Glossary: P

Definitions in this project whose names start with P. 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

p def

Extend the positive spherical profile by zero outside its first two roots. · OAI.Bernoulli.DeSilvaJerison.p

p def

Field order: label count, cutoff, phase count, restart time, observation count, return cap, capacity-bit allowance. · OAI.ContingencyTables.CountingPhaseScalarProgram.p

p def

The source map in the obstruction proof, with its faithful completed section-cone target. · OAI.SourceConeMorphism.p

p def

Independent sampling probability for each ordered pair. · OAI.TalagrandDiscreteConvexity.Covering.NumericalBounds.p

p def

The exact marginal letter frequency p=129/16513. · OAI.TorsionFreeZeroDivisors.WordEstimate.p

p0 def

Source zero quotient: retain precisely the path-multiplication terms. · OAI.ExternalTensor.Section4.SourceFiltration.p0

p2 def

The second filtration step and its source-one quotient. · OAI.ExternalTensor.Section4.SourceFiltration.p2

Pace inductive

Unary pace: entering the static system, or starting a new plateau · OAI.FourierLLogL.Dyadic.Procedure.Pace

pack def

A coordinate set of basis vectors of a subspace arranged as columns. · OAI.HadwigerCounterexample.pack

packBound def

One fixed universal integer, never depending on the finite list, the old identifiers, or the old chart count. · OAI.PlanarL1.Columns.Model.SurvivingGroups.packBound

packData def

Proof-level organization of exactly the finite bits physically read. · OAI.MaxCutGames.Foundations.Complexity.PoweringMachineRow.packData

packData def

Proof-level organization of exactly the finite bits physically read. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringMachineRow.packData

packed def

Input, output, then zero work space; this same layout is used by the classical subroutines inside the coherent controller. · OAI.ExactQuantumFactoring.BooleanNetwork.packed

packedMajorantSum def

Exact constant-piece evaluation: no traversal of the d residues. · OAI.ExactQuantumFactoring.OrderTrial.packedMajorantSum

packedOutputBit def

A designated target bit of the packed clean oracle output. · OAI.ExactQuantumFactoring.packedOutputBit

packedPieceSum def

Each piece sum is computed with five closed power sums plus its length. · OAI.ExactQuantumFactoring.OrderTrial.packedPieceSum

packedSpan def

Length of a list of windows packed with a fixed gap between consecutive windows, with no leading or trailing gap. · OAI.Problem310.WindowRecursion.packedSpan

Packet structure

A packet stores exactly the finite parent-chain graph, extended only on assignment. · OAI.CriticalZ3.Coarse.Packet

packet def

The actual angular packet, in the volume preserving coordinates of section 2. · OAI.ThreeTorus.packet

packetL2 def

The L2 packet vector, with its coefficient already included. · OAI.LipschitzHilbert.PacketLatticeData.packetL2

packetL2 def

The L2 packet vector, with its coefficient already included. · OAI.LipschitzHilbert.SourceLabeledLattice.packetL2

packetSpectralMeasure def

Packet spectral measure is the precise pushforward stated in (2.3). · OAI.ThreeTorus.packetSpectralMeasure

packetStretch def

Stretch inside one of the disjoint unit-time packets. · OAI.HarmonicCounterexample.FiniteControl.SmoothWord.packetStretch

packetVector def

A coefficient vector multiplying the original physical packet. · OAI.LipschitzHilbert.PacketLatticeData.packetVector

packetVector def

A coefficient vector multiplying the original physical packet. · OAI.LipschitzHilbert.SourceLabeledLattice.packetVector

PackingState structure

The bookkeeping invariant of a partial matching and its active graph. · OAI.AsymptoticallyMinimalLittlewood.Nibble.PackingState

packList def

A list of at most T chain darts can be stored in T option-valued slots. · OAI.TorsionFreeZeroDivisors.PatternCoding.packList

packMinus def

A represented minus behavior is packed without changing any observation. · OAI.PureTypeSystem.ObservationValues.packMinus

packRecord def

Consume a binary word, prepending one encoded reversed record. · OAI.RationalHitting.StackLanguage.packRecord

pad def

Appending untouched blank cells reserves quota without altering any already represented tape position. · OAI.FiniteAlgebraicEnvelopes.Compiler.Simulator.pad

pad def

Executable materialization of every padded rotation and predicate row. · OAI.MaxCutGames.Foundations.PCP.PreprocessingPaddingTables.pad

pad def

Actual regular padding by an arbitrary finite number of dummy vertices. · OAI.MaxCutGames.Foundations.PCP.VertexPadding.pad

pad def

Padding a finite table loses none of its entries. · OAI.TorsionFreeZeroDivisors.PatternCoding.pad

pad def

An exact common prefix and suffix are observed as ordinary independent complete traces. They neither change the score moment nor cost simulation error. This is the source's deterministic residual convention. · OAI.TraceReconstruction.OracleTracePair.pad

pad def

Executable materialization of every padded rotation and predicate row. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingPaddingTables.pad

pad def

Actual regular padding by an arbitrary finite number of dummy vertices. · OAI.UniqueGamesTheorem.Foundations.PCP.VertexPadding.pad

padArrayOperator def

An operator on the smaller array, extended by zero on all new coordinates. · OAI.KadisonSimilarity.padArrayOperator

Padded def

Strict padding in the actual floor-indexed band. · OAI.PlanarL1.Tiling.Padded

PaddedAt def

The closed metric ball is contained in the block containing x. · OAI.BoundedTreewidthL1.PaddedPartitions.PaddedAt

paddedBranch def

A branch extended to the common ambient sigma coordinate spaces. · OAI.MatrixMultiplication.AuxiliarySeparation.Character.paddedBranch

paddedCanonicalCounts def

A reserved word contributes a power of two for its unused suffix. · OAI.ContingencyTables.DensePrograms.paddedCanonicalCounts

PaddedCloud abbrev

Padding counts are supplied as executable natural-valued data, without choosing enumerations of the semantic clouds by their cardinalities. · OAI.MaxCutGames.Foundations.PCP.PreprocessingCloudIndex.PaddedCloud

PaddedCloud abbrev

Padding counts are supplied as executable natural-valued data, without choosing enumerations of the semantic clouds by their cardinalities. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingCloudIndex.PaddedCloud

paddedCloudEquiv def

The actual padded tail fiber contains old darts and exactly the dummy darts indexed by that tail, with transport along the sigma index equality. · OAI.MaxCutGames.Foundations.PCP.CloudPadding.paddedCloudEquiv

paddedCloudEquiv def

Local padded order: old cloud indices first, then increasing dummy index. · OAI.MaxCutGames.Foundations.PCP.PreprocessingCloudIndex.paddedCloudEquiv

paddedCloudEquiv def

The actual padded tail fiber contains old darts and exactly the dummy darts indexed by that tail, with transport along the sigma index equality. · OAI.UniqueGamesTheorem.Foundations.PCP.CloudPadding.paddedCloudEquiv

paddedCloudEquiv def

Local padded order: old cloud indices first, then increasing dummy index. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingCloudIndex.paddedCloudEquiv

PaddedDart abbrev

Additional darts are attached to their specified original vertex. · OAI.MaxCutGames.Foundations.PCP.DegreeReplacement.PaddedDart

PaddedDart abbrev

Additional darts are attached to their specified original vertex. · OAI.UniqueGamesTheorem.Foundations.PCP.DegreeReplacement.PaddedDart

paddedFamilyInTime def

At the physically computable padding level, the actual family run has a fixed polynomial bound in requested unary size. The separate ceiling-power machine supplies this exact level and preserves the requested size. · OAI.MaxCutGames.Foundations.Complexity.MachineExpanderFamily.paddedFamilyInTime

paddedFamilyInTime def

At the physically computable padding level, the actual family run has a fixed polynomial bound in requested unary size. The separate ceiling-power machine supplies this exact level and preserves the requested size. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineExpanderFamily.paddedFamilyInTime

paddedForestBits def

Padding to a common length preserves injectivity, since parsing stops at the exact end of the represented tree. · OAI.TwoPointCorrelations.paddedForestBits

paddedGraph def

Dummy darts are self-reversing loops with an always-accepting constraint. · OAI.MaxCutGames.Foundations.PCP.DegreeReplacement.paddedGraph

paddedGraph def

Whole-graph padding retains each original regularized constraint. · OAI.MaxCutGames.Foundations.PCP.Preprocessing.paddedGraph

paddedGraph def

The same graph, with the explicit initial/final-interval vertex ordering. · OAI.MaxCutGames.Foundations.PCP.PreprocessingPaddingTables.paddedGraph

paddedGraph def

Dummy darts are self-reversing loops with an always-accepting constraint. · OAI.UniqueGamesTheorem.Foundations.PCP.DegreeReplacement.paddedGraph

paddedGraph def

Whole-graph padding retains each original regularized constraint. · OAI.UniqueGamesTheorem.Foundations.PCP.Preprocessing.paddedGraph

paddedGraph def

The same graph, with the explicit initial/final-interval vertex ordering. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingPaddingTables.paddedGraph

paddedInTime def

A single concrete finite program takes the original unary request all the way to the exact forward family table. Both phase executions are supplied by proved machines; there is no construction or runtime hypothesis. · OAI.MaxCutGames.Foundations.Complexity.MachinePaddedExpanderFamily.paddedInTime

paddedInTime def

A single concrete finite program takes the original unary request all the way to the exact forward family table. Both phase executions are supplied by proved machines; there is no construction or runtime hypothesis. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachinePaddedExpanderFamily.paddedInTime

paddedLabelEquiv def

A padded label is a base-q integer whose digits are indexed by the explicit bounded-address equivalence. · OAI.MaxCutGames.Foundations.PCP.PoweringEnumeration.paddedLabelEquiv

paddedLabelEquiv def

A padded label is a base-q integer whose digits are indexed by the explicit bounded-address equivalence. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringEnumeration.paddedLabelEquiv

paddedLength def

The cutoff and padding are the standing parameters in Section 2. · OAI.ExactQuantumFactoring.paddedLength

paddedMatrix def

The same full sampled table used in the unconditional variation comparison. Its projection choices have not been supplied as extra compared coordinates. · OAI.MaxCutGames.Decoder.AdviceExperiment.paddedMatrix

paddedMatrix def

The same full sampled table used in the unconditional variation comparison. Its projection choices have not been supplied as extra compared coordinates. · OAI.UniqueGamesTheorem.Decoder.AdviceExperiment.paddedMatrix

PaddedOrder def

Zero padding follows every positive entry. · OAI.ExactQuantumFactoring.PaddedOrder

paddedSharedBranch def

A branch in the two common sigma coordinate spaces, with zero padding. · OAI.MatrixMultiplication.AuxiliarySeparation.paddedSharedBranch

paddedSize def

Leave empty clouds empty; pad every positive size to a geometric level. · OAI.MaxCutGames.Foundations.PCP.CloudPadding.paddedSize

paddedSize def

The algorithm returns the power it has already computed. · OAI.MaxCutGames.Foundations.PCP.PreprocessingLevels.paddedSize

paddedSize def

Leave empty clouds empty; pad every positive size to a geometric level. · OAI.UniqueGamesTheorem.Foundations.PCP.CloudPadding.paddedSize

paddedSize def

The algorithm returns the power it has already computed. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingLevels.paddedSize

paddedSliceEquiv def

The subtype equivalence leaves the actual matrix untouched. · OAI.MaxCutGames.Decoder.ProjectedColumnSlice.paddedSliceEquiv

paddedSliceEquiv def

The subtype equivalence leaves the actual matrix untouched. · OAI.UniqueGamesTheorem.Decoder.ProjectedColumnSlice.paddedSliceEquiv

paddedSum def

Charge a gap after every entry, including the final entry. · OAI.Problem310.WindowSpanSum.paddedSum

paddingCode def

A fixed vertex supplies a realized code for padding unused pivot slots. · OAI.MetricEntropyDuality.FixedLabels.paddingCode

paddingEquiv def

Successful enlarged tables are in exact bijection with original tables. · OAI.ContingencyTables.paddingEquiv

paddingInTime def

The actual padding execution stops at the family entry on the same physical level stack, and preserves both inactive finite states. · OAI.MaxCutGames.Foundations.Complexity.MachinePaddedOverlay.paddingInTime

paddingInTime def

The actual padding execution stops at the family entry on the same physical level stack, and preserves both inactive finite states. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachinePaddedOverlay.paddingInTime

paddingList def

Dummy coordinates, in increasing original vertex and local index order. · OAI.MaxCutGames.Foundations.PCP.PreprocessingRegularTables.paddingList

paddingList def

Dummy coordinates, in increasing original vertex and local index order. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingRegularTables.paddingList

paddingMass def

The finite sum over successive padding choices, with i the step index. · OAI.TwoPointCorrelations.paddingMass

paddingOrder def

Explicit list search/get indexing, with an erased cardinality cast. · OAI.MaxCutGames.Foundations.PCP.PreprocessingRegularTables.paddingOrder

paddingOrder def

Explicit list search/get indexing, with an erased cardinality cast. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingRegularTables.paddingOrder

padOperatorMatrix def

Entrywise zero extension of a square operator matrix. · OAI.KadisonSimilarity.padOperatorMatrix

padRight def

Add explicit zero slots, with <=m nodes. Duplicate outputs may share zero, but the simple linear upper bound is sufficient for the compiler. · OAI.ExactQuantumFactoring.BooleanNetwork.padRight

padTensorPolynomial def

Higher unused powers have the actual zero expression as coefficient. · OAI.ClosedSurfaceR4.JetPolynomial.Perturbation.padTensorPolynomial

padZero def

Add vacuous equations after the original finite list. · OAI.MaxCutGames.Inverse.RowErasureDescriptions.padZero

padZero def

Add vacuous equations after the original finite list. · OAI.UniqueGamesTheorem.Inverse.RowErasureDescriptions.padZero

Page structure

Package for a single executable prefix update, all higher-precision probes. · OAI.ExactDerandomization.Reed.Board.Page

pair def

Parallel computations share the input but retain disjoint new traces. · OAI.ExactQuantumFactoring.BooleanNetwork.pair

pair def

Pairing of names in the actual condition system. · OAI.PartitionConsistency.SetForcingNames.ConditionAction.Name.pair

pair def

First factor slow, second factor fast. · OAI.PowerSaving.Layout.pair

pairAverage def

The coefficient obtained by projecting a pair and dividing by two. · OAI.CirculantHadamard.pairAverage

pairAverage def

Equation (pair-average), with the two orientations counted separately. · OAI.MUB6.pairAverage

pairBits def

The explicit framing used for an input and its NP certificate. · OAI.DirectedFeedback.pairBits

pairBits def

A unary length frame followed by input bits and witness bits. · OAI.MaxCutGames.Foundations.Complexity.CookLevin.pairBits

pairBits def

A unary length frame followed by input bits and witness bits. · OAI.UniqueGamesTheorem.Foundations.Complexity.CookLevin.pairBits

pairCaps def

Individual marginal caps (mapW denotes the actual marginals) and a joint cap. · OAI.HadwigerCounterexample.pairCaps

pairCell def

Exact cell probabilities, with the diagonal excluded. · OAI.WeakPinned.Variance.pairCell

pairChartTransition def

The exact coordinate change underlying the inverse coupling. · OAI.PolycyclicRecognition.pairChartTransition

pairChoice def

Choose the left nonzero-denominator pair, otherwise the right. The costly candidate and recursive scan are paired ONCE before this inexpensive mux. · OAI.ExactQuantumFactoring.OrderTrial.pairChoice

pairCode def

Serialization of a typed pair. Only the first field needs a delimiter. · OAI.RationalHitting.StackLanguage.pairCode

pairCoefficient def

The cumulative coefficients, with division by zero harmless at the endpoints because the corresponding cumulative mass is proved zero. · OAI.ContingencyTables.pairCoefficient

PairComparison structure

Exact two-factor comparison; identity factors have width one and zero price. · OAI.ExactFourier.Packing.PairComparison

pairComplement def

An ordered pair and its untouched complement, used for exact identity padding. · OAI.ExactFourier.MatrixPrice.pairComplement

pairCoordinates def

The two affine coordinates of a point, enumerated by Fin 2. · OAI.Nagata.W06.LineArrangement.pairCoordinates

pairCoordinatesEquiv def

The difference and ratio coordinates parametrize exactly the ordered distinct nonzero pairs. · OAI.Ostmann.pairCoordinatesEquiv

pairCost def

The real-valued radial Coulomb interaction, used only away from the origin. · OAI.Problem356.CoulombCalculus.pairCost

pairCrossings def

The actual finite set of crossing points for a specified ordered edge pair. · OAI.Paper170Geometry.pairCrossings

pairCumulative def

The partial cross-entry sum in equation pair-cumulative. · OAI.ContingencyTables.pairCumulative

pairCutCount def

The exact edge count crossing between two distinct clusters. · OAI.OptimalMaxCut.Unweighted.pairCutCount

pairDensity def

The real even one-site density obtained by the literal auxiliary partial trace of a unit matched-pair vector. · OAI.CoordinateSweeps.SignedTensor.pairDensity

pairDivergence def

Divergence from one referenced leaf, with every old child retained. · OAI.DilutedSpinGlass.PrescribedTree.pairDivergence

pairDomainTranslation def

Domain translation, expressed in the target-normalized chart. · OAI.PolycyclicRecognition.pairDomainTranslation

Paired structure

Every uncut adjacency is a distinct-arc digon, and cuts are stable under its opposite gap. These are precisely the hypotheses obtained by cutting bad and marked gaps in PlanarRotation. · OAI.TorsionFreeZeroDivisors.PlanarRuns.CutSystem.Paired

pairedCoordinateAlgebra def

A concrete matrix algebra, with the common coordinate eigenvalue recorded as its first component. · OAI.DukePrimeDegree.pairedCoordinateAlgebra

pairedEntry def

A paired block selects exactly one entry from every pair. · OAI.Problem337.ResidueConstruction.pairedEntry

pairedEquations def

The two opposite queries give an explicit balanced list of equations. · OAI.MaxCutGames.Foundations.Hastad.EmptyContext.pairedEquations

pairedEquations def

The two opposite queries give an explicit balanced list of equations. · OAI.UniqueGamesTheorem.Foundations.Hastad.EmptyContext.pairedEquations

pairedFactorEquiv def

The source algebra is isomorphic to its concrete paired image. · OAI.KadisonKastler.pairedFactorEquiv

pairedFactorHom def

The actual paired representation on the two-fold Hilbert sum. · OAI.KadisonKastler.pairedFactorHom

pairedFlagMass def

None is the joint hidden flag; a visible flag reports the true orientation. · OAI.TraceReconstruction.FiniteLaw.pairedFlagMass

pairedFrequencyEquiv def

No multiplicity is lost when the two independent descendant histories are placed in the paired tree used by the arithmetic norm estimate. · OAI.Ostmann.pairedFrequencyEquiv

PairedMechanisms structure

Data of the paired mechanisms, with common submass proved for each side. All words and their actual trace laws will be substituted in these fields. · OAI.TraceReconstruction.PairedMechanisms

pairedPreCount def

Counts of literal side chunks, with one deterministic pure filler in each region. A bundle can contain any even number of paired chunks. · OAI.TraceReconstruction.pairedPreCount

pairedRepresentation def

The diagonal pair of two star representations. · OAI.KadisonKastler.pairedRepresentation

pairedTensorUnitary def

The tensor of two unitaries as a unitary of the completed tensor Hilbert space. · OAI.NuclearUltrapower.pairedTensorUnitary

pairedTracePotential def

Paired trace potential. At s=0 the assigned value is immaterial to its integral. · OAI.SharpLiebThirring.MatrixProof.pairedTracePotential

pairedVonNeumann def

The actual von Neumann algebra on the original two-fold Hilbert sum. · OAI.KadisonKastler.pairedVonNeumann

pairEq def

The indicator that two labels agree. · OAI.Problem335.IndependentCoordinates.pairEq

pairEquiv def

Distribution of a binary module product over the finite family of columns. · OAI.LittleFinitistic.FiniteColumnComplex.pairEquiv

pairExpect def

Two kernel paths share exactly their first d coordinates; the definition also includes the diagonal endpoint d=n. · OAI.DilutedSpinGlass.KernelTower.pairExpect

pairExponent def

Analysis scale for the number of low-filtered raw child pairs. · OAI.LowSpaceSubsetSum.pairExponent

pairExterior def

The actual ordered two-form alpha wedge beta, as a continuous form. · OAI.Mahler.pairExterior

PairFamily structure

A simultaneous pair family retains its actual disjoint ordered coordinate tuples. · OAI.ExactFourier.PairFamily

pairForm def

The alternating product of a bilinear form and an n-form, with an integer normalization convenient for the six-dimensional obstruction. · OAI.ArnoldCounterexample.Forms.pairForm

pairGradient def

The gradient of the real radial Coulomb interaction. · OAI.Problem356.CoulombCalculus.pairGradient

pairHeight def

Height evaluation of the forward map of a normalized pair. · OAI.PolycyclicRecognition.pairHeight

pairHeightL2 def

The L² class of height evaluation, using the canonical continuous-map inclusion. · OAI.PolycyclicRecognition.pairHeightL2

pairHeightProfile def

The continuous height profile at a fixed group element. · OAI.PolycyclicRecognition.pairHeightProfile

pairHessian def

The Hessian of the real radial Coulomb interaction. · OAI.Problem356.CoulombCalculus.pairHessian

pairHessian def

The inverse-distance Hessian at either chosen unit coordinate vector. · OAI.Problem356.LocalHessian.pairHessian

pairHistory def

One prescribed pair constraint, retaining the COMPLETE old law. · OAI.DilutedSpinGlass.PrescribedTree.pairHistory

pairHistoryMass def

Total absolute mass controlling one added color, with the entire old tree left present. The bound is independent of the realized reservoir size. · OAI.DilutedSpinGlass.PrescribedTree.pairHistoryMass

pairIdeal abbrev

The ideal generated by two simultaneous equations. · OAI.StableCoordinate.IteratedQuotient.pairIdeal

PairIndex abbrev

All independent ordered-pair sampling indices. · OAI.TalagrandDiscreteConvexity.Covering.PairIndex

pairing def

Bilinear pairing for the positive strip measure, without complex conjugation. · OAI.CriticalHoneycomb.FiniteFunctional.pairing

pairing def

The sphere obtained by pairing a polygon with a reversed copy. It is the literal base case of a cellular disk picture after adjoining its outside. · OAI.HyperbolicObstruction.PictureSwitch.PolygonSphere.pairing

pairing def

The positive invariant trace pairing on skew-Hermitian matrices. · OAI.KervaireSkewTrace.pairing

pairing def

The normalized Hilbert--Schmidt pairing. · OAI.PositiveEinsteinFour.Bianchi.Tensor.pairing

pairing def

The canonical bilinear pairing induced on homogeneous symmetric powers. · OAI.Problem346.PairingConstruction.pairing

pairing def

Pair an interior selected facet with its neighbor, and keep boundary facets fixed. · OAI.SharpLiebThirring.PLParity.FinitePairing.pairing

pairingLabelKernel def

On orthonormal labels, a pairing contraction is the product of equality indicators along its pairs. Repeating an indicator has no effect. · OAI.KadisonSimilarity.pairingLabelKernel

pairingPolynomial def

The actual pairing polynomial, with one bilinear factor per pair. An injective rank function chooses a representative of each pair. · OAI.KadisonSimilarity.pairingPolynomial

PairInteraction structure

A finite pair interaction allowing self-loops and arbitrary vertex and edge labels. · OAI.BufferedIsing.PairInteraction

pairInverseTargetTranslation def

Target translation by y⁻¹, expressed in the source-normalized chart. · OAI.PolycyclicRecognition.pairInverseTargetTranslation

pairKey def

Both complete leaf weights and every modular operation are charged. · OAI.LowSpaceSubsetSum.RepresentationKeys.pairKey

pairKey def

The canonical smaller group's variable list fixes the order in a pair key. The shared-variable mask is identical on the two sides. · OAI.ParityWL.Complexity.pairKey

pairLetters def

The alternating word containing n copies of the pair d,s. · OAI.KadisonSimilarity.DihedralCornerWords.pairLetters

pairLift def

The map from the genuine presented rank-two Coxeter group. · OAI.KLInvariance.Dihedral.pairLift

pairMask def

Local occupancy projection. Unlike an entire frontier mask this fixes only the two wires touched by one actual lozenge. · OAI.CriticalHoneycomb.Transfer.PlanarNormalization.pairMask

pairMatrix def

The matrix of a pair of distinct involutions. · OAI.KLInvariance.Dihedral.pairMatrix

pairMultiplicity def

A paired charge count, as a rational integer-valued polynomial. · OAI.HarmonicCounterexample.Certificate.pairMultiplicity

pairName def

A literal pairing name. · OAI.PartitionConsistency.SetForcingNames.pairName

pairNormingFunctional def

One actual norming functional for each difference of dense points. · OAI.ReflexiveFixedPoints.pairNormingFunctional

pairObservableHistory def

Actual one-new-path history for an arbitrary two-path observable (in particular the empirical spin overlap). All unused old replicas are retained. · OAI.DilutedSpinGlass.PrescribedTree.pairObservableHistory

PairOrbit def

The actual orbit closure of a normalized quasi-isometry pair. · OAI.PolycyclicRecognition.PairOrbit

pairOrder def

Coordinate-lex order, then indicator order for points and lines, then indicator order for families, finally lexicographic order of the pair. Discarding the non-full-line candidates does not change relative order. · OAI.KaplanskyCounterexample.Prescription.pairOrder

pairRanks def

The rank multiset, rather than a lexicographic ordering, permits the swaps in the recursive clauses of atomic forcing. · OAI.PartitionConsistency.SetForcingNames.AtomicForcing.pairRanks

pairRepresentation def

Direct sum of two star representations, with the Hilbert product norm. · OAI.KadisonSimilarity.pairRepresentation

Pairs def

Unordered two-element subsets, without an ordering or repetitions. · OAI.FiniteAlgebraicEnvelopes.TypeAAction.Pairs

pairs def

A Cartesian product that preserves repetitions in each input list. · OAI.MaxCutGames.Reduction.Explicit.pairs

pairs def

A Cartesian product that preserves repetitions in each input list. · OAI.UniqueGamesTheorem.Reduction.Explicit.pairs

pairSlotEquiv def

The recursive wedge slots list each coordinate's real slot before its imaginary slot. This equivalence records that interleaved order. · OAI.Mahler.pairSlotEquiv

pairSourceChart def

The second product chart parametrizes inverse-normalized pairs. It is obtained by swapping maps, with the inverse on the group coordinate. · OAI.PolycyclicRecognition.pairSourceChart

pairState def

Tensor states in the actual physical wire order. · OAI.ExactQuantumFactoring.Register.pairState

pairStratumEquiv def

The exact stratum bijection for the intersection with a coordinate Lagrangian: dimension h is neither approximated nor dropped. · OAI.SidorenkoCounterexample.pairStratumEquiv

pairSum def

Summing a labeled pair at occupancy one, on its actual remaining ground set. · OAI.ContingencyTables.BinaryContraction.pairSum

pairSwitch def

Switches on a family of disjoint two-point fibers. · OAI.CoordinateSweeps.pairSwitch

pairSystem def

An actual Coxeter system on the generated subgroup, not an extra assumption. · OAI.KLInvariance.Dihedral.pairSystem

pairTable def

A finite two-spin table extracted from a not necessarily normalized spin weight. · OAI.BufferedIsing.pairTable

pairTargetChart def

The first global product chart is target translation of a normalized pair. · OAI.PolycyclicRecognition.pairTargetChart

pairTransferDefect def

The stationary domain defect, with the proposed mean height subtracted. · OAI.PolycyclicRecognition.pairTransferDefect

pairTransferLift def

Extend a section transfer by target-height equivariance. · OAI.PolycyclicRecognition.pairTransferLift

pairTransport def

An actual Thompson-group witness, chosen only for a genuinely separated pair. · OAI.ThompsonNonamenability.pairTransport

pairUnion def

The generator produced by a sampled ordered pair. · OAI.TalagrandDiscreteConvexity.Covering.pairUnion

pairVariation def

Extract the difference of the two site and intercept perturbations. · OAI.Problem358.FaceDerivative.pairVariation

PairwiseCollisionBound def

The smallest probabilistic interface needed from a random modulus: every distinct pair has collision probability at most δ. The weights prob form a finite probability distribution on seeds. · OAI.LowSpaceSubsetSum.PairwiseCollisionBound

pairWord def

Interleave two equally long lists of letters in their original order. · OAI.KadisonSimilarity.pairWord

pairWord def

Two equal-sized blocks, indexed by a side bit and a common coordinate. · OAI.MaxCutGames.Foundations.PCP.CodeComposition.pairWord

pairWord def

Two equal-sized blocks, indexed by a side bit and a common coordinate. · OAI.UniqueGamesTheorem.Foundations.PCP.CodeComposition.pairWord

Pale structure

One fixed incoming port/s station op, varying endpoint ray index j of second. First score and MAX-prefix values supplied whenever first starts. Not depending on j. · OAI.ExactDerandomization.Chron.Moon.Weather.Pale

Palette structure

A finite fixed palette, all members genuinely on the unit circle. · OAI.ConeCoordinates.Palette

Palette structure

A finite fixed palette, all members genuinely on the unit circle. · OAI.SignedDisk.ConeCoordinates.Palette

palindromeCost def

Full alternating-cycle count along the ACTUAL sampled routing, recursively. At each child the labels are exactly those sent into it by the actual input coins. · OAI.CubeShuffle.palindromeCost

palindromeLevelCost def

The exact level cost in the actual palindrome recursion. It stops at the specified height, and sums across all independent child nodes. · OAI.CubeShuffle.palindromeLevelCost

palindromeLowCost def

The exact sum of FAC.1 node costs through height H. · OAI.CubeShuffle.palindromeLowCost

palindromeSlabCost def

Sum over a slab, padded by its actual zero costs when its levels exceed d. · OAI.CubeShuffle.palindromeSlabCost

pAmbient def

Total ambient functions, used only on the sphere by the convex-set argument. · OAI.Tingley.pAmbient

PanchenkoTalagrandFieldPairInput def

Same published marking result as PanchenkoTalagrandTensorPairInput, specialized to the separable vector terminal used after the cavity change of variables. No pressure, spin-moment, or limiting assertion is assumed. · OAI.InvariantIsing.PanchenkoTalagrandFieldPairInput

PanchenkoTalagrandRestrictedFieldPairInput def

The same published marking theorem, with a nonempty finite spin constraint retained inside the terminal log partition. · OAI.InvariantIsing.PanchenkoTalagrandRestrictedFieldPairInput

PanchenkoTalagrandTensorPairInput def

Conditional two-path marking law for finite hierarchical tensor fields. · OAI.InvariantIsing.PanchenkoTalagrandTensorPairInput

paperM def

Exact exponent from the TeX (natural logarithms). · OAI.PowerSaving.paperM

Parallel def

The zero set of the sum of the two Busemann coordinates of an axis. All splitting identities used below are proved from CN, not assumed. · OAI.HyperbolicObstruction.ConeData.Parallel

Parallel inductive

Parallel reduction is used only to establish the beta metatheory. · OAI.PureTypeSystem.Parallel

parallel def

Two arbitrary already verified subroutines acting independently on a pair. · OAI.ExactQuantumFactoring.BitStackProgram.Procedure.parallel

parallel def

zip in parallel; used with matching lengths. · OAI.PowerSaving.Slant.Reel.parallel

