAuxiliary def

Replace only the auxiliary history; all data determining t_s stay fixed. · OAI.PlanarL1.Columns.Model.OrderedData.withAuxiliary

withBase def

Updating only the base keeps every nonempty observation unchanged. · OAI.PureTypeSystem.ObservationValues.withBase

withDensity def

Densities relative to one finite reference, preserving the exact normalization. · OAI.TraceReconstruction.FiniteLaw.withDensity

withDominant def

A star as one dominant coordinate and a (possibly empty) finite side set. · OAI.KServer.OutputDynamics.withDominant

withExterior def

Enlarge the indexing types by one protected exterior polygon. Only the original target's free cone-walk cells may be split. All added side weights are zero, so the chain is exactly the interior cellular chain. · OAI.HyperbolicObstruction.PictureSwitch.WalkCellTarget.withExterior

withHeight def

Vertical motion on a chosen column representative. · OAI.PlanarL1.Columns.Model.Point.withHeight

Within def

M outputs y using at most S bits. · OAI.Fifths.TM.Within

Within def

A definite cutoff clock for an actually terminating command, not an assumed machine-computability predicate. · OAI.OptimalMaxCut.CounterMachine.Command.Within

withInputs def

Select or duplicate arbitrary existing inputs; these are wire names, not gates. · OAI.ExactFourier.LinearDAG.withInputs

withKernel def

Keep the common outer datum while sampling its conditional table law. · OAI.MaxCutGames.Decoder.AdviceLaw.withKernel

withKernel def

Keep the common outer datum while sampling its conditional table law. · OAI.UniqueGamesTheorem.Decoder.AdviceLaw.withKernel

withLargerParameter def

Increasing the permitted defect changes no maps or metrics. · OAI.RegularPointCoordinates.AnalyticIterationData.withLargerParameter

withLeaves def

Private leaves extend local translations exactly, retaining each private index; this is used before the owner pieces are combined into a game bijection. · OAI.WLIdentification.AffineBlocks.OwnedTranslations.withLeaves

withLeavesIso def

Color-preserving base isomorphisms extend to all private leaves. · OAI.WLIdentification.withLeavesIso

withNext def

First draw from P, then apply a channel, retaining the original. · OAI.FourierLLogL.Law.withNext

withOutputs def

Change the ordered output list without performing an arithmetic operation. · OAI.ExactFourier.LinearDAG.withOutputs

withOutside def

A single additional outside polygon, with a separate cell index. · OAI.HyperbolicObstruction.PictureSwitch.PolygonTarget.Boundary.withOutside

withPolynomial def

Reuse all geometric and reconstruction data. The uniform theorem below provides the polynomial estimate required by this constructor. · OAI.ClosedSurfaceR4.JetPolynomial.Perturbation.PolynomialSolveData.withPolynomial

withProfile def

Replacing the analyzing profile keeps all physical geometry. · OAI.LipschitzHilbert.PacketLatticeData.withProfile

withProfile abbrev

Replace only the profiles in the original labeled collection. · OAI.LipschitzHilbert.SourceLabeledLattice.withProfile

withProfile def

Change the profiles, retaining every site, center, carrier and scale. · OAI.LipschitzHilbert.SourceLatticePacket.withProfile

withQuotientHaar def

Replacing only the quotient Haar normalization leaves the intrinsic kernel measure and its conjugation invariance intact. · OAI.PolycyclicRecognition.CartanWordModel.MeasuredCone.withQuotientHaar

withSpatialAmplitude def

Normalize the record field itself to the fixed spatial function while preserving the same velocity loop and its proved geometric identities. · OAI.ClosedSurfaceR4.SurfaceVelocityFamily.Loop.withSpatialAmplitude

withTarget def

Replace the affine target while retaining all row and column data. · OAI.MaxCutGames.Inverse.RowErasureDescriptions.Description.withTarget

withTarget def

Replace the affine target while retaining all row and column data. · OAI.UniqueGamesTheorem.Inverse.RowErasureDescriptions.Description.withTarget

Witness abbrev

Data encoding exactly j distinct tails and one arbitrary parent per tail. The sub-event enforces the star/root restrictions. · OAI.CoordinateSweeps.Coverage.Witness

