Glossary: G

Definitions in this project whose names start with G. 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

G def

Affine normal derivative of the convexified linear weight. · OAI.ElasticitySchwartzCarleman.G

G abbrev

The group in the literal prescription: the coordinate gluing, its multiple HNN extension and the finite permutation/character block. · OAI.KaplanskyCounterexample.Prescription.Output.G

G abbrev

The exact group of the semidirect-product construction. · OAI.NuclearUltrapower.G

G def

Gluing to the identity outside the open tube, including its boundary. · OAI.ThompsonNonamenability.Analytic.CorrectionData.G

G def

The explicit correction, restricted to the shared actual closed unit ball. · OAI.ThompsonNonamenability.Analytic.G

g def

Integral of the entire cosine continuation, with removable value g 0 = angle. · OAI.CriticalHoneycomb.FiniteFunctional.g

g def

The thermal entropy per mode, with the convention 0 log 0 = 0. · OAI.EntropyPhotonNumber.g

g def

The thermal entropy per mode; Lean's Real.log 0 = 0 implements exactly the stated convention 0 log 0 = 0. · OAI.EntropyPhotonNumber.g

gallery def

Full gallery; no side branches have been dropped. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.ParentTemplate.gallery

Game structure

The generative presentation of a finite rational weighted Unique Game: choose a right question, then an incident edge. All actual games have this presentation after deleting zero-marginal right questions. · OAI.OptimalMaxCut.LongCode.Game

gameBits def

`gameBits game` applies the unary `encodeWords` representation to the complete natural-number field list from `gameWords`. · OAI.UniqueGamesTheorem.Foundations.Complexity.gameBits

gameEncoding def

Concrete forward-table output syntax with a checked parser. · OAI.MaxCutGames.Foundations.Complexity.gameEncoding

gameEncoding def

Concrete forward-table output syntax with a checked parser. · OAI.UniqueGamesTheorem.Foundations.Complexity.gameEncoding

gameWords def

`gameWords game` places a three-word header before the serialization of every constraint occurrence. · OAI.UniqueGamesTheorem.Foundations.Complexity.gameWords

gamma def

The signed coefficient at the exact uncontracted divergence vertex. · OAI.DilutedSpinGlass.PrescribedTree.gamma

gamma def

The positive logarithmic increment at level j. · OAI.MetricEntropyDuality.LogProfile.gamma

gamma def

Polarization of the minimal weak gradient on an infinitesimally Hilbertian space. · OAI.WeakHessian.gamma

gammaBadConnections def

Failure of the Gamma allowance for some actual coordinate path in the box. · OAI.GammaFPP.gammaBadConnections

GammaDifferentiabilityTarget def

The Gamma theorem: an actual deterministic passage-time limit norm exists for every positive shape and rate and every iid environment with that law. Its Frechet differentiability and literal unit-ball boundary charts are conclusions. No … · OAI.GammaFPP.GammaDifferentiabilityTarget

GammaEnvironment abbrev

An iid edge environment with Gamma marginal law. · OAI.GammaFPP.GammaEnvironment

gammaFEQuotient def

The elementary quotient in the trivial infinity-type equation. · OAI.CubicFirstMoment.gammaFEQuotient

GammaInverseFiniteOrder def

Weak reciprocal-Gamma growth, on one bounded strip only. · OAI.CubicFirstMoment.GammaInverseFiniteOrder

gammaMeanNorm def

Literal Euclidean-plane pullback of the actual mean seminorm. · OAI.GammaFPP.gammaMeanNorm

gammaQuantileExtended def

The extended Gamma quantile used for upper rounding takes the value infinity at u = 1. · OAI.GammaFPP.gammaQuantileExtended

GammaQuotientStripBound def

Small bounded-strip consequence of the uniform vertical Stirling formula, NIST DLMF (5.11.9), https://dlmf.nist.gov/5.11.E9. This concerns only the classical Gamma function, not an arithmetic sum. · OAI.CubicFirstMoment.GammaQuotientStripBound

gammaScaleLikelihood def

Likelihood for changing a Gamma rate from rate to rate / alpha. · OAI.GammaFPP.gammaScaleLikelihood

gammaShrinkLikelihood def

The likelihood with the parametrization alpha = 1 - delta. · OAI.GammaFPP.gammaShrinkLikelihood

gammaSphere def

A uniform direction with independent squared radius Γ(dim/2,1). · OAI.SubsphereCurrent.Mixture.gammaSphere

gammaWeight def

Normalized Gamma mixing density, used only on the positive half-line. · OAI.SharpLiebThirring.ScalarProof.gammaWeight

gap def

Width of the interval immediately above the indexed level. · OAI.ThompsonNonamenability.FiniteLevelChain.gap

GapChain def

A finite list of consecutive cells starts at a and ends at b. The empty list represents a gap of length zero. · OAI.ThompsonNonamenability.GapChain

gapD def

d-index of slack group number i, descending. · OAI.ExactDerandomization.Est.gapD

gapIndex def

The gap index is determined solely by the ordered root-component darts. The two ends of every exterior matched pair have the same gap index. · OAI.FKCRT.Chords.gapIndex

GapInstance structure

Output includes a strictly positive integer threshold. · OAI.DirectedFeedback.GapInstance

gapMapCertificate def

Actual 3SAT gap-map runtime follows from the one concrete round certificate; initialization, variable repetition, and the final CNF converter are supplied. · OAI.MaxCutGames.Foundations.Complexity.MachineTableIteration.gapMapCertificate

gapMapCertificate def

Actual 3SAT gap-map runtime follows from the one concrete round certificate; initialization, variable repetition, and the final CNF converter are supplied. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineTableIteration.gapMapCertificate

gapOperator def

Polynomial of a centered positional operator after removing from both incident gaps the one unit needed for either direction of movement. · OAI.TraceReconstruction.gapOperator

GapReduction structure

A strict YES/NO gap reduction implemented by a finite-alphabet machine with polynomial runtime in the raw input bit length. · OAI.DirectedFeedback.GapReduction

GapReduction structure

The rational gap bounds and the finite machine are fixed before the input. The bounds compare the maximum cut divided by the positive output scale. · OAI.OptimalMaxCut.GapReduction

GapRefines def

Every cell in the refined list lies in an original cell. · OAI.ThompsonNonamenability.GapRefines

GapUniverse abbrev

Canonical slot universe split at the tagged positions. Actual order is only needed to identify each component with its before/between/after interval. · OAI.TraceReconstruction.GapUniverse

Gate def

A minimal vertex outside a head. Gates in a tree form an antichain. · OAI.DiamondDistortion.Ancestry.Gate

Gate inductive

One charged scalar gate, referencing only available values. · OAI.ExactFourier.Gate

Gate structure

An inverse and/or one added control. This is a fixed set of 20 gate names, not an unbounded-control or arbitrary-angle gate model. · OAI.ExactQuantumFactoring.Gate

Gate def

A first vertex outside an ancestral set (including unattached roots). · OAI.ForestSpace.Gate

Gate def

A gate is an old generator for precisely the part of a right cone lying outside its word's first-letter block. · OAI.PartitionConsistency.ProbeAlgebra.Word.Gate

Gate def

Exact gates for continue/exit operations at free lambdas. · OAI.PureTypeSystem.ObservationValues.Gate

Gate inductive

A one-qubit unitary or an unbounded-arity Toffoli gate. · OAI.QAC.Gate

gateCap def

The cap is the cost of the actual least old right multiple, never that of an arbitrarily larger old multiple (costs need not be monotone). · OAI.PartitionConsistency.ProbeAlgebra.Word.gateCap

gateCoefficient def

The combined coefficient of all wires from a fixed predecessor. Wire multiplicity is absorbed into this coefficient, not charged as gates. · OAI.Problem335.gateCoefficient

gateIndex def

Index of a newly emitted vector gate. · OAI.ExactFourier.gateIndex

gateSet def

The represented finite set of all gates of H in K. · OAI.DiamondDistortion.Ancestry.gateSet

GateTower structure

A continuous concrete tower with the two successor alternatives of thm:monoid. Terminal segments may be constant. The successor alternatives are recorded in the final divisibility relation, as allowed by conservative embeddings in the … · OAI.PartitionConsistency.DiagramOrder.GateTower

gateWeight def

Square mass at gates, and zero elsewhere. · OAI.ForestSpace.gateWeight

gather def

Gather a finite list before running the dependent local continuation. · OAI.EditApproximation.FiniteQuery.gather

gather def

Two stack words become one prefix pair (on y); work slot t initially empty and returned empty. · OAI.Fifths.SM.gather

gatheredLocalCurrentCalls def

Actual oracle requests in the two gathers. Each list is sorted and contains no duplicates; overlap between the two stages is harmless. · OAI.EditApproximation.gatheredLocalCurrentCalls

gatherPaths def

Zip four independently indexed paths into one path of quadruple labels. · OAI.Problem335.Pairings.gatherPaths

gaugedFactors def

Change of diagonal frame at each successive prefix of a word. · OAI.PolycyclicRecognition.gaugedFactors

gaugeEquiv def

A generator-coordinate equivalence, meaningful over the full ring. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.gaugeEquiv

gaugeEquiv def

Right multiplication by the lift correction is an exact group isomorphism. · OAI.PolycyclicRecognition.NonabelianFactorSystem.gaugeEquiv

gaugeGenerators def

These are the old generator coordinates in new boundary base frames: all side occurrences and BOTH seam ends change together. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.gaugeGenerators

gaugeTransport def

The complexification of the real transport expression in the profile gauge. · OAI.DefocusingNLS.gaugeTransport

gaugeWeightedLaplacian def

The complexification of the weighted scalar Laplace expression. · OAI.DefocusingNLS.gaugeWeightedLaplacian

gaussCell def

The equal-weight rule on an actual interval, with its physical weights. · OAI.ContinuumCoulomb.gaussCell

gaussian def

A Gaussian with arbitrary amplitude, exponent, and center. · OAI.BoltzmannNonuniqueness.gaussian

GaussianLipschitzVarianceInput def

The finite-dimensional Gaussian Poincaré inequality specialized to a globally Lipschitz function. The Euclidean norm is essential: a coordinatewise bound for the supremum norm would introduce a dimension factor. · OAI.InvariantIsing.GaussianLipschitzVarianceInput

gaussianMassConvolutionConstant def

The fixed three-quarter decay is compatible with the proved slow inverse growth, and dominates every Gaussian shell coefficient. · OAI.RegularPointCoordinates.gaussianMassConvolutionConstant

gaussianOffDiagonalEnvelope def

The envelope after a polynomial lower bound for small ball volume. · OAI.RegularPointCoordinates.gaussianOffDiagonalEnvelope

gaussianOperator def

Gaussian logarithmic integration, with ordinary expectation at exponent zero. · OAI.InvariantIsing.gaussianOperator

gaussianRadial def

The actual one-dimensional radial Gaussian moment. · OAI.PettyProjection.gaussianRadial

gaussianRotate def

Polarization on one independent standard Gaussian pair. · OAI.IndependentProducts.gaussianRotate

gaussianScale def

The canonical coefficient action, with its group-algebra result type fixed. · OAI.CirculantHadamard.BinaryEvaluation.gaussianScale

gaussianSchur def

The Schur action obtained by one Gaussian average. · OAI.SharpLiebThirring.MatrixProof.gaussianSchur

gaussianShellFactor def

The first shell contains the pole; subsequent shells have a Gaussian factor at their inner radius. · OAI.RegularPointCoordinates.gaussianShellFactor

gaussianSquareLaw def

The law of the square of a standard real Gaussian. · OAI.IndependentProducts.gaussianSquareLaw

gaussianTail def

Gaussian-tail probability, in real rather than extended-nonnegative-real notation. · OAI.IndependentProducts.gaussianTail

gaussianTailEnergy def

Energy beyond a finite set of Fourier frequencies. · OAI.DefocusingNLS.gaussianTailEnergy

gaussianVolumeMomentConstant def

The local Gaussian volume sum has finite total mass in every fixed dimension. It will control bounded-density curvature sources. · OAI.RegularPointCoordinates.gaussianVolumeMomentConstant

gaussianWardCorrection def

The exact correction obtained from inserting the derivative of a Gaussian Hamiltonian into a normalized finite Gibbs average. The kernel need not be symmetric: its first endpoint is the differentiated one. · OAI.InvariantIsing.gaussianWardCorrection

gaussNode def

The two normalized abscissae, indexed for the tensor-product rule. · OAI.ContinuumCoulomb.gaussNode

gaussPoint def

The actual support point, reconstructed by the derivative of the support value. · OAI.AffineBernstein.gaussPoint