ParallelFrameCurveAt structure

The local velocity and parallel-frame equations in the fixed chart centered at the current base point. · OAI.PositiveEinsteinFour.Geometry.ParallelFrameCurveAt

parallelFrameEquiv def

The inverse at time t is the actual horizontal curve at time -t. · OAI.PositiveEinsteinFour.Geometry.parallelFrameEquiv

parallelFrameHomeomorph def

The actual parallel flow gives homeomorphisms, with inverse at negative time. · OAI.PositiveEinsteinFour.Geometry.parallelFrameHomeomorph

parallelFrameVectorField def

Horizontal motion in the kth frame direction in a fixed chart. · OAI.PositiveEinsteinFour.Geometry.parallelFrameVectorField

ParallelJacobiData structure

Ordinary parallel-frame Jacobi data and the polar formula. The matrix curve is the actual curvature along this same flow, and RegularJacobi contains its ordinary Jacobi equation and initial conditions. · OAI.PositiveEinsteinFour.Geometry.ParallelJacobiData

ParallelMetricData structure

Only raw flow, parallel differentiation and normalized Ricci data remain in the classical frame-calculus input. · OAI.PositiveEinsteinFour.Geometry.ParallelMetricData

parallelNetwork def

Independent fair butterflies in parallel with completely arbitrary fixed bijections at either end. All certificate fields are proved from the explicit Boolean query algorithms and deterministic butterfly geometry. · OAI.CubeShuffle.parallelNetwork

ParallelReflection def

The only reflection input on the oriented measure is the ordinary parallel-transport relation, valid independently of the measure. · OAI.PositiveEinsteinFour.Geometry.ParallelReflection

Param structure

A fixed positive word with polynomial diagonal parameters and its exact derivative. · OAI.ExactFourier.PositiveGeneration.Param

parameter def

Each parameter is an element of the actual coordinate field, by the proved recovery formula and coordinate-field equality. · OAI.EuclideanRamsey.NineField.parameter

parameter def

The factorial series starts at 1!. · OAI.GrahamSpherical.parameter

parameter def

This is m0 itself, with no recursive occurrence in its seven properties. · OAI.PureTypeSystem.Labelled.SpecializedCode.parameter

parameter def

The positive Laurent parameter lies in the Rees algebra by monotonicity. · OAI.StableCoordinate.FilteredRees.parameter

parameter def

The unique global nearest parameter, selected after proving attainment. · OAI.ThompsonNonamenability.Analytic.RegularCurveData.parameter

parameter def

The source's ramp of distance to the center set, equal to one for the empty set. · OAI.UniformKServer.HeavyEditRamp.parameter

parameter def

The source's ramp of distance to the center set, equal to one for the empty set. · OAI.UniformKServer.HeavyPilot.parameter

parameterCurryEquiv def

The natural Hom bijection on every globe-and-pasting parameter. · OAI.Thomason.Construction.StrictOmegaCategory.GrayMapping.parameterCurryEquiv

parameterDomain def

Start and step representatives of the global counting argument. · OAI.QuantitativeVanDerWaerden.CirclePartition.parameterDomain

parameterDomainHom def

On an abelian group the domain-action type tag is also a group homomorphism, so dense generators transfer to the Koopman action. · OAI.DukePrimeDegree.parameterDomainHom

parameterEquiv def

All solutions of GS=1 on this matrix chart, with no missing rank-open constraints on S, form exactly the free affine parameter space. · OAI.IntegralCharacterVarieties.ExactSequenceChart.Chart.parameterEquiv

parameterEquiv def

Change from surjective/injective presentation coordinates to the actual quotient-domain and subspace-target parameter. · OAI.MaxCutGames.Inverse.KMSAffineRestrictionPresentation.parameterEquiv

parameterEquiv def

Change from surjective/injective presentation coordinates to the actual quotient-domain and subspace-target parameter. · OAI.UniqueGamesTheorem.Inverse.KMSAffineRestrictionPresentation.parameterEquiv

parameterFactor def

The entire dictionary factor family, with its original two coordinates and no truncation in the number of types. · OAI.DilutedSpinGlass.HeterogeneousMarks.parameterFactor

parameterIso def

Exact agreement with the actual chain model on every chosen parameter. · OAI.Thomason.Construction.StrictOmegaCategory.GrayChainExtension.parameterIso

parameterMap def

Independent field entries specify a binary-linear map into F³. · OAI.MaxCutGames.Quadratic.parameterMap

parameterMap def

Independent field entries specify a binary-linear map into F³. · OAI.UniqueGamesTheorem.Quadratic.parameterMap

parameterMapEquiv def

Every binary-linear map is determined by its freely chosen field-valued basis entries, with no rejection or nonuniform multiplicity. · OAI.MaxCutGames.Quadratic.parameterMapEquiv

parameterMapEquiv def

Every binary-linear map is determined by its freely chosen field-valued basis entries, with no rejection or nonuniform multiplicity. · OAI.UniqueGamesTheorem.Quadratic.parameterMapEquiv

parameterMatrix def

Substitute genuine polynomial coordinate motions into every matrix entry. · OAI.Nagata.Workers.W14.parameterMatrix

parameterOperator def

The derivative in the parameter variable at the center. · OAI.Problem356.LocalHessian.parameterOperator

parameterProjection def

Collapse on each actual finite parameter, with the unit inverted. · OAI.Thomason.Construction.StrictOmegaCategory.GrayChainExtension.parameterProjection

Parameters structure

Fixed matrix-test parameters, before the separate bipartite subdivision and repetition. Existence is proved by the v2 spectral/decoding argument. · OAI.MaxCutGames.Decoder.MatrixGap.Parameters

Parameters structure

The two source pilot templates use D_H=51200, R_L=4D_H, D_L=100R_L. The small numerical parameters remain free so the later travel absorption can choose them once and for all. · OAI.UniformKServer.TierPilot.Parameters

Parameters structure

Fixed matrix-test parameters, before the separate bipartite subdivision and repetition. Existence is proved by the spectral/decoding argument. · OAI.UniqueGamesTheorem.Decoder.MatrixGap.Parameters

parameters def

Only these countably many old elements are needed to specify a probe cell; factors along the path belong to every conservative subobject that contains the displayed endpoints. · OAI.PartitionConsistency.ConservativeMonoids.Probe.parameters

parameters def

Explicit equation (2), in terms of √3 * s̃. · OAI.PositiveEinsteinFour.Bianchi.parameters

parameters def

Encode the matching typed plain payload without changing its tag. · OAI.PureTypeSystem.ObservationValues.ActualPlainKey.parameters

parameters def

The concrete diagram being extended. · OAI.Thomason.Construction.StrictOmegaCategory.GrayChainExtension.parameters

parameterSlice def

Restriction to the line (h,u,v,w)=(t,0,0,0). · OAI.AbhyankarSathaye.CommutingDerivations.parameterSlice

parameterSlope def

Minus the derivative in beta, exactly as defined after eq:explicit-rank. · OAI.KServer.RankFunctions.parameterSlope

parameterUnit def

The invertible Laurent parameter of fiber scaling. · OAI.StableCoordinate.DeterminantLiftWeights.parameterUnit

parameterUnitComponent def

The unit comparison at one actual globe or binary pasting. · OAI.Thomason.Construction.StrictOmegaCategory.GrayChainExtension.parameterUnitComponent

parameterUnitIso def

The actual unit is an isomorphism on every chosen finite parameter. · OAI.Thomason.Construction.StrictOmegaCategory.GrayChainExtension.parameterUnitIso

parameterWord def

A finite control word with genuine smooth scalar amplitudes. · OAI.HarmonicCounterexample.FiniteControl.parameterWord

parametricAreaDensity def

Source (affine-area), using a fixed ambient volume basis. · OAI.AffineBernstein.parametricAreaDensity

parametricFrame def

The augmented tangent frame, with its transverse column last. · OAI.AffineBernstein.parametricFrame

parametricInverseMetric def

Inverse metric of an actual parametrization and an actual test function. · OAI.AffineBernstein.parametricInverseMetric

parametricSecondForm def

The literal second form with a conormal frozen at the point in question. · OAI.AffineBernstein.parametricSecondForm

parametrization def

The concrete producer is exhaustive and has an explicit inverse. Neither surviving external side germs nor their previous matchings are quotient out. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.CyclicInterface.parametrization

paramFill def

A family of loop-space paths determines a family of actual square maps. · OAI.HyperbolicObstruction.SquarePathFilling.paramFill

ParamPolyProgram structure

A single finite program, polynomial on each fixed parameter fibre. The parameter is genuine input data, not an infinite family of finite controls. · OAI.ParityWL.Complexity.ParamPolyProgram

paramQuotient def

The product quotient is an actual compact closed quotient, ensuring continuous families of fillings without a choice-continuity assumption. · OAI.HyperbolicObstruction.SquarePathFilling.paramQuotient

params def

The three literal parameters determine exactly a partial affine map. · OAI.FiniteAlgebraicEnvelopes.PartialAffine.Data.params

Parcel def

direct untransformed data-station program assigned at one valid card. · OAI.ExactDerandomization.SiteX.Studio.Parcel

parent def

The parent of a global index. · OAI.ReflexiveFixedPoints.OrderedGroupConcat.parent

parentAt def

A total parent index for the natural-number recursion. Only the first frontier.size stages are used to construct the finite level. · OAI.ReflexiveFixedPoints.CandidateContext.parentAt

ParentClosed def

Vertices already absorbed into the disk form a parent-closed set. · OAI.PlanarL1.HeightTree.ParentClosed

parentEmbed def

Embedding parent shifts in the actual product of child input spaces. · OAI.MaxCutGames.Gadget.parentEmbed

parentEmbed def

Embedding parent shifts in the actual product of child input spaces. · OAI.UniqueGamesTheorem.Gadget.parentEmbed

parentGroupType def

Flattened deterministic parent type, before any random partition. · OAI.TraceReconstruction.parentGroupType

parentLogical def

Parent logical translation, the second aggregate coordinate. · OAI.MaxCutGames.Gadget.parentLogical

parentLogical def

Parent logical translation, the second aggregate coordinate. · OAI.UniqueGamesTheorem.Gadget.parentLogical

ParentMap structure

A partial parent map strictly decreasing the position index. · OAI.TorsionFreeZeroDivisors.BlockConstraints.ParentMap

ParentPath structure

A finite parent route in a replacement, before any global sewing. The only fields are endpoint equalities in finite permitted-vertex tables. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.PortPatch.ParentPath

parentPath def

Every gallery, including arbitrary splitting/merging and adjacent passages, has its input-to-output parent boundary route. All unconsumed branch ports remain exposed by patch; no branch is deleted here. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VariableGallery.parentPath

parentPath def

The parent address of an edge in the full deterministic numerical index tree. In particular it does not contain the edge's own child label. · OAI.TraceReconstruction.OccupiedParityTree.Edge.parentPath

ParentReturn abbrev

Every parent return retains the full states and candidates, including candidates subsequently assigned coefficient zero. · OAI.ReflexiveFixedPoints.CandidateContext.ParentReturn

parentShiftReferenceBad def

The common union is literal, with no dependence on an imaginary phase. · OAI.LipschitzHilbert.parentShiftReferenceBad

parentShiftReferenceData def

none denotes the parent square; some shift denotes its continuing path. · OAI.LipschitzHilbert.parentShiftReferenceData

parentShifts def

The parent shift space: precisely the child shifts with zero first aggregate coordinate. · OAI.MaxCutGames.Gadget.parentShifts

parentShifts def

The parent shift space: precisely the child shifts with zero first aggregate coordinate. · OAI.UniqueGamesTheorem.Gadget.parentShifts

ParentTemplate structure

Verified local data, with port tables and parent corner exhaustion. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.ParentTemplate

parentTemplate def

The very same band, with all branch ports and all colours. The fresh-name proof supplies exhaustion, rather than replacing the band by an unrelated cyclic template with identical endpoints. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.RealizedBand.parentTemplate

parentValue def

A node's parent intercept uses no noise from the current depth. The root convention below merely fills unused clamping slots with the fixed input. · OAI.PlanarL1.Columns.Model.Point.RadialTree.parentValue

parity def

The source quotient onto C₂, for every realization including all unmatched rows. No infinitude or asphericity hypothesis is used here. · OAI.HyperbolicObstruction.LetterPresentation.parity

parityCharacter def

The character of the color-shift group indexed by the functional τ S. · OAI.MaxCutGames.Fourier.MatrixParity.parityCharacter

parityCharacter def

The character of the color-shift group indexed by the functional τ S. · OAI.UniqueGamesTheorem.Fourier.MatrixParity.parityCharacter

ParityStatement def

The literal measured-output, polynomial-total-qubit parity lower bound. · OAI.QAC.ParityStatement

paritySuccess def

The actual weighted satisfaction probability of the parity equations. · OAI.MaxCutGames.Clean.IncidenceGap.paritySuccess

paritySuccess def

The actual weighted satisfaction probability of the parity equations. · OAI.UniqueGamesTheorem.Clean.IncidenceGap.paritySuccess

parityTail def

The genuine one-sided parity tail of the angular decomposition. · OAI.TriangularBilliards.Analysis.CircleAction.parityTail

parityValue def

The attained optimum over all assignments to the names. · OAI.MaxCutGames.Clean.IncidenceGap.parityValue

parityValue def

The attained optimum over all assignments to the names. · OAI.UniqueGamesTheorem.Clean.IncidenceGap.parityValue

parkingBox def

This contains the whole four-shear trajectory, not only its endpoints. · OAI.ForcedComputation.PlanarRouting.parkingBox

parse def

Parse any word into its unique consecutive D blocks. · OAI.DerivedBar.Multibar.parse

parseClause def

`parseClause` runs `parseLiteral` three times and packages the results as the length-three vector `#v[a,b,c]`. · OAI.UniqueGamesTheorem.BinaryEncoding.parseClause

parseClauses def

Fuel bounds the number of clause iterations, not the size of any name. · OAI.MaxCutGames.BinaryEncoding.parseClauses

parseClauses def

Fuel bounds the number of clause iterations, not the size of any name. · OAI.UniqueGamesTheorem.BinaryEncoding.parseClauses

parseForestBits def

Fuel is the maximal number of nested nodes plus one. · OAI.TwoPointCorrelations.parseForestBits

parseFrame def

`parseFrame` reads zero or more data bits, each introduced by `true`, until a `false` terminator, then returns the digits and untouched suffix. · OAI.UniqueGamesTheorem.BinaryEncoding.parseFrame

parseLiteral def

`parseLiteral` consumes the first bit as polarity, delegates the remaining input to `parseName`, and returns one `Literal` plus the suffix. · OAI.UniqueGamesTheorem.BinaryEncoding.parseLiteral

parseName def

Equality with the canonical digits rejects alternate padded encodings. · OAI.MaxCutGames.BinaryEncoding.parseName

parseName def

Equality with the canonical digits rejects alternate padded encodings. · OAI.UniqueGamesTheorem.BinaryEncoding.parseName

parseOfCount def

Parse a word of a specified count without changing its block data. · OAI.DerivedBar.Multibar.parseOfCount

parserEmbedding def

Place the parser's input in an existing work bank, its private storage in a fresh bank, and its output in a fresh counter register. · OAI.Superstring.BitCode.parserEmbedding

Parses def

A linear-time prefix scanner for an explicit encoding. It consumes exactly one encoded value, pushes its reversal onto an accumulator, and restores every private stack to empty, independently of the untouched input suffix. · OAI.Superstring.BitCode.Parses

partGraph def

Restrict f to the disjoint source and target pieces. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.SliceData.partGraph

partialAtOne def

Evaluate the remaining p-primary variable at one. · OAI.CirculantHadamard.PrimeCharacterEvaluations.partialAtOne

partialAtRoot def

Evaluate the remaining full p-primary variable at the selected pth root. · OAI.CirculantHadamard.PrimeCharacterEvaluations.partialAtRoot

PartialBlock def

A crop meets but does not contain an actual finite block support. · OAI.SeparableQuotient.ActualSpace.PartialBlock

PartialBudget structure

Literal domain/length/context premises of the finite compiler's bounded theorem. These are conditions on the given partial program, not a total replacement. · OAI.FiniteAlgebraicEnvelopes.Compiler.PartialBudget

partialCharacter def

A fourth-root character on the first factor, retaining the second factor. · OAI.CirculantHadamard.BinaryBlocks.partialCharacter

partialEval def

Actual partial evaluation, allowing the integer-to-complex coefficient map. · OAI.CirculantHadamard.BinaryBlocks.partialEval

partialEvaluation def

Leave precisely the full p-primary cyclic coordinate unevaluated. · OAI.CirculantHadamard.PrimeCharacterEvaluations.partialEvaluation

partialIsoCornerEquiv def

Conjugation identifies the initial and final corners of a partial isometry. · OAI.BoundedHochschild.KadisonRingrose.partialIsoCornerEquiv

partialIsometryBergCompletion def

An explicit angular path completing a partial isometry through a common spare range. Only its quarter-turn endpoint is required to extend v. · OAI.WeakPureInfiniteness.partialIsometryBergCompletion

partialIsometryBergRotation def

Rotate from the initial corner toward the final corner by angle θ. · OAI.WeakPureInfiniteness.partialIsometryBergRotation

partialIsometryColumns def

The columns are multiplication by the partial isometries after the isometric inclusion of the initial corner. · OAI.KadisonSimilarity.partialIsometryColumns

partialIsometryCompletion def

The chosen two-swap completion, with fixed initial support p and spare isometry r. This is a polynomial in the partial intertwiner v. · OAI.WeakPureInfiniteness.partialIsometryCompletion

partialIsometryCornerEquiv def

Compress a partial isometry whose initial and final projections are the two range projections. · OAI.KadisonKastler.partialIsometryCornerEquiv

partialIsometryCornerMap def

Conjugation by the partial isometry is an actual ambient corner element. · OAI.KadisonSimilarity.partialIsometryCornerMap

partialIsometryDilationPath def

A continuous path from one to the symmetry exchanging the two images. · OAI.CoarseAssembly.Analytic.partialIsometryDilationPath

partialIsometrySwap def

Swap the orthogonal initial and final spaces of a square-zero partial isometry, and act identically on their common complement. · OAI.WeakPureInfiniteness.partialIsometrySwap

PartialLift structure

A continuous partial lift on the (n-1)-skeleton, represented by a total function only to avoid transported subtype terms. Values outside the skeleton carry no condition and are never used. · OAI.GroupRingTopology.Cellular.PartialLift

partialLift def

A set-theoretic partial inverse to the realized inclusion. · OAI.Thomason.Construction.RealizationPrism.partialLift

partialLiftCocone def

Partial lifts form a cocone of underlying sets; no continuity of the partial map is asserted. · OAI.Thomason.Construction.RealizationPrism.partialLiftCocone

partialMap def

Finite-stage maps, with the initial map kept separate from the corrections. · OAI.ClosedSurfaceR4.ExactCorrection.partialMap

partialNormal def

The manuscript's partial normalizer. It is defined even for a divergent program; no arbitrary value is inserted for an undefined call. · OAI.FiniteAlgebraicEnvelopes.Compiler.Word.partialNormal

partialRestrict def

Fix the first product block, leaving the second block free. · OAI.MaxCutGames.Inverse.KMSFourthMoment.partialRestrict

partialRestrict def

Fix the first product block, leaving the second block free. · OAI.UniqueGamesTheorem.Inverse.KMSFourthMoment.partialRestrict

partialTrace def

The literal finite partial trace; it is a sum of positive submatrices. · OAI.CoordinateSweeps.PositiveTensor.partialTrace

ParticleInvariant def

The quadratic subset-sum invariant controlling particle bounds. Subtractions are real. In particular the bound at the full set is exactly zero. · OAI.BoundedTreewidthL1.ParticleInvariant

ParticleReachable inductive

Finite update histories; there is no bound on the number of updates. · OAI.BoundedTreewidthL1.ParticleReachable

particleSwitches def

Particle observations use the prescribed terminal vertex, not a hat-density surrogate. This is the frozen spatial location carried throughout the process. · OAI.ActualMarkTrees.particleSwitches

particleSwitches def

Particle observations use the prescribed terminal vertex, not a hat-density surrogate. This is the frozen spatial location carried throughout the process. · OAI.SignedDisk.ActualMarkTrees.particleSwitches

particleTrace def

The exact zero-error combined symmetric trace carried by a particle list. · OAI.BoundedTreewidthL1.particleTrace

Partition abbrev

A partition is represented by its block-label map. The fibers are the blocks; using vertices as labels represents every partition of a finite nonempty vertex set. Unused labels impose no extra block condition. · OAI.BoundedTreewidthL1.PaddedPartitions.Partition

Partition structure

A finite dyadic partition of the entire cube. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Partition

partition def

Finite exponential partition function, based on a probability law p. · OAI.MatchingEntropy.partition

PartitionAdmissible def

One simultaneous condition for all actual parent and child intervals. · OAI.ThompsonNonamenability.PartitionAdmissible

PartitionBound def

An interval-partition inequality and its constant-16 difference-vector version. Intervals may be unbounded or singletons. · OAI.SeparableQuotient.ActualSpace.PartitionBound

partitionColumnRuns def

Omitted labels stand alone; consecutive regular labels form one segment. · OAI.TwoPointCorrelations.partitionColumnRuns

partitionCompositionMean def

The actual finite uniform average, not an assumed moment functional. · OAI.Problem335.partitionCompositionMean

partitionDiagram def

Conversion from Mathlib's partition type to its Young diagram. · OAI.Saxl.partitionDiagram

partitionEntropyRate def

Entropy rate of one partition. The indexing N+1 avoids division by zero. · OAI.DukePrimeDegree.partitionEntropyRate

partitionEquiv def

This is the actual partition of doubled cells into small views and large pairs. · OAI.ContingencyTables.LargePairEnumeration.partitionEquiv

partitionEquiv def

A partition of ω₁ into countably many unbounded fibers. · OAI.SeparableQuotient.Colors.partitionEquiv

partitionFamilyColor def

The family color is zero whenever the actual partition is inadmissible. · OAI.ThompsonNonamenability.partitionFamilyColor

partitionFinsetEquiv def

Split a finite set into its restrictions to the fibers of a finite coloring. · OAI.LowSpaceSubsetSum.partitionFinsetEquiv

PartitionOn structure

A full-line partition of one finite affine slab. · OAI.KaplanskyCounterexample.FiniteGeometry.PartitionOn

partitionProductFiberEquiv def

Fixing all component totals is the product of the individual product fibers. · OAI.Ostmann.partitionProductFiberEquiv

partitionRemainder def

All discarded pieces are supported in the complement of the rounded pieces. · OAI.KadisonKastler.partitionRemainder

Partner abbrev

Each retained coordinate receives an independent slot in Fin 3. · OAI.MaxCutGames.Soundness.PartnerSampling.Partner

Partner abbrev

Each retained coordinate receives an independent slot in Fin 3. · OAI.UniqueGamesTheorem.Soundness.PartnerSampling.Partner

partValue def

Used only to state the bounded correctness proof of the partial algorithm. It has no place in its definition or in its claimed effective implementation. · OAI.FiniteAlgebraicEnvelopes.Compiler.partValue

pass def

Both computations are evaluated. · OAI.PowerSaving.RAM.Bill.pass

Passage inductive

Only these two passage permutations are allowed in a surface diagram. An arbitrary permutation is factored into several vertices, never one vertex. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.Passage

passage def

A continuation or interchange has precisely two ports. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.passage

passageBranchKernel def

General smooth jitter kernel. The source's old-frequency variable is the second coordinate of F; the first is the new slow-coordinate good arc. · OAI.ThreeTorus.passageBranchKernel

passageFamily def

A nonsingular two-parameter form of the exact passage lift (both signs). · OAI.ThreeTorus.passageFamily

passageFourierInput def

The precise old-amplitude Fourier coefficients in the passage chart. · OAI.ThreeTorus.passageFourierInput

passageGrade def

Fine list positions, with exactly the prescribed pair merged for an adjacent interchange. This retains the entire A⊕B identified quotient. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.LocalRanks.passageGrade

PassagePackets structure

All input data used by the proved simultaneous passage theorems. No asymptotic estimate, transport, density, or conclusion is a field. · OAI.ThreeTorus.PassagePackets

Passages structure

An incidence remembers its occurrence and which (oriented) half-edge it uses. Passages pair incidences at a vertex; path endpoints are fixed points. · OAI.TorsionFreeZeroDivisors.IncidenceStages.Passages

passages def

The incidence passage data is derived from the actual adjacency and nonbacktracking conditions, not assumed as an extra vertex-count estimate. · OAI.TorsionFreeZeroDivisors.PathIncidences.passages

passageSlope def

Rounding is in the slow Fourier slope, not a translation of the fast variable. · OAI.ThreeTorus.passageSlope

passageTime def

The infimum of costs among candidate trails. It is a finite minimum whenever the lattice distance is less than R. · OAI.GammaFPP.FiniteLocal.passageTime

passed def

Filtering the actual chronological query at each clock tick. No collection of answers for unselected/counterfactual moduli occurs in this predicate. · OAI.ExactQuantumFactoring.NodeMachine.passed

passed def

Filtering the actual chronological query at each clock tick. No collection of answers for unselected/counterfactual moduli occurs in this predicate. · OAI.ExactQuantumFactoring.SplitMachine.passed

passIndex def

On a collapsed triangular face the two nonconstant sides are paired; on an uncollapsed face every side is a live terminal. · OAI.ArtinCAT0.TriangleMesh.passIndex

paste def

Evaluate the local function belonging to the unique Whitney strip. · OAI.LipschitzHilbert.SourceWhitney.paste

pasteCube def

Assemble a configuration on a coordinate subset and its complement. · OAI.Problem313.pasteCube

pasted def

The actual pasted named output on representatives. · OAI.PlanarL1.Columns.Model.Point.RadialTree.pasted

pasteTable def

The table for composition along dimension n: retain common lower data, take the exterior boundaries at n, and add the higher chains. · OAI.Thomason.Construction.DirectedComplex.Cell.pasteTable

pastingIncidence def

The actual two-boundary incidence of the pasted-globe basis. · OAI.Thomason.Construction.RightCylinderBasis.pastingIncidence

pastingIso def

The canonical comparison of the actual pasted globes is a strict isomorphism. · OAI.Thomason.Construction.PastingGlobeBasis.pastingIso

pastingLower def

Lower the common dimension of the two universal composable cells. · OAI.Thomason.Construction.StrictOmegaCategory.GrayMapping.pastingLower

pastingUnitIso def

Strict abelianization also recovers the actual binary pasting parameter. · OAI.Thomason.Construction.PastingGlobeBasis.pastingUnitIso

patch def

Local incidence recipe. It contains all ten allowed vertices and all ten internal seams, with the eight boundary ports left exposed. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.ComplementaryPatch.patch

patch def

Variable-length path of split/merge/adjacent events. All side branches are part of its explicit interface, not discarded by the path. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VariableGallery.patch

patch def

Measurable full-lattice patch; no choice of maximizing radius occurs. · OAI.PhysicalCells.patch

patch def

A kept main cell, with the literal interpolated proposal and trapezoidal shield, is an actual invariant-preserving measurable set update. The old chart agreement is used only on its real update domain. · OAI.PlanarL1.Columns.Model.MainCell.patch

patch def

Literal secondary clipping on the actual closed seed-ball union, in both ordinary and admitted ordered orientations, is an invariant-preserving patch. · OAI.PlanarL1.Columns.Model.SecondaryCell.patch

patch def

Metadata on packed indices; never computed by inversion, only used for proofs. · OAI.PowerSaving.Grove.Shard.patch

patch def

Measurable full-lattice patch; no choice of maximizing radius occurs. · OAI.SignedDisk.PhysicalCells.patch

patchAssemblyCoefficient def

One explicit coefficient covers all directional and higher-jet bounds of the finite patch field. · OAI.RegularPointCoordinates.patchAssemblyCoefficient

patchCentralFactor def

The precise scale factor for the strictly interior central mass. · OAI.RegularPointCoordinates.patchCentralFactor

patchCoreBump def

An actual smooth core-two, support-three cutoff on the coordinate space. · OAI.RegularPointCoordinates.patchCoreBump

patchCutoffError def

The error produced by differentiating the cutoff functions. · OAI.RegularPointCoordinates.patchCutoffError

patchEnvelopeFactor def

The scaling ratio from the source radius to the enclosing dyadic mass. · OAI.RegularPointCoordinates.patchEnvelopeFactor

patchField def

The field used by the renormalization, written as a finite patch sum. · OAI.RegularPointCoordinates.patchField

patchHomotopy def

Homotopies on a closed subset glue to the stationary homotopy outside it, provided they are stationary on the frontier. · OAI.GroupRingTopology.Cellular.patchHomotopy

patchingCoefficients def

The last coefficient is an endpoint; all earlier coefficients are internal. · OAI.Generator.Tracial.patchingCoefficients

PatchIterationGeometry structure

Geometric patch data before the coordinate iteration. The assumptions are on cutoffs, positive mass operators, displacement energies and their cumulative budgets; no estimate on the resulting field is assumed. · OAI.RegularPointCoordinates.PatchIterationGeometry

patchJetBase def

A fixed base absorbs the overlap and center-displacement constants. · OAI.RegularPointCoordinates.patchJetBase

patchLocalizationFactor def

The metric radius forced by the normalized support and coarse separation. · OAI.RegularPointCoordinates.patchLocalizationFactor

patchMass def

The unweighted mass of the active patches. · OAI.RegularPointCoordinates.patchMass

patchOverlapBound def

A dimensional overlap bound in the half-error volume regime. · OAI.RegularPointCoordinates.patchOverlapBound

patchPrincipal def

The principal part of the derivative of the patch field. · OAI.RegularPointCoordinates.patchPrincipal

Path def

Continuous finite-order jet curves on a closed short-time interval. · OAI.ForcedComputation.JetVolterra.Path

Path def

A fixed normed-space structure for continuous paths on the closed time interval. · OAI.ForcedComputation.WeaklySingular.Path

Path inductive

Finite paths, with edges carrying their actual named fields. · OAI.HyperbolicObstruction.MasterGraph.Path

Path def

Full root-to-leaf paths, retaining distinct child-index histories. · OAI.LeanBlast.SecondKahnKalai.ProbabilityTree.Path

Path structure

The actual nondecreasing sequence occurring in a probe datum. The transition and terminal factors will be derived, not additionally chosen. · OAI.PartitionConsistency.ProbeAlgebra.Path

Path inductive

A finite column word with an allowed band-switch height at every step. Endpoint heights are supplied separately; the nil word is a vertical segment. · OAI.PlanarL1.Columns.Model.Path

Path def

A point map surrounded by frame changes. The path identity applies to arbitrary initial arrays. losses accounts for decreases in dimension (shifts free). · OAI.PowerSaving.Binary.Path

Path inductive

An explicit hierarchical-graph walk. Up edges append one symbol and down edges delete the first symbol. Transitivity retains every edge of both walks. · OAI.Superstring.Hierarchical.Path

Path inductive

An ordinary port path; no reduction assumption, and loops and parallel edges remain distinct. This is used for the exact graph relators. · OAI.TorsionFreeZeroDivisors.GraphPresentation.Path

Path def

A finite sequence represented as successive products, in reveal order. · OAI.UniqueGamesTheorem.Gadget.AdaptivePaths.Path

path def

Only the first five stages change the state. · OAI.BoltzmannNonuniqueness.FiveGainMap.path

path def

A total path; indices beyond the tree depth repeat the common leaf. · OAI.Problem348.Construction.SampledSelection.path

pathAccepts def

Check all base constraints along the recorded walk. The reach certificates select genuine vertices in the two full-radius neighborhood balls. · OAI.MaxCutGames.Foundations.PCP.PoweringTest.pathAccepts

pathAccepts def

Check all base constraints along the recorded walk. The reach certificates select genuine vertices in the two full-radius neighborhood balls. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringTest.pathAccepts

pathAction def

The actual descended length-two action, using the proved relation ideal. · OAI.LittleFinitistic.QuadraticEncoding.Representation.pathAction

pathAt def

Recover a whole path from a labeled leaf in a sampled tree. · OAI.DilutedSpinGlass.PrescribedTree.pathAt

pathBasis def

The actual ordered length-two path basis, with the later arrow first. · OAI.LittleFinitistic.RelationBasisCoefficients.pathBasis

pathBetween def

The unique auxiliary trajectory joining two slots. · OAI.CoordinateSweeps.Grid.pathBetween

pathChain def

Convert a literal nonempty dart path to the dependent chain encoding. · OAI.TorsionFreeZeroDivisors.PathMultiplicity.pathChain

pathCharge def

Sum a node charge only at and after the particle's birth. · OAI.IntrinsicMarks.pathCharge

pathCharge def

Sum a node charge only at and after the particle's birth. · OAI.SignedDisk.IntrinsicMarks.pathCharge