Witness structure

A literal surviving partner. This is tested before any group coin or orientation is drawn, and never asserts success of an update. · OAI.PlanarL1.Columns.Model.RetainedSelection.Witness

witness def

A bad decoded edge with matching opinions at its two true endpoints. · OAI.MaxCutGames.Foundations.PCP.PoweringTest.witness

witness def

One local existential witness for each formula and parameter tuple. The selection is external: it makes no choice function in the object structure. · OAI.PartitionConsistency.CountableModel.witness

witness def

A bad decoded edge with matching opinions at its two true endpoints. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringTest.witness

Witnesses def

The main path and all attached witnesses are segments of one recorded word. · OAI.TwoPointCorrelations.PrimeWordEncoding.Witnesses

witnesses def

Local existential witnesses, together with conditions which force that there are no witnesses. A generic must meet this uniformly dense set. · OAI.PartitionConsistency.ForcingQuotient.AtomicData.witnesses

WitnessFamily structure

All finite feasible witnesses with their one fixed choice of terminal type. The types remain fixed (up to renaming) during proof-context extensions. · OAI.PureTypeSystem.Labelled.WitnessFamily

WitnessIndex def

A small index set of precisely the domain-entry/condition pairs for which there is a local symmetric witness. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.WitnessIndex

witnessLabels def

A list of event labels determined only by the ordered chunk indices and by the number of selected witnesses in each position. · OAI.GammaFPP.CrossingBridge.witnessLabels

witnessRecordBudget def

The bound includes the main length and number of witnesses as well as all prime slots. · OAI.TwoPointCorrelations.witnessRecordBudget

WitnessRelationSlot abbrev

Two labels, a partner witness, and four bounded endpoints. Internal relations leave the unused entries arbitrary. · OAI.TwoPointCorrelations.WitnessRelationSlot

WitnessSystemData structure

The mode records the two internal and the two comparison alternatives. Every selected witness has only bounded numerical-index metadata. · OAI.TwoPointCorrelations.WitnessSystemData

WLdim def

The infimum of the positive identifying dimensions; this set is nonempty for finite G. · OAI.WLIdentification.WLdim

Wood abbrev

natural-address matrices (only read in extent) · OAI.PowerSaving.Grove.Wood

Word inductive

(TrivSqZeroExt D X) pure bar word with precisely r factors in the square-zero ideal. · OAI.DerivedBar.SquareZero.Word

Word abbrev

The formalization proves $\mathsf L=\mathsf{RL}=\mathsf{BPL}$ for binary languages in an explicit Turing-machine model with logarithmic workspace. The randomized classes use polynomial time: $\mathsf{RL}$ never accepts a no-instance and … · OAI.ExactDerandomization.Word

Word abbrev

Finite signed words, with true denoting a generator and false its inverse. · OAI.FiniteAlgebraicEnvelopes.Word

Word structure

Finite words, ordered by extension, including the empty root. · OAI.ForestSpace.Word

Word def

Fixed-length binary words, with a product decomposition at every successor. · OAI.NuclearUltrapower.CuntzWords.Word

Word inductive

The literal normal forms x z₁ ... zⱼ of equation eq:mono-merger-forms; words with no new letter are the old monoid. · OAI.PartitionConsistency.MergerAlgebra.Word

Word abbrev

Literal irreducibles for the reductions s(bZ) ↦ h(s)Z. · OAI.PartitionConsistency.ProbeAlgebra.Word

Word abbrev

Fixed-length words, with symbols at every position arbitrary. · OAI.TreeEdit.Word

word def

Full affine coefficients of the composition in address order. · OAI.CorSetReference.word

word def

Same right-to-left convention as Diagram.boundaryWord. · OAI.IntegralCharacterVarieties.BoundaryBand.word

word def

A representative word, chosen only to turn a group relator into a polynomial. · OAI.LittleFinitistic.GroupAlgebraPresentation.word

word def

The same complete 2s-ary tree code, with leaves canonically indexed by branch words instead of choosing an arbitrary bijection with Fin k. · OAI.OptimalMaxCut.TreeCode.word

word def

The associated walk in the active column graph, used for planar separation. · OAI.PlanarL1.Columns.Model.Path.word

word def

The complete balanced layer schedule. · OAI.Problem335.BalancedSchedule.word

word def

Literal canonical output, including dimension and accepted-list length. · OAI.RationalHitting.StackLanguage.Runtime.word

word def

Source words have exactly the recursively prescribed lengths, over the fixed initial alphabet. Reindexing is the fixed arbitrary labeling allowed by the manuscript, not a new conditioning at any level. · OAI.SmoothObstruction.WordHierarchy.word