Gear structure

Particular indexing programs are proved uniformly over all f through this finite syntax interface. · OAI.PowerSaving.RAM.Gear

generalCasimirCurvatureForm def

The same local Casimir expression accepts complete Weyl matrices. · OAI.PositiveEinsteinFour.HighestHessian.generalCasimirCurvatureForm

generalCoefficient def

Both matrix indices are kept in the actual curvature coefficient. · OAI.PositiveEinsteinFour.CurvatureContraction.generalCoefficient

generalCoefficientMap def

The coefficient is a fixed bounded linear function of the Weyl block. · OAI.PositiveEinsteinFour.Geometry.generalCoefficientMap

generalCurvaturePairing def

The polarized arbitrary-matrix curvature operator. · OAI.PositiveEinsteinFour.HighestHessian.generalCurvaturePairing

generalDerivativeSlot def

The two actual derivative slots, before taking a trace-free coefficient. · OAI.PositiveEinsteinFour.HighestHessian.generalDerivativeSlot

generalMatrixAction def

The actual trace-free anticommutator, acting on the two matrix indices. · OAI.PositiveEinsteinFour.Bianchi.generalMatrixAction

generalMatrixActionSquare def

The operator on a matrix obtained by contracting two infinitesimal matrix rotations with the full Weyl coefficient. · OAI.PositiveEinsteinFour.HighestHessian.generalMatrixActionSquare

generalMixedAction def

Mixed derivative/matrix action, retaining the full matrix of coefficients. · OAI.PositiveEinsteinFour.Bianchi.generalMixedAction

generalOriginalCurvatureForm def

The actual original curvature form, with the same contractions as the four geometric Green identities. · OAI.PositiveEinsteinFour.HighestHessian.generalOriginalCurvatureForm

generalRawCurvatureCommutator def

The two raw curvature actions in the commutator of a covariant Weyl gradient, with an arbitrary matrix in the Weyl bundle slot. · OAI.PositiveEinsteinFour.CurvatureContraction.generalRawCurvatureCommutator

Generated def

Generation at a stalk by genuine global sections. · OAI.KLInvariance.MomentGraph.Sheaf.Generated

Generated inductive

The actual family generated using finitely many constructors. · OAI.ReflexiveFixedPoints.Generated

generated def

The finitely generated subgroup, rather than the whole unrestricted wreath product. · OAI.FiniteAlgebraicEnvelopes.GroupInput.generated

generatedAlgebra def

The actual unital closed C*-subalgebra generated by the specified set. · OAI.Kaplansky.generatedAlgebra

generatedChartEquiv def

A literal Rees-Proj affine chart is the usual algebra of section fractions. · OAI.ReesProj.generatedChartEquiv

generatedDiagonal def

The norm-closed algebra generated by the diagonal projections. · OAI.KadisonKastler.generatedDiagonal

generatedDihedralAlgebra def

The finite dihedral von Neumann algebra, expressed through its bounded L² closure; no normal-form density is imposed as an extra premise. · OAI.KadisonSimilarity.TracialCornerGeometry.generatedDihedralAlgebra

generatedEmbeddingMatrices def

The matrix tower lives on the same native GNS space as the recovered operator, and on the exact range of the embedded free factor. · OAI.KadisonSimilarity.generatedEmbeddingMatrices

generatedFiniteFactorModel def

No matrix presentation, matrix-size divergence, or source-trace properties are assumed separately from the constructed diffuse subfactor. · OAI.KadisonSimilarity.generatedFiniteFactorModel

generatedGNSCompression def

Compress an ambient operator to the intermediate generated algebra. · OAI.KadisonSimilarity.generatedGNSCompression

generatedGNSInclusion def

The source embeds in the intermediate algebra's GNS space. · OAI.KadisonSimilarity.generatedGNSInclusion

generatedModification def

A presentation by an arbitrary chosen set of generators of the center. · OAI.StableCoordinate.ModificationPieces.generatedModification

GeneratedSelfMap abbrev

A genuine generated selfmap, carrying the proof of its origin. · OAI.ReflexiveFixedPoints.GeneratedSelfMap

generatedTestAlgebra def

The auxiliary algebra generated by the second factor and the test symmetry. · OAI.KadisonSimilarity.TracialCornerGeometry.generatedTestAlgebra

generatedUltrapowerOperator def

The concrete operator after Hilbert-ultrapower and generated-algebra compression. No operator on the generated algebra is chosen independently. · OAI.KadisonSimilarity.generatedUltrapowerOperator

generatedVonNeumannAlgebra def

The actual von Neumann algebra generated by a unital star algebra. · OAI.KadisonKastler.generatedVonNeumannAlgebra

Generation def

Generation is stated up to conversion, without any functionality premise. · OAI.PureTypeSystem.HasType.Generation

Generation def

Generation is stated up to conversion, without any functionality premise. · OAI.PureTypeSystem.Labelled.HasType.Generation

Generator inductive

Every frame and every transport has its own occurrence index. Handle generators are retained even on closed or multiply incident facets. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.Generator

Generator abbrev

Radial edges except for the one edge collapsed at the root of each component. · OAI.TorsionFreeZeroDivisors.GraphConePresentation.Generator

generator def

The basis element at the distinguished generator of the cyclic group. · OAI.CirculantHadamard.CyclicPolynomial.generator

generator def

The distinguished root in its integral order. · OAI.CirculantHadamard.CyclotomicLocalBase.generator

generator def

The distinguished generator e of the principal left ideal. · OAI.ExternalTensor.generator

generator def

The canonical group generators are unitaries in the full completion. · OAI.NuclearUltrapower.FullGroupModel.generator

generator def

The new generator represented by a letter, with terminal identity. · OAI.PartitionConsistency.ProbeAlgebra.Word.generator

generator def

The standard skew generator in the i,j plane. · OAI.PettyProjection.Spherical.generator

generator def

The coordinate images in a principal quotient. · OAI.StableCoordinate.HomogeneousQuotientWeight.generator

generator def

The original diagram cells as quotient cells. · OAI.Thomason.Construction.StrictOmegaCategory.ColimitConstruction.generator

generatorAdjointWord def

A finite product of actual operator exponentials of generator adjoints. · OAI.PolycyclicRecognition.generatorAdjointWord

GeneratorBudgetContract def

lem:generators: one budget and one increasing family for the entire ℤ path. · OAI.MatroidProphet.GeneratorBudgetContract

generatorCoefficients def

The coefficient part of the five canonical Laurent generator images. · OAI.StableCoordinate.SourceLocalizedDerivation.generatorCoefficients

GeneratorDivides def

An exponent pair divisible by one of the listed monomial generators. · OAI.Nagata.Workers.W30.GeneratorDivides

generatorGroupWord def

A finite word using only the specified generator exponentials. · OAI.PolycyclicRecognition.generatorGroupWord

generatorOrbitSpan def

The linear span of all conjugated generator directions. · OAI.PolycyclicRecognition.generatorOrbitSpan

generators def

Only the two cone objects are generators. · OAI.LittleFinitistic.TwoConeVerdier.ConePair.generators

generatorSchemeWord def

Fixed conjugating words with one variable generator inserted in each. · OAI.PolycyclicRecognition.generatorSchemeWord

generatorWord def

The original edge is based using the fixed two-edge spanning tree. · OAI.Release075.BlockPresentation.generatorWord

GenericFilter abbrev

A forcing filter is an order ideal for the opposite order. · OAI.PartitionConsistency.ForcingQuotient.GenericFilter

genericityError def

Strictly positive genericity error meeting the MaxCutGames.Gadget.Harmonic.harmonic bad-rank budget. · OAI.MaxCutGames.Gadget.Parameters.genericityError

genericityError def

Strictly positive genericity error meeting the harmonic bad-rank budget. · OAI.UniqueGamesTheorem.Gadget.Parameters.genericityError

genericMatrix def

The actual rational-coordinate matrix in any faithful polynomial realization. · OAI.EuclideanRamsey.NineRational.genericMatrix

genericPoint def

A nonzero positive homogeneous element makes (0) a genuine point of Proj. · OAI.ProjDomain.genericPoint

genericTable def

No row is recomputed, reordered, or selected by an abstract enumeration. · OAI.MaxCutGames.Foundations.Complexity.FinalCNFTableAdapter.genericTable

genericTable def

No row is recomputed, reordered, or selected by an abstract enumeration. · OAI.UniqueGamesTheorem.Foundations.Complexity.FinalCNFTableAdapter.genericTable

genuineComputedTensor def

The actual all-integer tensor of genuine Q_T with O, retaining both outside B-actions. · OAI.LittleFinitistic.OrdinarySimulation.genuineComputedTensor

genuineComputedTensorIso def

Agreement with the preserved full signed-comparison input in every integer degree. · OAI.LittleFinitistic.OrdinarySimulation.genuineComputedTensorIso

genuineFiniteComplexIso def

The actual finite bimodule tensor agrees with the existing full computation, preserving both B-actions. · OAI.LittleFinitistic.OrdinarySimulation.genuineFiniteComplexIso

genuineSourceBasis def

The full actual section basis indexed by all literal manuscript columns. · OAI.Nagata.Workers.W12.genuineSourceBasis

genuineSourceJetMap def

The actual linear jet map for the genuine source sections. · OAI.Nagata.Workers.W12.genuineSourceJetMap

genuineSourceSections def

The genuine global section space at the actual nine marked ray lifts. · OAI.Nagata.Workers.W12.genuineSourceSections

genuineSourceShiftComparison def

The actual ordinary tensor Q_T tensor O is isomorphic to P[b] as B-bimodule complexes. Only the manuscript support and indexing hypotheses are used. · OAI.LittleFinitistic.OrdinarySimulation.genuineSourceShiftComparison

GenusZero def

Combinatorial genus zero, expressed directly on the actual side-pairing. This definition neither asserts nor replaces a topological realization. · OAI.HyperbolicObstruction.PictureSwitch.PictureMap.GenusZero