PathChosen structure

Purely geometric sampling facts, separate from the host's matrix values. · OAI.Problem348.Construction.HostSampling.PathChosen

PathClosed def

A set contains the unique path between any two of its vertices. · OAI.HarmonicArtin.ParabolicIntersections.TreeAction.PathClosed

pathCongruence def

Connectivity in the marked graph is a congruence of the original algebra, not just of its admissible unary maps. · OAI.FiniteCongruence.pathCongruence

pathCoordinates def

Every literal boundary position is exactly one surviving picture occurrence. · OAI.TorsionFreeZeroDivisors.FinitePicture.Data.pathCoordinates

pathCost def

Recursion counts repeated traversals with multiplicity. · OAI.GammaFPP.LoopGraph.pathCost

pathCost def

The unrefined costs of the vertical pieces of an ACTUAL permitted path. · OAI.PlanarL1.Columns.Model.Activity.pathCost

pathDeficit def

Sum of the nonnegative deficits on the path incident to a leaf. · OAI.ReflexiveFixedPoints.pathDeficit

PathDisjoint def

Literal pairwise disjointness along every full root-to-leaf path. · OAI.LeanBlast.SecondKahnKalai.ProbabilityTree.PathDisjoint

pathDisplacement def

Sum the actual displacement at the proper ancestors of the selected node. · OAI.ReflexiveFixedPoints.TreeEvaluation.pathDisplacement

pathEdges abbrev

The same physical support used by the crossing and spatial events. · OAI.GammaFPP.BK.pathEdges

pathEdges def

The finite set of multiplier coordinates on a path. · OAI.IndependentProducts.pathEdges

pathEdges def

Edges of the path with n+1 vertices. · OAI.Problem335.SwitchRestoration.pathEdges

pathExpansion def

Positive expansion for a centered particle factor. This is a sum over NONEMPTY subsets of its stages; no independence of particle factors is used. · OAI.CoordinateSweeps.MomentExpansion.pathExpansion

pathFlux def

Product-rule expansion over all terminal label histories. The carried mass and its derivative are updated by the exact Leibniz rule. · OAI.SignedAssembly.pathFlux

pathFlux def

Product-rule expansion over all terminal label histories. The carried mass and its derivative are updated by the exact Leibniz rule. · OAI.SignedDisk.SignedAssembly.pathFlux

pathFunctional def

Path partial sums are bounded and converge pointwise. · OAI.SeparableQuotient.ActualSpace.pathFunctional

pathGeometricMoment def

The product of independent geometric local moments for a fixed pairing. · OAI.Problem335.MomentPairing.pathGeometricMoment

PathGeometry structure

All properties here are literal shortest-path/bag geometry. · OAI.BoundedTreewidthL1.AnchorExperiment.PathGeometry

pathHomotopy def

Conversely a path in a fixed-endpoint path space is a relative homotopy. · OAI.Thomason.Construction.PathSpace.pathHomotopy

PathIn def

Every vertex of a finite path, including the final vertex, is tested. · OAI.GammaFPP.IntervalTravelBridge.PathIn

PathIndex def

Paths are indexed by their full raw data, not by an arbitrary chosen expression witnessing membership. The witness is retained when used. · OAI.SeparableQuotient.Norming.PathIndex

pathL1 def

The integrable path product as an actual L1 element. · OAI.IndependentProducts.pathL1

pathLabel def

An inner color partitions the shared C₁ while retaining the outer color. · OAI.SeparableQuotient.Norming.pathLabel

pathLabelledMap def

A concrete genus-zero path map with every positive edge size. · OAI.FKCRT.pathLabelledMap

pathLabels def

The list of arc labels along a full path, in root-to-leaf order. · OAI.LeanBlast.SecondKahnKalai.ProbabilityTree.pathLabels

PathLength def

Finite path length, used solely to prove measurability of reachability. · OAI.CriticalZ3.PathLength

pathLength def

Riemannian length of a path on the unit parameter interval. · OAI.HarmonicCounterexample.pathLength

pathLine def

The stage-decorated line used by the auxiliary trajectory. · OAI.CoordinateSweeps.Grid.pathLine

pathMap def

Right multiplication by a path a: i→j maps B e_j to B e_i. Only the relation a e_i = a is needed for membership in the target ideal. · OAI.ExternalTensor.pathMap

pathMap def

A bounded path map constructed by completion of path cosets. · OAI.SeparableQuotient.ActualSpace.pathMap

pathMass def

Uniform nearest-neighbour path counting probability. · OAI.FKCRT.SimpleWalk.pathMass

pathMean def

The first-moment normalization for a fixed V/U layer partition. · OAI.Problem335.MomentPairing.pathMean

pathModuleMap def

Left multiplication by a length-two path, as a linear family of module maps. · OAI.LittleFinitistic.QuadraticEncoding.pathModuleMap

pathNat def

Right multiplication on the left tensor factor, natural in the right module. · OAI.ExternalTensor.Section4.pathNat

pathOccupation def

Occupation update for a variable path, in operator-composition order. · OAI.Problem335.pathOccupation

pathOfHomotopy def

A genuine relative path homotopy is a path in the compact-open path space. · OAI.TorsionFreeZeroDivisors.LoopTracks.pathOfHomotopy

pathPatterns def

The compact checker's exact pattern recursion, including extensions of components which do not themselves meet the ordering restriction. · OAI.CycleClique.Construction.pathPatterns

pathPatternsFuel def

Structurally recursive evaluator for the same domain. The fuel is bounded by the number of remaining clique vertices. · OAI.CycleClique.Construction.pathPatternsFuel

pathPrefix def

The nonroot prefix ending at position j. · OAI.IndependentProducts.pathPrefix

pathPrefix def

The node immediately before the kth decision along a nondefault path. · OAI.Problem310.FirstDefault.pathPrefix

pathProbability def

This declaration is an exact comparator-selected statement. Read the Lean statement below for its mathematical content. · OAI.KServer.pathProbability

pathProbability def

Probability of a finite label string under the successive behavioral choices. · OAI.KServer.pathProbability

pathProduct def

Root product is the empty product 1; every other product uses the path prefixes. · OAI.IndependentProducts.pathProduct

PathPropertiesContract def

lem:paths: both paths, all integer endpoints, and the true least birth. · OAI.MatroidProphet.PathPropertiesContract

pathReward def

Additive reward along a reversed history, with its true previous state. · OAI.SignedDisk.SignedRuns.pathReward

pathReward def

Additive reward along a reversed history, with its true previous state. · OAI.SignedRuns.pathReward

PathSaturated def

Every path shortcut includes all shorter shortcuts along the same tree path. · OAI.HarmonicArtin.ParabolicIntersections.TreeAction.PathSaturated

pathShape def

Every actual loop-free path through the rectangle supplies an exact occupancy shape; no hypothetical shape data is needed. · OAI.CriticalHoneycomb.Transfer.RasterGrid.pathShape

pathShift def

Path translations for H on ascending physical wires followed by a classically controlled translation on a separate additive register. · OAI.ExactQuantumFactoring.Triangular.pathShift

pathShift def

The signed occupation shift of a layered path. · OAI.Problem335.pathShift

pathShiftMatrix def

A matrix whose column is a weighted sum of path shifts. · OAI.Problem335.pathShiftMatrix

pathSpace def

Paths with the compact-open topology. · OAI.Thomason.Construction.TopologicalInterval.pathSpace

pathSpelling def

The literal edge word traversed by a generator word. Free reduction of this word is harmless for all nonabelian edge labelings. · OAI.EilenbergGanea.pathSpelling

PathState abbrev

The first coordinate records which entries of the vector have been written. Saturation keeps the state space finite even after the horizon. · OAI.ContingencyTables.RealKernelApproximation.PathState

pathSynthesis def

The map B_n is exactly the absolutely convergent signed sum of root paths. · OAI.BoundedTreePotentials.pathSynthesis

PathSystem structure

Literal path/comparison data before the separator and averaging step. Unlike BoundedSystem it makes no quantitative smallness or pairing assumption. · OAI.TorsionFreeZeroDivisors.SampleGraph.PathSystem

pathTruncate def

Pointwise truncation of continuous jet curves. · OAI.ForcedComputation.JetVolterra.pathTruncate

pathUnion def

Union of all arc labels along a specified full path. · OAI.LeanBlast.SecondKahnKalai.ProbabilityTree.pathUnion

pathValue def

Evaluate a length-two path b a in the original algebra. · OAI.LittleFinitistic.RelationBasisAction.pathValue

pathVector def

Path products viewed in the closed span itself. · OAI.IndependentProducts.pathVector

pathWeight def

Product of the manuscript's normalized local geometric weights. · OAI.Problem335.MomentPairing.pathWeight

pathWeight def

The product of the actual edge weights on the path from the root. · OAI.ReflexiveFixedPoints.WeightedTree.pathWeight

PathWithin def

Paths all of whose edges and vertices lie in a specified region. For the reflexive path the endpoint condition is supplied separately. · OAI.HarmonicArtin.ParabolicIntersections.PathWithin

Pattern inductive

Exact positive pattern: k forward calls, interleaved with arbitrary monomials. · OAI.ExactFourier.PositiveGeneration.Pattern

Pattern structure

Independent table entry types, with their coordinate supports and critical costs. The table model of Sections 2 and 4 is a concrete instance. · OAI.HyperbolicObstruction.SparsePattern.Pattern

Pattern abbrev

A pattern is a tuple of sequences of bounded binary rows, modulo eventual equality. The order later used on this type is fixed, not recomputed from rows. · OAI.PartitionConsistency.TailSelector.TailData.Pattern

pattern def

Evaluating at the standard representative is harmless: the preceding lemma proves independence of the representative. · OAI.PartitionConsistency.TailSelector.TailData.pattern

pattern def

The orthogonal projection specifying precisely the mismatch set F. It is used only for F contained in the counted support. · OAI.QAC.Count.pattern

PatternEquiv structure

Equality of the symbolic data, with an explicit position-preserving index map. · OAI.TwoPointCorrelations.LabeledPrimeWord.PatternEquiv

PatternFamilyClaim def

Source lem:pattern-family, with its fixed-before-focal-bits quantifiers. The source uses κ=2^100; this statement also covers every κ≥2. · OAI.MatroidProphet.PatternFamilyClaim

patternNumberFuel def

Count components recursively without constructing a single long flattened list. This keeps kernel reductions shallow. · OAI.CycleClique.Construction.patternNumberFuel

patternRelationIndex def

The address is computed from the converter's actual enumerated pattern. · OAI.MaxCutGames.Foundations.Complexity.FinalCNFPattern.patternRelationIndex

patternRelationIndex def

The address is computed from the converter's actual enumerated pattern. · OAI.UniqueGamesTheorem.Foundations.Complexity.FinalCNFPattern.patternRelationIndex

patterns def

All actual pivot-mark vectors over all interleavings and all movable marks. · OAI.MatroidProphet.MarkedPivots.patterns

patterns def

The enumeration visits only the fixed-size truth table, never proof assignments. · OAI.MaxCutGames.Foundations.PCP.VerifierToCNF.patterns

patterns def

The enumeration visits only the fixed-size truth table, never proof assignments. · OAI.UniqueGamesTheorem.Foundations.PCP.VerifierToCNF.patterns

patternTerm def

Genuine iterated tensors over k for a prescribed D/X pattern, ending in N. No ordinary tensor over D is used in these bar terms. · OAI.DerivedBar.Multibar.patternTerm

patternYoungConstant def

The Young coefficient associated with one active subset. · OAI.SLEExactGauge.Multipoint.patternYoungConstant

PaulsenSimilarityInput def

The quantitative published Paulsen input, specialized to a single bounded unital homomorphism. Paulsen, Completely bounded homomorphisms of operator algebras (1984), p. 226. It is an explicit hypothesis interface. · OAI.KadisonSimilarity.PaulsenSimilarityInput

pay def

Charge some work and optionally a new word bound. · OAI.PowerSaving.RAM.Bill.pay

payload def

Decode a plain frame's unique optional domain coordinate. · OAI.PureTypeSystem.ObservationValues.ActualPlainKey.payload

PayloadFits def

Absence is mandatory at inactive domains. There are no latent values. · OAI.PureTypeSystem.ObservationValues.PayloadFits

PaymentIndex abbrev

Outgoing and incoming exceptional-port incidences of selected top turns. · OAI.HyperbolicObstruction.FanLink.IndexedFanData.PaymentIndex

pbar def

Parameter antiholomorphic derivative in fixed physical coordinates, without changing the physical point or differentiating any moving plane coordinates. · OAI.ElasticityParameter.pbar

pCore def

The ambient image of the p-core of J. · OAI.PCoreCancellation.pCore

Pdir def

Minus the semiclassical conjugated Laplacian for the real/imaginary parts α,β of the phase. The last term vanishes exactly for null phases. · OAI.ElasticityNegativeOrder.Pdir

peakCells def

The old and new arrays may coexist during one extension, together with the recursive digit stack. All are linear in the ambient input length. · OAI.LowSpaceSubsetSum.PermutationArray.peakCells

peek def

Only read entries within 'wide'; other values are ignored. · OAI.PowerSaving.Slant.Zone.peek

peekChain def

Each statement reads only the head of one supplied polarity field. The other signature fields and the entire ambient-register component are copied. · OAI.MaxCutGames.Foundations.Hastad.SourceSignLoad.peekChain

peekChain def

Each statement reads only the head of one supplied polarity field. The other signature fields and the entire ambient-register component are copied. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceSignLoad.peekChain

peel def

token extraction (destination) · OAI.ExactDerandomization.Sub.Bow.peel

peelAt def

Peeling a genuine interior face adjacent to the outside deletes one paired edge and retains every other indexed relator boundary and label. · OAI.HyperbolicObstruction.WordPictures.WordPicture.peelAt

penalty def

The selected-root penalty of an actual integer free assignment. · OAI.ContingencyTables.RootedAnnealing.penalty

pendingSplitBranch def

Pending subtrees are visited in preorder. Depth-zero branches require no split coordinate, so they are never stored in the frontier. · OAI.Ostmann.pendingSplitBranch

pentagon def

The pentagon coloring: red edges have cyclic difference one. · OAI.CycleClique.Construction.pentagon

percolates def

The event that at least one open cluster is infinite. · OAI.CriticalPercolation.BondGraph.percolates

Perfect def

Exactly one selected edge is incident to every vertex. · OAI.MatchingFPRAS.Perfect

perfectBlockCuts def

Mark every entry of a perfect block, forcing its first run to start afresh. · OAI.TwoPointCorrelations.perfectBlockCuts

PerfectMatching abbrev

A perfect matching represented by its fixed-point-free involution. · OAI.KadisonSimilarity.PerfectMatching

PerfectPower def

The actual first rejection test in the appendix. · OAI.ExactQuantumFactoring.Primality.PerfectPower

PerfectRow def

Every slot in a perfect row is lit and belongs to a nonsingleton label. · OAI.TwoPointCorrelations.PerfectRow

perimeter def

De Giorgi perimeter, allowed to be infinite; no regularity of E is imposed. · OAI.CubicTorus.perimeter

periodBCF def

Every bounded, boundedly supported vector function has a bounded periodization on the discrete quotient F/Z. · OAI.PlaneFiveColor.InvariantMean.periodBCF

periodCentralSumConstant def

Explicit central-anchor summation constant for doubled period gaps. · OAI.TraceReconstruction.periodCentralSumConstant

periodDifferencePrefactor def

Explicit source rank-difference prefactor. It depends on the period count, slot length, derivative order and Bernoulli parameter, not the selected period tuple. · OAI.TraceReconstruction.periodDifferencePrefactor

PeriodGrid structure

A finite rectangle of complete level-j period blocks. · OAI.DoublingHilbert.PeriodGrid

periodicArc def

A major arc with all integer translates retained. Restricting this set to a unit interval handles the arc crossing the endpoint zero without loss. · OAI.Problem337.MinorArc.periodicArc

periodicBilinear def

Apply a continuous bilinear map pointwise to continuous periodic maps. · OAI.ClosedSurfaceR4.CovarianceCorrector.periodicBilinear

periodicEntryApprox def

The actual factorial-period approximation to one countable matrix unit. · OAI.KadisonKastler.FullMatrixCorner.periodicEntryApprox

PeriodicExistenceInput def

Periodic existence for continuation, as in Levermore–Masmoudi (2010), Theorem 4.1, including separate renormalized gain and loss integrability. · OAI.BoltzmannNonuniqueness.PeriodicExistenceInput

periodicFilledValue def

Fill on one sheet and translate on every other sheet. · OAI.CircleDomainRigidity.FiniteTransfer.periodicFilledValue

periodicGridKey def

The integer label of the half-open periodic grid cell containing a point. · OAI.Problem310.periodicGridKey

periodicKey def

The periodic key of a real point in a grid of mesh 1 / q. · OAI.Problem310.GridSeparation.periodicKey

periodicStage def

The full matrix stage of factorial size inside the containing algebra. · OAI.KadisonKastler.FullMatrixCorner.periodicStage

periodicVonNeumann def

The actual von Neumann algebra generated by the increasing periodic matrix stages. · OAI.KadisonKastler.FullMatrixCorner.periodicVonNeumann

periodize def

Periodize by a locally finite sum. Unlike a raw fract expression this also makes smoothness at the identified endpoints explicit. · OAI.ThreeTorus.periodize

periodizedReal def

The actual period-one lattice sum used to fold an unfolded temporal density. · OAI.ThreeTorus.periodizedReal

periodLift def

Literal quotient-valued periodization. · OAI.PlaneFiveColor.InvariantMean.periodLift

periodMap def

F/Z embeds in the ordinary circle; no ordinary topology is imposed on its source. · OAI.PlaneFiveColor.InvariantMean.periodMap

periodOccurrence def

A period-spaced suboccurrence of a fixed long occurrence. · OAI.Superstring.Counts.periodOccurrence

periodPerm def

A text period permutes, rather than fixes, the multiple aligned passages. · OAI.Superstring.Hierarchical.Threading.periodPerm

PeriodRankRegion def

The literal rank-clipping rectangles used in the manuscript, with doubled coarse gaps to avoid half-integer natural indices. Only the first and internal gaps are needed to determine the period tuple. · OAI.TraceReconstruction.PeriodRankRegion

periodSlot def

Actual defect slot in a period, with zero-based period and slot indices. · OAI.TraceReconstruction.periodSlot

periodSum def

Periodization is defined for all functions; bounded support below ensures that these sums are finite and uniformly bounded. · OAI.PlaneFiveColor.InvariantMean.periodSum

perm def

Insert exactly one separately indexed point after every point satisfying p. · OAI.HyperbolicObstruction.PictureSwitch.PermInsertion.perm

permAction def

The coordinate permutation action in the exact finite auxiliary group. · OAI.KaplanskyCounterexample.FiniteBlock.permAction

permanentPairing def

The normalized permanent of a pairing on two indexed families. · OAI.Problem346.permanentPairing

permanentPairingLeft def

The permanent pairing as a multilinear map in its left family. · OAI.Problem346.permanentPairingLeft

permanentPairingMultilinear def

Both families of the permanent pairing are multilinear. · OAI.Problem346.permanentPairingMultilinear

permanentPairingRight def

The permanent pairing as a multilinear map in its right family. · OAI.Problem346.permanentPairingRight

permFn def

Uniform radix on axes in layout J. · OAI.PowerSaving.Layout.permFn

permittedFaces def

Faces of the original simplex are exactly the full subposets of subdivision faces using only their permitted original vertices. · OAI.Thomason.Construction.SubdivisionBarycentric.permittedFaces

permPartition def

Cycle-type partition, with only the definitional `card (Fin n)=n` transport. · OAI.Saxl.permPartition

permutationContraction def

The coordinate formula for the trace contraction of a permutation tensor. Each index follows the successor convention in the manuscript. · OAI.KadisonSimilarity.permutationContraction

permutationContractionMultilinear def

The contraction is multilinear in its matrix variables. · OAI.KadisonSimilarity.permutationContractionMultilinear

permutationCoupling def

Insert one uniformly chosen location, then permute the remaining locations. The insertion bijection is Mathlib's explicit swap-based permutation code. · OAI.LowSpaceSubsetSum.permutationCoupling

permutationCycleCount def

The number of cycles of a permutation, with singleton cycles included. · OAI.KadisonSimilarity.permutationCycleCount

PermutationCycles abbrev

The finite set of cycles of a permutation, including fixed points. · OAI.KadisonSimilarity.PermutationCycles

permutationEquiv def

The full forward and inverse tables already prove the permutation laws. · OAI.MaxCutGames.Explicit.MachineOutputContract.permutationEquiv

permutationEquiv def

The full forward and inverse tables already prove the permutation laws. · OAI.UniqueGamesTheorem.Explicit.MachineOutputContract.permutationEquiv

permutationLift def

Coxeter's relations are sufficient, not just necessary: any adjacent family satisfying the three local laws gives a canonical finite permutation action. · OAI.FiniteAlgebraicEnvelopes.Adjacent.permutationLift

permutationMatrix def

A reversible Boolean operation, viewed as an exact permutation matrix. · OAI.ExactQuantumFactoring.permutationMatrix

PermutationTable structure

Finite forward and inverse tables make totality and invertibility explicit. An eventual external encoding only needs to serialize the forward table. · OAI.MaxCutGames.Foundations.Target.PermutationTable

PermutationTable structure

Finite forward and inverse tables make totality and invertibility explicit. An eventual external encoding only needs to serialize the forward table. · OAI.UniqueGamesTheorem.Foundations.Target.PermutationTable

permutationTest def

Distinct labels assigned to the vertices of a permutation give its test tuple of matrix units. · OAI.KadisonSimilarity.permutationTest

permutationUnit def

A permutation and its inverse form a unit in the full relation monoid. · OAI.TwoWayComplementation.permutationUnit

permute def

Ordinary tensor permutation, with the manuscript's product convention. · OAI.CoordinateSweeps.OrdinaryTensor.permute

permute def

Permute all members of a partition, preserving their cube tails. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Partition.permute

permute def

Label permutations act on the entire diagram, not on its monoid positions. · OAI.PartitionConsistency.DiagramForcing.Diagram.permute

permute def

The action fixes the chosen root diagram literally, and leaves the monoid index and ideal unchanged. · OAI.PartitionConsistency.LocalDiagramSystem.Vertex.permute

permute def

Simultaneous transport of labels in both coordinates, as in the paper. · OAI.PartitionConsistency.ProductForcing.permute

permute def

Permuting coordinates is unitary on ℓ², including for arbitrary index sets. · OAI.TorsionFreeBass.Regular.permute

permutedCharacter def

The three cyclic orders followed by their reflected orders. · OAI.MatrixMultiplication.AuxiliarySeparation.Character.permutedCharacter

permuteOrderIso def

The same label permutations used for diagrams are order automorphisms of Cohen forcing. They can therefore act diagonally on the product. · OAI.PartitionConsistency.CohenForcing.Condition.permuteOrderIso

permuteOrderIso def

The label action preserves and reflects the forcing order. · OAI.PartitionConsistency.DiagramForcing.Diagram.permuteOrderIso

permutePorts def

Relabeling time coordinates is a bijection on actual port words. · OAI.MaxCutGames.Foundations.PCP.PoweringWalks.permutePorts

permutePorts def

Relabeling time coordinates is a bijection on actual port words. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringWalks.permutePorts

permuteSigns def

Permutations act on diagonal signs by permuting their coordinates. · OAI.KadisonKastler.permuteSigns

perpCoordinate def

Squared norm of the distinguished perpendicular component. · OAI.SubsphereCurrent.Mixture.perpCoordinate

perpendicularPart def

Perpendicular displacement from the line in a unit direction. · OAI.Falconer.perpendicularPart

perSlotCleanup def

Retained global headers are excluded from cleanup. · OAI.MaxCutGames.Foundations.Hastad.SourceRuntimeModel.perSlotCleanup

perSlotCleanup def

Retained global headers are excluded from cleanup. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceRuntimeModel.perSlotCleanup

perturb def

A genuine inverse for an explicitly estimated perturbation of an invertible matching map. This is the Neumann step in source (4.15), with no bijectivity hypothesis on the perturbed map. · OAI.SmoothObstruction.BoundaryNeumann.perturb

perturbationHomeomorph def

A small globally Lipschitz perturbation of the identity is an actual bi-Lipschitz homeomorphism, obtained by the contraction theorem. · OAI.Bernoulli.Shape.perturbationHomeomorph

perturbedMean def

Total (unnormalized) pressure of the literal countable reservoir. The entropy restores the counting-spin partition function from uniform spins. · OAI.DilutedSpinGlass.SizeCoupling.perturbedMean

perturbLog def

Actual log weight after a finite family of perturbations of one type. The base includes the physical Hamiltonian and every other fixed type. · OAI.DilutedSpinGlass.KernelTower.perturbLog

PetersenPositiveRicciInput def

The one positive Ricci entry needed to show the Einstein constant is positive; it is the contraction of three strictly positive sectional terms. · OAI.PositiveEinsteinFour.Geometry.PetersenPositiveRicciInput

PetersenScalingInput def

Constant metric rescaling, in actual chart tensors and actual distance. No Einstein normalization or model conclusion is included. · OAI.PositiveEinsteinFour.Geometry.PetersenScalingInput

pfaffianFour def

The Pfaffian with respect to the fixed orientation of four coordinates. · OAI.PositiveEinsteinFour.QuaternionFrame.pfaffianFour

Phase abbrev

Primitive phases with stable layer indices. · OAI.Superstring.Hierarchical.Threading.Phase

phase def

Exact polygon phase, with clockwise orientation as in the forward shift. · OAI.ExactQuantumFactoring.OrderTrial.phase

phase def

Character with cycles-per-unit real frequency. · OAI.FourierLLogL.Period.phase

phase def

The exponent occurring in the finite Fourier projection. · OAI.MatrixMultiplication.AuxiliarySeparation.phase

phase def

Rotational scalar on a weight block. · OAI.PolycyclicRecognition.ComplexCartan.phase

phase def

The standard real-frequency complex phase. · OAI.Problem337.HyperbolaCorrelation.phase

phase def

Period-one complex phase, with the same normalization as the standard cyclic character and the circle-method Fourier sum. · OAI.Problem337.RealPhase.phase

phase def

Positive-sign integral Fourier phase on the unit interval. · OAI.Problem337.ThreePrimeContinuousFourier.phase

phase def

The standard real additive phase, with the 2π convention. · OAI.Problem337.VaughanBilinear.phase

phase def

Necessary analytic transfer from §2. We use finite positive measures for the comparison form; the eventual quadratic inequality also handles real signed linear combinations, without imposing spacing on the configuration. · OAI.TriangularUniversal.phase

phaseApprox def

Continuous lower approximations to phase on a unitary spectrum. · OAI.KervaireSpectral.phaseApprox

phaseBin def

A canonical finite bin index for the phase of an integer. · OAI.Problem337.RationalPhaseSpacing.phaseBin

phaseChar def

Wrap grid characters as an AddChar. · OAI.FourierLLogL.Dyadic.phaseChar

phaseCLM def

The real-linear complex phase functional. · OAI.ElasticityCarleman.phaseCLM

phaseCoefficientInjection def

Fixed continuous linear injection from angular coefficients to the lower-left phase block. It does not depend on histories or parameters. · OAI.HarmonicCounterexample.LinearODE.phaseCoefficientInjection

phaseConfigs def

The two physical configurations used for observing one phase. · OAI.ContingencyTables.CompanionObservationQueriesProgram.phaseConfigs

phaseConstant def

The manuscript's absolute phase constant 2⁻⁶⁰. · OAI.LipschitzHilbert.PhaseCombinatorics.phaseConstant

phaseEquiv def

Every primitive phase and aligned passage occurs exactly once. This is the finite bookkeeping behind splitting a long turn into primitive layers. · OAI.Superstring.Hierarchical.Threading.phaseEquiv

phaseFrame def

Finite phase average of actual pure product states. · OAI.CoordinateSweeps.OrdinaryTensor.phaseFrame

phaseFrequencies def

The recursive choices allow any finite lower bound depending on the already selected frequencies. · OAI.ThreeTorus.phaseFrequencies

phaseGoodBlock def

Good derivative indices in the integer cell labelled by k. · OAI.Problem337.phaseGoodBlock

phaseInTime def

Literal execution of the fixed program; all six input fields are preserved. · OAI.MaxCutGames.Explicit.MachineSubdivisionDynamicRows.phaseInTime

phaseInTime def

Exact physical row execution, preserving the five saved fields and restoring scratch. This is a phase theorem; it does not assume its own execution. · OAI.MaxCutGames.Explicit.MachineSubdivisionRows.phaseInTime

phaseInTime def

Timed witness with the exact sum of the five concrete phase counts. · OAI.MaxCutGames.Reduction.MachineTemplateAddress.phaseInTime

phaseInTime def

Literal execution of the fixed program; all six input fields are preserved. · OAI.UniqueGamesTheorem.Explicit.MachineSubdivisionDynamicRows.phaseInTime

phaseInTime def

Exact physical row execution, preserving the five saved fields and restoring scratch. This is a phase theorem; it does not assume its own execution. · OAI.UniqueGamesTheorem.Explicit.MachineSubdivisionRows.phaseInTime

phaseInTime def

Timed witness with the exact sum of the five concrete phase counts. · OAI.UniqueGamesTheorem.Reduction.MachineTemplateAddress.phaseInTime

PhaseLaurent def

An actual finite expansion; every coefficient is proved measurable and bounded. The finite coefficient type can vary with the expression. · OAI.ThreeTorus.PhaseLaurent

phaseMeasure def

Exactly normalized area in the open table times uniform angular measure. · OAI.TriangularBilliards.phaseMeasure

phaseOracle def

Predicate phase kickback is a clean oracle, two S gates, and the same clean XOR oracle again, all over the fixed gate alphabet. · OAI.ExactQuantumFactoring.BooleanNetwork.phaseOracle

phaseOverlapSum def

Unnormalized correlation of the exactly prepared quarter-step state. · OAI.ExactQuantumFactoring.OrderTrial.phaseOverlapSum

phasePolynomial def

Polynomial formed from the actual per-row and counted-loop transition budgets. Fixed degree parameters determine its coefficients. · OAI.MaxCutGames.Foundations.Complexity.MachineOverlayTable.phasePolynomial

phasePolynomial def

Polynomial formed from the actual per-row and counted-loop transition budgets. Fixed degree parameters determine its coefficients. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineOverlayTable.phasePolynomial

PhasePositive def

Nonnegative multiples of an exact endpoint phase. · OAI.CriticalHoneycomb.Transfer.PlanarNormalization.PhasePositive