word def

The actual signed labels, at every position of every active chain. · OAI.TorsionFreeZeroDivisors.ActualStageRealization.word

word def

The literal nonempty word of original darts in one oriented marked chain. · OAI.TorsionFreeZeroDivisors.ChainPaths.word

word def

The actual word on an underlying block, in its chain orientation. · OAI.TorsionFreeZeroDivisors.RepeatedBlocks.word

wordAt def

Level-i symbols beginning at an integer time. · OAI.SmoothObstruction.PhaseDynamics.wordAt

wordBall def

Words of length at most n in a finite set of left moves. · OAI.CoarseAssembly.Congruence.wordBall

wordBall def

Products of at most n letters from T. Choosing T symmetric gives the usual intrinsic word ball of radius n in the subgroup generated by T. · OAI.PolycyclicRecognition.wordBall

wordBallLeftShift def

The genuine group-defined shift, with its range restricted by the proved chart bound. · OAI.PolycyclicRecognition.wordBallLeftShift

wordBallScaleTransition def

The same group point in the two word charts, restricted to a fixed coordinate ball. Larger physical scales contract this ball. · OAI.PolycyclicRecognition.wordBallScaleTransition

wordBasisOfDet def

An invertible coordinate selection gives precisely the actual word basis needed for the finite family, including dimension zero. · OAI.PolycyclicRecognition.wordBasisOfDet

WordBasisSelections def

All ordered bases selected from S, retaining the actual word labels. · OAI.PolycyclicRecognition.WordBasisSelections

wordBlocks def

Consecutive fixed-width words; construction duplicates a word network at most w times and so has a direct polynomial-size bound. · OAI.ExactQuantumFactoring.BitArithmetic.wordBlocks

wordCellSupport def

The selected integer log cell clipped to its original shell. · OAI.Ostmann.wordCellSupport

wordChainEquiv def

An honest free-generator equivalence gives an invertible integral edge-chain map; the path chains, not only their augmented exponent sums, are transported. · OAI.EilenbergGanea.wordChainEquiv

wordChartLeftShift def

The chart change determined by left multiplication by k⁻¹. · OAI.PolycyclicRecognition.wordChartLeftShift

wordCol def

The five columns carrying the word coordinates in the fixed anchor. · OAI.Problem348.AnchorBridge.wordCol

wordConstant def

A word of constants uses at most one Boolean node per output bit. · OAI.ExactQuantumFactoring.BitArithmetic.wordConstant

wordCoordinate def

Literal increasing edge coordinates shared by every repetition. · OAI.TorsionFreeZeroDivisors.RepeatedBlocks.wordCoordinate

wordCoordinateSupport def

Support restricted to the injectively numbered singleton coordinates. · OAI.TwoPointCorrelations.wordCoordinateSupport

wordCornerEquiv def

Exact finite rooted corner-substitution bijection. This is not an assumed block law, and it preserves all zero-size corner inserts. · OAI.FKCRT.Inventory.wordCornerEquiv

wordCost def

Positive costs satisfying both of the required last-letter bounds. · OAI.PartitionConsistency.MergerAlgebra.Word.wordCost

wordCount def

The concrete multinomial coefficient, defined as a finite fiber cardinality. · OAI.TalagrandThreshold.wordCount

wordCycleEquiv def

Invertible change of free generators preserves the full cycle module. · OAI.EilenbergGanea.wordCycleEquiv

wordDegreeBound def

A fixed degree bound for the finite list of actual words. · OAI.PolycyclicRecognition.CartanWordModel.wordDegreeBound

wordDiagram def

Multiply the single-letter diagrams in input order, using identity for the empty word. · OAI.TwoWayComplementation.wordDiagram

wordDisplacement def

Total displacement, with each visit counted in order. · OAI.TwoPointCorrelations.wordDisplacement

wordDivisorPrimeSupport def

Distinct prime factors in all the step divisors, both tuples and padding. · OAI.TwoPointCorrelations.wordDivisorPrimeSupport

wordEquiv def

Full actual-word / normalized connected-core ordered-tree bijection. · OAI.FKCRT.Inventory.CoreTree.wordEquiv

wordEquiv def

An executable bijection from fixed-length port words to their ranks. · OAI.MaxCutGames.Foundations.PCP.PoweringEnumeration.wordEquiv

wordEquiv def

An executable bijection from fixed-length port words to their ranks. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringEnumeration.wordEquiv

wordFiber def

All words on a fixed finite domain having the specified profile. · OAI.TalagrandThreshold.wordFiber

wordFiberEquiv def

Reindexing gives an equivalence between words of the same profile. · OAI.TalagrandThreshold.wordFiberEquiv

wordHeight def

Word heights with the source's parameters θ = φ = 1/4. · OAI.PartitionConsistency.MergerAlgebra.Word.wordHeight

wordHeightCost def

Extension of the old height and cost to the actual merger monoid. · OAI.PartitionConsistency.MergerAlgebra.Word.wordHeightCost