GenusZero def

Euler equality componentwise, i.e., genus zero for all components. · OAI.KaplanskyCounterexample.Planar.Ribbon.GenusZero

GenusZero def

Combinatorial genus zero, including disconnected maps and isolated darts. This is a definition in permutation data, not an asserted sphere embedding. · OAI.TorsionFreeZeroDivisors.MapEuler.GenusZero

Geodesic def

A shortest edge path in the unit-edge graph. · OAI.Release075.Geodesic

geodesicHilbertFlow def

The actual unitary geodesic dynamics on the oriented double. · OAI.TriangularBilliards.geodesicHilbertFlow

GeodesicMetricExistenceInput def

The implication (i)⇒(ii) of Cornulier–de la Harpe (2016), Proposition 4.B.9, with the chain witnesses retained. · OAI.PolycyclicRecognition.GeodesicMetricExistenceInput

GeometricAction def

Proper and cocompact isometric action. Properness of the metric space and nonemptiness are separate standing hypotheses in the theorem below. · OAI.ArtinCAT0.GeometricAction

GeometricAction structure

A proper, cocompact isometric action of a discrete group. · OAI.HyperbolicObstruction.GeometricAction

geometricAction def

The genuine geometric reflection action. Each relation has been proved; this definition does not presuppose a faithful representation. · OAI.KLInvariance.Realization.geometricAction

geometricBound def

A total geometric majorant. Resonant frequencies use the trivial bound; the definition never interprets division by zero as cancellation. · OAI.Problem337.VaughanBilinear.geometricBound

geometricBoxIntegrand def

The lag is displayed separately from the geometric endpoint scale. · OAI.JointDickman.geometricBoxIntegrand

geometricExpansionValue def

The corresponding product-geometric factorial-moment expression. · OAI.Problem335.geometricExpansionValue

geometricFactorialMonomial def

Independent geometric factorial moments, with a possibly different mean per coordinate. · OAI.Problem335.geometricFactorialMonomial

GeometricFastFamily def

Quantitative finite-accuracy data supplied by the actual primitive construction. · OAI.ClosedSurfaceR4.FiniteOrderSmoothing.SmoothingAtlas.GeometricFastFamily

GeometricFastFamilyWithProperty def

Retain a proved property of the actual finite-accuracy approximant. · OAI.ClosedSurfaceR4.FiniteOrderSmoothing.SmoothingAtlas.GeometricFastFamilyWithProperty

geometricIntervalConstant def

The constant controlling intervals with exponentially decaying lags. · OAI.RegularPointCoordinates.geometricIntervalConstant

geometricKey def

The literal box label and integer squared-norm band. · OAI.QuantitativeVanDerWaerden.geometricKey

geometricMean def

Product-geometric evaluation of a nonnegative factorial expansion. · OAI.Problem335.FactorialExpansion.geometricMean

geometricPowers def

A summable family of geometricPowers of a Hahn series x. If x has non-positive orderTop, then return a junk value given by pretending x = 0. · OAI.RationalHitting.HahnGeometric.geometricPowers

geometricRadii def

Global geometric extension of the finite descending radius grid. · OAI.ActualLabels.geometricRadii

geometricRadii def

Global geometric extension of the finite descending radius grid. · OAI.SignedDisk.ActualLabels.geometricRadii

GeometricTrack structure

Physical, unnormalized data for a single actual root-to-endpoint cache. The probabilistic tests are discharged from the constructed ensemble below. · OAI.BoundedTreewidthL1.GeometricTrack

geometricValue def

The finite value to which occupation comparison bounds an expansion. · OAI.Problem335.FactorialExpansion.geometricValue

geometricWeight def

Product of the geometric means of the paired longitudinal derivatives. · OAI.Problem358.TriangularDensity.geometricWeight

Geometry structure

The pointwise geometry used by the periodic coefficient recursion. · OAI.ClosedSurfaceR4.PeriodicExpansion.Geometry

geometry def

Pulling back through the actual second jet supplies the differentiated constraint Y_y=C, rather than adding that constraint as a new input. · OAI.ClosedSurfaceR4.SurfaceVelocityFamily.Loop.geometry

GeometryScales structure

All elementary small-mesh inequalities used in the geometric dichotomy. · OAI.QuantitativeVanDerWaerden.Parameters.GeometryScales

getPair def

For complete generality this returns zero on pairs with equal tags. · OAI.HadwigerCounterexample.getPair

ghpEDist def

The common-isometric-embedding definition of the source GHP distance. · OAI.FKCRT.ghpEDist

giantPairPhase def

The two cofactor phases before inserting the logarithmic coordinates. · OAI.Ostmann.giantPairPhase

giantProgressionCutoff def

The giant comparison includes internal prime squares; the external bulk comparison keeps bulkProgressionCutoff L. · OAI.Ostmann.giantProgressionCutoff

giantReciprocalFactor def

Reuse the existing reciprocal factor after scaling the polynomial by its lower cell endpoint. The variation budget is independent of the center. · OAI.Ostmann.giantReciprocalFactor

GiantReversal structure

The data retained at one ancestor substitution. · OAI.Ostmann.GiantReversal

GiantRows structure

Coefficients of the two current giants in the two top giants. · OAI.Ostmann.GiantRows

gibbs def

The Gibbs probability law, normalized by its partition function. · OAI.LogConcaveSampling.gibbs

gibbsAverage def

Finite normalized Gibbs expectation, for an arbitrary nonzero positive prior. · OAI.InvariantIsing.gibbsAverage

gibbsDensity def

The unnormalized Gibbs density relative to Euclidean volume. · OAI.LogConcaveSampling.gibbsDensity

gibbsKernelCorrection def

The Gaussian correction written in terms of a contracted observable and kernel. Its three arguments retain the observable and both endpoints. · OAI.InvariantIsing.gibbsKernelCorrection

gibbsPairAverage def

Independent replicas sampled from the same finite Gibbs law. · OAI.InvariantIsing.gibbsPairAverage

gInv def

The inverse of increasing g : [0,∞) → [0,∞), expressed by its lower level cut. Only its values on [0,∞) occur in the main theorem. · OAI.EntropyPhotonNumber.gInv

givenAction def

The presented algebra acts on Y by precisely its specified generator endomorphisms. · OAI.LittleFinitistic.QuadraticEncoding.givenAction

Gives def

Access condition with a numeric allowance of cells (bits accounting of TM.Within). · OAI.Fifths.Single.Gives

glimpse def

local output from packed data at a sector address. · OAI.PowerSaving.Grove.glimpse

global def

Evaluate an sheaf derivation on the global coordinate functions. · OAI.AffineDerivation.global

GlobalAdmissible def

One condition for the whole actual parent/child family at a group element. · OAI.ThompsonNonamenability.GlobalAdmissible

globalCoefficients def

Retain the actual coefficient values when a supported local family has already been proved smooth after extension by zero. · OAI.ClosedSurfaceR4.LocalPeriodicExpansion.globalCoefficients

globalCorrelation def

A real correlation on all of the actual group. · OAI.ThompsonNonamenability.globalCorrelation

globalGaussianShellBound def

The pole volume normalizes a kernel bound on any dyadic shell, even outside an almost Euclidean region. · OAI.RegularPointCoordinates.globalGaussianShellBound

globalIndex def

Input position in the actual next table, with port varying fastest. · OAI.MaxCutGames.Foundations.PCP.ExpanderTableEnumeration.globalIndex

globalIndex def

Input position in the actual next table, with port varying fastest. · OAI.UniqueGamesTheorem.Foundations.PCP.ExpanderTableEnumeration.globalIndex

globalize def

Attach the actual endpoints and edge identifier to a local raw address. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Queries.globalize

globalize def

Attach the actual endpoints and edge identifier to a local raw address. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Queries.globalize

GlobalKey abbrev

Distinct table vertices and the three side tags never share storage. Unused ambient query positions are harmless explicit padding. · OAI.MaxCutGames.Foundations.Hastad.SourceOccurrences.GlobalKey

GlobalKey abbrev

Distinct table vertices and the three side tags never share storage. Unused ambient query positions are harmless explicit padding. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceOccurrences.GlobalKey

GlobalMinimizer def

Nonnegative global minimizer, with all centers and all positive radii. · OAI.Bernoulli.GlobalMinimizer

GlobalPlaneScalarSolution def

Bounded smooth solutions on every finite closed time cylinder. · OAI.ForcedComputation.VelocityDetector.GlobalPlaneScalarSolution

globalRestrictionFiberEquiv def

The actual zero-extended fibers of the restricted and joint representations are identified by the identity on ambient vectors. · OAI.KadisonSimilarity.globalRestrictionFiberEquiv

globalSlot def

One angular choice for the ENTIRE infinite history. At all diagonal legs it is the distinguished background choice none. · OAI.HarmonicCounterexample.AngularStream.globalSlot

globalTapeEquiv def

Exact arithmetic numbering of every physical global tape. · OAI.MaxCutGames.Foundations.Complexity.PoweringGlobalEnumeration.globalTapeEquiv

globalTapeEquiv def

Exact arithmetic numbering of every physical global tape. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringGlobalEnumeration.globalTapeEquiv

GlobalTorusScalarSolution def

Smoothness and the equation on every finite closed time cylinder. · OAI.ForcedComputation.VelocityDetector.GlobalTorusScalarSolution

globalTrialEquiv def

The literal global schedule is indexed by a fixed pair/block bijection. · OAI.TraceReconstruction.globalTrialEquiv

globalTrialIndex def

One actual global-pair orientation, ordered by its pre-region block and its local identifier there. · OAI.TraceReconstruction.globalTrialIndex

globeCoyonedaIso def

The free globe represents this bounded-cell functor, naturally. · OAI.Thomason.Construction.StrictOmegaCategory.FilteredColimit.globeCoyonedaIso

globeHomEquiv def

Representability on every free globe. · OAI.Thomason.Construction.StrictOmegaCategory.GrayMapping.globeHomEquiv

globeIso def

The strict globe and its positive-table realization are isomorphic. · OAI.Thomason.Construction.DirectedGlobeComplex.globeIso

globeModelIso def

Restriction of the global extension to globes agrees naturally with its original chain model. · OAI.Thomason.Construction.StrictOmegaCategory.GrayChainExtension.globeModelIso

globeRestrictionIso def

The extended cylinder has the concrete signed right-cylinder values and maps on the actual finite globes. · OAI.Thomason.Construction.StrictOmegaCategory.GrayChainExtension.globeRestrictionIso

globes def

Keep the globe objects and all of their strict maps in the dense presentation. · OAI.Thomason.Construction.StrictOmegaCategory.GlobePasting.globes

globesInclusionIso def

The restricted inclusion is exactly the original globe inclusion. · OAI.Thomason.Construction.StrictOmegaCategory.GlobePasting.globesInclusionIso

globeToAbelianization def

The universal top cell supplies the reverse positive chain map. · OAI.Thomason.Construction.DirectedGlobeComplex.globeToAbelianization

globeUnitIso def

The unit comparison is an actual strict isomorphism on each globe. · OAI.Thomason.Construction.DirectedGlobeComplex.globeUnitIso

globeUnitNerveIso def

The unit induces a literal isomorphism of the actual Street nerves. · OAI.Thomason.Construction.DirectedGlobeComplex.globeUnitNerveIso

gloved def

The contour order is an invariant of the effective graph. · OAI.ExactDerandomization.Sub.Quill.gloved

glow def

Unit linearization of an output triple · OAI.FourierLLogL.Dyadic.Procedure.Paths.Beam.glow