PhaseSpace abbrev

Finite independent root phases. The modulus is strictly larger than any occupancy, so it distinguishes every histogram without an analytic limit. · OAI.CoordinateSweeps.OrdinaryTensor.PhaseSpace

phaseSum def

The finite polynomial before the extra factor of the fast scale. · OAI.ClosedSurfaceR4.PeriodicExpansion.phaseSum

phaseSumRe def

The real coordinate of the seven-ramp spline derivative summed exactly. · OAI.ExactQuantumFactoring.OrderTrial.phaseSumRe

phaseSystem def

The actual conjugated differential operator, with a local cutoff that is identically one on the support, not an abstract operator hypothesis. · OAI.ElasticityNegativeOrder.phaseSystem

phaseTest def

A finite Laurent expression, with coefficients depending measurably on its original circle variable. · OAI.ThreeTorus.phaseTest

phaseTimePolynomial def

Header, at most N rows with guards, and the final output reversal. · OAI.MaxCutGames.Foundations.Complexity.FinalCNFMachine.Program.phaseTimePolynomial

phaseTimePolynomial def

Header, at most N rows with guards, and the final output reversal. · OAI.UniqueGamesTheorem.Foundations.Complexity.FinalCNFMachine.Program.phaseTimePolynomial

phaseTriangle def

A continuous triangular cutoff on the chosen angle interval. · OAI.KervaireSpectral.phaseTriangle

phaseUnitary def

In a fixed eigenbasis of B, the unitary exp(itB). · OAI.SharpLiebThirring.MatrixProof.phaseUnitary

phaseValue def

Holevo value of the explicit 2^n equiprobable product signals. · OAI.GAD.phaseValue

phaseVector def

The genuine Cartan rotations belong to this subgroup. · OAI.PolycyclicRecognition.ComplexCartan.phaseVector

phaseVelocity def

The derivative generator has the normalization used for fractional phase. · OAI.KervaireTraceCalculus.phaseVelocity

phaseWord def

Exit window of a sorted layer at a specified primitive phase. · OAI.Superstring.Hierarchical.Threading.phaseWord

phaseWordBase def

The base word after the independent diagonal phase. · OAI.KadisonKastler.phaseWordBase

phi def

The actual cutoff profile, with a separate value at infinite distance. · OAI.MetricEntropyDuality.LogProfile.phi

phi def

This declaration is an exact comparator-selected statement. Read the Lean statement below for its mathematical content. · OAI.SpinAngle.TypeAngle.phi

phi def

The actual minimum over all nonempty proper cuts, not a relaxation. · OAI.UniformSparsestCut.Graph.phi

physical def

Forget all auxiliary marks. · OAI.DilutedSpinGlass.HeterogeneousMarks.physical

physical def

The conjugated physical stress-divergence, without treating its shifted derivatives as derivations. · OAI.ElasticityPhysicalAlgebra.physical

physicalAngularError def

The original positive error sum evaluated at a physical point. · OAI.LipschitzHilbert.SourceLabeledLattice.physicalAngularError

physicalAug def

The scalar component is the actual shifted divergence. · OAI.ElasticityPhysicalEstimate.physicalAug

physicalBandOnlineDraw def

The exact action sampler retains the order of the literal physical band. · OAI.EditApproximation.physicalBandOnlineDraw

physicalChildMass def

The old or genuinely delayed proper-child projection cost, in the same physical scheduled covariance family as the parent. · OAI.DilutedSpinGlass.UniversalDictionary.physicalChildMass

physicalCircuitMatrix def

Chronological order: the rightmost matrix acts first. · OAI.QAC.physicalCircuitMatrix

physicalCoefficient def

The actual canceled-anchor covariance coefficient of one independent site/mark insertion into the complete physical Poisson reservoir. · OAI.DilutedSpinGlass.ConcreteReservoir.physicalCoefficient

PhysicalColumn def

Quantified actual physical CGO corrections on the ball, with the full physical divergence in H1. This is a conclusion, never an input assumption. · OAI.ElasticityCGO.PhysicalColumn

physicalCoordinateEquiv def

Every finite enumeration of the large cells gives the same physical coordinate space, with each original doubled cell appearing once. · OAI.ContingencyTables.LargePairEnumeration.physicalCoordinateEquiv

physicalError def

The exact zero-trace correction, not an assumed quasimode remainder. · OAI.Elasticity.physicalError

physicalFamilyCoarseBitWork def

Bit arithmetic is executed once for each fresh coarse source request. · OAI.EditApproximation.physicalFamilyCoarseBitWork

physicalFamilyCoarseInputWork def

Source scans and a constant setup charge for each first coarse request. · OAI.EditApproximation.physicalFamilyCoarseInputWork

PhysicalFiniteFunctional def

Representation of the physical bridge mass by the finite strip functional. · OAI.CriticalHoneycomb.FiniteFunctional.PhysicalFiniteFunctional

physicalHistoryCovariance def

The actual charge-weighted, sharing-constrained overlap history in the complete prescribed-tree physical reservoir, centered by its old test. · OAI.DilutedSpinGlass.UniversalDictionary.physicalHistoryCovariance

physicalLM def

Exact shifted stress-divergence as a linear map, for duality on its range. · OAI.ElasticityPhysicalDual.physicalLM

physicalLocalMedianUpdateTable def

The executable local pass: scale and round each full-copy value, take the upper median, and clamp each result to the common finite seed type. · OAI.EditApproximation.physicalLocalMedianUpdateTable

physicalMatrixCovariance def

Centered exact-matrix multioverlap history of the actual physical root. The root alphabet includes every sampled marker, and its entire old test is retained. This definition uses no analytic decorrelation identity. · OAI.DilutedSpinGlass.UniversalDictionary.physicalMatrixCovariance

physicalMediatorHamiltonian def

Central antiferromagnetic bonds with their explicit three-Delta singlet shifts, plus the physical original-spin and spoke interactions. · OAI.ContinuumCoulomb.physicalMediatorHamiltonian

physicalMediatorPenalty def

The physical central-bond sum, with its singlet scalar shifts. · OAI.ContinuumCoulomb.physicalMediatorPenalty

physicalMiddleRegion def

The actual physical middle strip in the source's local coordinates. · OAI.SLEExactGauge.RibbonCoordinates.physicalMiddleRegion

PhysicalParameters abbrev

Phase, twelve translations, and logarithmic blowup time. · OAI.DefocusingNLS.PhysicalParameters

physicalParentStoppedBoundAt def

The exact physical stopped-path statement at one manuscript scale. · OAI.LipschitzHilbert.PacketLatticeData.physicalParentStoppedBoundAt

physicalParentStoppedBoundAt def

The exact physical stopped-path statement at one manuscript scale. Separating its dependent data from the eventual proof keeps elaboration local. · OAI.LipschitzHilbert.SourceLabeledLattice.physicalParentStoppedBoundAt

physicalPassed def

The on-support passing event in the retained concrete physical registers. An efficient retrospective Boolean validator is still needed before using it as a circuit-controlled phase; this definition does not provide one by fiat. · OAI.ExactQuantumFactoring.NodeMachine.physicalPassed

physicalPassed def

The on-support passing event in the retained concrete physical registers. An efficient retrospective Boolean validator is still needed before using it as a circuit-controlled phase; this definition does not provide one by fiat. · OAI.ExactQuantumFactoring.SplitMachine.physicalPassed

physicalPath def

Forget the selected auxiliary mark rows. · OAI.DilutedSpinGlass.physicalPath

physicalPolarL2 def

The pullback is a genuine L² isometry, including for nonsmooth completed data. · OAI.DefocusingNLS.physicalPolarL2

physicalPolarTest def

An L¹ polar test is continuous on the actual physical C0 realization. · OAI.DefocusingNLS.physicalPolarTest

physicalPole def

The actual nearest pole is 64/9, with every third pole cancelled. · OAI.CriticalHoneycomb.FiniteFunctional.physicalPole

physicalProgramDrawSelections def

The actual complete selection program evaluated on its global array. · OAI.EditApproximation.physicalProgramDrawSelections

physicalProgramOnlineGuard def

The literal online query uses its one rounded center at each allowed scale, with the scale window tested against the literal initial table. · OAI.EditApproximation.physicalProgramOnlineGuard

physicalProgramRefresh def

Refresh exactly one online band and one shared whole-cell coordinate. · OAI.EditApproximation.physicalProgramRefresh

physicalProgramResampleOnline def

Replacing exactly one parent's arrays leaves every other raw parent selection function unchanged. The replacement arrays retain their literal finite coordinate types. · OAI.EditApproximation.physicalProgramResampleOnline

physicalProgramSelections def

The actual complete selection interface contains exactly the finite online and shared whole-cell candidates produced by the streams. · OAI.EditApproximation.physicalProgramSelections

physicalProjection def

The actual physical solution with the same trace as the trial field. · OAI.Elasticity.physicalProjection

physicalResidual def

The variational residual, acting on all zero-trace test fields. · OAI.Elasticity.physicalResidual

physicalSchwartzTest def

A Schwartz test acts continuously on the physical C0 realization. · OAI.DefocusingNLS.physicalSchwartzTest

physicalShapeAverage def

Regular depth-cube average of the literal physical covariance energy. · OAI.DilutedSpinGlass.UniversalDictionary.physicalShapeAverage

physicalShapeCovariance def

Literal exact prescribed-shape covariance, using the original physical site mean at every signed extension of the complete tilted reservoir. · OAI.DilutedSpinGlass.UniversalDictionary.physicalShapeCovariance

physicalSingletonEnergy def

The manuscript's singleton conditional spatial covariance density, under the actual tilted reservoir, averaged over roots and the d-coordinate prefix. · OAI.DilutedSpinGlass.UniversalDictionary.physicalSingletonEnergy

physicalStoppedBoundAt def

The exact physical stopped-path statement at one manuscript scale. Separating its dependent data from the eventual proof keeps elaboration local. · OAI.LipschitzHilbert.SourceLabeledLattice.physicalStoppedBoundAt

physicalSynthesis def

The finite physical adjoint sum over the original labeled packets. · OAI.LipschitzHilbert.SourceLabeledLattice.physicalSynthesis

PhysicalTerminalDisks structure

Actual terminal disks, including their smooth boundary parametrizations. The specified neighborhoods are in the quotient sphere. · OAI.CircleDomainRigidity.FiniteTransfer.PhysicalTerminalDisks

PhysicalTest structure

A real bounded physical test. The dense test family will be constructed explicitly; this structure only records the hypotheses of the one-stage lemma. · OAI.ThreeTorus.PhysicalTest

physicalToTorus def

The coordinate change is a genuine isometry on the full L2 spaces. · OAI.ThreeTorus.physicalToTorus

physicalTreeMatrixCovariance def

Literal centered history for the multileaf old overlap, with the actual variable-marker root alphabet. · OAI.DilutedSpinGlass.UniversalDictionary.physicalTreeMatrixCovariance

physicalTreeSelectorValue def

The original physical packet sum, before restriction to its common angular-support set. · OAI.LipschitzHilbert.SourceLabeledLattice.physicalTreeSelectorValue

physicalTupleSum def

Grouping at each distinct prime preserves the original physical product, including when a prime occurs in several different roles. · OAI.Ostmann.physicalTupleSum

PhysicalWord structure

Fixed chronological smooth control data and a nonempty compact parameter ball. Slot count is positive; a globally zero family is not substituted. · OAI.HarmonicCounterexample.LinearODE.PhysicalWord

pi def

Independent product of finitely many finite conditional laws. · OAI.DilutedSpinGlass.FiniteLaw.pi

pi def

Finite direct sums represented by products, as used in the plane localization. · OAI.KLInvariance.MomentGraph.Sheaf.pi

pi def

Fresh independent draws, allowing a different finite law at each index. · OAI.LowSpaceSubsetSum.FiniteLaw.pi

pi def

Independent fresh seeds for a finite array of calls. Each coordinate may have a different output type and a different number of internal attempts. · OAI.LowSpaceSubsetSum.SamplerCoupling.pi

pickPair def

A deterministic pair selection using only pre-launch data. Classical choice here is a fixed function of the finite eligible-pair set, not of the fresh noise. · OAI.BoundedTreewidthL1.ParticleSchedule.pickPair

picture def

All endpoints are furnished by literal graph paths, not by equality of labels. Inserting the single exterior marker gives an actual finite picture. · OAI.TorsionFreeZeroDivisors.WordBlocks.LiftedMatching.picture

PictureMap structure

Oriented polygonal side-pairings (darts are individually indexed sides). face turns around each polygon; edge pairs sides with opposite orientation. Vertices are the cycles of face * edge, so no abstract vertex incidences or Euler … · OAI.HyperbolicObstruction.PictureSwitch.PictureMap

pictureSubgroup def

Elements represented by actual finite genus-zero diagrams. The subgroup and its normality are proved by gluing, reflecting and attaching a whisker. · OAI.HyperbolicObstruction.WordPictures.pictureSubgroup

Piece abbrev

A complete bounded incidence piece consists of its valuation class and both vertices of every bit pair used by this block. · OAI.WLIdentification.AffineBlocks.Piece

piece def

The actual degree-n vector space of a submodule, expressed in the fixed ambient homogeneous piece. · OAI.KLInvariance.Hilbert.piece

piece def

The span of the monomials of integer source weight at most d. · OAI.StableCoordinate.Filtration.piece

piece def

Images of homogeneous pieces in the actual ideal quotient. · OAI.StableCoordinate.HomogeneousQuotient.piece

piece def

The image definition of a weight piece in a homogeneous principal quotient. · OAI.StableCoordinate.HomogeneousQuotientWeight.piece

piece def

The cone piece on one original simplex of the ordinary nerve. · OAI.Thomason.Construction.ConePrism.piece

piece def

The prism on an original simplex, mapped into the global simplicial cylinder. · OAI.Thomason.Construction.RealizedCylinder.piece

piece def

Transport of a local piece to the right Gray cylinder. · OAI.Thomason.Construction.RelativePrismData.piece

PieceLaw def

Law of a local piece when normalization is its positive mean; used as proof-carrying relation to any specified law. · OAI.FourierLLogL.Dyadic.Thermal.PieceLaw

pieceMap def

The quotient map, restricted to one homogeneous piece. · OAI.StableCoordinate.HomogeneousQuotient.pieceMap

pieceSupport def

Each target piece is supported both where the target was supported and where its partition coefficient is supported. · OAI.ClosedSurfaceR4.PhasePartitions.CompactPhasePartition.pieceSupport

piMultiplier def

Literal unit-valued multiplier of the degree-nine marked theta product. · OAI.Nagata.W06.TorusConfiguration.piMultiplier

Pin structure

Pin data: stored subspaces of directions with assigned images. We store a full linear map for convenience; only its restriction to the pin directions at each place is used. Counts charge only for the restrictions. · OAI.HadwigerCounterexample.Pin

pin def

Marked vertices carry the optional prescribed root. Internal vertices never gain spurious pins. · OAI.TorsionFreeZeroDivisors.ChainRealization.pin

pin def

At most one prescribed root. All other vertices remain unpinned. · OAI.TorsionFreeZeroDivisors.RootPins.pin

pinCandidates def

Arbitrary fixed named fields, not pairwise-distinct numerical letters. · OAI.HyperbolicObstruction.TableModel.pinCandidates

pinchingFrameVolumeRatio def

The volume loss from the radial normalization ball to the mass-test ball. · OAI.RegularPointCoordinates.pinchingFrameVolumeRatio

pinchingMassMatrix def

These matrices retain the full original Bochner measure and use the mean at a fixed dyadic offset. Their mass radius is one third of that mean radius. · OAI.RegularPointCoordinates.RegularBilinearData.pinchingMassMatrix

pinchingMassUnitFactor def

The original pinching radius is a fixed multiple of the mean radius. · OAI.RegularPointCoordinates.pinchingMassUnitFactor

pinchingMatrix def

Supermatrix of the sum of rank-one pinching maps for a vector family. · OAI.MUB6.Weiner.pinchingMatrix

pinConstant def

Field and bond terms depending only on fixed sites. They cancel from all normalized laws, but recording them makes the elimination exact. · OAI.BufferedIsing.pinConstant

pinExtension def

Extension by the actual common pins, in contrast to the deleted zero-field comparison graph. · OAI.BufferedIsing.pinExtension

pinExtensionEquiv def

Fixing sites is exactly elimination of their spin variables. This finite bijection does not replace their interactions by deletion. · OAI.BufferedIsing.pinExtensionEquiv

pinField def

The finite-field contribution induced by the actual prescribed spins. Neither these induced fields nor the original fields survive in qZero. · OAI.BufferedIsing.pinField

pinMean def

Conditional mean at a fixed-spin site; defined on actual remaining spins. · OAI.BufferedIsing.DSS.pinMean

piOneTransport def

Transport the actual cubical first homotopy group along a path. · OAI.Thomason.Construction.HomotopyMaps.piOneTransport

pipelineBound def

A fixed fivefold composition of explicit polynomials. · OAI.ContinuumCoulomb.QuantumOrderedXZ.pipelineBound

piSection def

The genuine holomorphic marked section, typed with the literal unit multiplier Q. · OAI.Nagata.W06.TorusConfiguration.piSection

piSingleEquiv def

A finite product with only one possibly nonzero factor is that actual factor. · OAI.LittleFinitistic.BimoduleTotal.piSingleEquiv

pitch def

dry, single-bit price · OAI.FourierLLogL.Dyadic.Procedure.Fork.pitch

pivot def

The largest element of bad P s, or bottom if there are none. · OAI.FiniteChain.pivot

pivot def

This is exactly (v * right - w * left) / s, with integer clearing of all previously inserted denominators. · OAI.Ostmann.ClearedHistoryValue.pivot

pivot def

Reversing one elimination. With no lower bound choose the least upper bound; with no bounds choose zero, exactly as in the source. · OAI.UniformKServer.FourierMotzkin.pivot

pivotConstituentEquiv def

A single pivot atom contributes exactly its original number of prime constituents, even after the earlier copy steps. · OAI.Ostmann.pivotConstituentEquiv

PivotControl def

The quantitative cancellation test at every actual internal node. · OAI.Ostmann.MovingSlotData.PivotControl

PivotDependencyScheme structure

Only ancestors occur in a coefficient; their depth is strictly smaller. The order is the preorder in which split products are exposed. · OAI.Ostmann.PivotDependencyScheme

pivotDiagonal def

The diagonal determined by the pivots. The predecessor sum has at most one term; its use also covers the first node without a separate convention. · OAI.PositiveEinsteinFour.Jacobi.pivotDiagonal

pivotEquiv def

Isolate any actual path edge. Previously traversed ports are replaced by their arrival ports, so the change of coordinates remains invertible. · OAI.MaxCutGames.Foundations.PCP.PoweringWalks.pivotEquiv

pivotEquiv def

Isolate any actual path edge. Previously traversed ports are replaced by their arrival ports, so the change of coordinates remains invertible. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringWalks.pivotEquiv

placedInTime def

Static placements of the complete computation into a larger caller preserve every complement tape and consume exactly the same runtime budget. · OAI.MaxCutGames.Foundations.Complexity.MachinePaddedExpanderFamily.placedInTime

placedInTime def

Direct embedding of the entire helper into another finite program. · OAI.MaxCutGames.Foundations.Complexity.MachineRegularFamily.placedInTime

placedInTime def

Static placements of the complete computation into a larger caller preserve every complement tape and consume exactly the same runtime budget. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachinePaddedExpanderFamily.placedInTime

placedInTime def

Direct embedding of the entire helper into another finite program. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRegularFamily.placedInTime

placedLoadUnaryInTime def

A caller shares this complete tape arena and state, mapping only labels and the final continuation. All saved occurrence IDs remain available to its body. · OAI.MaxCutGames.Reduction.MachineSourceTuple.placedLoadUnaryInTime

placedLoadUnaryInTime def

A caller shares this complete tape arena and state, mapping only labels and the final continuation. All saved occurrence IDs remain available to its body. · OAI.UniqueGamesTheorem.Reduction.MachineSourceTuple.placedLoadUnaryInTime

placedLookupInTime def

Same-tape placement into the caller's finite control. Choosing the ambient type to be τ × Unit gives the shared state (τ × Unit) × Option Bool used by the field-template machinery. The lookup's final transition enters exit. · OAI.MaxCutGames.Reduction.SourceEquationLookup.placedLookupInTime

placedLookupInTime def

Same-tape placement into the caller's finite control. Choosing the ambient type to be τ × Unit gives the shared state (τ × Unit) × Option Bool used by the field-template machinery. The lookup's final transition enters exit. · OAI.UniqueGamesTheorem.Reduction.SourceEquationLookup.placedLookupInTime

placementHSsq def

Counting-measure Hilbert--Schmidt square on the actual free placements. · OAI.CoordinateSweeps.Grid.Holes.placementHSsq

placementKernel def

The path-weight alternating placement matrix Q, with counting measure and s, not s-h, in its normalization, exactly as 04-sparse:eq7. · OAI.CoordinateSweeps.Grid.Holes.placementKernel

plain def

Undecorated full tree wiring. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Diagram.plain

plain def

A scaled lookup; used for monomial slots. · OAI.PowerSaving.Slant.Stroke.plain

plainApply def

Plain application, separated to keep dependent prefix casts abstract. · OAI.PureTypeSystem.ObservationValues.plainApply

plainGate abbrev

No inverse/control flag is used in these elementary fresh-bit operations. · OAI.ExactQuantumFactoring.plainGate

plainLambda def

The actual plain prefix operation, with the unique optional payload. · OAI.PureTypeSystem.ObservationValues.plainLambda

Planar def

Planarity means existence of a crossing-free topological plane embedding. · OAI.Barnette.Planar

planarDensity def

The vertical primitive density, written with squared norms so smoothness at the zero is explicit for integral m. · OAI.SymmetricMahler.planarDensity

PlanarDiagram structure

A finite genus-zero comparison diagram, with its edge type retained. · OAI.KaplanskyCounterexample.Planar.PlanarDiagram

planarEmbed def

The planar sublattice used only for the elementary strict upper threshold. · OAI.CriticalZ3.planarEmbed

planarError def

A concrete nonnegative error function, valid also for every smaller m≥2. · OAI.SymmetricMahler.planarError

planarFormIntegral def

Real line integral of P dx + Q dy, with parameter interval [0,1]. · OAI.CircleDomainRigidity.planarFormIntegral

planarFourierSchwartz def

The planar transform remains Schwartz, with the product norm. · OAI.LipschitzHilbert.planarFourierSchwartz

PlanarJetOracle structure

The numerical information supplied with the effectively specified input planar field. All bounds are global input-derivative bounds. · OAI.ForcedComputation.VelocityDetector.PlanarJetOracle

planarLeftStrip def

The negative outer cell of a planar affine strip, with null ties omitted. · OAI.Problem358.planarLeftStrip

planarLevelChart def

Restrict ambient regular-level coordinates to an actual real chart of the level subtype. · OAI.CircleDomainRigidity.planarLevelChart

PlanarMap def

Genus zero for every connected component, expressed without an embedding or an unproved Jordan-curve theorem. This uses the universal Euler upper bound. · OAI.Release075.PermCycles.PlanarMap

planarPart def

The finite-coordinate part of a sphere set. · OAI.CircleDomainRigidity.planarPart

planarPrimitive def

The original vertical integral, before its equality with the radial integral. · OAI.SymmetricMahler.planarPrimitive

planarRightStrip def

The positive outer cell of a planar affine strip, with null ties omitted. · OAI.Problem358.planarRightStrip

PlanarScalarCoarea def

Federer (1969), §3.2.11, specialized to a real C¹ function on an open subset of the Euclidean plane. · OAI.CircleDomainRigidity.PlanarScalarCoarea

PlanarSchoenflies def

Published input: the planar Schoenflies extension theorem in exactly its circle-to-plane form. · OAI.CircleDomainRigidity.PlanarSchoenflies

PlanarSpan inductive

Finite linear combinations of sliced planar diagrams with no incoming occupied strand. The count equality fixes the exact frontier arity. · OAI.CriticalHoneycomb.Transfer.PlanarNormalization.PlanarSpan

planarStrip def

A central strip in the square, written in slice coordinates. · OAI.Problem358.planarStrip

planarSynthesisSchwartz def

The Schwartz-map representative of the manuscript's fixed synthesis. · OAI.LipschitzHilbert.planarSynthesisSchwartz

PlanarVariations structure

The first two variational equations for a transition map. The second time argument is elapsed time; the first is its initial phase. · OAI.ForcedComputation.PlanarVariations

PlanarWord inductive

A sliced planar diagram, with all vacant strands suppressed. The local moves are exactly cup creation and adjacent cap contraction; any number of identity tiles can be inserted without changing this word. · OAI.CriticalHoneycomb.Transfer.PlanarNormalization.PlanarWord

Plane abbrev

The Euclidean plane, not the product space with its max norm. · OAI.CenteredDiskEndpoint.Plane

Plane abbrev

The fixed reference plane, with its induced topology and measurable structure. · OAI.Problem358.MovingPlaneEnvelope.Plane

Plane abbrev

The Euclidean plane, with its Euclidean norm. · OAI.SignedDisk.CenteredDiskEndpoint.Plane

Plane abbrev

The Euclidean plane, not the product space with its max norm. · OAI.SignedDisk.CenteredDiskEndpoint.Plane

plane def

The selected row and column have one shared coordinate. Their complement is exactly the Cartesian product of the two unshared data sets. · OAI.ExactFourier.OnePort.plane

PlaneBasis structure

Canonical angular endpoints of a plane; no Coxeter-system assertion is built into this data. The latter will be proved from the inversion criterion. · OAI.KLInvariance.TitsSpace.PlaneBasis

planeConormalChartTransport def

The exact inverse image of the global conormal on this chart. · OAI.SourcePullbackChart.planeConormalChartTransport

planeCoordinate def

Ordered affine coordinates of an actual complex-plane point. · OAI.Nagata.Workers.W17.planeCoordinate

planeCoordinateIsometry def

The two coordinate conventions used by the geometric and jet formulas. · OAI.ClosedSurfaceR4.JetPolynomial.planeCoordinateIsometry

planeCover def

The exact three standard plane charts. · OAI.SourcePullbackChart.planeCover

planeCurveEquationEquiv def

Full equivalence between genuine positive effective ideal cycles and all positive-degree projective equation classes. Repeated components are preserved. · OAI.Nagata.Workers.W10.planeCurveEquationEquiv

planeDefect def

The finitely supported change in root inversions, with the distinguished root removed, grouped by actual root planes. · OAI.KLInvariance.TitsSpace.planeDefect

PlaneEmbedding structure

A topological plane drawing of a simple graph, with no crossings. · OAI.Barnette.PlaneEmbedding

planeEmbedding def

The transverse plane in the physical normalization. · OAI.RuledApproximation.planeEmbedding

PlaneEquationClass abbrev

Nonzero homogeneous equations modulo nonzero complex scalar. · OAI.Nagata.W16.PlaneEquationClass

planeEval def

Evaluation in two actual polynomial indeterminates. · OAI.Nagata.W01.planeEval

planeEval def

Evaluation of an actual bivariate polynomial at affine coordinates (U,Y). · OAI.Nagata.W06.LineArrangement.planeEval

PlaneForms abbrev

The ordinary vector space of homogeneous plane forms of degree n. · OAI.Nagata.Workers.W24.PlaneForms

planeGenerator def

Generator with xᵢ'=xⱼ and xⱼ'=-xᵢ. · OAI.InvariantIsing.planeGenerator

planeHomeomorph def

The geometric identification of any two planes transverse to the same direction. · OAI.Problem358.MovingHyperplaneChart.planeHomeomorph

PlaneJet abbrev

Point and Taylor-order indices, allowing a different required order at each point, including order zero. · OAI.Nagata.W27.PlaneJet

planeJetExponent def

The actual two-variable exponent selected by a finite plane jet row. · OAI.Nagata.W27.planeJetExponent

planeJetMatrix def

The actual finite matrix of affine jet expressions. · OAI.Nagata.W27.planeJetMatrix

planeLength def

The actual finite count of ambient inversions belonging to a root plane. No Coxeter presentation of the plane subgroup is assumed here. · OAI.KLInvariance.TitsSpace.planeLength

planeLift def

The fixed orthonormal-plane inclusion, written as a continuous linear map. · OAI.ClosedSurfaceR4.VelocityFrame.planeLift

planeMap def

The original plane map sends x to u^p and y to v^p+1; its inclusion in the original five-root field is exactly the source one. · OAI.KummerNode.planeMap

planeMeasure def

Canonical volume on a subspace, translated into the ambient space. · OAI.Problem358.IntrinsicSurface.planeMeasure

PlaneMonomial abbrev

The affine monomials of total degree at most d, indexing the usual degree-d homogeneous forms after setting the third coordinate to one. · OAI.Nagata.W27.PlaneMonomial

planeNonzeroEquiv def

Nonzero elements of a plane are precisely its ambient nonzero vectors. · OAI.KaplanskyCounterexample.FiniteGeometry.planeNonzeroEquiv

planeNormalOnLine def

The literal normal polynomial restricted coefficientwise to component i. · OAI.Nagata.Workers.W24.planeNormalOnLine

planePoint def

The ordinary affine point (x,y). · OAI.Nagata.Workers.W30.planePoint

planePolynomialEquiv def

The actual Rees chart is the affine line over its actual plane chart. · OAI.SourcePullbackChart.planePolynomialEquiv

planePolynomialEval def

Evaluate an actual two-variable polynomial on the complex affine plane. · OAI.Nagata.Workers.W28.planePolynomialEval

PlanePrimeComponent def

An actual homogeneous nonzero prime principal ideal of the ternary coordinate ring. No equation or multiplicity is assumed in this type. · OAI.Nagata.W15.PlanePrimeComponent

planeRotation def

A genuine orthogonal rotation; its formula is the usual two-plane rotation, not an assumed measure-invariant flow. · OAI.AffineBernstein.planeRotation

planeRotation def

The exponential of the plane generator is orthogonal. · OAI.InvariantIsing.planeRotation

PlaneSimple def

The canonical simple-root criterion stated on the actual ambient roots. · OAI.KLInvariance.TitsSpace.PlaneSimple

PlaneSpatialBounds def

Only the orders needed for local tail and H² estimates are retained. The bound may depend on the finite time interval and the coefficients. · OAI.ForcedComputation.VelocityDetector.PlaneSpatialBounds

planeSphereHomologyOneIso def

Integral H₁ of a plane sphere, normalized by the ordered negative/positive components of its chosen stereographic overlap. · OAI.Kaplansky.planeSphereHomologyOneIso

planeSplit def

The actual characteristic plane and its transverse kernel are complementary. No orthonormal-coordinate or slicing-existence premise is imposed. · OAI.ElasticityCoordinates.planeSplit

planesThroughEquiv def

The number of planes through a nonzero vector is independent of it. · OAI.KaplanskyCounterexample.FiniteGeometry.planesThroughEquiv

planeSubgroup def

The manuscript's maximal plane reflection subgroup, defined using actual normalized roots, not by assuming a rank-two presentation. · OAI.KLInvariance.TitsSpace.planeSubgroup

planeSubset def

Return from a subset of labelled vertices to the original plane, retaining all distances and its exact cardinality. · OAI.WeakPinned.planeSubset

planeTransitionCoefficient def

The pullback of the projective coordinate x_j/x_i to V_i. · OAI.SourcePullbackChart.planeTransitionCoefficient

PlaneTree inductive

Ordered rooted trees; the empty child list is an actual leaf. · OAI.FKCRT.TreeCoding.PlaneTree

planeValue def

Evaluate a genuine bivariate polynomial on its two complex coordinates. · OAI.Nagata.Workers.W30.planeValue

planInTime def

A genuine execution certificate for an arbitrary concrete bounded plan. · OAI.MaxCutGames.Foundations.Complexity.PoweringPlanBudget.planInTime

planInTime def

A linear bound in the largest unary input field; plan size is fixed. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Emitter.planInTime

planInTime def

A genuine execution certificate for an arbitrary concrete bounded plan. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringPlanBudget.planInTime

planInTime def

A linear bound in the largest unary input field; plan size is fixed. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Emitter.planInTime

planMain def

A finite main label also exists for an empty command list. · OAI.MaxCutGames.Foundations.Complexity.PoweringMachineRowBody.planMain

planMain def

A finite main label also exists for an empty command list. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringMachineRowBody.planMain