WordHyperbolic def

Four-point word hyperbolicity, for some finite generating set. · OAI.HyperbolicObstruction.WordHyperbolic

WordHyperbolic def

A finitely generated group with a unit-edge Cayley graph whose geodesic triangles are uniformly thin. Connectedness of this Cayley graph asserts that S generates. · OAI.Release075.WordHyperbolic

wordLeafCoordinate def

The actual word-state cyclic coordinate, indexed by the same function leaves as the spectral tree's finite leaf set. · OAI.TreeEdit.BinaryLower.FiniteTree.wordLeafCoordinate

WordLevelMatrixActions structure

Raw action data on the ambient word Hilbert space. Both ordinary and adjoint inclusion covariance are explicit, as supplied by reducing free-word levels. No operator block vanishing is included. · OAI.KadisonSimilarity.WordLevelMatrixActions

wordLevelRepeatedOperator def

The canonical degree-wise operator is a concrete compression to one unit repeated word. Its independence of that word is proved below. · OAI.KadisonSimilarity.wordLevelRepeatedOperator

wordLinearPart def

The degree-one part of a word as a linear map in its input letters. · OAI.PolycyclicRecognition.wordLinearPart

wordMap def

A tensor power of a single letter-linear map, in ordinary word coordinates. · OAI.Saxl.wordMap

wordMaskEquiv def

All words occur once, so subset counting keeps the actual uniform law. · OAI.MaxCutGames.Foundations.PCP.CayleyTailCount.wordMaskEquiv

wordMaskEquiv def

All words occur once, so subset counting keeps the actual uniform law. · OAI.UniqueGamesTheorem.Foundations.PCP.CayleyTailCount.wordMaskEquiv

wordMatrix def

Stored chronologically, so a new step left-multiplies earlier steps. · OAI.ExactFourier.wordMatrix

wordMetric_linearChains def

Prefixes of a shortest representing word give unit-step chains of length exactly the word-metric distance. · OAI.PolycyclicRecognition.wordMetric_linearChains

wordMonomial def

Literal weight in the letter-size variable. Specializing r²=u gives exactly the source weight t^flex u^edges. · OAI.FKCRT.Inventory.wordMonomial

WordMultiplicity abbrev

The multiplicity of words with k centered base-algebra letters. · OAI.KadisonSimilarity.WordMultiplicity

wordMux def

A word mux shares previously computed word outputs via fanout; it does not recompute the operand circuit separately at every bit. · OAI.ExactQuantumFactoring.BitArithmetic.wordMux

wordNodeEquiv def

Preorder nodes indexed at the true total word length. · OAI.FKCRT.Inventory.CoreTree.wordNodeEquiv

wordPeriod def

Prescribed period at a list prefix. · OAI.TreeEdit.BinaryLower.FiniteTree.wordPeriod

wordPeriods def

The actual leaf periods, in the list-path presentation used by the words. · OAI.TreeEdit.BinaryLower.FiniteTree.wordPeriods

wordPerm def

The underlying ordinary slot permutation; gh applies h first. · OAI.CoordinateSweeps.SignedTensor.wordPerm

wordPerm def

On words, precomposition by a position permutation is a bijection. · OAI.Saxl.wordPerm

WordPicture structure

Finite, genuinely genus-zero van Kampen side-pairings, with a literal outside word and enumerated indexed relator boundaries on every interior face. There is no disk-existence or isoperimetric field. · OAI.HyperbolicObstruction.WordPictures.WordPicture

wordPicture def

Reverse bridge from a literal genus-zero target-cell diagram to an area-counted word picture. Every local premise is an actual attaching polygon membership, not a filling or diagram-existence assumption. · OAI.HyperbolicObstruction.PictureSwitch.CellPicture.wordPicture

wordPrefix def

Ordered product of the first m letters of a word. · OAI.PolycyclicRecognition.wordPrefix

wordPrefix def

The first r complete chunks of the balanced schedule. · OAI.Problem335.BalancedSchedule.wordPrefix

wordPrimeSupport def

The distinct prime factors of the whole tuples in a word. · OAI.TwoPointCorrelations.wordPrimeSupport

wordProfile def

The number of positions of each color in a word on a finite domain. · OAI.TalagrandThreshold.wordProfile

wordProfiles def

The finite set of profiles attained by words of length t. · OAI.TalagrandThreshold.wordProfiles