GLPS1988Proposition3 def

A sequential specialization of GLPS (1988), Proposition 3, with the evolution-equation extension in Remark (2), p. 119. Boundedness and uniform integrability are both contained in UniformIntegrable. The hypothesis has no relation to the … · OAI.BoltzmannNonuniqueness.Published.GLPS1988Proposition3

glue def

Prefix a (possibly empty) fragment to the first sector, creating a sector only when the fragment is nonempty. · OAI.FiniteAlgebraicEnvelopes.Compiler.Path.glue

glue def

Gluing is with the actual source topology, not a replacement quotient. · OAI.HyperbolicObstruction.ClosedCoverGluing.glue

glue def

Identifying two classes, without making any other identifications. · OAI.HyperbolicObstruction.PictureSwitch.glue

glue def

Sample values on the sacrificed mask, and realized values elsewhere. · OAI.MatroidProphet.General.glue

glue def

Glue local potentials on disjoint components, with value zero outside them. · OAI.Problem356.ComponentPotential.glue

glue def

Glue the actual maps on the two cylinder pieces. · OAI.Thomason.Construction.StrictOmegaCategory.GrayMapping.glue

glueCross def

The cross block implementing gluing of forms on two subspaces. Its restrictions to the common subspace agree with the prescribed forms. · OAI.SidorenkoCounterexample.glueCross

Glued def

The source glued component. · OAI.PartitionConsistency.LocalDiagramSystem.System.Glued

gluedEquiv def

A component equivalence obtained through a root fiber. The independent proof below shows that it agrees with transport at every other vertex. · OAI.PartitionConsistency.LocalDiagramSystem.ReindexIso.gluedEquiv

gluedFamily def

Union of the shifted local families, with levels numbered 0, ..., N. · OAI.Problem326.Affine.gluedFamily

gluedProfile def

Glue the row's two actual profiles using their agreement on overlap. · OAI.PolycyclicRecognition.CartanWordModel.AffineNeighborRow.gluedProfile

glueFaceHomotopies def

The actual global continuous prism. · OAI.HyperbolicObstruction.FinitePolyhedron.Voltage.glueFaceHomotopies

glueGraph def

Keep the two copies of the interface separate and join them by edges. This subdivision avoids parallel edges without changing boundary paths. · OAI.OneWayLiveness.Pairing.glueGraph

glueNeighborProfile def

Retain the first profile on the first cell and the transported second profile elsewhere. Compatibility makes the values agree on the overlap. · OAI.PolycyclicRecognition.glueNeighborProfile

gluePrism def

A homotopy prescribed continuously on each closed cell glues once the literal formulas coincide on intersections. · OAI.HyperbolicObstruction.ClosedCoverGluing.gluePrism

glueWeights def

The two-part conditional product, with zero denominator interpreted as zero. Only compatible pairs receive mass. · OAI.BoundedTreewidthL1.glueWeights

gnsHilbert def

The usual GNS Hilbert space, with its existing canonical structures. · OAI.WeakPureInfiniteness.gnsHilbert

gnsLeftSequence def

The actual uniformly bounded family of left multipliers. · OAI.KadisonSimilarity.gnsLeftSequence

gnsTracialGeometry def

The concrete tracial geometry is obtained directly from the GNS construction, rather than assumed as separate representation data. · OAI.KadisonSimilarity.gnsTracialGeometry

gnsTracialHomIsometry def

The GNS isometry induced by a trace-preserving star homomorphism. · OAI.KadisonSimilarity.gnsTracialHomIsometry

gnsTracialVector def

The canonical algebra vector in the completed GNS space. · OAI.KadisonSimilarity.gnsTracialVector

goldbachSieveDensity def

The exact finite local root density, normalized as an arithmetic function. · OAI.Problem337.goldbachSieveDensity

GoldbachSieveRoot def

Residue classes removed by the binary Goldbach sieve. · OAI.Problem337.GoldbachSieveRoot

goldbachSieveRootCount def

Local number of roots of x * (u - x) modulo d. · OAI.Problem337.goldbachSieveRootCount

goldbachSieveRootEquiv def

The Chinese remainder theorem separates the local root conditions. · OAI.Problem337.goldbachSieveRootEquiv

goldbachSieveRootLCMEmbedding def

Reduction modulo two possibly noncoprime moduli is injective when the source modulus is their least common multiple. This remains true on roots. · OAI.Problem337.goldbachSieveRootLCMEmbedding

goLeft def

gray positive step at a digit: 'up' = increase overall number, 'rev' = digit reversed · OAI.ExactDerandomization.Counter.goLeft

Good def

Closed sector between the literal two palette axes. · OAI.ConeCoordinates.Good

Good structure

Circuits are specification-only objects prior to uniformity hypotheses. · OAI.Fifths.Circuit.Good

Good structure

Full-tape agreement (padding by blank), plus inclusion of both lists in the allowed tape window. · OAI.Fifths.WorkSplit.Good

Good def

The source good-rank condition for this literal collection shape. · OAI.HyperbolicObstruction.IncidenceTemplate.Occurrences.Shape.Good

Good def

Goodness is evaluated on the fixed input tree, using local nonfailure target masses before any pruning or reweighting. · OAI.LeanBlast.SecondKahnKalai.Good

Good def

The accumulated product on every prefix reached by a path state. · OAI.OneWayLiveness.PathNFA.Good

Good def

A good condition actually decides a distinguishing pattern; arbitrary unassigned row values are never used for a stable selection. · OAI.PartitionConsistency.TailSelector.TailData.Good

Good def

A center triple belongs to one of the two intended contact families. · OAI.Problem356.CenterCertificate.Good

Good def

Closed sector between the literal two palette axes. · OAI.SignedDisk.ConeCoordinates.Good

Good def

A finite set of forward chord inequalities. Centered integer intervals are precisely these sets pulled back by the negative iterate of a homeomorphism. · OAI.SmoothObstruction.GoodStationary.Good

Good inductive

Admissibility uses only boundary equations forced by the diagram. · OAI.Thomason.Construction.StrictOmegaCategory.ColimitConstruction.Good

Good structure

The manuscript's deterministic hypotheses for the bounded first moment. Every condition concerns only the unlabeled graph, paths or comparisons. · OAI.TorsionFreeZeroDivisors.BoundedPattern.Data.Good

Good def

An actual distinct-arc digon gap, read solely from the two permutations. · OAI.TorsionFreeZeroDivisors.PlanarRotation.Good

Good def

Validity as a chain of actual nonbacktracking directed edges. · OAI.TorsionFreeZeroDivisors.WeightedPortMoore.Good

good def

Positive cube-free integers up to N. · OAI.Problem337.CubeFreeDensity.good

GoodArc def

The entire increasing good arc, including its relation to the lifted support interval. The excluded length is exactly 2*δ. · OAI.ThreeTorus.PassagePackets.GoodArc

GoodAt structure

Full endpoint property for a total system. · OAI.SeparableQuotient.CoherentClosures.GoodAt

GoodBound def

Good-set estimate. We state it at exponential heights exp d with d = 2^j. · OAI.FourierLLogL.GoodBound

GoodCell def

The exact cell predicate needed by transported_guard_frequency. · OAI.Problem348.Construction.HostSampling.GoodCell

GoodCollisionBlock structure

The arithmetic data of a good collision in a single counting block. The positive-index block endpoints will be supplied by dyadic subdivision. · OAI.TotientAsymptotic.GoodCollisionBlock

goodCone def

The actual normalized positive-cone part on one cell. · OAI.CommonPrices.goodCone

goodCone def

The actual normalized positive-cone part on one cell. · OAI.SignedDisk.CommonPrices.goodCone

goodCosetMass def

The exact good Fourier mass in a witness coset; unlike the response law, this analysis-only quantity is allowed to mention the witness. · OAI.MaxCutGames.Decoder.PrivateStrategy.goodCosetMass

goodCosetMass def

The exact good Fourier mass in a witness coset; unlike the response law, this analysis-only quantity is allowed to mention the witness. · OAI.UniqueGamesTheorem.Decoder.PrivateStrategy.goodCosetMass

GoodDimension def

The universal property used in the definition of d_p(n,D). · OAI.SubpolynomialLp.GoodDimension

GoodEndpoint def

The same leading minor is used at the old and perturbed point. · OAI.SmoothObstruction.EndpointFrame.GoodEndpoint

goodExcess def

The positive excess on executions having a good seed. · OAI.EditApproximation.goodExcess

goodExpansion def

Combined centered expansion, with the actual same variable on a shared line. · OAI.CoordinateSweeps.MomentExpansion.goodExpansion

GoodExposure def

The good atoms are those whose (equivalently, every) center route has a default. · OAI.Problem310.CenterRouteExposure.GoodExposure

goodFirstIndicator def

Indicator of the first accepted proposal passing a second test. · OAI.MaxCutGames.Foundations.CorrelatedSampling.goodFirstIndicator

goodFirstIndicator def

Indicator of the first accepted proposal passing a second test. · OAI.UniqueGamesTheorem.Foundations.CorrelatedSampling.goodFirstIndicator

goodFirstMass def

Probability that the first accepted proposal satisfies an additional test. For two threshold samplers, take accept = left || right and good = left && right. first_union_common gives deterministic agreement on this event. · OAI.MaxCutGames.Foundations.CorrelatedSampling.goodFirstMass

goodFirstMass def

Probability that the first accepted proposal satisfies an additional test. For two threshold samplers, take accept = left || right and good = left && right. first_union_common gives deterministic agreement on this event. · OAI.UniqueGamesTheorem.Foundations.CorrelatedSampling.goodFirstMass

goodFraction def

The fraction of indices whose nonnegative error is below a threshold. · OAI.EuclideanFiveColor.AverageErrors.goodFraction

GoodLaplacianCutoff structure

An actual Sobolev cutoff with bounded weak Laplacian. · OAI.RegularPointCoordinates.GoodLaplacianCutoff

goodMesh def

A finite measurable subset of the good set. Sampling one extra endpoint keeps all sample times simple multiples of the mesh width. · OAI.TriangularBilliards.goodMesh

goodNumeratorAux def

Finite backwards construction, indexed first by the number of remaining steps so that its recursion is structurally terminating. · OAI.Problem337.goodNumeratorAux

GoodOutput def

`GoodOutput C x bits` requires a `TreeOutput` and additionally requires `bits = []` whenever the input has exactly one vertex. · OAI.AlgorithmicThinTrees.GoodOutput

goodPairRoot def

Roots in a negative pair which are both positive before the ascent. · OAI.KLInvariance.TitsSpace.goodPairRoot

goodParameters def

Actual good pairs in the unconditioned product sample space. · OAI.MaxCutGames.Inverse.KMSBasisComparison.goodParameters

goodParameters def

Actual good pairs in the unconditioned product sample space. · OAI.UniqueGamesTheorem.Inverse.KMSBasisComparison.goodParameters

goodParametersEquiv def

Equivalence of the two explicit presentations of the good sampling event. · OAI.MaxCutGames.Inverse.KMSBasisComparison.goodParametersEquiv

goodParametersEquiv def

Equivalence of the two explicit presentations of the good sampling event. · OAI.UniqueGamesTheorem.Inverse.KMSBasisComparison.goodParametersEquiv

goodPatternGain def

The signed gain can be chosen before the local quartet tolerance. This amount absorbs every original pattern and paired-frequency count. · OAI.Ostmann.goodPatternGain