PlannedBlock structure

A concrete outgoing plan. Its local walks include each constituent base turn exactly once in the reserve. · OAI.Superstring.Hierarchical.LayerCollection.PlannedBlock

planNew def

Joint nominal rank modulo pins. · OAI.HadwigerCounterexample.planNew

plateau def

The plateau selected by the constant branch, as finite affine inequalities. · OAI.Problem326.BarrierConstruction.AffineBarrierData.plateau

PLDrawing structure

A literal finite polyline drawing. Simplicity is a proved property of the finite segment chains, not an added planarity promise. · OAI.PlanarL1.Topology.PLDrawing

plethysmAction def

The underlying endomorphism of the canonical plethysm action. · OAI.Problem346.plethysmAction

plethysmDualEquivOf def

The duality chart for a plethysm is the composite of its inner and outer one-level duality charts. · OAI.Problem346.PlethysmDual.plethysmDualEquivOf

plethysmEquiv def

The canonical action of a linear automorphism on a plethysm. · OAI.Problem346.plethysmEquiv

plethysmFormMultilinear def

Flatten the a rows of b vector arguments of a form on SymPow b V. · OAI.Problem346.plethysmFormMultilinear

plethysmMap def

The map on a plethysm induced by a linear map on the original space. · OAI.Problem346.plethysmMap

plusCoordinates def

The same action in the twenty-one actual free coordinates. · OAI.PositiveEinsteinFour.HighestHessian.plusCoordinates

plusGradientTensor def

The concrete tensor to which the finite Bianchi estimates apply. · OAI.PositiveEinsteinFour.Geometry.plusGradientTensor

plusKernel def

The Kronecker sum of the four actual infinitesimal rotation actions. · OAI.PositiveEinsteinFour.TraceHessian.plusKernel

plusLambdaTable def

The unrestricted plus-side analogue of the minus lambda table. · OAI.PureTypeSystem.FreeObservations.plusLambdaTable

plusList def

The first ten entries are internal seams; the last four are the exposed U,Q,A,B positive-endpoint ports. Each is a whole ordered seam list. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.ComplementaryPatch.plusList

plusPolynomial def

The actual polynomial used for the total-plus spectral projection. · OAI.PositiveEinsteinFour.HighestHessian.plusPolynomial

plusPos def

The plus class; at depth h this is the shared leaf class. · OAI.Problem348.TreePositions.plusPos

plusShear def

Inverse shear, adding the transformed remainder. · OAI.StableCoordinate.Linearization.plusShear

plusWeyl def

The self-dual Weyl block in the normalized Ricci=3g convention. · OAI.PositiveEinsteinFour.Geometry.plusWeyl

plusWeylGradient def

The actual plus-block covariant gradient in an orthonormal frame. · OAI.PositiveEinsteinFour.Geometry.plusWeylGradient

plusWeylHessian def

The actual plus-block covariant Hessian, before any trace decomposition. · OAI.PositiveEinsteinFour.Geometry.plusWeylHessian

plusWeylThirdDerivative def

Three successive covariant derivatives, retaining all derivative slots. · OAI.PositiveEinsteinFour.Geometry.plusWeylThirdDerivative

ply def

Polynomial evaluated at points of the full grid. · OAI.FourierLLogL.Dyadic.Thermal.ply

PNeNP def

Separation of deterministic polynomial time from finite-verifier nondeterministic polynomial time. · OAI.MetricKMedian.Complexity.PNeNP

PoincareBound def

Direct Poincare inequality, avoiding an infimum convention for the constant. · OAI.LeanBlast.KLS.PoincareBound

Point structure

Points are represented by the unique open cube containing them. · OAI.EilenbergGanea.CubeRealization.Point

Point inductive

Endpoints are retained at this stage and identified by the metric quotient. · OAI.HarmonicArtin.ParabolicIntersections.GraphRealization.Point

Point abbrev

The disjoint union used for gluing; its summands are the actual exact fibers of the inclusion system. · OAI.PartitionConsistency.LocalDiagramSystem.System.Point

Point structure

Points before quotienting, in one component of the intrinsic band. · OAI.PlanarL1.Columns.Model.Point

point def

The normalized embedded lattice attached to a full lattice and an ordering of the real embeddings. · OAI.DukePrimeDegree.FullLattice.point

point def

The center and the two opposite pairs used in the local construction. · OAI.Problem356.CenterCertificate.point

point def

The real vertex obtained by rescaling the half-integer grid. · OAI.SharpLiebThirring.CubeFlags.Flag.point

point def

Affine extension of the face barycenters across the actual face nerve. · OAI.Thomason.Construction.SubdivisionBarycentric.point

pointAppend def

Append the primal value at the last coordinate. · OAI.MaxCutGames.Inverse.KMSAnalytic.pointAppend

pointAppend def

Append the primal value at the last coordinate. · OAI.UniqueGamesTheorem.Inverse.KMSAnalytic.pointAppend

pointCode def

The arbitrary pairing of the remaining ordinary letters is fixed once, using a cardinality equivalence; the first three partner the extras. · OAI.TorsionFreeZeroDivisors.Types.pointCode

pointCoordinates def

Canonical coordinates in the unique open cell. · OAI.EilenbergGanea.CubeRealization.pointCoordinates

pointDegree def

Incidence degree before alteration, with the marked point removed from its line. · OAI.Release075.pointDegree

pointed def

A space with a chosen actual point. · OAI.Thomason.Construction.PointedHomotopyGroups.pointed

pointedMap def

A continuous map with an equality of the chosen basepoints. · OAI.Thomason.Construction.PointedHomotopyGroups.pointedMap

PointedMatrixProduct structure

A concrete pointed matrix product in its faithful product-state representation. This is construction data, not a classification. · OAI.KadisonKastler.PointedMatrixProduct

PointedMatrixSites structure

A coordinate description of the product of two-dimensional matrix states. Only distinct-site moments are needed: repeated entries at a site are reduced by the matrix-unit identities. · OAI.KadisonKastler.PointedMatrixSites

pointedPowerFibreEquiv def

Splitting off one site is a genuine Hilbert-space equivalence. · OAI.KadisonKastler.pointedPowerFibreEquiv

pointedPowerOperator def

Apply a four-dimensional operator at one site of the Hilbert product. · OAI.KadisonKastler.pointedPowerOperator

pointedPowerSplit def

A configuration is its label at one site and the vacuum-filled tail. · OAI.KadisonKastler.pointedPowerSplit

pointedProfile def

Shift the fixed integer cutoff to zero while retaining every recorded level. · OAI.HarmonicArtin.ParabolicIntersections.pointedProfile

PointedRescalingPrecompactness def

Rescalings at a point admit a pointed-convergent subsequence with a proper metric-space limit. · OAI.RegularPointCoordinates.PointedRescalingPrecompactness

PointedSequenceConverges def

Pointed metric convergence allowing the physical centers to move. · OAI.RegularPointCoordinates.PointedSequenceConverges

pointEquiv def

Free coordinates of the actual partner, indexed by the exact raw coefficient slots. · OAI.MaxCutGames.Soundness.PartnerMapCoordinates.pointEquiv

pointEquiv def

A polynomial algebra automorphism induces an affine-space automorphism on points; both directions are explicitly given by polynomial evaluation. · OAI.StableCoordinate.ZeroLocusInterface.pointEquiv

pointEquiv def

Free coordinates of the actual partner, indexed by the exact raw coefficient slots. · OAI.UniqueGamesTheorem.Soundness.PartnerMapCoordinates.pointEquiv

pointEquivOfIsCompl def

The coordinates supplied by a complement to a nonzero point. · OAI.MaxCutGames.Inverse.KMSPointComplement.pointEquivOfIsCompl

pointEquivOfIsCompl def

The coordinates supplied by a complement to a nonzero point. · OAI.UniqueGamesTheorem.Inverse.KMSPointComplement.pointEquivOfIsCompl

pointFiberToPullback def

The original point fibre maps canonically into every pullback containing that point. · OAI.Thomason.Construction.pointFiberToPullback

pointHomology def

The actual integral H₀ class of a point, expressed as a map out of ℤ. · OAI.Kaplansky.pointHomology

pointIdeal def

The ideal of polynomials vanishing at an affine point. · OAI.Nagata.AffineMultiplicity.pointIdeal

pointInterpolation def

Straight interpolation in the actual coordinates of a realized cube. · OAI.Thomason.Construction.RealizedCube.pointInterpolation

pointIrrep def

The remaining symmetric-group type transported to the actual input placement. · OAI.CoordinateSweeps.PlacementInduction.Chart.pointIrrep

pointLambda def

These are exactly lambda,z from fourier:frequencies, restricted to real t. · OAI.SharpPlanar.IntegerIntervals.pointLambda

pointLaw def

A fixed valid fallback gives a total policy even on impossible row inputs. · OAI.MaxCutGames.Decoder.VisiblePolicies.pointLaw

pointLaw def

A fixed valid fallback gives a total policy even on impossible row inputs. · OAI.UniqueGamesTheorem.Decoder.VisiblePolicies.pointLaw

pointLift def

Extend an injection while retaining the newly fixed coordinate. · OAI.MaxCutGames.Inverse.KMSAnalytic.pointLift

pointLift def

Extend an injection while retaining the newly fixed coordinate. · OAI.UniqueGamesTheorem.Inverse.KMSAnalytic.pointLift

PointLocalRing abbrev

The genuine local ring of affine space at the indicated rational point. · OAI.Nagata.AffineMultiplicity.PointLocalRing

pointMap def

The point map contravariantly induced by a homomorphism of polynomial algebras. · OAI.StableCoordinate.AmbientMaps.pointMap

pointMap def

The map on affine points contravariantly induced by a polynomial substitution. · OAI.StableCoordinate.ZeroLocusInterface.pointMap

pointMatrices def

Reading the universal transports along an ring-valued point. · OAI.IntegralCharacterVarieties.FiniteMatrixPresentation.pointMatrices

pointMean def

Exact mean, including the missing self-mark when the point belongs to the grid. · OAI.Release075.pointMean

pointOfRepresentative def

X representatives occupy 0,...,6 and the four Y representatives 7,...,10. · OAI.HyperbolicObstruction.IncidenceTemplate.pointOfRepresentative

pointOrbitStabilizer def

The subgroup of the orbit group fixing the lattice represented by g. · OAI.DukePrimeDegree.pointOrbitStabilizer

pointPad def

The lower-dimensional embedding into the old coordinates. · OAI.MaxCutGames.Inverse.KMSAnalytic.pointPad

pointPad def

The lower-dimensional embedding into the old coordinates. · OAI.UniqueGamesTheorem.Inverse.KMSAnalytic.pointPad

pointRestrict def

Fix one primal column, with no averaging or normalization. · OAI.MaxCutGames.Inverse.KMSAnalytic.pointRestrict

pointRestrict def

Fix one primal column, with no averaging or normalization. · OAI.UniqueGamesTheorem.Inverse.KMSAnalytic.pointRestrict

Points abbrev

Local indexed hypothesis. Global pool and program counter may be any encoded state of the corresponding semantic type. · OAI.Fifths.Flow.Points

pointsEquiv def

Exact equation quotient over any full coefficient ring. · OAI.IntegralCharacterVarieties.EquationQuotient.pointsEquiv

pointsEquiv def

Exact representability over arbitrary coefficient rings, including the full ring of integers. No reduction, localization or rank condition is dropped. · OAI.IntegralCharacterVarieties.FiniteMatrixPresentation.pointsEquiv

pointsEquiv def

The finite algebra representing the split-frame incidence scheme. This is not an abstract affine-bundle interface: all entries and all relations are in the explicit quotient algebra, and its ring-valued points are identified. · OAI.IntegralCharacterVarieties.FlagCircuit.Data.pointsEquiv

pointsEquiv def

The equivalence works over an arbitrary commutative algebra, in particular the full ring of integers, and retains forward AND inverse entries. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.pointsEquiv

pointShuffle def

Put the last coordinate into the fixed block through e, keeping the free block unchanged. · OAI.MaxCutGames.Inverse.KMSFourthMoment.pointShuffle

pointShuffle def

Put the last coordinate into the fixed block through e, keeping the free block unchanged. · OAI.UniqueGamesTheorem.Inverse.KMSFourthMoment.pointShuffle

pointShuffleTransport def

Express a function in the shuffled coordinates. · OAI.MaxCutGames.Inverse.KMSFourthMoment.pointShuffleTransport

pointShuffleTransport def

Express a function in the shuffled coordinates. · OAI.UniqueGamesTheorem.Inverse.KMSFourthMoment.pointShuffleTransport

pointsOn def

Ordinary labels carried by a line type. · OAI.TorsionFreeZeroDivisors.Types.pointsOn

pointThetaMultiply def

Multiplication by the actual marked theta section to its kth power. · OAI.Nagata.W07.pointThetaMultiply

pointwiseIterationCoefficient def

A common coefficient for the linear, inverse, squared, and recurrence errors. · OAI.RegularPointCoordinates.pointwiseIterationCoefficient

poissonAverage def

Poisson total count and iid countable labels. Marks remain in their finite, label-dependent prior rows and are not part of this quenched root law. · OAI.DilutedSpinGlass.CountableReservoir.poissonAverage

poissonBoundaryApprox def

The scalar factor is exp(-i(t+ir)) = exp(r) exp(-it). · OAI.KadisonKastler.poissonBoundaryApprox

poissonCharacterField def

Scalar characters, regarded as a concrete field of operators. · OAI.KadisonKastler.poissonCharacterField

poissonCommutatorConstant def

The coefficient is independent of the center and scale. · OAI.RegularPointCoordinates.poissonCommutatorConstant

poissonEulerGradientConstant def

The coefficient is independent of the center and scale. · OAI.RegularPointCoordinates.poissonEulerGradientConstant

PoissonHarmonicLimit structure

Precisely the weak pairings retained from the local strong/weak Sobolev convergence theorem, with the physical radius factors explicit. · OAI.RegularPointCoordinates.PoissonHarmonicLimit

poissonRootAverage def

Root averaging followed by the genuine Poisson count average. · OAI.DilutedSpinGlass.poissonRootAverage

poissonSpan def

The last two components are the Poisson replacements used in a same-scale two-center comparison. · OAI.RegularPointCoordinates.poissonSpan

poissonSpanLaplacian def

Both replacements have the same constant Laplacian; the harmonic coordinate components have zero Laplacian. · OAI.RegularPointCoordinates.poissonSpanLaplacian

poissonVectorAverage def

Poisson averaging, written with the standard Cauchy random variable. · OAI.KadisonKastler.poissonVectorAverage

poissonWeight def

The archival Mathlib Poisson weights, not a replacement count model. · OAI.DilutedSpinGlass.poissonWeight

poke abbrev

Canonical toss calls for stationary/moving candidates · OAI.ExactDerandomization.Swirl.Bud.poke

polarAt def

The polar of a convex body translated by an interior center. · OAI.GeneralMahler.polarAt

polarAt def

(K-z)^o from the manuscript. · OAI.GeneralMahler.polarAt

PolarConeModelInputs structure

The six published inputs actually used to realize a specified cone. Lee's nilpotent existence/integration and Le Donne's exponential Haar statements retain their original Lean propositions. · OAI.PolycyclicRecognition.CartanWordModel.PolarConeModelInputs

polarize def

A polarization family retains all geometry and twice the profile bound. · OAI.LipschitzHilbert.SourceLatticePacket.polarize

polarizedDerivative def

The operator formed by polarizing candidate quadratic derivatives. · OAI.RegularPointCoordinates.polarizedDerivative

polarModulus def

Positive square root used only for this polar factor. · OAI.KadisonSimilarity.polarModulus

polarPartialIsometry def

The actual partial isometry, extended by zero. · OAI.KadisonSimilarity.polarPartialIsometry

polarRangeIsometry def

Isometry from the initial closed range to the ambient Hilbert space. · OAI.KadisonSimilarity.polarRangeIsometry

polarSamplingTemplates def

The exact translated templates used by the three probe chains. · OAI.PolycyclicRecognition.polarSamplingTemplates

polarSlice def

Slice t=1 of the dual, a reflected polar. · OAI.GeneralMahler.Body.polarSlice

polarTensor def

Polar tensor in unnormalized Cartesian coordinates. Its radial/Hopf/ horizontal eigenvalues are respectively 1,D,A. · OAI.HarmonicCounterexample.Berger.polarTensor

PolarTiledSamplerData structure

A single source partition retains its exact count and all three actual samplers. The logarithmic chain coordinates are supplied by the proved box construction, not chosen as additional input. · OAI.PolycyclicRecognition.CartanWordModel.PolarTiledSamplerData

polarUnitary def

The polar retraction on the open set of invertible elements. · OAI.NuclearUltrapower.polarUnitary

poleRotationFormula def

This formula gives smooth dependence on the requested pole whenever the pole is separated from the antipode of p. · OAI.Falconer.poleRotationFormula

policies def

Actual observation-local stochastic policies, in the exact advice space used by the clean-coordinate experiment. Public A is fixed before questions. · OAI.MaxCutGames.Decoder.ActualAdviceUpper.policies

policies def

Actual observation-local stochastic policies, in the exact advice space used by the clean-coordinate experiment. Public A is fixed before questions. · OAI.UniqueGamesTheorem.Decoder.ActualAdviceUpper.policies

Policy abbrev

This declaration is an exact comparator-selected statement. Read the Lean statement below for its mathematical content. · OAI.KServer.Policy

Policy abbrev

A behavioral policy knows the entire past request/choice history and the new request. It is a single rule for all horizons. Pairing the past requests and choices is equivalent to the histories X^t × [k]^(t-1) in the manuscript. · OAI.KServer.Policy

Policy structure

A single online policy: its row depends on prior requests and choices and the current request, never on future requests or the horizon. · OAI.UniformKServer.Policy

policy def

Null histories are filled by a fixed legal label, not by any future input. · OAI.KServer.HiddenMachine.policy

PolicyJetData structure

Data retained during an angular episode. Its descriptor list and split are computed from the inherited positive interval. · OAI.Falconer.PolicyJetData

Poly abbrev

Sparse integral polynomials, before combining repeated labels. · OAI.NuclearUltrapower.SOS.Poly

Poly def

Polynomial time means a concrete finite clean stack program and actual step bound, through Procedure. It is not machine advice or an oracle. · OAI.RationalHitting.StackLanguage.Poly

Poly def

Uniform polynomial boundedness of a fixed word expression. This is a proven realizer property, not an oracle or a unit-cost word instruction. · OAI.WLIdentification.WordExpressions.Poly

PolyAt def

Uniform polynomial bounds over arbitrary auxiliary parameters. Only the chosen length is charged, so this also records worst-case bounds over all finite loop indices and all networks satisfying a supplied count invariant. · OAI.ExactQuantumFactoring.PolyAt

PolyBound def

A worst-case bound for all input lengths, with nonnegative coefficients. · OAI.ExactQuantumFactoring.PolyBound

PolyCan def

One algorithm; time is polynomial in (S+2), integer bounds polynomial in (B+2). Often S polylogarithmic in B. · OAI.PowerSaving.RAM.PolyCan

PolycyclicTorsionFreeIndexInput def

The torsion-free consequence of Robinson 5.4.15(i). · OAI.PolycyclicRecognition.PolycyclicTorsionFreeIndexInput

PolyEmits def

A fixed polynomial syntax skeleton transports literal coefficient emitters through substitution. The outer polynomial's depth is fixed; input coefficients are generated by the supplied charged Boolean-stack procedures. · OAI.ExactQuantumFactoring.NetworkEmission.PolyEmits

PolyExpr inductive

Bounded polynomial arithmetic is itself finite rational syntax. · OAI.ExactQuantumFactoring.PolyExpr

PolyExprPoly def

Every fixed-index coefficient of this family has polynomial syntax. · OAI.ExactQuantumFactoring.PolyExprPoly

Polygon structure

A cyclic simple macro polygon with at least three sides, expressed by its actual vertices and successor. The normal-form splitting must produce this data; all template bounds below are derived. · OAI.HyperbolicObstruction.MacroGraph.Polygon

Polygon structure

A cyclicly ordered embedded boundary polygon. · OAI.IntegralCharacterVarieties.FiniteCycles.Polygon

polygon def

The two sides of a genuine relator disk and its distinguished exterior. Every side is separately indexed, even when its letter is repeated. · OAI.HyperbolicObstruction.WordPictures.WordPicture.polygon

PolygonallyJoined def

A finite chain of straight segments contained in a specified set. · OAI.PlanarL1.Topology.PolygonallyJoined

PolygonalParametrization structure

A finite affine parametrization of the actual path. · OAI.Paper170.PolygonalParametrization

PolygonGroup abbrev

The actual indexed polygon words. Radial edges are not silently equated or turned into extra letter relators. This bouquet presentation has extra free factors relative to the graph fundamental group, but the original letter relations still … · OAI.HyperbolicObstruction.ConeSkeleton.PolygonGroup

polygonGroupCover def

Canonical cover data for this presentation, with actual prefix-product coordinates and the proved cone potentials. · OAI.HyperbolicObstruction.ConeSkeleton.polygonGroupCover

Polygons structure

An indexed polygonal attaching system, before topological realization. · OAI.HyperbolicObstruction.IntegralIncidence.Polygons

polyLinear def

Coefficientwise transport by a rational linear functional. · OAI.EuclideanFiveColor.AlgebraicCayley.polyLinear

polynomial def

The standard lower-half KL recursion, on the actual Bruhat interval. Its reciprocity will be proved from the genuine R-kernel, not stipulated. · OAI.KLInvariance.CanonicalP.polynomial

polynomial def

The degree-at-most-two defining polynomial with the displayed coefficients. · OAI.LittleFinitistic.QuotientAction.polynomial

polynomial def

One simultaneous polynomial restriction of the whole source tensor. · OAI.MatrixMultiplication.AuxiliarySeparation.Sector.polynomial

polynomial def

A finite exponential polynomial on the normalized unit circle. · OAI.Problem337.ThreePrimeContinuousEnergy.polynomial

polynomial def

Coordinate polynomial of a coefficient array and an assignment of slots to blocks. · OAI.Problem346.SlotPolynomial.polynomial

polynomial def

Evaluate a real polynomial using the explicit integer spectral decomposition. · OAI.QAC.Count.polynomial

polynomial def

A finite-support construction of exp(T D)(a). · OAI.StableCoordinate.LNDExponential.polynomial

PolynomialAdditiveAction structure

A polynomial additive-group action on an affine algebra, in contravariant form. In coassoc, the outer polynomial variable is the second parameter and the inner variable is the first parameter. · OAI.StableCoordinate.CoactionLND.PolynomialAdditiveAction

polynomialAdjugate def

A polynomial differential adjugate; the middle polarization uses the same scalar determinant, so all three components admit one uniform construction. · OAI.ElasticityBoundarySymbol.polynomialAdjugate

PolynomialBoundedChainPacking def

Polynomial packing of bounded-height chains in the Lie model. · OAI.PolycyclicRecognition.PolynomialBoundedChainPacking

polynomialBudget def

A dominating polynomial; every fixed polynomial nonnegative on natural inputs is bounded by one of this form after increasing c and e. · OAI.UniformKServer.polynomialBudget

PolynomialCertificate structure

Finite arithmetic certificate interface for the actual selected weak sonic Taylor series, retaining the exact normalized septic source equation. · OAI.SepticProfile.SonicContinuation.PolynomialCertificate

polynomialChartSeriesEquiv def

Formal coordinates on the completion of an étale polynomial chart. · OAI.CartierSections.polynomialChartSeriesEquiv

polynomialClock def

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

polynomialCoordinateEquiv def

Genuine invertible polynomial coordinate substitution. · OAI.Nagata.W01.polynomialCoordinateEquiv

PolynomialCoordinateGroup def

A separate type for the group structure supplied by a coordinate law. · OAI.PolycyclicRecognition.PolynomialCoordinateGroup

polynomialCoordinateHeightHom def

Regard an additive coordinate height as a homomorphism on the polynomial group. Its additivity is supplied by the preceding limit proof. · OAI.PolycyclicRecognition.polynomialCoordinateHeightHom

polynomialDegree def

Fixed exponent; the enormous fixed graph constants remain unevaluated. · OAI.MaxCutGames.Foundations.PCP.PCPIteration.polynomialDegree

polynomialDegree def

Fixed exponent; the enormous fixed graph constants remain unevaluated. · OAI.UniqueGamesTheorem.Foundations.PCP.PCPIteration.polynomialDegree

polynomialEmbedding def

Renaming the polynomial variables into the independent-variable field. · OAI.EuclideanRamsey.NineMain.polynomialEmbedding

polynomialEndpointScheme def

The expanded conjugate scheme, evaluated as a polynomial in every parameter, whether or not that parameter defines a group law. · OAI.PolycyclicRecognition.polynomialEndpointScheme

polynomialEpsilon def

Choose the uniform ε before any endpoint coefficient arrays are supplied. · OAI.FiniteFactor.polynomialEpsilon

PolynomialEquiv structure

A polynomial isomorphism, with inverse identities in the polynomial rings, not just identities of functions on the base field. · OAI.StableCoordinate.AmbientMaps.PolynomialEquiv

polynomialFamilyDegree def

The largest number of powers in a finite dependent family. · OAI.ClosedSurfaceR4.JetPolynomial.Perturbation.polynomialFamilyDegree

polynomialFunction def

Ambient evaluation of a polynomial, using the Euclidean coordinates. · OAI.PettyProjection.Spherical.polynomialFunction

PolynomialGapReduction structure

A fixed-alphabet gap reduction from 3SAT, including its actual finite-machine polynomial runtime, with the two approximation thresholds expressed over ℝ. · OAI.MaxCutGames.Integration.RealTarget.PolynomialGapReduction

PolynomialGapReduction structure

The full output form asserted in Theorem 1.1, including both real gaps, the actual polynomial-time machine, and one fixed binary alphabet. · OAI.MaxCutGames.Integration.TranslationTarget.PolynomialGapReduction

PolynomialGapReduction structure

A fixed-alphabet gap reduction from 3SAT, including its actual finite-machine polynomial runtime, with the two approximation thresholds expressed over ℝ. · OAI.UniqueGamesTheorem.Integration.RealTarget.PolynomialGapReduction

PolynomialGapReduction structure

The full output form asserted in Theorem 1.1, including both real gaps, the actual polynomial-time machine, and one fixed binary alphabet. · OAI.UniqueGamesTheorem.Integration.TranslationTarget.PolynomialGapReduction

polynomialGradient def

The continuous linear map given by the two actual polynomial partials. · OAI.Nagata.Workers.W28.polynomialGradient

polynomialGrading abbrev

The actual polynomial grading, with each variable in degree one, on integer degrees as used for shifted stalks. · OAI.KLInvariance.Graded.polynomialGrading

PolynomialGroupLaws def

Identities obeyed by a group in exponential coordinates, including its actual straight-line one-parameter subgroups. · OAI.PolycyclicRecognition.PolynomialGroupLaws

polynomialInclusion def

The polynomial inclusion into the doubled-variable fraction field. · OAI.EuclideanRamsey.NineRational.polynomialInclusion

polynomialLift def

The two-parameter polynomial lift. · OAI.RationalHitting.QuantumLift.polynomialLift

PolynomialLinearRepresentation def

The local linearized operators act on the same global polynomial metric. · OAI.ClosedSurfaceR4.FiniteOrderSmoothing.SmoothingAtlas.PolynomialLinearRepresentation

PolynomialManyOne structure

Actual polynomial-time many-one reduction with both promise maps. · OAI.ContinuumCoulomb.PolynomialManyOne

polynomialMap def

Evaluation of a weighted polynomial gives an actual map from the weight-k total space to the weight-d total space over the same curve. · OAI.Nagata.W16.AssociatedBundle.polynomialMap

polynomialMap def

The coefficient realization of P_M(K), with its exact degree bound. · OAI.RationalHitting.SmallExpansion.polynomialMap

polynomialMap def

Evaluation on the five independent determinant-presentation variables. · OAI.StableCoordinate.FrameLift.Frame.polynomialMap

polynomialMap def

Polynomial evaluation before taking the degeneration quotient. · OAI.StableCoordinate.ModificationEmbedding.polynomialMap

polynomialMap def

Evaluation in the explicit four-variable polynomial presentation. · OAI.StableCoordinate.Presentation.polynomialMap

polynomialNormalCoordinates def

Tangential coordinate and actual polynomial defining equation. · OAI.Nagata.Workers.W28.polynomialNormalCoordinates

polynomialOnLine def

Restriction of a multivariable polynomial to a complex line through zero. · OAI.KahlerUniformization.polynomialOnLine

polynomialOtherCoordinates def

The alternative tangential coordinate, used when the other partial is nonzero. · OAI.Nagata.Workers.W28.polynomialOtherCoordinates

PolynomialPackage def

Complete first principal group, with the advertised exact total degree. · OAI.StableCoordinate.PolynomialPackage

polynomialPhysical def

The actual Fourier-transformed isotropic physical elasticity operator on polynomial amplitudes, tangential covector (1,0), inward normal t. · OAI.ElasticityBoundarySymbol.polynomialPhysical

polynomialPiece abbrev

The weighted pieces on the five-variable polynomial presentation. · OAI.StableCoordinate.Obstruction.polynomialPiece

polynomialPresentation def

Proposition 2.1, stable under adjoining any number of independent variables. · OAI.StableCoordinate.Presentation.polynomialPresentation

PolynomialPromise structure

A single fixed exponent bounds lattice extent, source weights and inverse gap, as in the polynomial-weight spatially sparse source branch. · OAI.ContinuumCoulomb.BinaryHeisenberg.PolynomialPromise

PolynomialQuadraticRepresentation def

The same local polynomials represent the global quadratic variation. · OAI.ClosedSurfaceR4.FiniteOrderSmoothing.SmoothingAtlas.PolynomialQuadraticRepresentation

polynomialRelationIdeal def

Every relation is tested by evaluation at every point of the punctured plane. · OAI.Nagata.Workers.W25.polynomialRelationIdeal

polynomialRestriction def

Actual polynomial restrictions, as continuous functions on the sphere. · OAI.PettyProjection.Spherical.polynomialRestriction

polynomialRootCellFlag def

This is the actual support flag, defined from the inequalities, rather than an assumption that the supports are regular on the cells. · OAI.Ostmann.polynomialRootCellFlag

polynomialRotation def

Infinitesimal linear coordinate motion: (M x) · ∇. · OAI.HarmonicCounterexample.Angular.polynomialRotation

polynomialRotation def

Polynomial rotation differentiation, a derivation preserving degree. · OAI.PettyProjection.Spherical.polynomialRotation

polynomials def

The uncompleted graded section algebra in a rational trivialization. · OAI.SectionCompletion.polynomials

polynomialSectionBasisFromCoefficients def

Conditional assembly from actual block coefficient maps and actual delta vectors. Each analytic input is exposed; no branch basis is supplied as a field. · OAI.Nagata.Workers.W10.polynomialSectionBasisFromCoefficients

polynomialSections def

Actual finite-degree polynomials in the fibre coordinate with coefficients in the above spaces. The degree bound is imposed coefficientwise and therefore includes the zero polynomial without a special degree convention. · OAI.Nagata.CoefficientSpaces.polynomialSections

PolynomialSimulation def

An alphabet-independent polynomial bound for deterministic simulation. · OAI.OneWayLiveness.PolynomialSimulation

polynomialSpace def

The corresponding subspace inside the actual bivariate polynomial ring. · OAI.Nagata.W01.polynomialSpace

PolynomialThreeSATReduction structure

The demanded Cook–Levin endpoint: both the SAT equivalence and an actual machine producing the fixed existing unary-token 3CNF encoding. This structure contains obligations; its inhabitance is not assumed or asserted here. · OAI.MaxCutGames.Foundations.Complexity.CookLevin.PolynomialThreeSATReduction