WordRelator inductive

Literal boundaries, retaining every side index of its graph walk or row. The arbitrary fan/top walks will later be reduced to a finite list of defining cycles; the disk theorem here has no area estimate. · OAI.HyperbolicObstruction.LetterPresentation.WordRelator

wordRep def

The position action satisfies `g e_a = e_(a ∘ g⁻¹)`. · OAI.Saxl.wordRep

wordRestrict def

Restrict letters on a specified position set, and sum over all other letters. This is precisely contraction with the all-ones functional at deleted positions. · OAI.Saxl.wordRestrict

wordRow def

The 32 rows carrying all binary words in the fixed anchor. · OAI.Problem348.AnchorBridge.wordRow

Words abbrev

Words with exactly k odd entries. This is the k-parity sector of the literal signed slot space, not a formal unknown representation. · OAI.RowColumn.Signed.Words

words def

Binary tails in their literal lexicographic order, with a before c. · OAI.FiniteAlgebraicEnvelopes.Bootstrap.LexBinary.words

words def

The finite word ball over an explicit finite support. · OAI.FiniteAlgebraicEnvelopes.Bootstrap.Uniform.words

words def

Enumerate all words of length k over an explicitly listed alphabet. · OAI.MaxCutGames.Reduction.Explicit.words

words def

The word labels of this selected basis. · OAI.PolycyclicRecognition.WordBasisSelections.words

words def

Exactly the finite source word-prefix set, explicitly enumerated. · OAI.UniformKServer.EffectiveLP.words

words def

Enumerate all words of length k over an explicitly listed alphabet. · OAI.UniqueGamesTheorem.Reduction.Explicit.words

wordSeries def

The full genuine word generating series, with no convergence premise. · OAI.FKCRT.Inventory.wordSeries

wordSet def

Rotation words and reflection-coset words, as elements of the algebra. · OAI.KadisonSimilarity.DihedralWordAlgebra.wordSet

wordSharedRole def

Table, query, scan, reverse identifier, scratch, selected endpoint. · OAI.MaxCutGames.Foundations.Complexity.PoweringMachineTapes.wordSharedRole

wordSharedRole def

Table, query, scan, reverse identifier, scratch, selected endpoint. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringMachineTapes.wordSharedRole

WordSheetsGood def

All the states at which a support leg is traversed are in the good set. · OAI.PolycyclicRecognition.WordSheetsGood

WordsLE abbrev

Words of length at most the specified cap. · OAI.TreeEdit.BinaryLower.WordsLE

WordSpace abbrev

Coefficients of position tensors on an alphabet of size `d`. · OAI.Saxl.WordSpace

WordStep inductive

Only forward calls to the specified A are allowed. · OAI.ExactFourier.WordStep

WordSubset abbrev

The subset indices left after grouping the ordered labels of a cut. · OAI.KadisonKastler.WordSubset

wordTensor def

Tensor regrouping by positions. · OAI.Saxl.wordTensor

wordTransferLeafValue def

At a leaf, the state contains the original word whose Fourier factor is evaluated. Internal states are never queried by the leaf list. · OAI.Ostmann.wordTransferLeafValue

wordU def

Actual cheap subgroup vector at a list prefix. · OAI.TreeEdit.BinaryLower.FiniteTree.wordU

wordValue def

Little-endian binary words; zero has the empty word. · OAI.RationalHitting.StackLanguage.wordValue

WordVertex def

A main vertex is the endpoint of a prefix, including both extreme endpoints. · OAI.TwoPointCorrelations.WordVertex

wordVertices def

The finite set of suffix vertices of a free word, including both endpoints. · OAI.EilenbergGanea.wordVertices

wordW def

Actual distinguished subgroup vector at a list prefix. · OAI.TreeEdit.BinaryLower.FiniteTree.wordW

wordWeight def

Source P(W), with empty products equal to one. · OAI.TorsionFreeZeroDivisors.WordEstimate.wordWeight

work def

Sum of the actual supplied emitter costs, with no counter work included. · OAI.MaxCutGames.Foundations.Complexity.MachineTupleOdometer.NestedCycle.work

work def

Sum of the actual supplied emitter costs, with no counter work included. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineTupleOdometer.NestedCycle.work