GoodPhaseChart structure

All geometric data of one compactly bounded local phase chart. · OAI.ClosedSurfaceR4.PhaseGeometry.GoodPhaseChart

goodPredicate def

The generic selector predicate is now the graph of the actual normalized row-advice selector, rather than an unspecified class of candidate witnesses. · OAI.MaxCutGames.Decoder.SelectedPolicies.goodPredicate

goodPredicate def

The generic selector predicate is now the graph of the actual normalized row-advice selector, rather than an unspecified class of candidate witnesses. · OAI.UniqueGamesTheorem.Decoder.SelectedPolicies.goodPredicate

GoodReplacement def

Goodness assumptions on the values grafted at declarations. · OAI.PureTypeSystem.ObservationValues.GoodReplacement

GoodResource def

The resource event is determined before the extra prime, prefixes, or separating families. Counts are taken on raw occurrences or actual entry keys. · OAI.LowSpaceSubsetSum.RepresentationMoments.GoodResource

goodSections def

Positive, finite fibers form a measurable parameter set. · OAI.Problem358.NormalizedSections.goodSections

goodSetExtension def

The finite McShane extension, clamped at the chosen upper bound. · OAI.JointDickman.goodSetExtension

GoodSets def

The exact recursive good-set specification needed for both effects of the construction: closure counts bad residues, and decomposition propagates lists. · OAI.Problem337.DensityResidueFamily.GoodSets

GoodSpacing def

A single universal integer spacing. It is independent of the graph, its edge lengths, the number of scales, and every finite random grid. · OAI.PlanarL1.Parameters.GoodSpacing

goodSplitDivisors def

Proper divisors whose two new denominators avoid all fixed denominators. · OAI.Problem337.goodSplitDivisors

GoodTable def

A successful table has all the finite mathematical certificates needed by the bounded Newman argument. No totality of the program is part of this. · OAI.FiniteAlgebraicEnvelopes.Bootstrap.GoodTable

GoodTransport structure

Exact tree transport at a fixed root expectation. Only good observations are equated; raw environments and values are otherwise unrestricted. · OAI.PureTypeSystem.ObservationValues.GoodTransport

goodUnion def

A matrix belongs to the erased union exactly when at least one row map provides good advice there. · OAI.MaxCutGames.Inverse.RowErasure.goodUnion

goodUnion def

A matrix belongs to the erased union exactly when at least one row map provides good advice there. · OAI.UniqueGamesTheorem.Inverse.RowErasure.goodUnion

GoodUpTo def

The exact source diagram:rank conclusion at any fixed finite size. · OAI.HyperbolicObstruction.IncidenceTemplate.Occurrences.GoodUpTo

GoodVertex def

`GoodVertex r v` is precisely the conjectured cardinal inequality `|N₁⁺(v)| ≤ |N₂⁺(v)|`. It does not require either neighborhood to be nonempty, so every sink is automatically good. · OAI.SeymourSecondNeighborhood.GoodVertex

GoodWitnessConditions def

The four conditions in extraction.tex, equation good-conditions. The coordinates and scales are those of the common original endpoint x. · OAI.TotientAsymptotic.GoodWitnessConditions

goStutter def

Uniform form of step: give halting states self-loop actions writing the scanned bits back. · OAI.ExactDerandomization.Table.TM.goStutter

GotsmanLinialStatement def

The average-sensitivity bound for positive dimension and degree at most the dimension. · OAI.LeanBlast.GotsmanLinial.GotsmanLinialStatement

gPiece def

The degree-n submodule in the actual principal quotient G. · OAI.StableCoordinate.Obstruction.gPiece

GPV2009CarlemanInput def

The application-specific nonnegative form of the published Carleman identity. For densities the zero transverse-normal set is Lebesgue null. The statement does not assert equality for arbitrary singular incoming measures. · OAI.BoltzmannNonuniqueness.Published.GPV2009CarlemanInput

Gr abbrev

The genuine associated graded algebra, modeled as the Rees special fiber. · OAI.StableCoordinate.FilteredRees.Gr

grade0Vertex1Iso def

The actual source-zero quotient at vertex one is the first adjacent-arrow cone. · OAI.LittleFinitistic.GradedVertexAcyclic.grade0Vertex1Iso

grade1VertexIso def

The actual source-one quotient is the adjacent-arrow cone after taking vertex two. · OAI.LittleFinitistic.GradedVertexAcyclic.grade1VertexIso

gradedAlgebra def

The quotient algebra carries the induced grading, with no restriction on negative degrees or on the grading monoid. · OAI.StableCoordinate.HomogeneousQuotient.gradedAlgebra

gradedCoordinateBracket def

Retain exactly the structure constants which preserve total degree. · OAI.PolycyclicRecognition.gradedCoordinateBracket

gradedDerivation def

The induced derivation on the genuine associated graded ring. · OAI.StableCoordinate.FilteredRees.gradedDerivation

gradedGenerator def

The part of a vector whose coordinate degree equals the prescribed generator degree. · OAI.PolycyclicRecognition.gradedGenerator

gradedGeneratorLinearMap def

The degree-one part of exponential coordinates is linear in the original horizontal vector. · OAI.PolycyclicRecognition.gradedGeneratorLinearMap

GradeDiagonal def

Change of bases on the named graded fibers. · OAI.IntegralCharacterVarieties.MatrixIso.GradeDiagonal

gradedMap def

The natural ring map from the source polynomial ring into the actual associated graded algebra. · OAI.StableCoordinate.GradedPresentationMap.gradedMap

gradedPresentationMap abbrev

The actual polynomial map onto the source's associated graded algebra. · OAI.StableCoordinate.AssociatedGradedPresentation.gradedPresentationMap

gradedTensor def

The associated graded tensor, whose two branches still share the first leg. · OAI.MatrixMultiplication.AuxiliarySeparation.DeterminantFiltration.gradedTensor

gradeEquiv def

Preservation of split prefixes makes the named row and column refinement maps invertible over the coefficient ring. · OAI.IntegralCharacterVarieties.NamedBandGrades.gradeEquiv

gradeMap def

The induced map on a named graded quotient in the original coordinates. · OAI.IntegralCharacterVarieties.NamedBandGrades.gradeMap

gradient def

Cartesian gradient and Hessian, with the same coordinate order as the frozen definition of the Laplace--Beltrami operator. · OAI.HarmonicCounterexample.Berger.gradient

gradientBlock def

A fixed derivative direction in the covariant derivative of curvature. · OAI.PositiveEinsteinFour.Geometry.gradientBlock

gradientCoupling def

The coupling in the Bianchi projector (6-C)/10. · OAI.PositiveEinsteinFour.Bianchi.gradientCoupling

gradientCurve def

Orthogonal exponential is a globally defined smooth deformation of K. · OAI.ArnoldCounterexample.CriticalBound.gradientCurve

gradientDivergence def

This is the coordinate divergence of the matrix gradient. · OAI.PositiveEinsteinFour.Geometry.gradientDivergence

gradientEnergy def

Half the squared Euclidean gradient, expressed in the frozen orthonormal coordinates. · OAI.AffineBernstein.gradientEnergy

gradientGenerator def

The actual total plus generator on a Bianchi gradient. · OAI.PositiveEinsteinFour.Bianchi.gradientGenerator

gradientKernel def

A Cartesian heat derivative needs no additional cutoff. · OAI.ForcedComputation.PlaneScalarMild.gradientKernel

gradientL2 def

The gradient in the existing Lebesgue measurable space of FourSpace. · OAI.PositiveEinsteinFour.Geometry.ScalarEuclideanSobolev.gradientL2

gradientNorm def

The gradient L1 norm in Euclidean norm. · OAI.CenteredDiskEndpoint.gradientNorm

gradientNorm def

The gradient L1 norm in Euclidean norm. · OAI.SignedDisk.CenteredDiskEndpoint.gradientNorm

gradientOperator def

Convolution by one Cartesian derivative of the Gaussian. · OAI.ForcedComputation.PlaneHeat.gradientOperator

gradientProjection def

The polynomial expression for the orthogonal Bianchi projector. · OAI.PositiveEinsteinFour.Bianchi.gradientProjection

gradientTensor def

Reorder the derivative slot of a symmetric matrix gradient. · OAI.PositiveEinsteinFour.Geometry.gradientTensor

gradientVectorMap def

Reindexing the derivative and matrix components is continuous. · OAI.PositiveEinsteinFour.Geometry.gradientVectorMap

gradingPhase def

The spectral phase I^k of the degree grading. · OAI.LeanBlast.GotsmanLinial.gradingPhase

graftBand def

A actual gallery at the two old ends, including genuine transverse seams, is spliced by the same-q inverse-cut three-cycle. The endpoint lists are those of the OLD seam, so each original child occurrence survives in its own slot. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.PortAssembly.graftBand

graftBoundaryEnumeration def

The entire actual graft boundary, based at the original old occurrence on EVERY old circle, and at literal mirror/strip occurrences on new circles. Unlike fromPorts, this equivalence retains the old marking and circle labels. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.graftBoundaryEnumeration

graftCycle def

The SAME seam handle cut has exactly this three-cycle, not identity closure: the outer ends lengthen its old boundary, and the mirror gallery self-closes as the new inner boundary. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.graftCycle

graftLift def

Replace each leaf by the tree attached to its image. This is cabling of a bijection, defined on the actual dependent leaf sets. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Tree.graftLift

graftMap def

Whole dependent substitution of a bijection and bijections in every fibre. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Tree.graftMap

graftSideEquiv def

Sides have complete positive-port coordinates. The arity is the ACTUAL arity of that port, not merely a count inferred from endpoint names. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.graftSideEquiv

graftWiring def

Literal whole-port inverse-band wiring on the disjoint union of the old vertices and ALL band vertices. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.graftWiring

gramCharges def

The three charges produced by integrating out the shared column. · OAI.MUB6.gramCharges

gramNormalizer def

The inverse square roots of the four residual energies give an orthonormal frame, with no eigenvector selection. · OAI.PositiveEinsteinFour.Geometry.gramNormalizer

gramState def

A purification letter z is an actual one-slot substate after partial trace. · OAI.RowColumn.MatrixState.gramState

Graph structure

A nonempty, finite, simple, undirected, unweighted graph. · OAI.LargeIndependentSets.Graph

Graph structure

A finite simple unweighted graph with its full Boolean adjacency table. · OAI.OptimalMaxCut.Graph

Graph structure

One chosen orientation of each edge of a finite simple graph. · OAI.Paper170.Graph