PolynomialThreeSATReduction structure

The demanded Cook–Levin endpoint: both the SAT equivalence and an actual machine producing the fixed existing unary-token 3CNF encoding. This structure contains obligations; its inhabitance is not assumed or asserted here. · OAI.UniqueGamesTheorem.Foundations.Complexity.CookLevin.PolynomialThreeSATReduction

polynomialThreshold def

The polynomial threshold function with the nonnegative-at-zero convention. · OAI.LeanBlast.GotsmanLinial.polynomialThreshold

polynomialUnitsEquiv def

Evaluation at a point identifies polynomial units with coefficient-ring units. · OAI.StableCoordinate.polynomialUnitsEquiv

polynomialValue def

Evaluate a real polynomial at the signs represented by a cube vertex. · OAI.LeanBlast.GotsmanLinial.polynomialValue

polynomialWord def

A word evaluated directly by the polynomial multiplication. · OAI.PolycyclicRecognition.polynomialWord

polynomialZeros def

The literal zero set in the full finite assignment space. · OAI.MetricEntropyDuality.polynomialZeros

polynomialZeroSet def

Common zero set of a family of actual polynomials. No topological or geometric consequence is built into this definition. · OAI.Nagata.W27.polynomialZeroSet

polyOf def

correspondence lemma for polynomials in dense coefficient representation · OAI.PowerSaving.polyOf

PolyProgram structure

Polynomial cost is witnessed by an actual finite tree program. · OAI.ParityWL.Complexity.PolyProgram

PolySchemaPoly def

Substitution-stable, fixed-depth polynomial schemas. In particular this controls the literal syntax of every fixed coefficient, not just its denotation. · OAI.ExactQuantumFactoring.PolySchemaPoly

polytabloid def

The standard column-antisymmetrized row tabloid, with complex coefficients. · OAI.Saxl.polytabloid

Polytime def

A finite-stack, finite-alphabet Turing machine with a polynomial step bound. · OAI.UniformSparsestCut.Polytime

pop def

Destructively peek (return none on empty). · OAI.Fifths.SM.pop

Popa1995ConjugateFreenessInput def

Local published specialization for a fixed irreducible inclusion of separable II₁ factors A⊂B and a fixed separable subfactor ρ(R)⊂A. The two free algebras are concrete ranges, rather than unspecified copies. · OAI.KadisonSimilarity.Popa1995ConjugateFreenessInput

popaCornerEquiv def

An algebra equivalence induces an equivalence between corresponding corners. · OAI.BoundedHochschild.KadisonRingrose.popaCornerEquiv

popularSet def

The arc of total angular length short / long about the long-side direction. The field is separately required to have unit norm. · OAI.LipschitzHilbert.Rectangle.popularSet

populatedKernelPoints def

The retained set is a union of entire original fibers. · OAI.Ostmann.populatedKernelPoints

Port structure

A midpoint port, oriented into the triangle domain. · OAI.CriticalHoneycomb.Port

port def

Initial and terminal half-edge of an occurrence, respectively. · OAI.TorsionFreeZeroDivisors.PathIncidences.port

portalAt def

The fixed enumeration remembers whichever portal occurrence generated a name. Duplicated portals still produce the same name. · OAI.PlanarL1.Columns.Model.Point.RadialTree.portalAt

PortalData structure

Two fixed geometric portals, with all bounds required by the actual interpolation hierarchy. Repeated portals encode the one-portal case. · OAI.PlanarL1.Columns.Model.Point.PortalData

PortAssembly structure

Whole ordered ports, rather than unrelated side germs, attach to seam endpoints. This is the finite incidence condition for surface diagrams. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.PortAssembly

portAt def

Enumerate the genuine ends of the old whole seams. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.PortAssembly.portAt

portAt def

Both ends of each newly assembled seam, with their original local ports. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.PortWiring.portAt

portBits def

A prefix of the emitted rows, defined without any execution hypothesis. · OAI.MaxCutGames.Foundations.Complexity.MachineRegularVertexBlock.portBits

portBits def

A prefix of the emitted rows, defined without any execution hypothesis. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRegularVertexBlock.portBits

portColor def

Ordered list form of a local decoration; none is still the parent. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.Decoration.portColor

portEquiv def

Ordered seam columns, not an unlabelled flag: first expand each named child into its atoms and then identify with the parent named atoms. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.ComplementaryPatch.portEquiv

portEquiv def

Literal ordered child basis into the parent's basis, preserving atom names. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.AtomicData.portEquiv

PortFrame structure

Released radial port tails; unlike PortMove this does not prescribe the selected route, which may already have passed through several bends. · OAI.PlanarL1.Topology.PortFrame

portFrame def

The rectangular seam frame. Both child and parent bases come from named occurrences; all reindexings are fixed by the source incidence data. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.portFrame

portGraph def

The actual reversible regular port graph represented by the vectors. · OAI.MaxCutGames.Foundations.PCP.PortTables.portGraph

portGraph def

The actual reversible regular port graph represented by the vectors. · OAI.UniqueGamesTheorem.Foundations.PCP.PortTables.portGraph

PortMove structure

The exact geometric output needed when retubing a tree with exterior stems. The old disk may be discarded, but no new curve leaves it except along the selected stem segment. · OAI.PlanarL1.Topology.PLDrawing.PortMove

portNeighborEquiv def

A port has exactly two neighbors: its original vertex and its reverse port. They lie in disjoint summands even when the original edge is a graph loop. · OAI.TorsionFreeZeroDivisors.PortSubdivision.PortGraph.portNeighborEquiv

portOrder def

Internal cloud ports precede the one inherited-constraint port. · OAI.MaxCutGames.Foundations.PCP.PreprocessingRegularTables.portOrder

portOrder def

Internal cloud ports precede the one inherited-constraint port. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingRegularTables.portOrder

PortPatch structure

A finite open template is given by explicit enumerations of all vertex ports. Internal wires match equal indices; the remaining ports are the interface. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.PortPatch

portRel def

The hard-wall two-port graph: the two exterior markers have only their prescribed incident edge; internal adjacency remains unchanged. · OAI.CriticalHoneycomb.Transfer.FiniteConfigurations.portRel

portRelation def

Boundary ports: false is left and true is right. · OAI.TwoWayComplementation.Diagram.portRelation

Ports abbrev

Merge two endpoints of raw frame. Assign ports according to their endpoint. · OAI.HadwigerCounterexample.Ports

Ports def

explicitly chosen root-order depending on size branch · OAI.PowerSaving.Selection.Ports

Ports abbrev

A pair of public bit-stack ports and a private register bank. · OAI.Superstring.BitCode.Ports

portsEquiv def

The two sides use exactly the same slot/lane labels. · OAI.OneWayLiveness.CellNetwork.portsEquiv

portSide def

Each local port remembers its entire parent/child list at that precise endpoint. Facet labels may repeat without identifying any side coordinates. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.portSide

portVertex def

A boundary port and its incident vertex. · OAI.PlanarL1.Topology.Drawing.portVertex

PortWiring structure

A wiring joins whole ordered ports of equal arity with opposite endpoint orientations. No side-germ is allowed to migrate to another seam on its own. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.PortWiring

portWord def

Recording ports from the physical walk recovers each directed macro-edge. · OAI.TorsionFreeZeroDivisors.WeightedPortMoore.portWord

Pos abbrev

Literal field positions in source diagram:rank. · OAI.HyperbolicObstruction.IncidenceTemplate.Occurrences.Shape.Pos

Pos abbrev

Actual amalgamated row positions; all positions still refer to the original arrays, not to newly independent copies of an array. · OAI.HyperbolicObstruction.StarModel.Pos

posDual abbrev

Paper C* = D. · OAI.GeneralMahler.posDual

Poset abbrev

The paper explicitly allows adjoining a fixed largest condition. · OAI.PartitionConsistency.ProductForcing.Poset

posetCode def

A genuine small set of condition codes, obtained by wellordering the condition type. The codes carry no forcing or satisfaction information. · OAI.PartitionConsistency.SetForcingNames.posetCode

posetCosieveIso def

A finite cosieve is its actual full one fibre, not merely an equivalent replacement category. · OAI.Thomason.Construction.FiniteReflectionInput.posetCosieveIso

posetCosieveIso def

A poset sieve categorifies to exactly the full one fibre of membership. · OAI.Thomason.Construction.NormalizedSimplexIdentities.posetCosieveIso

posetEvaluation def

The actual finite-dimensional strict evaluation into ordered labels. · OAI.Thomason.Construction.FiniteReflectionInput.posetEvaluation

posetEvaluation def

Evaluation of categorified poset cells in their underlying ordered labels. · OAI.Thomason.Construction.NormalizedSimplexIdentities.posetEvaluation

posetFiberMap def

The corresponding canonical map in the actual positive finite category. · OAI.Thomason.Construction.FiniteReflectionInput.posetFiberMap

posetFiberMap def

The canonical strict map into a categorified sieve's actual full fibre. · OAI.Thomason.Construction.NormalizedSimplexIdentities.posetFiberMap

posetLabelCocone def

The compatible ordered labels form a cocone on the oriental presentation. · OAI.Thomason.Construction.NormalizedSimplexIdentities.posetLabelCocone

posetLeg def

The oriental map specified by a monotone list of vertices in a poset. · OAI.Thomason.Construction.NormalizedSimplexIdentities.posetLeg

posetMapIso def

Categorification carries an order isomorphism to an actual strict isomorphism. · OAI.Thomason.Construction.NormalizedSimplexIdentities.posetMapIso

posetNerveZeroEquiv def

In degree zero the ordinary nerve of a poset is its set of elements. · OAI.Thomason.Construction.NormalizedSimplexIdentities.posetNerveZeroEquiv

posetObjectsEquiv def

Poset objects retain their canonical names after finite categorification. · OAI.Thomason.Construction.FiniteReflectionInput.posetObjectsEquiv

posetObjectsEquiv def

The object set of the actual categorified poset nerve is the original poset. · OAI.Thomason.Construction.NormalizedSimplexIdentities.posetObjectsEquiv

PosetPoint abbrev

The ordinary point used in the contraction, with its canonical simplex iso. · OAI.Thomason.Construction.PosetPoint

posetSieveIso def

A poset sieve categorifies to exactly the full zero fibre of membership. · OAI.Thomason.Construction.NormalizedSimplexIdentities.posetSieveIso

Position structure

Geometric part of a retained endpoint, before witness selection. Both old identifiers refer to the actual closed unrestricted radius-five ball. · OAI.PlanarL1.Columns.Model.SurvivingGroups.Position

position def

Normalized clipped position in the source radius interval [r,2*r]. · OAI.KServer.PartitionProbabilities.position

position def

The variable-position sample, injected into the full host axis. · OAI.Problem348.HostSampledPath.position

position def

Position of a coordinate in an ordered finite closure; no ordinal label is stored in the code. · OAI.SeparableQuotient.PathCoding.position

position def

Zero-based cell coordinate. · OAI.TwoWayComplementation.ChainNode.position

positionalRankDifference def

Simultaneous literal positional operators applied to the actual conditional retained-subset kernel. · OAI.TraceReconstruction.positionalRankDifference

positionalWeight def

Coefficients of the unchanged, centered first, and centered second one-step positional operators. · OAI.TraceReconstruction.positionalWeight

positionEmbedding def

The literal increasing compression map, not a fresh random matching. · OAI.FKCRT.Inventory.positionEmbedding

positionEquiv def

The inverse equivalence is used only to prove the probability identity; the forward noise lookup is ordinary finite-list indexing. · OAI.MaxCutGames.Integration.NoiseTables.Table.positionEquiv

positionEquiv def

The inverse equivalence is used only to prove the probability identity; the forward noise lookup is ordinary finite-list indexing. · OAI.UniqueGamesTheorem.Integration.NoiseTables.Table.positionEquiv

positionGroup def

Group label, ignoring the within-group coordinate. · OAI.Problem348.AnchorRigidity.positionGroup

positionMode def

The mode label of a raw anchor position, and none off the anchors. · OAI.Problem348.AnchorRigidity.positionMode

positionNode def

Extend the variable node index to all positions. · OAI.Problem348.Construction.positionNode

positionOrder abbrev

Explicit order structures, not global instances: the same raw positions carry different row and column orders. · OAI.Problem348.TreePositions.positionOrder

positions def

Explicit contiguous coordinate chart, including zero-length blocks. · OAI.CoordinateSweeps.OrderedBlocks.positions

positions def

An explicit computable enumeration, without an arbitrary finite-type equivalence. Slots occur first, second, third within each clause coordinate. · OAI.MaxCutGames.Foundations.Hastad.SourceSignLoad.positions

positions def

Integer edge indices whose half-integer centers belong to a real interval. · OAI.TorsionFreeZeroDivisors.BlockAlignment.positions

positions def

An explicit computable enumeration, without an arbitrary finite-type equivalence. Slots occur first, second, third within each clause coordinate. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceSignLoad.positions

positionsWhere def

Positions at which the recorded label satisfies a specified predicate. · OAI.QuantitativeVanDerWaerden.RecordedSupport.positionsWhere

Positive inductive

A positive telescope ending in an all-data terminal. The source uses constant telescopes; the no-normal-proof argument also permits dependency. · OAI.PureTypeSystem.Labelled.Expr.Positive

Positive abbrev

The positive cone of the algebra, with its norm-subspace topology. · OAI.TraceConeClassification.Positive

positive def

The positive boundary orientation follows the seam parameter on the parent and goes against it on every child. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.positive

positive def

The positive monoid consists of sums of actual cell generators. · OAI.Thomason.Construction.StrictOmegaCategory.Abelianization.positive

positiveAction def

Conjugation on the actual root lines, represented by positive roots. · OAI.KLInvariance.TitsSpace.positiveAction

PositiveApproximation def

The positive compression premise, stated on the actual coefficients. · OAI.MetricEntropyDuality.SignedCompression.PositiveApproximation

positiveAtlas def

The centers and outer functions are unchanged. Only the square partition is normalized, retaining every support exactly. · OAI.ClosedSurfaceR4.FiniteOrderSmoothing.SmoothingAtlas.positiveAtlas

positiveAtom def

The positively oriented, O-based atom. Nonfaces give zero. · OAI.Release075.BlockPresentation.positiveAtom

positiveBaseEquiv def

The positive diagonal base change sending s to the all-ones base point. · OAI.AffineBernstein.positiveBaseEquiv

positiveBaseForm def

The positive base functional occurring in normalized model caps. · OAI.AffineBernstein.positiveBaseForm

positiveBidegreeSet def

Factors applied first in the intermediate-space argument. · OAI.Problem335.positiveBidegreeSet

positiveBidegreeU def

Total multiplication degree of the first group of factors. · OAI.Problem335.positiveBidegreeU

positiveBidegreeV def

Total differentiation degree of the first group of factors. · OAI.Problem335.positiveBidegreeV

PositiveBilinearMeasure structure

The finite-dimensional restriction of a positive, symmetric, measure-valued quadratic form, such as the local Bochner measure. · OAI.RegularPointCoordinates.PositiveBilinearMeasure

PositiveBranchChildrenMatch def

The child identifications along each positive transverse branch. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.PositiveBranchChildrenMatch

PositiveBranchFlagsHold def

The framed flag equations on each positive transverse branch. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.PositiveBranchFlagsHold

PositiveBranchFrameMatch def

Compatibility of the transported frames across a positive transverse branch. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.PositiveBranchFrameMatch

PositiveBranchFrameValuesAgree def

Equality of the two frame values on a positive transverse branch. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.PositiveBranchFrameValuesAgree

PositiveBranchMirrorAttached def

The attachment of the reflected port of a positive transverse branch. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.PositiveBranchMirrorAttached

PositiveBranchOriginalAttached def

The attachment of the original port of a positive transverse branch. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.PositiveBranchOriginalAttached

PositiveBranchSameSeam def

The two ports of each positive transverse branch attach to the same seam. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.PositiveBranchSameSeam

PositiveBranchSideValuesTrivial def

The side values on each positive transverse branch are trivial. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.PositiveBranchSideValuesTrivial

positiveCarrier def

The coordinates at which a vector is strictly positive. · OAI.Tingley.positiveCarrier

positiveClip def

Positive clipping, with a nonnegative upper level. · OAI.RegularPointCoordinates.positiveClip

positiveColumnFiberEquiv def

The positive high coefficient fiber has exactly its source theta indices. · OAI.Nagata.Workers.W10.positiveColumnFiberEquiv

PositiveContraction def

Positivity via an adjoint square is equivalent to C*-positivity; the norm bound is the actual contraction condition. · OAI.Kaplansky.PositiveContraction

positiveCopy def

A fresh nonaccepting initial state copies the original initial row. · OAI.OneWayLiveness.Machine.positiveCopy

positiveCornerPowers def

The positive compressed rotation powers spanning the centered dihedral corner. · OAI.KadisonSimilarity.DihedralCornerWords.positiveCornerPowers

PositiveDataWrapper inductive

A fixed all-positive data path, with its original dummy argument values. · OAI.PureTypeSystem.Labelled.PositiveDataWrapper

PositiveDimensionFunctional structure

The elementary dimension properties used to construct a compact content. This is an auxiliary interface, not a quasitrace or a linearity assumption. · OAI.Kaplansky.PositiveDimensionFunctional

PositiveEigenvaluesUnique def

A necessary condition for having at most one positive eigenvalue, even when multiplicities are counted. Two distinct positive native eigenvalues already contradict this condition, so no inertia convention is required. · OAI.DistinctLifts.PositiveEigenvaluesUnique

positiveEntry def

Diagonal matrix entries do not name defect classes. · OAI.ContingencyTables.CompanionMultiplierMatrixProgram.positiveEntry

positiveEntryDensity def

The positive representative of the entrywise Radon–Nikodym derivative. · OAI.ZeroKey.positiveEntryDensity

PositiveEvaluation def

Positivity of the actual ordered evaluation determinant. Used below for both regular and zero-at-origin jet bases, including the mixed second chain. · OAI.CriticalHoneycomb.FiniteFunctional.StripAsymptotic.PositiveEvaluation

positiveExtension def

Restrict the ambient vector space to the genuinely positive edge coordinates. · OAI.MatchingEntropy.LooplessGraph.positiveExtension

positiveFrequencyAlgebra def

Decomposable operators preserving the actual positive-frequency cut. Commutation with characters is included so the published causal multiplier theorem applies after Fourier conjugation. · OAI.KadisonKastler.positiveFrequencyAlgebra

positiveFunctionalCLM def

The bounded functional underlying a positive functional. · OAI.BoundedHochschild.KadisonRingrose.positiveFunctionalCLM

positiveHeightConePreimage def

Pullback cone with its zero height functional removed. · OAI.PolycyclicRecognition.CartanWordModel.positiveHeightConePreimage

PositiveHolomorphicBisectionalCurvature def

Pointwise strict positivity, for every pair of real metric-unit vectors. There is no uniform lower bound or upper curvature bound in this definition. · OAI.KahlerUniformization.SmoothKahlerMetric.PositiveHolomorphicBisectionalCurvature

positiveHomotopy def

Positive degree-minus-one h of the manuscript, packaged with its equation. · OAI.ExternalTensor.Section4.positiveHomotopy

PositiveIndegree def

`PositiveIndegree r` requires an incoming witness `u → v` for each vertex `v`. In the reduction this property follows from strong connectivity of a minimal counterexample and survives the transitive-tournament blowup. · OAI.SeymourSecondNeighborhood.PositiveIndegree

PositiveInterlacing def

Strict positive interlacing, with the exact characteristic factorization. · OAI.CriticalHoneycomb.FiniteFunctional.StripAsymptotic.PositiveInterlacing

PositiveIntervalDensity structure

A positive continuous density on an open bounded interval. Endpoint values are irrelevant. · OAI.Problem358.PositiveIntervalDensity

PositiveJordanParametrization structure

A positively oriented regular parametrization of the actual Jordan boundary. Positivity is specified by the real argument lift around interior points, independently of any integration formula. · OAI.CircleDomainRigidity.PositiveJordanParametrization

PositiveKernelHilbert abbrev

The separated Hilbert completion of the finite kernel vectors. · OAI.BoundedHochschild.KadisonRingrose.PositiveKernelHilbert

PositiveKernelPreSpace def

A type copy keeps the kernel seminorm separate from any norm on the free coefficient space. Its completion automatically removes the null vectors. · OAI.BoundedHochschild.KadisonRingrose.PositiveKernelPreSpace

positiveKernelProjection def

Kernel projection of a positive measurable field, with no norm restriction. · OAI.KadisonKastler.WeakOperatorRepresentative.positiveKernelProjection

positiveLaplaceKernel def

The exponentially decreasing kernel on the positive frequency half-line. · OAI.KadisonKastler.positiveLaplaceKernel

positiveLimitSet def

Limit points of a state family as a positive real parameter tends to zero. · OAI.Problem326.positiveLimitSet

positiveMassConfigurations def

Distinct configurations with strictly positive actual cell masses. · OAI.Problem358.CellGeometry.positiveMassConfigurations

positiveNormalPart def

The positive normal part, realized by a projected GNS vector. · OAI.BoundedHochschild.KadisonRingrose.positiveNormalPart

positiveOrientationMatrix def

Coefficients of the positive two-form action. · OAI.PositiveEinsteinFour.QuaternionFrame.positiveOrientationMatrix

positiveOrthant def

The manuscript's positive base orthant. · OAI.AffineBernstein.positiveOrthant

PositiveOutsidePath def

A positive-parameter outside path: exactly d internal vertices, all outside X. The boundary graph excludes the parameter-zero edge. · OAI.CycleClique.Construction.PositiveOutsidePath

positivePeriod def

A positive real period as an actual nonzero complex number. · OAI.Nagata.Workers.W05.positivePeriod

positivePhaseCoefficient def

The degree-zero convention allows the positive coefficients to be defined by removing a first label. · OAI.KadisonKastler.positivePhaseCoefficient

positivePhaseCoefficientSequence def

The actual positive coefficient bounded sequence. · OAI.KadisonKastler.positivePhaseCoefficientSequence

positivePhaseWordUnitary def

Positive word with its last free-unitary factor removed. · OAI.KadisonKastler.positivePhaseWordUnitary

positivePivotKey def

Coefficients include their positive-integer support, so the impossible zero-modulus case has an empty coefficient family. · OAI.Ostmann.positivePivotKey

PositivePlaneEquation structure

A positive-degree equation class. The Cartier-divisor correspondence is not included as a structure field or an assumption. · OAI.Nagata.W16.PositivePlaneEquation

positivePrefix def

A positive prefix, also when its length is zero. · OAI.TwoPointCorrelations.positivePrefix

PositivePrefixGrowth def

The geometric one-step conclusion for the invariant convex set. · OAI.Tingley.PositivePrefixGrowth

PositiveRateETH def

Positive-rate ETH: one positive forbidden rate, with arbitrary fixed polynomial input factor. This is not the weaker uniform premise below. · OAI.ParityWL.Complexity.PositiveRateETH

positiveRealComplexification def

Positivity alone suffices for the complexification; no unit or normalized state hypothesis is required. · OAI.WeakPureInfiniteness.positiveRealComplexification

PositiveRoot def

Genuine positive roots, with their actual coordinate sign. · OAI.KLInvariance.TitsSpace.PositiveRoot

positiveRootFactorHomeomorph def

The explicit root-times-remainder coordinates used in the positive-root product induction. · OAI.DukePrimeDegree.positiveRootFactorHomeomorph

positiveRootGeneratedGroup def

The subgroup generated by the root lines in S. · OAI.DukePrimeDegree.positiveRootGeneratedGroup

PositiveRootPattern def

Outside the chosen positive roots the matrix equals the identity. · OAI.DukePrimeDegree.PositiveRootPattern

positiveRootRightFactorHomeomorph def

With the root factor last, its coordinate lines are actual right root orbits. These are the coordinates used for disintegration. · OAI.DukePrimeDegree.positiveRootRightFactorHomeomorph

positiveSides def

The literal positive indexed attaching word, with no quotienting of repeated labels or cyclic starting positions. · OAI.HyperbolicObstruction.PictureSwitch.PolygonTarget.Boundary.positiveSides

positiveSingularPart def

The remaining positive part, realized by the complementary GNS vector. · OAI.BoundedHochschild.KadisonRingrose.positiveSingularPart

positiveSourceGradientBound def

An explicit, nonoptimal uniform constant for the normalized source. · OAI.RegularPointCoordinates.positiveSourceGradientBound

PositiveSpan inductive

Positive combinations of genuine planar tensor diagrams. · OAI.CriticalHoneycomb.Transfer.PlanarNormalization.PositiveSpan

positiveSphereDilation def

The positive radial parametrization, packaged as a homeomorphism so change of variables is valid even before proving integrability. · OAI.MahlerStokes.positiveSphereDilation

positiveSquareRatio def

The square perspective on the nonnegative cone, with its zero value at zero denominator. Used on actual admissible Lipschitz variations. · OAI.Bernoulli.Halfspace.positiveSquareRatio

positiveStageWeights def

Mix with the uniform vector on all available indices. · OAI.Tingley.positiveStageWeights

PositiveStrictDisjointPaths def

Positive survival probabilities separate finite quantile costs from infinity cells. · OAI.GammaFPP.BK.PositiveStrictDisjointPaths

PositiveSubunitAVR def

The volume-ratio limit is finite, positive, and strictly below one. · OAI.HarmonicCounterexample.PositiveSubunitAVR

positiveSupport def

Support of the positive part, as an actual measurable field. · OAI.KadisonKastler.WeakOperatorRepresentative.positiveSupport

PositiveTable structure

A finite positive table with a common augmentation on its two sides. Sums of atoms have this form even when their augmentation exceeds one. · OAI.Thomason.Construction.DirectedComplex.Cell.PositiveTable

PositiveTrace structure

A finite positive trace. This definition is used only for the finite-rank part of the Powers--Størmer argument; no trace is assumed in the main theorem. · OAI.NuclearUltrapower.PositiveTrace

positiveTriangular def

Positive invertible upper triangular real matrices, as an actual subgroup. · OAI.PolycyclicRecognition.positiveTriangular

positiveTripleSum def

The displayed A.13 sum: its original positive total order is retained. · OAI.MaxCutGames.Appendix.A13Reduction.positiveTripleSum

positiveTripleSum def

The displayed A.13 sum: its original positive total order is retained. · OAI.UniqueGamesTheorem.Appendix.A13Reduction.positiveTripleSum

positiveTruncation def

The actual clipped Sobolev function. · OAI.RegularPointCoordinates.positiveTruncation

positiveValue def

The spectral value in a fixed matrix size, on the genuine positive cone. · OAI.Kaplansky.CuntzDimensionState.positiveValue

positiveWeightCone def

The pointed two-generator cone with its vertex omitted. · OAI.PolycyclicRecognition.positiveWeightCone

positiveWeightRay def

The ray with its vertex omitted. Generator weights are nonzero. · OAI.PolycyclicRecognition.positiveWeightRay

PositiveWord def

Exact positivity at every successive departure site. · OAI.TwoPointCorrelations.PositiveWord

posPow def

The positive-part powers used by the compact cutoff. · OAI.HenonLaneEmden.Cutoff.posPow

posPower def

Positive part to a natural power; the endpoint is fixed to zero. · OAI.PettyProjection.Spherical.posPower

possible def

All potential unordered prescribed matching pairs within a vertex set. Keeping infeasible ones is a harmless overcount. · OAI.TorsionFreeZeroDivisors.DiameterCounting.possible

postA5Sum def

Actual A.5 descendants before recombining the first hybrid restriction. · OAI.MaxCutGames.Appendix.A5Induction.postA5Sum

postA5Sum def

Actual A.5 descendants before recombining the first hybrid restriction. · OAI.UniqueGamesTheorem.Appendix.A5Induction.postA5Sum

postcompose def

Postcomposition acts boundedly on every derivative in a finite jet. · OAI.ForcedComputation.BoundedSpatialJets.postcompose

postcompose def

Postcomposition permutes all linear injections, including their uniform finite distribution. · OAI.MaxCutGames.Gadget.Enlargement.postcompose

postcompose def

Postcomposition permutes all linear injections, including their uniform finite distribution. · OAI.UniqueGamesTheorem.Gadget.Enlargement.postcompose

postcomposeCLM def

Apply one fixed bounded linear map to a jet-valued function. · OAI.ForcedComputation.BoundedSpatialJets.postcomposeCLM

posteriorAnchorSeries def

The literal initial anchor after conditioning the paired side outputs. · OAI.TraceReconstruction.posteriorAnchorSeries

posteriorJointAnchors def

The actual two-sibling anchor multiplier. Global bits are shared by the two siblings until AFTER averaging; their local bits are independent. · OAI.TraceReconstruction.posteriorJointAnchors

posteriorLaw def

The actual probability measure in the displayed conditional-kernel formula. The disintegration identities are proved separately. · OAI.FiniteFactor.CouplingParameters.posteriorLaw

posteriorMass def

This is the conditional label weight from conditional-kernel. · OAI.FiniteFactor.CouplingParameters.posteriorMass

posteriorSmoothingCost def

The exact signed smoothing cost read off the visible metadata alone. · OAI.TraceReconstruction.posteriorSmoothingCost

Potential structure

Actual consistency of the labelled graph in a group, not a metric configuration or a conclusion of the sampling argument. · OAI.HyperbolicObstruction.MasterGraph.Potential

potential def

A degree-two Hamiltonian near the unit fixed quadric, with restriction f. · OAI.ArnoldCounterexample.Field.potential

potential def

The root-inclusive scalar prefix sum. Rootless rows will set f([])=0. · OAI.BoundedTreePotentials.TreeCalculus.potential

potential def

potential via bounded recursion. · OAI.ExactDerandomization.Base.Source.potential

potential def

Newton potentials are kept as nonnegative extended integrals until their finiteness has separately been supplied by the PDE. · OAI.HenonLaneEmden.LayerTonelli.potential

potential def

Every master match has a genuine group-valued potential in the literal letter presentation. Only its actual top and square relators are used. · OAI.HyperbolicObstruction.LetterPresentation.potential

potential def

Exact bounded affine potential used in the manuscript. · OAI.KServer.RankTracking.potential

potential def

Graph-distance potentials extended to the two endpoints of each split edge by their shortest distance through either of that edge's endpoints. · OAI.PlanarL1.Weighted.EnlargedTree.potential

potential def

The actual polynomial F = c E₀ - Θ² from equation (23). · OAI.PositiveEinsteinFour.Potential.potential

potential def

The prescribed value of the supporting potential at each center. · OAI.Problem356.CenterCertificate.potential

potential def

The convex potential used by a finite-affine transport map. · OAI.Problem358.FiniteAffine.potential

potential def

A finite maximum of affine functions, with slopes y and intercepts c. · OAI.Problem358.MaxPotential.potential

potential def

The maximum of the three affine branches. · OAI.Problem358.Sharpness.potential

potential def

The exact source syntax-tree potential. Its recursive child conditions retain the active-open-codomain guard on a counted domain. · OAI.PureTypeSystem.PlainObservations.potential

potential def

The exact closed-prefix vector potential, updated without ever taking the absolute size of an old large-radius envelope. · OAI.SignedDisk.SignedRuns.potential

potential def

The exact closed-prefix vector potential, updated without ever taking the absolute size of an old large-radius envelope. · OAI.SignedRuns.potential

potential def

The diagonal is deliberately absent from overlapLength's domain: the transformed singleton tails have infinite diagonal overlap. · OAI.WeakPinned.Hierarchy.Depth.potential

potentialEnvelope def

Deterministic envelope obtained by iterating the expectation recursion. · OAI.LeanBlast.SecondKahnKalai.potentialEnvelope

potentials def

The actual five local potentials of the Coulomb construction, with the same central quadratic for both branches. · OAI.Problem356.FiveComponentGeometry.potentials

potentialSum def

The signed three-coordinate dual potential associated with one marginal potential. · OAI.Problem356.Certificate.potentialSum

potentialSum def

The sum of a nonnegative supporting potential over the three coordinates. A bounded signed potential can be shifted by a constant before using this sum. · OAI.Problem356.DualCertificate.potentialSum

potentialVisits def

The sum of ambient entry counts over all task nodes, for charging by depth. · OAI.LowSpaceSubsetSum.potentialVisits

pow def

Taking a power multiplies the degree, including the zeroth power. · OAI.Nagata.W16.NonzeroHomogeneousForm.pow

power def

Iterated tensor functors. · OAI.CartierImageFiltration.power

power def

The power-set name uses every HS normalized subset candidate and the conditions which force that candidate to be a subset. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.power

power def

The root split over the two prescribed sides has exactly the source saving. · OAI.TorsionFreeZeroDivisors.StageRoot.power

powerAction def

The n-fold ideal action, as a natural transformation. · OAI.CartierImageFiltration.powerAction

powerAddIso def

Reassociate ideal powers without changing their order. · OAI.CartierImageFiltration.powerAddIso

powerBasis def

The genuine finite integral power basis of ℤ[eta]. · OAI.CirculantHadamard.CyclotomicLocalBase.powerBasis

powerBody def

Repeated multiplication, preserving the binary base in register 8 and consuming a unary exponent in register 9. Used in the Taylor input matrices. · OAI.RationalHitting.StackLanguage.powerBody