Works def

Uniform relation on port stacks. Context X,J,L,Y are finite in uses that witness computation in the main theorem. · OAI.Fifths.Works

Works def

A code keeps representing a genuine exact-fiber value at every stronger Cohen condition. This is a predicate on the already defined name collection, not an extra mathematical hypothesis about that collection. · OAI.PartitionConsistency.ProductForcing.ChainCode.Works

workTapes def

An executable complete cleanup list from an explicit finite enumeration. · OAI.MaxCutGames.Foundations.Complexity.MachineDrainMany.workTapes

workTapes def

An executable complete cleanup list from an explicit finite enumeration. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineDrainMany.workTapes

World structure

A state of the actual construction in its current physical units. Both label spaces are enlarged at every step, even in the no-change mode. · OAI.PlanarL1.Columns.Model.World

WOTCarrier abbrev

The carrier equipped with the subspace topology from ambient WOT. · OAI.Generator.WOTCarrier

WOTClosed def

Weak operator closedness, not operator norm closedness. · OAI.Generator.WOTClosed

WOTClosed def

Weak operator closedness of a concrete unital star subalgebra. · OAI.RelativeGeneration.WOTClosed

wotMap def

A homomorphism viewed on the WOT carrier, with ambient WOT codomain. · OAI.Generator.wotMap

wotSelfAdjointBall def

Self-adjoint contractions of the given algebra, with the weak operator topology. · OAI.KadisonKastler.wotSelfAdjointBall

WOTStarAlgEquiv structure

An algebra isomorphism continuous in both directions for the concrete weak operator topologies. · OAI.Generator.WOTStarAlgEquiv

WPath inductive

bounded transitive closure · OAI.ExactDerandomization.Relay.WPath

writeEncodedSlots def

The delimiter is pushed first, then the optional unary one; writing slots in reverse order gives the canonical field order on the destination stack. · OAI.MaxCutGames.Foundations.Complexity.PoweringMachineRow.writeEncodedSlots

writeEncodedSlots def

The delimiter is pushed first, then the optional unary one; writing slots in reverse order gives the canonical field order on the destination stack. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringMachineRow.writeEncodedSlots

writeSlots def

The slot list is a fixed part of the syntax; each instruction reads only the finite internal state. Push order is reversed on the stack. · OAI.MaxCutGames.Foundations.Complexity.MachineFixedBlockMap.writeSlots

writeSlots def

The slot list is a fixed part of the syntax; each instruction reads only the finite internal state. Push order is reversed on the stack. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineFixedBlockMap.writeSlots

writesWeight def

Number of deferred writes along the longest branch. · OAI.RationalHitting.TMTime.Alphabet.writesWeight

writeWork def

Update only the simulated work track; all auxiliary tag fields are retained. · OAI.FiniteAlgebraicEnvelopes.Compiler.Simulator.writeWork

written def

The word written by a hierarchical walk: exactly one symbol per up edge. · OAI.Superstring.Hierarchical.Path.written

wronskian def

The Wronskian with derivative orders 0, ..., h - 1. · OAI.RationalHitting.Expansion.wronskian

wShift def

Small-angle link at W, with u,v interchanged. · OAI.Release075.MarkedLineData.wShift

wstar def

The manuscript's W*(s), as the intersection of all weakly closed unital star subalgebras containing s. This does not invoke a double-commutant theorem. · OAI.Generator.wstar

wstar def

The weakly closed unital star algebra generated by a set of operators. · OAI.RelativeGeneration.wstar

wt def

Local normalized subdensity (as mass weights, counting-grid). Its total is m/α. · OAI.FourierLLogL.Dyadic.Thermal.wt

wulff def

Intersection of the actual supporting constraints, with no evenness or convexity condition on the constraint function. · OAI.PettyProjection.wulff

wulffCoefficients def

Compact coefficient set of the maximum formula. · OAI.PettyProjection.wulffCoefficients

W₀ abbrev

The literal uncompleted fiber-product model, not a postulated total space. · OAI.KummerSourceModel.W₀