Graph structure

An undirected rational-capacity graph. Zero capacities encode absent edges. There are no input demand weights: all distinct pairs have demand one. · OAI.UniformSparsestCut.Graph

graph def

The graph, regarded as an affine subset of R^{n+1}. · OAI.AffineBernstein.graph

graph def

`x.graph` turns every inhabitant of `x.Edge` into its own indexed edge copy and assigns it the endpoints of its pair index. · OAI.AlgorithmicThinTrees.BinaryInput.graph

graph def

The undirected replacement graph. · OAI.DiamondDistortion.Diamond.graph

graph def

The family is a recursive rotation algorithm when its fixed base is given. · OAI.MaxCutGames.Foundations.PCP.ExpanderFamily.graph

graph def

The actual constraint graph, with its equality and inherited original tests. · OAI.MaxCutGames.Foundations.PCP.Regularization.graph

graph def

Distinct vertices are adjacent when a selected label agrees. · OAI.MetricEntropyDuality.PartitionGraph.graph

graph def

graph f produces the finset of pairs (f i, i) equipped with the lexicographic order. · OAI.Nagata.Workers.W10.LocalTupleSort.graph

graph def

The actual HS name for the bounded relation on X × Y. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.BinaryRelation.graph

graph def

A nearby affine plane written as a graph in one fixed transverse direction. The denominator is nonzero precisely when the direction remains transverse. · OAI.Problem358.AffinePlaneGraph.graph

graph def

The one-skeleton in an abstract graph cone after its root radial edge is collapsed. Physical graph edges and all other radial edges are retained. · OAI.TorsionFreeZeroDivisors.ConePicture.graph

graph def

The underlying simple graph of the actual permutation map. Ignoring loops and parallel copies does not discard any adjacency between distinct vertices. · OAI.TorsionFreeZeroDivisors.MapEuler.graph

graph def

The induced nearest-neighbor graph on the vertices of a finite polygon (or, more generally, any finite set of lattice vertices). · OAI.TriangularSurface.Lattice.graph

graph def

The label tuple is numbered by its explicit base-q radix code. · OAI.UniqueGamesTheorem.Explicit.ProductOutput.graph

graph def

The family is a recursive rotation algorithm when its fixed base is given. · OAI.UniqueGamesTheorem.Foundations.PCP.ExpanderFamily.graph

graph def

The actual constraint graph, with its equality and inherited original tests. · OAI.UniqueGamesTheorem.Foundations.PCP.Regularization.graph

graphAffineFunction def

Restriction of an arbitrary ambient affine function to the original graph. · OAI.AffineBernstein.graphAffineFunction

graphAlgebra def

All label-decreasing maps, made into a finite list of unary total operations. · OAI.FiniteCongruence.graphAlgebra

graphBits def

Full explicit adjacency matrix, row-major, with binary vertex count. · OAI.LargeIndependentSets.graphBits

graphBundleCoordinates def

Coordinates on the ORIGINAL hyperplane-sum total space restricted to any subset of an actual graph chart. The fiber is evaluated on an explicit nonvanishing graph representative. · OAI.Kaplansky.graphBundleCoordinates

graphBundleNormalization def

Local normalization in integral fiber homology of the actual bundle pair. Choice of graph frame and gluing are proved separately. · OAI.Kaplansky.graphBundleNormalization

graphBundlePullbackRelativeIso def

Pullback coordinates retain the same fiber and removed zero section. · OAI.Kaplansky.graphBundlePullbackRelativeIso

graphChain def

All three literal subdivisions of an indexed signed graph edge. · OAI.HyperbolicObstruction.FinitePolygonModel.Data.graphChain

graphChartNeighborsEmbedding def

Inclusion into actual full neighbors preserves the subspace. · OAI.MaxCutGames.Inverse.MatrixChart.graphChartNeighborsEmbedding

graphChartNeighborsEmbedding def

Inclusion into actual full neighbors preserves the subspace. · OAI.UniqueGamesTheorem.Inverse.MatrixChart.graphChartNeighborsEmbedding

graphComponent def

Sending each rotation vertex to its true connected component. · OAI.HyperbolicObstruction.PictureSwitch.PictureMap.graphComponent

graphComponentEquiv def

The map and the underlying simple graph have literally the same components. · OAI.TorsionFreeZeroDivisors.MapEuler.graphComponentEquiv

graphCutStub def

The closed half of a deleted edge, viewed inside the cut graph. · OAI.EuclideanFiveColor.graphCutStub

graphDenominatorLinear def

The variable part of the graph-overlap denominator. · OAI.Kaplansky.ComplexGrassmann.graphDenominatorLinear

graphDiagonalUnitary def

Two unitary coordinate actions form an actual Hilbert-space unitary. · OAI.KadisonKastler.graphDiagonalUnitary

graphDistance def

The infimum of lengths of actual walks between the two vertices. · OAI.PlanarL1.graphDistance

graphDistance def

Weighted shortest-path distance, as in the first paragraph of the paper. The main statement requires connectivity and strict positivity on every edge. · OAI.PlanarL1.graphDistance

graphEdgeEquiv def

No graphs or edge sets are omitted or multiply represented. · OAI.LeanBlast.SecondKahnKalai.graphEdgeEquiv

graphEdgePoint def

Affine parameterization of an edge, including its two vertices. · OAI.EuclideanFiveColor.graphEdgePoint

graphEdges def

The non-loop edge set of an ordinary graph on Fin n. · OAI.LeanBlast.SecondKahnKalai.graphEdges

graphEdgesEquiv def

Edge-set encoding preserves the ordinary unlabelled edge count. · OAI.LeanBlast.SecondKahnKalai.graphEdgesEquiv

graphEDist def

The intrinsic extended distance: infimum of lengths of C¹ paths in the domain. Using a single C¹ path gives the same intrinsic distance as piecewise C¹ paths, by smooth endpoint reparameterization and concatenation. · OAI.AffineBernstein.graphEDist

graphEmbedding def

The graph of the reconstruction parameter as an injective linear map. · OAI.MaxCutGames.Appendix.DerivativeInduction.graphEmbedding

graphEmbedding def

The graph of the reconstruction parameter as an injective linear map. · OAI.UniqueGamesTheorem.Appendix.DerivativeInduction.graphEmbedding

GraphEquiv structure

An isomorphism of directed graphs, with no order-reflection requirement. · OAI.KLInvariance.MomentGraph.GraphEquiv

graphEquiv def

The first coordinate identifies the graph with its domain. · OAI.MaxCutGames.Inverse.MatrixChart.graphEquiv

graphEquiv def

The first coordinate identifies the graph with its domain. · OAI.UniqueGamesTheorem.Inverse.MatrixChart.graphEquiv

graphEquiv₁ def

graphEquiv₁ f is the natural equivalence between Fin n and graph f, mapping i to (f i, i). · OAI.Nagata.Workers.W10.LocalTupleSort.graphEquiv₁

graphEquiv₂ def

graphEquiv₂ f is an equivalence between Fin n and graph f that respects the order. · OAI.Nagata.Workers.W10.LocalTupleSort.graphEquiv₂

graphEvaluation def

The actual linear family whose zero set is the local incidence locus. · OAI.Kaplansky.Incidence.graphEvaluation

graphFace def

The three positive graph segments. · OAI.HyperbolicObstruction.FinitePolygonModel.Data.graphFace

GraphFlag abbrev

The two flags at each edge, together with the labeled legs. · OAI.PixtonChow.GraphFlag

graphHighInverseScalar def

The inverse of the complementary graph resolvent, clipped at the cut. · OAI.KadisonKastler.graphHighInverseScalar

GraphHolds def

Density closure of the literal entry relation. · OAI.PartitionConsistency.ProductForcing.ChainCode.GraphHolds

graphHyperplaneSectionCoordinates def

The actual K hyperplane sections, expressed in the genuine evaluation trivialization over a projective graph chart. · OAI.Kaplansky.ComplexGrassmann.graphHyperplaneSectionCoordinates

graphHyperplaneTrivialization def

Genuine bundle coordinates obtained by evaluating a dual-line vector on the nonvanishing graph representative. · OAI.Kaplansky.ComplexGrassmann.graphHyperplaneTrivialization

GraphInput structure

Each unordered edge is represented once, by increasing endpoints. · OAI.MatchingFPRAS.GraphInput

graphInverseMetric def

The literal inverse-Hessian norm of a restricted ambient affine covector. · OAI.AffineBernstein.graphInverseMetric

graphIso def

The restricted dart system realizes precisely the induced simple graph. · OAI.TorsionFreeZeroDivisors.MapRestrict.graphIso

graphLeafMove def

Linear transfer of a leaf coordinate along its unique incident edge. · OAI.EuclideanFiveColor.graphLeafMove

graphLinear def

A graph chart with one coordinate determined by a linear equation. · OAI.ProjectionCounterexample.graphLinear

graphLineFrame def

A continuously varying independent frame for a graph-chart line. · OAI.Kaplansky.ComplexGrassmann.graphLineFrame

graphLineVector def

A nonvanishing representative of a line in a graph chart. · OAI.Kaplansky.ComplexGrassmann.graphLineVector

graphLoopClass def

The analytic class of the parametrized boundary loop at this scale. · OAI.CoarseAssembly.graphLoopClass

graphMomentCost def

The nonnegative cost appearing before graph-vector clipping. · OAI.KadisonKastler.graphMomentCost

graphNeighbors def

The ordinary finite neighbor set, with its classical finiteness data fixed. · OAI.TorsionFreeZeroDivisors.BallCount.graphNeighbors

graphNormal def

The corresponding unnormalized normal has coordinate i equal to one. · OAI.ProjectionCounterexample.graphNormal

graphNullhomotopy def

A coherent system of vertex paths contracts the entire graph map. · OAI.GroupRingTopology.Cayley.graphNullhomotopy

graphOfEdges def

A set of edges, interpreted as an ordinary graph on all n vertices. · OAI.LeanBlast.SecondKahnKalai.graphOfEdges

graphOrderIso def

The lattice is isomorphic to the full congruence lattice of the graph algebra. · OAI.FiniteCongruence.graphOrderIso

graphPairing def

Retain all literal edge copies whose endpoints are adjacent in T. Fixed darts are stubs; no parallel edge is silently identified. · OAI.HyperbolicObstruction.PictureSwitch.PictureMap.graphPairing

graphParameter def

The graph map from the fixed compressed image into the removed summand. · OAI.MaxCutGames.Appendix.DerivativeInduction.graphParameter

graphParameter def

The graph map from the fixed compressed image into the removed summand. · OAI.UniqueGamesTheorem.Appendix.DerivativeInduction.graphParameter

graphPathLength def

Length on [0,1] for g_x(v,v) = ‖v‖² + (Du_x v)², the metric induced by the Euclidean graph embedding x ↦ (x,u(x)). The product norm on Lean's ordinary product type is NOT used (it would be the maximum norm). · OAI.AffineBernstein.graphPathLength

graphPermutation def

The permutation on cloud vertices induced by the actual outer rotation. · OAI.MaxCutGames.Foundations.PCP.ZigzagGraphs.graphPermutation

graphPermutation def

The permutation on cloud vertices induced by the actual outer rotation. · OAI.UniqueGamesTheorem.Foundations.PCP.ZigzagGraphs.graphPermutation

graphPlan 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.graphPlan

graphPlane def

The usual graph chart, transported by an arbitrary linear splitting. · OAI.Kaplansky.ComplexGrassmann.graphPlane

graphProjection def

Projection to the coordinate hyperplane omitting the i-th original base coordinate, represented in its standard n coordinates. · OAI.AffineBernstein.graphProjection

graphQuotient def

The quotient coordinate of a graph t-plane, (u,v) ↦ v-bu. · OAI.Kaplansky.Incidence.graphQuotient

graphRealization def

The geometric realization of a simple graph in its probability simplex. The equal-endpoint option includes isolated vertices. · OAI.EuclideanFiveColor.graphRealization

graphReciprocalFlip def

The reciprocal graph symmetry, viewed as a real-linear isometry. · OAI.KadisonKastler.graphReciprocalFlip

graphResolventCutoff def

This product kills both singular endpoints and still has dense range. · OAI.KadisonKastler.graphResolventCutoff