PowerCandidate def

The actual HS candidates inside the small set of normalization codes. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.PowerCandidate

powerCharacter def

A genuine cyclic character associated to an element whose nth power is one. · OAI.CirculantHadamard.CyclicPolynomial.powerCharacter

powerComparison def

Iterated tensor actions commute with restriction through the restriction comparison. · OAI.CartierImageFiltration.powerComparison

poweredGraph def

An actual powered constraint graph, with a common raw alphabet at all vertices. Its reversal law is witnessed by the orientation bit. · OAI.MaxCutGames.Foundations.PCP.PoweringTest.poweredGraph

poweredGraph def

An actual powered constraint graph, with a common raw alphabet at all vertices. Its reversal law is witnessed by the orientation bit. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringTest.poweredGraph

poweredPolynomialTimeOfCertificates def

Both actual transfer loops between the machines are included by MachineSequential.composeBits; their work contributes to its polynomial. · OAI.MaxCutGames.Foundations.PCP.RoundComputation.poweredPolynomialTimeOfCertificates

poweredPolynomialTimeOfCertificates def

Both actual transfer loops between the machines are included by MachineSequential.composeBits; their work contributes to its polynomial. · OAI.UniqueGamesTheorem.Foundations.PCP.RoundComputation.poweredPolynomialTimeOfCertificates

powerFinEquiv def

The order is recursive and interleaved, not an arbitrary finite equivalence. · OAI.Mahler.powerFinEquiv

powerGenerators def

The generator of a powered port is the xor of its original port word. · OAI.MaxCutGames.Foundations.PCP.CayleySpectral.powerGenerators

powerGenerators def

The generator of a powered port is the xor of its original port word. · OAI.UniqueGamesTheorem.Foundations.PCP.CayleySpectral.powerGenerators

PoweringCertificate abbrev

The exact powering interface at the fixed parameters of the round. · OAI.MaxCutGames.Foundations.PCP.RoundComputation.PoweringCertificate

PoweringCertificate abbrev

The exact powering interface at the fixed parameters of the round. · OAI.UniqueGamesTheorem.Foundations.PCP.RoundComputation.PoweringCertificate

PowerInput def

Polynomial form of the homogeneous power input, used only as an intermediate algebraic rewrite of the displayed sixth and cubic estimates. · OAI.DiamondDistortion.RecursiveApplications.PowerInput

powerIso def

Every actual native iterate on a stalk is the original-X bar-derived iterate. · OAI.LittleFinitistic.SelectedBarComparison.powerIso

powerMap def

Restriction of exterior vectors, using the given semilinear restriction maps of the module presheaf. · OAI.ActualExterior.powerMap

powerObjIso def

A natural endofunctor comparison identifies every actual object-function iterate. · OAI.LittleFinitistic.BimoduleTotalPresentation.powerObjIso

powerOrbit def

Distinct powers in one period; this is an embedding, not an oracle for r. · OAI.ExactQuantumFactoring.OrderTrial.powerOrbit

PowerRate def

Logarithmic order along a genuine diverging positive scale. · OAI.TraceReconstruction.PowerRate

powerSumValue def

Five closed power-sum formulas, not an enumeration up to the endpoint. · OAI.ExactQuantumFactoring.OrderTrial.powerSumValue

powerThreshold def

The exponent of this threshold depends only on a lower bound for the positive exponent, not on either numerical constant. · OAI.ClosedSurfaceR4.ExactCorrection.powerThreshold

powerToImage def

Powers surject onto the corresponding iterated image. · OAI.CartierImageFiltration.powerToImage

powerTransgressionPrimitive def

The ordered primitive alpha wedge beta wedge (d alpha)^k. · OAI.Mahler.powerTransgressionPrimitive

powerTwoStep opaque

Emit the fixed scale constant by twenty doublings. This keeps the finite machine description small instead of hardcoding a million pushes. · OAI.ContinuumCoulomb.MediatorUnaryProgram.powerTwoStep

PowerUpper def

A robust upper order, with arbitrary strictly positive exponent slack. · OAI.TraceReconstruction.PowerUpper

PP def

The Partition Principle. The reverse injection is not required to be a section of the original surjection. Both X and Y range over all sets. · OAI.PartitionConsistency.SetLanguage.PP

pProbe def

The finite-prefix count difference p_n^T(X). · OAI.InfiniteMatroidCounterexample.pProbe

PPTBadResidual def

The elementary residual facts supplied by maximal-prefix selection. This is an internal grouping invariant, not a published-input assumption. · OAI.TotientAsymptotic.PPTBadResidual

pptComparisonScale def

The part of the comparison bound independent of the two fixed integer factors. · OAI.TotientAsymptotic.pptComparisonScale

pptPrefixesAt def

Only prefixes selected at index J enter its reciprocal mass. · OAI.TotientAsymptotic.pptPrefixesAt

PPTPrefixWitness def

A residual nontrivial preimage whose prime factors fit after the already removed prefix. At index zero there is no preceding bound. · OAI.TotientAsymptotic.PPTPrefixWitness

pptRemoveHeadEmbedding def

Delete the leading coordinate from an ordered embedding fixing zero. · OAI.TotientAsymptotic.pptRemoveHeadEmbedding

PPTResidualGridWitness structure

Arithmetic evidence retained from the actual terminal-prefix construction for one integer. Each field is supplied by ppt_selected_grid_data; the structure contains no estimate for a finite family. · OAI.TotientAsymptotic.PPTResidualGridWitness

pptSuffixProduct def

The concrete suffix of a finite ordered candidate-prime list. · OAI.TotientAsymptotic.pptSuffixProduct

pr def

Weight of an event. · OAI.HadwigerCounterexample.pr

pr def

ret true for inverse signs · OAI.PowerSaving.Spectral.pr

pRadius def

The first radial radius. The isometry is an explicit common parameter even though this first contact only uses the domain center. · OAI.Tingley.pRadius

preBudget def

Initial setup, the actual vertex loop, and the bridge to cleanup. · OAI.MaxCutGames.Foundations.Complexity.PoweringPolynomialBudget.preBudget

preBudget def

Initial setup, the actual vertex loop, and the bridge to cleanup. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringPolynomialBudget.preBudget

precisionScaleData opaque

A fixed integer bounds the finite-difference precision independently of the input instance. · OAI.ContinuumCoulomb.MoserDifferenceBudget.precisionScaleData

precompLinear def

Precomposition by a fixed selfmap is a linear map on ambient-valued maps. · OAI.ReflexiveFixedPoints.precompLinear

precompLocal def

Precomposition in the genuine module of local linear maps. · OAI.CoherentDual.precompLocal

precompose def

Reinterpret the domain, with its entire previous encoding still on tape. · OAI.ExactQuantumFactoring.BitStackProgram.Procedure.precompose

predAt def

Circulant acts by difference of row & column · OAI.PowerSaving.Spectral.predAt

predecessorRoots def

The previously encountered minimal ancestral directions of letters which are dependent on the next direction. Taking the minimum recursively computes the first minimal ancestor in the occurrence poset. · OAI.EilenbergGanea.TraceWords.predecessorRoots

predicateFilter def

A deferred guess predicate, rather than a circuit that constructs the canonical CRT string. Both branches use the same retained completion coin. · OAI.ExactQuantumFactoring.Completion.predicateFilter

predicateGame def

Pass a Boolean verifier predicate to the already checked deterministic local-reconstruction and incidence-extraction interface. · OAI.MaxCutGames.Soundness.RepeatedGameBounds.Simulation.predicateGame

predicateGame def

Pass a Boolean verifier predicate to the already checked deterministic local-reconstruction and incidence-extraction interface. · OAI.UniqueGamesTheorem.Soundness.RepeatedGameBounds.Simulation.predicateGame

PredictableLinearHeads structure

Exact predictable timing: P_i has the same value for the two children of a history of length i-1. The support and nesting clauses are retained. · OAI.BoundedTreePotentials.CostHeads.PredictableLinearHeads

predictionCell def

A finer cell, restricted to the current root region. · OAI.LipschitzHilbert.predictionCell

predictionCellSup def

Supremum on one step cell. · OAI.LipschitzHilbert.predictionCellSup

predictionLimit def

The actual retained parameter inserted into the completed prediction family. · OAI.ReflexiveFixedPoints.CandidateLedger.predictionLimit

predictionMixture def

The actual probability mixture used to approximate cell averages. · OAI.LipschitzHilbert.predictionMixture

predictionNormalizer def

The normalization denominator from §9. · OAI.LipschitzHilbert.predictionNormalizer

predictionStepSup def

Actual cell supremums, constant on the finer partition. · OAI.LipschitzHilbert.predictionStepSup

predictionWeakLimit def

The actual weak limit of already evaluated prediction outputs. · OAI.ReflexiveFixedPoints.Prediction.predictionWeakLimit

predictionWeakLimit def

Compactness chooses limits of the already evaluated finite-frontier outputs. · OAI.ReflexiveFixedPoints.Prediction.Prefix.predictionWeakLimit

preDist def

Real ultrafilter distance; boundedness proves convergence rather than assuming existence of a limiting metric. · OAI.HyperbolicObstruction.MetricLimit.preDist

predLiftMap def

The data parameter binder, not an assumed semantic universal. · OAI.PureTypeSystem.Labelled.RelationalCode.predLiftMap

predual def

The coordinate predual. · OAI.SeparableQuotient.NormConstruction.NormingSet.predual

predualComponentEquiv def

Exact ℓ₂ decomposition of the completed predual into completed blocks. · OAI.BoundedTreePotentials.Segments.predualComponentEquiv

predualCoordinate def

The coordinate functional regarded as an element of the actual predual. · OAI.SeparableQuotient.NormConstruction.NormingSet.predualCoordinate

predualFunctional def

Evaluation by x, made complex-linear in the algebra argument. · OAI.Generator.predualFunctional

predualFunctionalIsometry def

A conjugate-linear isometric embedding of the actual given predual. · OAI.Generator.predualFunctionalIsometry

predualNormalEquiv def

Identification of the predual with the normal functionals. · OAI.BoundedHochschild.KadisonRingrose.predualNormalEquiv

PredualNull def

Bounded blocks are null on the coordinate predual. This intermediate property is stated separately so that the path argument works for any such sequence, not just a chosen block representation. · OAI.SeparableQuotient.ActualSpace.PredualNull

predualRangePullback def

Pull a dual functional back through the conjugate-linear range equivalence. · OAI.Generator.predualRangePullback

preEvaluate def

Structural recursion retaining every stored role choice. The annotation child at a lambda is immaterial to its observation value, but is retained in the input tree for the fundamental normalization lemma. Representability follows from the … · OAI.PureTypeSystem.ObservationValues.preEvaluate

Prefix structure

A dyadic brick has only finitely many nonempty coordinate prefixes. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Prefix

prefixAndFinishInTime def

Composition retains both actual execution witnesses. The cleanup witness is supplied by SourceRuntimeFinish.finishInTime in the caller's real program. · OAI.MaxCutGames.Foundations.Hastad.SourceRuntimeSpace.prefixAndFinishInTime

prefixAndFinishInTime def

Composition retains both actual execution witnesses. The cleanup witness is supplied by SourceRuntimeFinish.finishInTime in the caller's real program. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceRuntimeSpace.prefixAndFinishInTime

prefixArithmeticDegree def

Fixed degree of the prefix/guard arithmetic implementation. · OAI.EditApproximation.prefixArithmeticDegree

prefixBasis def

Embed a basis simplex into the old face. · OAI.Thomason.Construction.SimplexComplex.prefixBasis

prefixBitMachine def

A fixed-bit prefixer uses one finite stack, one label, and one internal state. The bit is fixed in the program; it is not a second, freely accessible input. · OAI.MaxCutGames.Foundations.Complexity.prefixBitMachine

prefixBitMachine def

A fixed-bit prefixer uses one finite stack, one label, and one internal state. The bit is fixed in the program; it is not a second, freely accessible input. · OAI.UniqueGamesTheorem.Foundations.Complexity.prefixBitMachine

prefixBitPolyTime def

A real machine certificate: one transition performs the push and halts. TM2 counts execution of one finite statement as one transition. Here that statement contains exactly one stack push. · OAI.MaxCutGames.Foundations.Complexity.prefixBitPolyTime

prefixBitPolyTime def

A real machine certificate: one transition performs the push and halts. TM2 counts execution of one finite statement as one transition. Here that statement contains exactly one stack push. · OAI.UniqueGamesTheorem.Foundations.Complexity.prefixBitPolyTime

prefixClosure def

The finite ancestral closure, not a same-level truncation. · OAI.BoundedTreePotentials.Segments.prefixClosure

prefixCoefficient def

Coefficients in the exact degree-six reflected Vandermonde expansion. · OAI.HarmonicCounterexample.Certificate.prefixCoefficient

prefixCoordinateEquiv def

Finite multiplication in successive endpoint coordinates, retaining the first coordinate as the first increment. · OAI.PolycyclicRecognition.prefixCoordinateEquiv

prefixCovarianceEnergy def

Conditional covariance density at the branching depth, averaged over the actual shared prefix kernels. · OAI.DilutedSpinGlass.KernelTower.prefixCovarianceEnergy

prefixDomain def

The valid coordinate prefixes are the projections of the body's interior. · OAI.Problem358.FinitePrefix.prefixDomain

prefixEnergyAt def

The squared spatial covariance density under the conditional continuation law after a prefix of length d. This version allows the fixed physical height. · OAI.DilutedSpinGlass.KernelTower.prefixEnergyAt

prefixEquiv def

Splitting all-path tables at their first frame. · OAI.PureTypeSystem.PlainSpaces.Schema.prefixEquiv

prefixes def

Prefixes contain genuine future zero controls, rather than the base point of an endpoint chart (whose control word need not vanish). · OAI.HarmonicCounterexample.CausalConstruction.prefixes

prefixFace def

The k-th face of the full flag contains precisely the first k+1 vertices. · OAI.Thomason.Construction.SubdivisionBarycentric.prefixFace

prefixGraph def

Adjacent copies in the cyclic contour of an ancestral subtree. · OAI.PlanarL1.HeightTree.prefixGraph

prefixInTime def

Actual header initialization followed by every product row and odometer step. · OAI.MaxCutGames.Explicit.MachineProductRuntime.prefixInTime

prefixInTime def

The prefix contains every actual header read, fresh-name initialization, header emission, complete four-row body, and occurrence-counter test. · OAI.MaxCutGames.Explicit.MachineSubdivisionRuntime.prefixInTime

prefixInTime def

Compose the actual stages up to any fixed position in the pair list. · OAI.MaxCutGames.Foundations.Complexity.MachineFieldProfile.prefixInTime

prefixInTime def

Place the actual finite initialization prefix inside an ambient program. · OAI.MaxCutGames.Foundations.Complexity.MachineOdometerInit.prefixInTime

prefixInTime def

The complete actual prefix, with both initialization and every tuple body supplied by their constructed finite programs. · OAI.MaxCutGames.Reduction.MachineAddressGame.prefixInTime

prefixInTime def

Actual header initialization followed by every product row and odometer step. · OAI.UniqueGamesTheorem.Explicit.MachineProductRuntime.prefixInTime

prefixInTime def

The prefix contains every actual header read, fresh-name initialization, header emission, complete four-row body, and occurrence-counter test. · OAI.UniqueGamesTheorem.Explicit.MachineSubdivisionRuntime.prefixInTime

prefixInTime def

Compose the actual stages up to any fixed position in the pair list. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineFieldProfile.prefixInTime

prefixInTime def

Place the actual finite initialization prefix inside an ambient program. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineOdometerInit.prefixInTime

prefixInTime def

The complete actual prefix, with both initialization and every tuple body supplied by their constructed finite programs. · OAI.UniqueGamesTheorem.Reduction.MachineAddressGame.prefixInTime

prefixKernel def

The next-coordinate conditional law, completed by a point on null prefixes. · OAI.Problem358.FinitePrefix.prefixKernel

PrefixLabel def

The manuscript's finite set S, including the identity prefix ratio as a label. · OAI.Generator.PrefixLabel

prefixLabels def

Labels visible in the observation prefix of the currently available history. · OAI.MatroidProphet.prefixLabels

prefixLaw def

The actual first-coordinate marginal of normalized volume on the body. · OAI.Problem358.FinitePrefix.prefixLaw

prefixMiss def

Prefix miss probabilities for independent records in chronological order. · OAI.KServer.PartitionProbabilities.prefixMiss

PrefixNormers structure

Finite prefix normers for a zero-based sequence. The existential formulation is equivalent to a maximum over the finite set. · OAI.SeparableQuotient.PrefixNormers

prefixNormers def

The recursive choices satisfy exactly the prefix hypotheses. · OAI.SeparableQuotient.NormerStep.prefixNormers

prefixPath def

Constant-height switches connect all copies on a common root prefix. · OAI.PlanarL1.HeightTree.Layout.prefixPath

prefixProduct def

Chronological prefix products: the first edge acts first. · OAI.IntegralCharacterVarieties.SurfaceSurgery.prefixProduct

prefixRegion def

The prefix inequalities for a fixed tail threshold, including index zero. · OAI.TotientAsymptotic.prefixRegion

prefixReplay def

Replay every observed arrival in the core history, while forcibly rejecting all positions strictly before K. At suffix arrivals the complete past history (including the rejected prefix) is supplied, never a future weight. · OAI.MatroidProphet.SecretaryReplay.prefixReplay

PrefixResult structure

Actual initialization followed by every row, ending just before reversal. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Loop.PrefixResult

PrefixResult structure

Actual initialization followed by every row, ending just before reversal. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Loop.PrefixResult

prefixScalar def

The scalar prefix change of basis leaves the suffix literally fixed. Its prefix-to-suffix block is zero because the convolution is lower triangular. · OAI.ExactFourier.CoefficientTime.prefixScalar

prefixSection def

The remaining section after a fixed coordinate prefix. · OAI.Problem358.FinitePrefix.prefixSection

prefixSet abbrev

Compatibility name for the source prefix set; only the analysis uses a full order. · OAI.MatroidProphet.Secretary.prefixSet

prefixSet def

Labels in the first K positions of an arrival order. This also handles K = 0 and K ≥ n without a special convention. · OAI.MatroidProphet.SecretaryReplay.prefixSet

prefixSet def

Colors have zero-based indices; the actual color level is one larger. · OAI.TalagrandThreshold.Selector.prefixSet

prefixSimplex def

The full ascending flag, regarded as a simplex of the face-poset nerve. · OAI.Thomason.Construction.SubdivisionBarycentric.prefixSimplex

prefixTasks def

The task count includes empty completed tasks and dropped full prefixes. · OAI.LowSpaceSubsetSum.prefixTasks

prefixTime def

Specification of the trace length; this function is not part of control. · OAI.MaxCutGames.Foundations.Complexity.MachineCloudRank.prefixTime

prefixTime def

A quadratic prefix allowance because the number of original rows is at most the length of their fully encoded input table. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Loop.prefixTime

prefixTime def

Specification of the trace length; this function is not part of control. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineCloudRank.prefixTime

prefixTime def

A quadratic prefix allowance because the number of original rows is at most the length of their fully encoded input table. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Loop.prefixTime

prefixTrace def

Prefix requests before deterministic activation use source label 1 (zero in Fin k); the raw suffix then uses the computed exact-bit epochs. · OAI.UniformKServer.ComputedInstance.prefixTrace

prefixTransport def

Parallel transport from the distinguished starting fiber. The first edge acts first, matching Diagram.boundaryWord exactly. · OAI.IntegralCharacterVarieties.DiskGauge.prefixTransport

prefixUnion def

Labels committed by the deterministic choices on the removed prefix. · OAI.LeanBlast.SecondKahnKalai.ProbabilityTree.prefixUnion

PrefixValid structure

Exactly the finite prefix conditions, with unused raw entries ignored. Padding raw data beyond length imposes no mathematical restriction. · OAI.SeparableQuotient.Norming.PrefixValid

prefixVertices def

The actual values at indices zero through n; repeated values are allowed. · OAI.Tingley.prefixVertices

prefixWeights def

The observed weights, extended by zero away from the observation prefix. Repeated labels in malformed histories are summed; genuine histories have none. · OAI.MatroidProphet.prefixWeights

prefixWires def

Every previously retained bit, not merely the original N, is read-only for an appended fresh transition. The transition may use them as controls. · OAI.ExactQuantumFactoring.prefixWires

PrefixWitness def

The selected sorted prime-factor prefix, including both separation facts. · OAI.Problem337.DivisorClasses.PrefixWitness

prefixWords def

The output headers are included even before the first old row is read. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.LoopState.prefixWords

prefixWords def

The output headers are included even before the first old row is read. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.LoopState.prefixWords

prefixZeroExtend def

Extend a finite vector by zero, using the actual natural-number indices. · OAI.Tingley.prefixZeroExtend

PreGood structure

Coherence conditions below β. · OAI.SeparableQuotient.CoherentClosures.PreGood

preimageHomeomorph def

The inverse-image subcomplex realizes to the actual topological inverse-image subspace. · OAI.Thomason.Construction.RealizationPrism.preimageHomeomorph

preimageMap def

Restrict alpha to Q and identify its image J with T. · OAI.FiniteLattice.CommonCore.preimageMap

Preliminary def

j is assigned to the class named by τ at the local vertex w. Both ordinal values in this definition are forced values of actual names. · OAI.PartitionConsistency.ProductForcing.ChainCode.Preliminary

preliminary def

The preliminary bijection uses the actual least-code order and the one fixed table. The displayed cardinal equality is precisely the point at which the exact-fiber theorem is used. · OAI.PartitionConsistency.ProductForcing.ChainCode.preliminary

PreliminaryData structure

The recursively selected unit blocks, before pathwise-Cauchy differences. No weak compactness hypothesis is smuggled into this structure. · OAI.SeparableQuotient.ActualSpace.PreliminaryData

preLo def

Exact integer endpoints of the inverse image; reflection does not incur any rounding loss on the source half-integer convention. · OAI.TorsionFreeZeroDivisors.AffineIntervals.preLo

preorder def

The full occurrence preorder; equal labels under different parents are retained as different list elements. This is only a correctness invariant, never used as the modulus-input physical circuit's initializer. · OAI.ExactQuantumFactoring.AuxiliaryTree.preorder

preorderLengths def

The deterministic list of incoming-window lengths in full preorder. · OAI.Problem310.WindowRecursion.preorderLengths

preorderLengths def

The window lengths in edge preorder; the first length is that of the incoming edge, followed by its complete descendant block. · OAI.Problem310.WindowSpanSum.preorderLengths

prepare def

The ceiling logs are computed once before the shared tables. · OAI.LowSpaceSubsetSum.PreparedIncidenceCells.prepare

prepare def

Shared arrays are returned once. The caller retains them through row initialization and all later reads. The mask equalities are proof arguments. · OAI.LowSpaceSubsetSum.RepresentationStoredArrays.prepare

prepareBlock def

The table construction runs once for this block. · OAI.LowSpaceSubsetSum.StoredSeparatorCheck.prepareBlock

prepareChunk def

Preprocessing is applied only to headed sectors. Headless pieces retain their literal words; no implicit boundary or identity generator is added. · OAI.FiniteAlgebraicEnvelopes.Compiler.Path.prepareChunk

prepareData def

Synthetic ranks use the new flags on the intersection of active sets. · OAI.KServer.SideFamilies.prepareData

preparedCellGroupSourceWithWork def

The same raw dyadic scale list and Fin-count indices as the original cell estimator, now with the cost of decoding every sampled coordinate. · OAI.EditApproximation.BinaryFraction.preparedCellGroupSourceWithWork

PreparedCore structure

Uniform subset bounds for the finite planar core. · OAI.WeakPinned.PreparedCore

PreparedCorrection structure

Uniform geometric radii and the order-dependent estimates used by the iteration. This package is obtained by preparedCorrection_nonempty. · OAI.ClosedSurfaceR4.FiniteOrderSmoothing.PreparedCorrection

preparedFlags def

The prepared input is concrete and uses only new flags on the common old/new active domain, as in §04 (not intersection of old and new flags). · OAI.KServer.AdaptiveAllocation.preparedFlags

PreparedGreedyPrefix def

A prefix ending either at zero or at a denominator in the target window. · OAI.Problem337.PreparedGreedyPrefix

PreparedKernelNumericBounds structure

Numeric register and loop bounds, independent of previously cached values. · OAI.EditApproximation.BinaryFraction.PreparedKernelNumericBounds

preparedLocalCellGroupsSourceWithWork def

Every admitted scale uses the source-aware whole-cell loop. · OAI.EditApproximation.BinaryFraction.preparedLocalCellGroupsSourceWithWork

PreparedOffsetData abbrev

A shared bit allowance, widths, actual bit lengths and offsets. · OAI.ContingencyTables.DensePrograms.PreparedOffsetData

preparedOracle def

Prepare a runtime-dependent input with a genuine clean Boolean oracle, then run the same fixed quantum program on that fresh input. All old bits x survive, and c is a network, not a large table of counterfactual outputs. · OAI.ExactQuantumFactoring.preparedOracle

preparedRowInTime def

Preparation and one actual embedded row call, with the two extra tapes retained. The row value is computed from the actual old and small tables. · OAI.MaxCutGames.Foundations.Complexity.MachineExpanderTable.preparedRowInTime

preparedRowInTime def

Preparation and one actual embedded row call, with the two extra tapes retained. The row value is computed from the actual old and small tables. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineExpanderTable.preparedRowInTime

preparedTapes def

Exact frame left by initialization, including the preserved input table. · OAI.MaxCutGames.Foundations.Complexity.PoweringInitializeFrame.preparedTapes

preparedTapes def

Exact frame left by initialization, including the preserved input table. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringInitializeFrame.preparedTapes

prepareFour def

Four preparations are shared by every dictionary check in this trial. · OAI.LowSpaceSubsetSum.StoredSeparatorCheck.prepareFour

prepareInTime def

Header reading, literal seed setup, actual Horner power computation, and all odometer digits are supplied by real execution traces of the same program. · OAI.MaxCutGames.Explicit.MachineProductHeaders.prepareInTime

prepareInTime def

From raw input headers to the running occurrence loop, including all finite arithmetic phases and literal output-header emission. · OAI.MaxCutGames.Explicit.MachineSubdivisionEmission.prepareInTime

prepareInTime def

Actual preparation from isolated name fields and unary sign fields. The conclusion preserves every tape and arbitrary caller metadata. · OAI.MaxCutGames.Foundations.Hastad.SourceSignaturePrepare.prepareInTime

prepareInTime def

Header reading, literal seed setup, actual Horner power computation, and all odometer digits are supplied by real execution traces of the same program. · OAI.UniqueGamesTheorem.Explicit.MachineProductHeaders.prepareInTime

prepareInTime def

From raw input headers to the running occurrence loop, including all finite arithmetic phases and literal output-header emission. · OAI.UniqueGamesTheorem.Explicit.MachineSubdivisionEmission.prepareInTime

prepareInTime def

Actual preparation from isolated name fields and unary sign fields. The conclusion preserves every tape and arbitrary caller metadata. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceSignaturePrepare.prepareInTime

preparePhase def

The exact phase-state preparation from Section 3.1: H on every wire, Z=S² on the most significant wire, and S on the next wire. · OAI.ExactQuantumFactoring.preparePhase

preparePhase def

Concrete context-loader/signature phase in the tuple-body program. · OAI.MaxCutGames.Foundations.Hastad.SourceTupleBody.preparePhase

preparePhase def

Concrete context-loader/signature phase in the tuple-body program. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceTupleBody.preparePhase

prepend def

Fold a nonempty raw run by adjacent pair replacement. We prove below that it agrees with the source's single call on the concatenated maximal run. · OAI.FiniteAlgebraicEnvelopes.Compiler.RawRuns.prepend

prependChosen def

Prepend the chosen incoming arc to a path of the chosen child. · OAI.LeanBlast.SecondKahnKalai.ProbabilityTree.prependChosen

prependHomotopy def

The path-space version of concatenating a variable path with a fixed homotopy. · OAI.Thomason.Construction.PathSpace.prependHomotopy

prependRecord def

Prefix a record in its original order, consuming the first stack. · OAI.RationalHitting.StackLanguage.prependRecord

preprocess def

Complete sector preprocessing, prior to the only partial operation: maximal full-raw-run evaluation. Headless sectors will be preserved by the parser. · OAI.FiniteAlgebraicEnvelopes.Compiler.Surface.preprocess

preprocess def

Same completed table as the source builder, with an additional cell trace; the cached payload count changes only when a row is stored. · OAI.LowSpaceSubsetSum.IncidenceBuildCells.preprocess

preprocess def

Both compatibility modes are stored for every local bit mask. · OAI.LowSpaceSubsetSum.SeparatorIncidenceTable.preprocess

preprocessing def

Repackage the checked marker-removal machine without changing any machine or word: the intermediate semantic value is the same formula. · OAI.MaxCutGames.BinaryRenameMachine.preprocessing

preprocessing def

Repackage the checked marker-removal machine without changing any machine or word: the intermediate semantic value is the same formula. · OAI.UniqueGamesTheorem.BinaryRenameMachine.preprocessing

PreprocessingCertificate abbrev

The exact pending preprocessing interface, including its output encoding. · OAI.MaxCutGames.Foundations.PCP.RoundComputation.PreprocessingCertificate

PreprocessingCertificate abbrev

The exact pending preprocessing interface, including its output encoding. · OAI.UniqueGamesTheorem.Foundations.PCP.RoundComputation.PreprocessingCertificate

prescribedMass def

Exact prescribed per-numerator mass, with the manuscript's ceiling. · OAI.ExactQuantumFactoring.OrderTrial.prescribedMass

prescribedPartition def

Actual finite grid tensor with all four boundary spin lists prescribed. West and north are built into the parametrization; east and south are imposed by their exact indicator, before performing any orientation sum. · OAI.CriticalHoneycomb.Transfer.RasterGrid.prescribedPartition