GraphRowApproximation abbrev

The individual finite-row approximation property used in the graph construction. This notation imposes no compatibility among the choices. · OAI.KadisonKastler.GraphRowApproximation

graphRowClipFactor def

The common positive left multiplier used to clip the entire row. · OAI.KadisonKastler.graphRowClipFactor

graphRowCost def

The fourth-moment cost controlling both rescaled row coordinates. · OAI.KadisonKastler.graphRowCost

graphRows def

Actual finite graph rows in increasing reverse-index order. · OAI.MaxCutGames.Foundations.Complexity.MachineDummyRows.graphRows

graphRows def

Actual finite graph rows in increasing reverse-index order. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineDummyRows.graphRows

graphRowSquare def

The positive left square of an operator row. · OAI.KadisonKastler.graphRowSquare

graphScaleDenominator def

The positive denominator relating the rescaled graph resolvents. · OAI.KadisonKastler.graphScaleDenominator

graphSizeDecidable def

Existence of a colored-graph witness at a fixed candidate size is decidable. · OAI.FiniteCongruence.graphSizeDecidable

graphTransition def

The exact change of graph slope: bottom block times inverse top block. · OAI.Kaplansky.ComplexGrassmann.graphTransition

graphTransport def

A full signed original edge is exactly its three path segments. · OAI.HyperbolicObstruction.FinitePolygonModel.Data.graphTransport

graphValue def

Only the middle third of each graph edge carries its original voltage. · OAI.HyperbolicObstruction.FinitePolygonModel.Data.graphValue

graphValue def

Affine graph of line i in the U-coordinate. · OAI.Nagata.W06.LineArrangement.graphValue

graphVariationArea def

Actual affine-area density of the variation, in its exact graph jets. · OAI.AffineBernstein.graphVariationArea

GraphVertex abbrev

The vertices of the actual rotation system, without collapsing darts or parallel edge copies. Only the associated simple adjacency ignores loops. · OAI.HyperbolicObstruction.PictureSwitch.PictureMap.GraphVertex

graphVertex def

Standard affine vertices. These are the manuscript's probability-simplex coordinates. · OAI.EuclideanFiveColor.graphVertex

graphWindows def

The genuine two-window graph event on a boundary-started orbit. · OAI.SmoothObstruction.WordHierarchy.graphWindows

GraphWitness def

A bounded graph entry is witnessed by a genuine local assignment, with forced equality to its literal label and its normalized class name. · OAI.PartitionConsistency.ProductForcing.ChainCode.GraphWitness

graphWitnessDecidable def

An executable fixed-size test, with no classical choice in its computational content. · OAI.FiniteCongruence.graphWitnessDecidable

Grassmann def

Actual complex s-planes in complex dimension s+k. · OAI.Kaplansky.ComplexGrassmann.Grassmann

grassmannChart def

A genuine open chart, with all s*k complex graph coordinates as its target. · OAI.Kaplansky.ComplexGrassmann.grassmannChart

grassmannNeighborsMapEquiv def

Ambient linear equivalence transports the actual neighbor subspaces. · OAI.MaxCutGames.Inverse.MatrixChart.grassmannNeighborsMapEquiv

grassmannNeighborsMapEquiv def

Ambient linear equivalence transports the actual neighbor subspaces. · OAI.UniqueGamesTheorem.Inverse.MatrixChart.grassmannNeighborsMapEquiv

GrayTruncationInput structure

Right-interval internal hom preserves the finite dimension bound. · OAI.Thomason.Construction.GrayTruncationInput

greatestElementPrism def

The greatest-element contraction satisfies precisely the hypotheses of the constructed relative chain prism, with empty stationary locus. · OAI.Thomason.Construction.greatestElementPrism

GreatestNew structure

A greatest element of the new part of the poset. It need not be greatest in the whole poset: old objects can lie above neither it nor the new part. · OAI.Thomason.Construction.SieveRetraction.GreatestNew

greatestVertex def

The recursively greatest original vertex of a subdivided vertex. · OAI.Thomason.Construction.RelativeLabelCarrier.greatestVertex

greedy def

Positive priority-greedy on the entire ground set, characterized by failure of the strictly higher-priority labels to span the current label. · OAI.MatroidProphet.CandidateSpecification.greedy

GreedyDenominator inductive

Denominators produced by multiplying by positive integers at most one more than the preceding denominator, without reducing fractions. · OAI.Problem337.GreedyDenominator

greedyFibre def

A fallback is computed for the current residual margins, then moved back through the entry-preserving completion equivalence. · OAI.ContingencyTables.SmallGraphProfiles.greedyFibre

greenMajorant def

Bounded discrete Green supersolution for the walk killed at -1. · OAI.FKCRT.SimpleWalk.greenMajorant

Grid structure

An ordered power-of-two grid. Its coordinate bounds are kept in the theorem, not hidden in a change to the definition of the process. · OAI.CoordinateSweeps.Grid

grid def

Equally spaced exponents, with an arbitrary incoming depth. · OAI.DilutedSpinGlass.PrescribedTree.grid

grid abbrev

Repeated simultaneous bisections, recording all colours with multiplicity. This is a genuine finite forest partition, not an assumed normal form. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Partition.grid

grid def

The fixed lexicographic three-dimensional integer grid, encoded by Fin H. · OAI.RationalHitting.Generator.grid

GridAddress abbrev

Coordinates of a finite family of dyadic tables. Different table names are different coordinates even when their numeric cell keys coincide. · OAI.Problem310.TrialAddresses.GridAddress

gridAssignment def

T_(a,b) is sent to grid a, and U_(a,b) to grid b. · OAI.EuclideanRamsey.NineGrid.gridAssignment

gridBadCenters def

Centers whose positive displacement of length δ crosses a boundary of the periodic grid with q cells. · OAI.Problem310.gridBadCenters

gridBinGood def

Grid-bin failure bound from a lower bound on the origin crossing probability. · OAI.GammaFPP.CrossingBridge.gridBinGood

GridBound def

Finite cut statement; asks for an L¹ majorant restricted to points where density steps stay below threshold. There is no dependence on a frequency or depth cutoff in the constant. · OAI.FourierLLogL.Dyadic.GridBound

gridChoice def

A table-based node rule, with no restriction on which deterministic first-success/default convention the rule uses. · OAI.Problem310.LocalPredicateFactor.gridChoice

gridDirichlet def

A rotated, finite Dirichlet kernel on the root grid. · OAI.AsymptoticallyMinimalLittlewood.gridDirichlet

gridDurations def

Successive durations at a deterministic increasing list of observation times. · OAI.FKCRT.SimpleWalk.gridDurations

gridExponents def

The interior grid exponents; the automatically adjoined root exponent is zero. · OAI.DilutedSpinGlass.UniversalDictionary.gridExponents

gridIndex def

The precise regular translated-grid label. · OAI.SmoothObstruction.GridMargin.gridIndex

gridLagrangeLine def

Restriction of the grid Lagrange basis along the parametrized affine line. · OAI.Release075.gridLagrangeLine

GridLaw structure

The only additional colour relation needed for all tree-partition coherence. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Tree.GridLaw

gridPoint def

Embed the twice-barycenter lattice at mesh size h. · OAI.SharpLiebThirring.CubeFlags.gridPoint

gridPrimeCellPartition def

The finite cells used for idealization are constructed directly from an equal-mesh logarithmic grid and every invertible residue class. · OAI.Ostmann.gridPrimeCellPartition

GridSample abbrev

The finite noise sample space used by the construction. · OAI.PlanarL1.Columns.Model.Point.RadialTree.GridSample

GridSample abbrev

Both independent axis constructions, before phase and rotation. · OAI.WeakPinned.Grid.GridSample

gridSize def

One input-size polynomial denominator, rather than a product of all vertex degrees. Both grid and unweighting error are uniform over all cuts. · OAI.OptimalMaxCut.LongCode.Game.gridSize

gridStableCenters def

Simultaneous stable centers for a finite list of earlier-grid/later-window pairs. · OAI.Problem310.gridStableCenters

gridTranspose def

The row/column enumeration of the same intersection blocks. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.gridTranspose

gridWeight def

Downward rounding is performed BEFORE aggregation. Thus every output mass has the same polynomially sized integer denominator; no product of all input vertex degrees has to be represented as a unary integer. · OAI.OptimalMaxCut.LongCode.Law.gridWeight

GridWire abbrev

Horizontal wires run west/east; vertical wires run south/north. The extra endpoint in each coordinate is literal, not periodic boundary data. · OAI.CriticalHoneycomb.Transfer.RasterGrid.GridWire

GromovPolynomialGrowthInput def

Gromov's Main Theorem (1981, p. 54), specialized to the finite symmetric word-ball formulation used by the height-kernel argument. · OAI.PolycyclicRecognition.GromovPolynomialGrowthInput

grothendieckConstant def

Constants in the bounded product algebra. · OAI.KadisonSimilarity.grothendieckConstant

grothendieckCorner00 def

The two diagonal corner maps used for the two states. · OAI.KadisonSimilarity.grothendieckCorner00

grothendieckEntry def

Extract an entry with its actual C-star matrix norm bound. · OAI.KadisonSimilarity.grothendieckEntry

grothendieckFamilyLinear def

Coordinatewise application, with the ultralimit seminorm on its target. · OAI.KadisonSimilarity.grothendieckFamilyLinear

grothendieckFamilyMap def

The family map into the completed Hilbert ultraproduct. · OAI.KadisonSimilarity.grothendieckFamilyMap

grothendieckMatrixMap def

The original map applied to the distinguished off-diagonal entry. · OAI.KadisonSimilarity.grothendieckMatrixMap

grothendieckOffDiagonal def

The distinguished off-diagonal corner. · OAI.KadisonSimilarity.grothendieckOffDiagonal

grothendieckRestrict def

Restrict a positive functional on bounded families to the constants. · OAI.KadisonSimilarity.grothendieckRestrict

grothendieckState11 def

Restrict the first matrix state to its supporting corner. · OAI.KadisonSimilarity.grothendieckState11

grothendieckUnitaryFamily def

A bounded family of unitaries is unitary in the bounded product. · OAI.KadisonSimilarity.grothendieckUnitaryFamily

groundEnergy def

Variational bottom over every normalized spinful fermionic H¹ state. · OAI.ContinuumCoulomb.groundEnergy

groundGraph def

A ground function is represented by a literal set of ordered pairs. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.groundGraph

Group def

Grothendieck group of the exact category underlying an abelian category. · OAI.CoherentK0.Group

Group structure

The geometric information attached to one actual surviving g-cell group. The seed collection is finite, lies in one broad component and one narrow height band, and keeps its TWO chart identifiers through the surviving-pair selection. The B … · OAI.PlanarL1.Columns.Model.SecondaryGeometry.Group

group def

Concrete permutation model generated by finite twisted dyadic tables. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.group

group def

Finset of vertices. · OAI.FourierLLogL.Dyadic.Node.group

group def

Group formation: exactly one finite proposal family per g-cell and pair of old identifiers. The update groups themselves add no chart boundary. · OAI.PlanarL1.Columns.Model.SurvivingGroups.group

group def

The actual return chosen while processing this parent. · OAI.ReflexiveFixedPoints.CandidateContext.group

group def

Each individual group is an increasing interval in the concatenation. · OAI.ReflexiveFixedPoints.OrderedGroupConcat.group

groupAdjointOperator def

The intrinsic bracket, viewed as a continuous operator on the model space. · OAI.PolycyclicRecognition.groupAdjointOperator

groupAlgebraEmbedding def

Input units in the simple algebra, whose internal identity is preserved. · OAI.FiniteAlgebraicEnvelopes.Bootstrap.Source.groupAlgebraEmbedding

groupBiproduct def

Regroup by weight using the proved fiber cardinality, not an assumed decomposition. · OAI.ExternalTensor.Binomial.groupBiproduct

GroupBornologous def

Uniform control of finite sets of group differences. · OAI.PolycyclicRecognition.GroupBornologous

GroupBornologous def

Uniform control of bounded differences, for the discrete group coarse structure. · OAI.PolycyclicRecognition.GroupBornologous

groupBox def

A Cartan group box is the exponential image of its scaled coordinate cube. · OAI.PolycyclicRecognition.CartanWordModel.groupBox

GroupCertificate structure

Exact data for a paired, nonmiddle Hopf-charge prefix at n=16,k=50000. · OAI.HarmonicCounterexample.Certificate.GroupCertificate

GroupCoarseEquivalence structure

Coarse inverse maps with errors contained in fixed finite sets. · OAI.PolycyclicRecognition.GroupCoarseEquivalence

GroupCoarseEquivalence structure

Coarse inverse maps with uniformly bounded errors, measured by finite sets of group differences. · OAI.PolycyclicRecognition.GroupCoarseEquivalence

groupCoarseEquivalence def

Finite metric balls turn the specified coarse inverse maps into the finite-difference comparison required by the cohomological inputs. · OAI.PolycyclicRecognition.MetricQuasiEquivalence.groupCoarseEquivalence

groupCoarseEquivalence def

An isomorphism is also a discrete coarse equivalence, with exact inverses. · OAI.PolycyclicRecognition.MulEquiv.groupCoarseEquivalence

groupCoarseEquivalenceOfMetricInverse def

Coarse inverse metric maps produce the concrete equivalence used by Li's group-cohomological invariance theorems. · OAI.PolycyclicRecognition.groupCoarseEquivalenceOfMetricInverse

groupCoarseEquivalenceOfQuasiIsometry def

A metric quasi-isometry yields an actual discrete-group coarse equivalence. The inverse is chosen from coarse surjectivity, and all its control estimates are derived from the lower inequality. · OAI.PolycyclicRecognition.groupCoarseEquivalenceOfQuasiIsometry

groupDomain def

The four rows in the compression table, with their actual index domains. · OAI.LowSpaceSubsetSum.CompressionSetup.groupDomain

groupedAverage def

The subgroup operator is an actual group-algebra average on the full set of slots, not an operator posited to factor. · OAI.CoordinateSweeps.SignedTensor.groupedAverage

groupedCoefficient def

The coefficient of a candidate within one parent group. · OAI.ReflexiveFixedPoints.Prediction.groupedCoefficient

groupedEquiv def

A bijective enumeration which keeps each label fibre contiguous. · OAI.QuantitativeVanDerWaerden.groupedEquiv

groupedIndex def

The position of an element in the concatenation of the label fibres. · OAI.QuantitativeVanDerWaerden.groupedIndex

groupedWeight def

Total weight of columns with the specified slot and tag. · OAI.MetricEntropyDuality.groupedWeight

groupEquiv def

Group branches by their weight, with the binomial coefficient as the exact fiber size. · OAI.ExternalTensor.Binomial.groupEquiv

groupExponential def

The intrinsic exponential, constructed without a Lie-theoretic existence input. · OAI.PolycyclicRecognition.groupExponential

groupExponential def

The intrinsic exponential, constructed without a Lie-theoretic existence input. · OAI.PolycyclicRecognition.SmoothLie.groupExponential

groupExtension def

The defining group extension of a factor system. · OAI.PolycyclicRecognition.FactorExtension.groupExtension

groupFootprint def

A conservative count of array entries, cached tournament nodes and constant-size row cursors, at any current state or replay snapshot. · OAI.LowSpaceSubsetSum.CompressionData.groupFootprint

groupIndex def

H,U,V are respectively none, some false, some true. · OAI.PermanentBorder.AmbientPolar.groupIndex

groupingMachine def

One concrete finite machine: five Boolean stacks, eight labels, and three internal states. Four destination stacks hold the four separated fields. · OAI.MaxCutGames.Foundations.Hastad.SourceMachine.groupingMachine

groupingMachine def

One concrete finite machine: five Boolean stacks, eight labels, and three internal states. Four destination stacks hold the four separated fields. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceMachine.groupingMachine

groupingMachineInTime def

The concrete machine's verified trace includes arbitrary unread input and arbitrary prior destination contents. No assumed algorithm remains here. · OAI.MaxCutGames.Foundations.Hastad.SourceMachine.groupingMachineInTime

groupingMachineInTime def

The concrete machine's verified trace includes arbitrary unread input and arbitrary prior destination contents. No assumed algorithm remains here. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceMachine.groupingMachineInTime

GroupInjective def

Group uniqueness only concerns anchors, so it permits many nonanchors. · OAI.Problem348.Construction.SelectedAnchors.GroupInjective

groupMemberWithWork def

Every scale is accumulated using the same finite child arrays. · OAI.EditApproximation.BinaryFraction.groupMemberWithWork

groupN def

The number of candidates actually selected for each parent. · OAI.ReflexiveFixedPoints.CandidateContext.groupN

groupNormalTrace def

The normal trace of an element, obtained from the actual Artin presentation. · OAI.EilenbergGanea.TraceWords.groupNormalTrace

groupOppositeEquiv def

Inversion identifies a group with the opposite group without any commutativity assumption. · OAI.HyperbolicObstruction.groupOppositeEquiv

groupPhase def

Exact scalar-rotated fractional phase of a group element. · OAI.KervaireRegular.groupPhase

groupScaleSourceCalls def

The exact number of source calls at one group scale. · OAI.EditApproximation.BinaryFraction.groupScaleSourceCalls

GroupsMonotone def

The within-mode group order enjoyed by an increasingly selected axis. · OAI.Problem348.HostAnchorRigidity.GroupsMonotone

groupTrace def

The unnormalized matrix trace, linear in its operator argument. · OAI.GroupRingDeterminant.groupTrace

groupTracialRepresentation def

The actual group algebra supplies the concrete tracial data; no group approximation, ICC, factor, or countability premise occurs. · OAI.UltrapowerBridge.groupTracialRepresentation

groupValue def

Read a local value using the unique group and position of a global index. · OAI.ReflexiveFixedPoints.FrontierExtension.groupValue

groupWeightDilation def

Exact weight dilation in the actual nilpotent group. · OAI.PolycyclicRecognition.CartanWordModel.groupWeightDilation

grow def

Attach a genuinely new child at the divergence vertex; keep every old child and every unary level. Child 0 is the new child for root attachment. · OAI.DilutedSpinGlass.PrescribedTree.grow

Growth structure

A partially swallowed tree, retaining real residual stems to the original frontier vertices in the supplied polygonal drawing. · OAI.PlanarL1.HeightTree.Growth

growth def

The growth function g_x, including its zero convention. Only nonnegative s are used. · OAI.CliqueFreeIndependence.WeightedGraph.growth

growth def

Natural-logarithm scale of all three asymptotic endpoints. · OAI.TreeEdit.BinaryLower.growth

growthProduct def

The product sequence from Section 6, indexed from zero. · OAI.MatrixMultiplication.AuxiliarySeparation.growthProduct

GSub def

The free-boundary gradient numerator for the explicit lower barrier. · OAI.Bernoulli.DeSilvaJerison.GSub

gSup def

For τ=-2 the auxiliary Legendre solution is elementary and rational. · OAI.Bernoulli.DeSilvaJerison.gSup

Guard def

A marker-outward transition is never usable on this bounded tape. · OAI.TwoWayComplementation.CompanionBridge.Guard

guard def

The outer loop consumes one real unary fuel bit. The zero delimiter is consumed on exit; its temporary finite register is reset on both branches. · OAI.MaxCutGames.Foundations.Complexity.MachineRegularTable.Fuel.guard

guard def

A single finite statement: inspect the counter, decrement a positive counter, and jump to the body or exit. The zero delimiter is never consumed. · OAI.MaxCutGames.Foundations.Complexity.MachineUnaryCounter.guard

guard def

The outer loop consumes one real unary fuel bit. The zero delimiter is consumed on exit; its temporary finite register is reset on both branches. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRegularTable.Fuel.guard

guard def

A single finite statement: inspect the counter, decrement a positive counter, and jump to the body or exit. The zero delimiter is never consumed. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineUnaryCounter.guard

GuardCell def

The protected cells of the whole selected path. · OAI.Problem348.GuardedPath.Selection.GuardCell

guardedEquation def

An actual guarded multihomogeneous extension of an affine chart equation. · OAI.Nagata.ProjectiveGeometry.guardedEquation

guardedExponent def

Insert the distinguished projective-coordinate exponent. The extra one is the guard: every term vanishes outside the chosen product affine chart. · OAI.Nagata.W27.guardedExponent

guardedHomogenize def

Guarded block homogenization of an arbitrary affine polynomial. · OAI.Nagata.W27.guardedHomogenize

guardedV def

Scalar suffix exact part with mask specified by chosen representative. · OAI.FourierLLogL.Dyadic.Procedure.Paths.Troupe.guardedV

guardStates def

Static movement of the bit register to the last product coordinate. · OAI.MaxCutGames.Foundations.Complexity.MachineExpanderTable.guardStates

guardStates def

Static movement of the bit register to the last product coordinate. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineExpanderTable.guardStates

Guess structure

Consistency with deterministic finite approximations for access. · OAI.ExactDerandomization.At.Blob.Guess

guessRange def

The precise interval sampled by the implementation. · OAI.LowSpaceSubsetSum.RepresentationSetup.guessRange

guessVerifier def

Concrete Boolean root-verification circuit with no run-time primality or factorization oracle. The integer input is retained separately from the guess. · OAI.ExactQuantumFactoring.guessVerifier

guessWords def

The specified sorted, zero-padded n²-bit root encoding. · OAI.ExactQuantumFactoring.guessWords

guideCoordinate def

The paper's access guide, before applying the ribbon coordinate map. · OAI.SLEExactGauge.RibbonCoordinates.guideCoordinate

guidePatch def

The fixed length-four patch appearing in the admissible ribbon class. · OAI.SLEExactGauge.RibbonCoordinates.guidePatch

GurskyLeBrunSectionalInput def

Equation (5) in the proof of the published Lemma 1, with strict sectional positivity and scalar curvature twelve. · OAI.PositiveEinsteinFour.Geometry.GurskyLeBrunSectionalInput

gX def

Generator images of the weighted degeneration presentation. · OAI.StableCoordinate.Obstruction.gX

g₀ def

Second generator of the pulled-back distinguished ideal. · OAI.StableCoordinate.Bundle.g₀