PrescribedTree inductive

Prescribed trees include a vertex at every depth, including unary vertices. Every internal vertex has at least one child; every leaf has the given height. · OAI.DilutedSpinGlass.PrescribedTree

prescribedTupleHeadEquiv def

Splitting off the first position marks a label once, even when that label occurs several times in the prescribed multiset. · OAI.KadisonKastler.prescribedTupleHeadEquiv

prescribedTupleOrderedEquiv def

In the distinct case, both tuple types have literally the same underlying label function. · OAI.KadisonKastler.prescribedTupleOrderedEquiv

prescribedTupleSigmaEquiv def

Forgetting the prescribed multiset gives all label functions exactly once when the multiset itself is also summed over. · OAI.KadisonKastler.prescribedTupleSigmaEquiv

prescriptions def

Each undirected actual edge is prescribed once, regardless of its number of occurrences in the paths. This prevents the independence fallacy. · OAI.TorsionFreeZeroDivisors.PatternEmbeddings.prescriptions

Present def

Row positions of a candidate may belong to the same array. Actual trials, not row positions, are the independent Bernoulli variables. · OAI.HyperbolicObstruction.BernoulliCandidates.Present

Presentation structure

Quotient construction of the exact infimum over all linear representations. No infimum or supremum is assumed to be attained. · OAI.DiamondDistortion.InfimalCost.Presentation

presentation def

The admissible decompositions use full ℓ² and ℓ¹, and the norm on their pair is exactly the linear (q=1) or quadratic (q=2) representation cost. · OAI.DiamondDistortion.PathModel.presentation

presentation def

The presentation is fixed from the finite data, before any quotient or putative separation homomorphism is considered. · OAI.Release075.MarkedLineData.presentation

presentation def

The actual quotient presentation homomorphism. · OAI.StableCoordinate.SourceFiltration.presentation

presentationEquiv def

Equivalence of the explicit finite presentation and the supplied infinite presentation. · OAI.LittleFinitistic.FinitePresentation.presentationEquiv

presentationEquiv def

Relabeling finite generators and relations preserves the actual presented algebra. · OAI.LittleFinitistic.PresentationReindex.presentationEquiv

presentationEquiv def

The zero fiber of the five-generator hypersurface presentation is affine three-space, expressed by an isomorphism of coordinate algebras. · OAI.StableCoordinate.ZeroFiber.presentationEquiv

PresentationIndex def

All literal member names of X together with all their supporting diagrams. No rank bound is placed on X or on its descendants. · OAI.PartitionConsistency.ProductForcing.PresentationIndex

presentationIso def

The original strict category is recovered by the small presentation. · OAI.Thomason.Construction.StrictOmegaCategory.GlobePasting.presentationIso

presentationLift def

Descent through the actual presentation, with every relation checked. · OAI.LittleFinitistic.QuadraticEncoding.presentationLift

presentationPartial def

The partial function associated to a presentation record is a literal orbit graph, not a selected representative in an evaluated universe. · OAI.PartitionConsistency.ProductForcing.presentationPartial

presentationWeight def

The exact negative integer weights on the presentation variables. · OAI.StableCoordinate.Obstruction.presentationWeight

presentedAction def

Transport the descended action through the actual presentation equivalence. · OAI.LittleFinitistic.QuotientAction.presentedAction

PresentedAlgebra abbrev

The algebra with exactly the displayed finite quadratic presentation. · OAI.LittleFinitistic.QuadraticEncoding.PresentedAlgebra

presentedLift def

The merger normal forms have the universal property of the finite relative monoid presentation used in the countable-base rearrangement. · OAI.PartitionConsistency.DiagramForcing.OverlapPrescription.presentedLift

presentedLift def

Universal evaluation for exactly the indexed probe presentation from the source: no gate, join, normal-form, or height law is imposed on the target monoid. · OAI.PartitionConsistency.ProbeAlgebra.Path.presentedLift

preservationSubsetMatrix def

Compression onto the centered GNS leg, including its scalar correction. · OAI.KadisonKastler.preservationSubsetMatrix

Preserves def

A machine preserves the distinguished bottom marker on every write. · OAI.FiniteAlgebraicEnvelopes.MachineRealization.TapeSafety.Preserves

PreservesBall def

The ball is preserved with its actual original shortest paths, not only with unquantified reachability. This is the induction invariant for padding. · OAI.BoundedTreewidthL1.Annulus.PreservesBall

PreservesCanonicalConvergence def

Preservation of canonical convergence for filters on the trace cone. The image-filter identity extends this property to nets in every index type. · OAI.TraceConeClassification.PreservesCanonicalConvergence

PreservesJordan def

Preservation of twice the Jordan product. Over the complex numbers this is equivalent to preservation of (x * y + y * x) / 2. · OAI.KadisonKastler.PreservesJordan

PreservesNondegenerate def

The hypothesis records preservation of dimension, rather than choosing new nondegenerate representatives of possibly degenerate images. · OAI.Thomason.Construction.NonsingularLastVertex.PreservesNondegenerate

PreservesProfiles def

A genuine property of a transition, not a field assumed at arbitrary limits. · OAI.PartitionConsistency.MonoidConstruction.PreservesProfiles

PreservesScheduledAnchorCode def

This is the code of the actual leaf path, independent of its labeled position inside that leaf. · OAI.Ostmann.PreservesScheduledAnchorCode

preservingAlgebra def

The elements preserving a given word span form an ordinary subalgebra. · OAI.KadisonSimilarity.CenteredWordSpan.preservingAlgebra

preservingOperatorAlgebra def

Bounded maps on the ambient operator algebra preserving P. · OAI.KadisonKastler.preservingOperatorAlgebra

presheaf def

The presheaf of Kähler differentials of all section rings. · OAI.ActualCotangent.presheaf

presheaf def

Objectwise exterior powers of the sections and their maps. · OAI.ActualExterior.presheaf

presheafOver def

Restriction of a module presheaf to the slice category. · OAI.ActualSheafTensor.presheafOver

presheafTensorCompOverIso def

Restriction commutes naturally with tensoring by a fixed presheaf. · OAI.ActualSheafTensor.presheafTensorCompOverIso

presheafTensorOverIso def

Tensor presheaves restrict objectwise exactly. · OAI.ActualSheafTensor.presheafTensorOverIso

preSpliceFamily def

Insert a finite family of separate doubled strips AFTER their old child occurrences. Several old contacts may lie on the same old boundary circle. · OAI.IntegralCharacterVarieties.BoundarySurgery.preSpliceFamily

pressure def

The literal marked-Poisson pressure, with every type in the same law. · OAI.DilutedSpinGlass.HeterogeneousMarks.pressure

pressure def

F_N, using an explicit uniform average over the independent index labels. The value at N=0 is irrelevant to the limit; valid system sizes are N>=1. · OAI.DilutedSpinGlass.pressure

pressure def

Finite-volume pressure, normalized per site and by the probability spin prior. · OAI.InvariantIsing.pressure

PreSystem abbrev

Finite closures on the strict initial segment at β. · OAI.SeparableQuotient.CoherentClosures.PreSystem

preTime def

Replace the vertex count by the input length in the actual setup/loop bound. · OAI.MaxCutGames.Foundations.Complexity.PoweringPolynomialBudget.preTime

preTime def

Replace the vertex count by the input length in the actual setup/loop bound. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringPolynomialBudget.preTime

PreTraceGNS def

Type synonym with the L² seminorm, not the original C*-norm. · OAI.TraceConeClassification.ExtendedTrace.PreTraceGNS

preUnitary def

Coordinate unitaries preserve the ultralimit seminorm. · OAI.NuclearUltrapower.HilbertUltraproduct.preUnitary

prev def

The previous boundary occurrence, within the same independent polygon. · OAI.HyperbolicObstruction.CyclicTriangleBoundary.prev

previous def

At time t, at most K earlier positions are constrained. None represents an absent constraint, whose equation is identically zero. · OAI.SmoothObstruction.GroupSampler.previous

previousNet def

Access only retained earlier wires. No history is recomputed. · OAI.ExactQuantumFactoring.NodeMachine.previousNet

price def

A Borel price built from exceptions, rather than a measurability-dependent selection of maximizing radii. · OAI.DyadicMajorant.price

price def

Uniform fee estimates intentionally very loose. · OAI.Fifths.Circuit.price

price def

A Borel price built from exceptions, rather than a measurability-dependent selection of maximizing radii. · OAI.SignedDisk.DyadicMajorant.price

primal def

aggregate primal (resp. channel) directions in a unit. · OAI.HadwigerCounterexample.primal

primalChange def

The invertible change of the ordered domain basis. · OAI.MaxCutGames.Inverse.KMSBasisInvariant.primalChange

primalChange def

The invertible change of the ordered domain basis. · OAI.UniqueGamesTheorem.Inverse.KMSBasisInvariant.primalChange

primalSeminorm def

The exact incomparable-segment primal norm, not its disjoint-segment analogue. · OAI.DiamondDistortion.IncomparableModel.primalSeminorm

primaryAngularMixedChar def

The actual finite part of theta_l * mixedCubic a b, with the necessary unit compensation at 3. Its values on primary elements equal the mixed cubic symbol, but its values on units depend on ell. · OAI.CubicFirstMoment.primaryAngularMixedChar

PrimaryEdge inductive

The sign is true for a domain edge and false for a positive edge. · OAI.PureTypeSystem.PrimaryEdge

primaryElementBall def

Primary element sums omit every ideal divisible by the prime over three. · OAI.CubicFirstMoment.primaryElementBall

primaryIdealBall def

The corresponding finite ideal set, explicitly restricted to ideals with a primary generator, hence prime to the ramified prime. · OAI.CubicFirstMoment.primaryIdealBall

primaryIdealPolynomial def

The primary ideal character polynomial with an arbitrary arithmetic coefficient function. · OAI.CubicFirstMoment.primaryIdealPolynomial

primaryMixedIdealChar def

The actual ideal coefficient, with ramified factors explicitly zero. · OAI.CubicFirstMoment.primaryMixedIdealChar

primaryMixedResidueChar def

The finite residue character agrees with the actual mixed cubic symbol on primary elements and is trivial on global units. It may still be imprimitive; conductor reduction is a separate step. · OAI.CubicFirstMoment.primaryMixedResidueChar

primaryPrimeCount def

The ordinary count of primary prime elements in the norm ball. · OAI.CubicFirstMoment.primaryPrimeCount

primaryPrimeMultiplicity def

The number of primary primes having a specified integer norm. · OAI.CubicFirstMoment.primaryPrimeMultiplicity

PrimaryPrimePNT def

The ordinary prime ideal theorem, in the normalization used here. The weighted first moment is proved below, rather than included in this input. · OAI.CubicFirstMoment.PrimaryPrimePNT

primeBin def

The prime bin (x^(k/J), x^((k+1)/J)]. · OAI.JointDickman.primeBin

primeChebyshev def

The logarithmically weighted sum over the actual primary prime elements. · OAI.CubicFirstMoment.primeChebyshev

primeCoefficient def

The nonzero-frequency local coefficient in the ternary prime problem. · OAI.Problem337.RamanujanEuler.primeCoefficient

primeCollisionWeight def

Logarithmic weight of the primes at which two natural numbers collide. · OAI.Ostmann.primeCollisionWeight

primeCount def

The global divisor pool uses twice the binary logarithm of its range. · OAI.Problem337.CompositeSupply.primeCount

PrimeCountingUpperBound def

Only the range t ≥ y of the usual Chebyshev upper estimate is needed. · OAI.JointDickman.PrimeCountingUpperBound

primeCRTFunction def

The actual function on the product modulus obtained from local prime functions. · OAI.Ostmann.primeCRTFunction

primeCutoffFactors def

The extra factor is the original top-prime cell cutoff, evaluated in the same positive real coordinate as the history polynomials. · OAI.Ostmann.primeCutoffFactors

primeDensityFourier def

Extend the local transform to an unrestricted prime index. The value at zero is immaterial; all applications index positive primes. · OAI.Ostmann.primeDensityFourier

primeDistanceSquared def

The squared pretentious distance from n ↦ n^(it) up to X. · OAI.JointDickman.PublishedInputs.primeDistanceSquared

primeEligible def

Exactly the prime test performed on an ell-bit candidate. · OAI.LowSpaceSubsetSum.primeEligible

primeExponentialSum def

The actual prime logarithmic exponential sum, with positive phase. · OAI.Problem337.ThreePrimeContinuousMinorArc.primeExponentialSum

primeFourierIntegrand def

The cubic integrand extracting the ordered prime triples of sum u. · OAI.Problem337.ThreePrimeFourier.primeFourierIntegrand

primeFourierSum def

The ordinary positive-sign prime exponential sum, at the zero-padded cyclic modulus 3*u+1. · OAI.Problem337.ThreePrimeFourier.primeFourierSum

primeFrequencies def

Positive prime frequencies up to the integer under representation. · OAI.Problem337.ThreePrimeContinuousFourier.primeFrequencies

primeLogIndex def

The cell (h,h+1] containing the logarithm of a prime. · OAI.Ostmann.primeLogIndex

primeLogVariation def

The exact partial-summation cost after removing the logarithmic weight. · OAI.CubicFirstMoment.primeLogVariation

primeLogWeight def

Logarithmic weight supported only on primes, unlike the von Mangoldt weight. · OAI.Problem337.primeLogWeight

primeLogWeight def

Prime logarithmic weights, zero padded in a cyclic group. · OAI.Problem337.ThreePrimeMinorArc.primeLogWeight

primeLowUnionKernel def

The nonnegative part discarded from the independent model. · OAI.JointDickman.primeLowUnionKernel

primeNormCoefficient def

Coefficients collected by integer norm, on a finite ambient prime set. · OAI.CubicFirstMoment.primeNormCoefficient

PrimePaddingObstruction def

Algebraic interface needed to rule out the two exceptional splits at a prime. · OAI.Problem337.PrimePaddingObstruction

primePairs def

First coordinates of the ordered representations of N by two primes. · OAI.Problem337.PrimePairSieve.primePairs

primePhaseSum def

The unweighted positive-sign prime exponential sum. · OAI.Problem337.ThreePrimeContinuousFourier.primePhaseSum

primePhysicalTest def

A giant uses the Fourier phase on favorable primes and zero elsewhere; every other prime position uses the exact normalized residue indicator. · OAI.Ostmann.primePhysicalTest

primePowerUnitCount def

A closed integer expression; no enumeration of residue classes. · OAI.ExactQuantumFactoring.primePowerUnitCount

primePrefixProduct def

Product of the first n primes, indexed from zero. · OAI.Problem337.primePrefixProduct

primeProductFiber def

The ordered prime tuples of length k whose product is N. · OAI.Problem337.primeProductFiber

primeProductMass def

The prime-product pushforward of independent Bernoulli z/p selection; this is the manuscript's ν_z on the finite allowed prime set. · OAI.JointDickman.primeProductMass

PrimeProductMertensInput def

Mertens' product theorem, with its normalized limiting constant: Montgomery–Vaughan, Multiplicative Number Theory I, Theorem 2.7(e). · OAI.JointDickman.PublishedInputs.PrimeProductMertensInput

primeProductModMass def

The probability mass of one residue for independent uniform prime samples. · OAI.Problem337.primeProductModMass

primeProductTransform def

The cofactor formula written in terms of the integer product, so that the order of the current prime list is absent. · OAI.Ostmann.primeProductTransform

primeRamanujanSum def

The prime-modulus Ramanujan sum, indexed by nonzero residues. · OAI.Problem337.ThreePrimeLocalFactor.primeRamanujanSum

primeRange def

The finite prime range used by all modular filters. · OAI.LowSpaceSubsetSum.primeRange

PrimeReciprocalInput def

The bounded-error consequence of published Mertens' second theorem, stated exactly as the existing library declaration except for this name. · OAI.TwoPointCorrelations.PrimeReciprocalInput

PrimeReciprocalInput def

The bounded-error consequence of published Mertens' second theorem, with a uniform bound for all real cutoffs at least two. · OAI.TwoPointCorrelations.PrimeReciprocalInput

PrimeReciprocalMertensInput def

The classical reciprocal-prime Mertens estimate, at its usual O(1/log x) strength: Montgomery–Vaughan, Multiplicative Number Theory I, Theorem 2.7(d). · OAI.JointDickman.PublishedInputs.PrimeReciprocalMertensInput

primeRelationContribution def

The contribution of precisely the terms containing this label. · OAI.TwoPointCorrelations.primeRelationContribution

primeRelationEvent def

The whole congruence, including its nondegeneracy test and controlling prime, ignores a variable outside the recorded support and control. · OAI.TwoPointCorrelations.primeRelationEvent

PrimeScale structure

The complete arithmetic choice of prime and prime-power base. · OAI.QuantitativeVanDerWaerden.Parameters.PrimeScale

primeSetExponent def

The squarefree product of a finite set of prime ideals. · OAI.CubicFirstMoment.primeSetExponent

PrimeSlotData structure

Each occurrence records its step, whether it belongs to the padding, and its numerical prime. Unused slots are avoided by taking the actual total length. · OAI.TwoPointCorrelations.PrimeSlotData

PrimeSquareTests def

The numerator at every node is tested before division by the node's frequency or by its compensation product. · OAI.Ostmann.MovingSlotData.PrimeSquareTests

primesUpTo def

The primes not exceeding an integer cutoff. · OAI.TwoPointCorrelations.primesUpTo

primeTripleLogSum def

The corresponding weighted sum over genuine prime triples. · OAI.Problem337.primeTripleLogSum

primeTupleCount def

Ordered tuples of h primes from P with product at most N. · OAI.JointDickman.primeTupleCount

primeWord def

The range check makes the AKS interface exact also for the invalid 0,1 entries occurring in arbitrary uniform guesses. · OAI.ExactQuantumFactoring.BitArithmetic.primeWord

PrimeWordEncoding abbrev

Slot count, complete equality-pattern code, and separate prime-pool assignments. · OAI.TwoPointCorrelations.PrimeWordEncoding

Primitive inductive

The five primitive gates exactly as in Section 2. · OAI.ExactQuantumFactoring.Primitive

Primitive structure

The least positive period is explicit, as required for primitive groups. · OAI.Superstring.Hierarchical.Primitive

primitive def

The explicit integral in eq:side-primitive-explicit. · OAI.KServer.SideTracker.primitive

primitive def

One finite-table operation, including a return bit chosen from scanned tops. All alphabets and control sets of this routine are visibly finite. · OAI.ParityWL.Complexity.primitive

PrimitiveAngularHeckeInput def

Precise published continuation/functional equation for the actual complete angular ideal coefficients. The dual has angle -ell and conjugate finite character. The exceptional principal case is excluded. · OAI.CubicFirstMoment.PrimitiveAngularHeckeInput

primitiveAtlasPolynomial def

A single primitive's mean polynomial in the fixed atlas. · OAI.ClosedSurfaceR4.FiniteOrderSmoothing.SmoothingAtlas.primitiveAtlasPolynomial

primitiveBlock def

The primitive alphabet consists of the literal strings 01 and 1. · OAI.TraceReconstruction.primitiveBlock

PrimitiveHeckeAnalyticData structure

The precise ordinary Hecke inputs needed for contour shifting: right-half-plane series, entireness, the regular-point functional equation, and polynomial growth in each of the required strips. See Watkins (2011) §3.6 and Thorner--Zaman … · OAI.CubicFirstMoment.PrimitiveHeckeAnalyticData

PrimitiveHeckeFiniteOrderData structure

Finite order is the weak growth statement supplied by the classical completed Hecke-function theory. Its conversion from the completed order-one bound is kept separate from the Phragmén–Lindelöf argument. · OAI.CubicFirstMoment.PrimitiveHeckeFiniteOrderData

PrimitiveMark abbrev

The four complete traces possible from the ordinary primitive 01. · OAI.TraceReconstruction.PrimitiveMark

primitiveMean def

The mean of the unnormalized primitive is also a bounded linear map. · OAI.ClosedSurfaceR4.PeriodicIntegralOperator.primitiveMean

primitivePeriodConstant def

Absolute constant in the conditional second-moment exponent. · OAI.TraceReconstruction.primitivePeriodConstant

PrimitiveRealZero structure

An actual primitive real character and an actual zero of its continued Dirichlet L-function, rather than an unrestricted correction coefficient. · OAI.Ostmann.PrimitiveRealZero

PrimitiveResidueCharacter def

No inducing modulus has smaller norm. This is the usual primitivity criterion for a finite ray character over a principal ideal domain. · OAI.CubicFirstMoment.PrimitiveResidueCharacter

PrimitiveResidueHeckeInput def

The standard analytic continuation and functional equation for a primitive nonprincipal finite Hecke character with trivial infinity type. The two Dirichlet series are literally those of the residue character and its complex conjugate on … · OAI.CubicFirstMoment.PrimitiveResidueHeckeInput

PrimitiveVector def

The denominator and all coordinates have no nontrivial common divisor. · OAI.QuantitativeVanDerWaerden.PrimitiveVector

principal def

A principal tensor acts on actual polynomial conjugated derivatives. · OAI.ElasticityBoundaryLayerOperator.principal

principal def

The intersection of all congruences containing the generating pair. · OAI.FiniteCongruence.Congruence.principal

principalCell def

The principal cell, built recursively rather than postulated. · OAI.Thomason.Construction.SimplexComplex.principalCell

principalColor def

Color a pair by its principal congruence, transported to the lattice. · OAI.FiniteCongruence.principalColor

principalCubeKernel def

The principal kernel after its exact three-to-one cube reindexing. · OAI.CubicFirstMoment.principalCubeKernel

principalDualInjection def

On an affine Cartier chart, the restriction of the sheaf dual embeds in the dual of the genuinely scalar-extended module. · OAI.CoherentDual.principalDualInjection

principalFilter def

The directed filter generated by one condition; it is NOT called generic and no meeting property is attributed to it. · OAI.PartitionConsistency.ForcingQuotient.principalFilter

principalIdealAssociate def

Associate class of a principal ideal; unsupported nonprincipal ideals map to zero. · OAI.Nagata.Workers.W30.principalIdealAssociate

principalIn def

Along a maximal oriented facet this is the incoming parent port. A branch parent at a split is paired to a CHILD, so cannot lie on such a facet. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.Kind.principalIn

PrincipalPackage def

Both principal groups must be proved together, without additional hypotheses. · OAI.StableCoordinate.PrincipalPackage

principalQuotientEquiv def

An ambient algebra equivalence carrying one equation to another induces an algebra equivalence of their hypersurface coordinate rings. · OAI.StableCoordinate.principalQuotientEquiv

principalRestriction def

The induced chart restriction on the genuine hypersurface quotient. · OAI.Nagata.W18.principalRestriction

principalUnitCoeff def

The coefficient of a principal unit. Its uniqueness is proved separately using the domain assumption and t ≠ 0. · OAI.CirculantHadamard.principalUnitCoeff

principalUnits def

Units whose difference from one is a multiple of the specified element. · OAI.CirculantHadamard.principalUnits

prioritySet def

The priority order in source compiler-normalization: entry first, success second, and arbitrary component member only when neither is present. · OAI.FiniteAlgebraicEnvelopes.Compiler.prioritySet

priorPaths def

All slots before time t, including the whole current cycle when t=i+1. · OAI.CubeShuffle.priorPaths

priorPaths def

All slots before time t, including the whole current cycle when t=i+1. · OAI.CubeShuffle.RoutingNetwork.priorPaths

prism def

A natural actual cell of the right cylinder associated to every cell. · OAI.Thomason.Construction.DirectedComplex.RightCylinder.prism

prism abbrev

The actual product of the two representable simplicial sets. · OAI.Thomason.Construction.PrismBarycentric.prism

prism def

The strict cylinder is defined using the proved unique assembly. · OAI.Thomason.Construction.RelativePrismData.prism

prism def

The constructed chain prism transported through the published comparison. · OAI.Thomason.Construction.RightGrayInput.prism

prismApp def

The generator formula on the actual right tensor chain groups. · OAI.Thomason.Construction.SimplexComplex.prismApp

prismBasis def

The image of a prism generator is the normalized tuple with the final upper endpoint appended. · OAI.Thomason.Construction.SimplexComplex.prismBasis

prismChainMap def

The positive relative prism is a genuine augmented directed chain map. · OAI.Thomason.Construction.SimplexComplex.prismChainMap

prismCocone def

Compatible topological prisms indexed by every simplex, including degenerate ones. · OAI.Thomason.Construction.RealizationPrism.prismCocone

prismSide def

Which endpoint occurs in the chosen face of the prism. · OAI.Thomason.Construction.DirectedComplex.RightCylinder.prismSide

prismTable def

A prism has one endpoint chain and the preceding mixed chain in every positive degree. · OAI.Thomason.Construction.DirectedComplex.RightCylinder.prismTable

privateH def

expected entropy remaining conditional on a signal channel · OAI.FourierLLogL.Law.privateH

privateLaw def

This is an actual finite distribution on actual answers, for every visible row table. The sampler's normalization is Parseval, not a hypothesis. · OAI.MaxCutGames.Decoder.PrivateStrategy.privateLaw

privateLaw def

This is an actual finite distribution on actual answers, for every visible row table. The sampler's normalization is Parseval, not a hypothesis. · OAI.UniqueGamesTheorem.Decoder.PrivateStrategy.privateLaw

privateOrigin def

The slice origin, expressed relative to the fixed visible row base. · OAI.MaxCutGames.Decoder.TransferredDecoding.privateOrigin

privateOrigin def

The slice origin, expressed relative to the fixed visible row base. · OAI.UniqueGamesTheorem.Decoder.TransferredDecoding.privateOrigin

privateOutput def

The decoder receives no left witness. The first-bit repair affects only zero-mass frequencies when the table is folded. · OAI.MaxCutGames.Decoder.PrivateStrategy.privateOutput

privateOutput def

The decoder receives no left witness. The first-bit repair affects only zero-mass frequencies when the table is folded. · OAI.UniqueGamesTheorem.Decoder.PrivateStrategy.privateOutput

privateSeedLaw def

The uniform output character followed by its actual Fourier frequency. · OAI.MaxCutGames.Decoder.PrivateStrategy.privateSeedLaw

privateSeedLaw def

The uniform output character followed by its actual Fourier frequency. · OAI.UniqueGamesTheorem.Decoder.PrivateStrategy.privateSeedLaw

prize def

Bucket equations sum over endpoint for each first port, then firsts. · OAI.ExactDerandomization.Chron.Moon.Weather.prize

prob def

Probability, in real rather than ENNReal coordinates, of a finite event. · OAI.HyperbolicObstruction.FiniteLaw.prob

prob def

Uniform probability on the complete finite product sample space. · OAI.MetricEntropyDuality.FiniteProductCounting.prob

prob def

The weight of a finite event. · OAI.QuantitativeVanDerWaerden.FiniteProbability.prob

Probability structure

A probability on the finite sample space Ω, with the event identities needed by the finite local lemma. · OAI.QuantitativeVanDerWaerden.FiniteLocalLemma.Probability

probability def

Finite event probability, with the product law on the full endpoint array. · OAI.CoordinateSweeps.Coverage.probability

probability def

Probability of the prescribed trajectory event, with the root definition. · OAI.CoordinateSweeps.Grid.Holes.probability

probability def

A literal finite arithmetic template for P. Its size is independent of Q, B,d,j,t; there is no enumeration of the phase-register range. · OAI.ExactQuantumFactoring.OrderTrial.Expressions.probability

probability def

Uniform finite probability, kept rational because the gadget noise is a pushforward of finite uniform choices. · OAI.MaxCutGames.Gadget.Enlargement.probability

probability def

Probability of a decidable event in the finite experiment. · OAI.MaxCutGames.Gadget.Harmonic.probability

probability def

Uniform finite probability, kept rational because the gadget noise is a pushforward of finite uniform choices. · OAI.UniqueGamesTheorem.Gadget.Enlargement.probability

probability def

Probability of a decidable event in the finite experiment. · OAI.UniqueGamesTheorem.Gadget.Harmonic.probability

probabilityAverage def

Equal-weight mixture of two probability measures. · OAI.FiniteFactor.probabilityAverage

probabilityCouplings def

The closed set of probability measures with two specified marginals. · OAI.Problem358.probabilityCouplings

probabilityDensityKernel def

A globally Markov version, using a fixed point mass at invalid density sections. · OAI.Problem358.probabilityDensityKernel

probabilityOperatorIntegral def

An actual bounded operator obtained by integrating its values on vectors. · OAI.KadisonKastler.probabilityOperatorIntegral

probabilitySimplex def

Nonnegative real weights with total mass one. · OAI.MatchingEntropy.probabilitySimplex

probabilitySimplexReindex def

Relabelling the coordinates is a homeomorphism of probability simplexes. · OAI.Tingley.probabilitySimplexReindex

ProbabilityTree inductive

Equal-depth trees carrying an actual strictly positive child law at every branch. · OAI.LeanBlast.SecondKahnKalai.ProbabilityTree

probabilityVelocityDistance def

Distance to the actual velocity subspace, measured using the fixed matrix norm. · OAI.DukePrimeDegree.probabilityVelocityDistance

probe def

The actual new probe after its conservative transport to a later cut. · OAI.PartitionConsistency.MonoidConstruction.Round.probe

ProbeCode structure

The A-independent syntax of the common probe type. · OAI.PureTypeSystem.Labelled.ProbeCode

ProbeData structure

A probe request has exactly the old parameters in the presentation; its path contains its countably many transition and terminal quotients by conservativity, not an additional saturation assumption. · OAI.PartitionConsistency.MonoidConstruction.ProbeData

probeDenseSet def

The explicit ground dense set in the probe argument for ordinal choice (source equation eq:probe-cohen-dense). It contains conditions incompatible with the common root and conditions extending one of the placed copies. · OAI.PartitionConsistency.CohenForcing.Condition.probeDenseSet

probeFor def

The source z_i: the body abstracts z:Z, then w:Single(z↓). No distinguished A or m occurs in this term. · OAI.PureTypeSystem.Labelled.RelationalCode.probeFor

probeLiminf def

Extended lower limiting value of the real probes. · OAI.InfiniteMatroidCounterexample.probeLiminf

ProbeProfile structure

The full profile of a probe, with downsets taken in the entire monoid rather than only in the base over which the probe was adjoined. · OAI.PartitionConsistency.DiagramOrder.ProbeProfile

Procedure structure

A completely explicit finite-state Boolean-stack subroutine, with the same clean input/output convention as the frozen TM2 uniformity target. · OAI.ExactQuantumFactoring.BitStackProgram.Procedure

Procedure structure

A quantitative implementation, with completely clean stack input and output. Both execution and result size are polynomially bounded. · OAI.RationalHitting.StackLanguage.Procedure

proceeds def

Normalized scalar transform along part (plateau jumps) on path x. · OAI.FourierLLogL.Dyadic.Procedure.Paths.proceeds

Processed structure

Ground=top has already been proved on U; U is predecessor closed and contains every strict predecessor of the component currently processed. · OAI.PureTypeSystem.Processed

processed def

The finite list is precisely the closed bands meeting the prescribed closed height window, not a truncation at individual points of a band. · OAI.PlanarL1.Tiling.processed

processor def

A finitely supported Mathlib TM2 has an actual finite-control processor. It will be initialized with its input on its designated stack. · OAI.UniformKServer.FlatTM2.processor

ProcessorFlowBounds def

Euclidean, rather than coordinate-maximum, norms are used in this clause. · OAI.ForcedComputation.ProcessorFlowBounds

ProcessorInput structure

The prescribed periodic planar processor and its effective derivative data. The force compiler invokes only this input oracle. · OAI.ForcedComputation.VelocityDetector.CompactCenter.ProcessorInput

ProcessorProperties structure

All clauses of Theorem 3.1, with the two equivalent chart interpretations and the finite compiler data kept explicit. · OAI.ForcedComputation.Recorder.Planar.ProcessorProperties

prod def

Independent auxiliary mark kernels are joined at each depth, not added as separately normalized replicas. · OAI.DilutedSpinGlass.KernelTower.prod

prod def

Product law for the independent master and six-letter arrays. · OAI.HyperbolicObstruction.FiniteLaw.prod

prod def

The separate action on the product is again nonsingular. · OAI.KadisonKastler.NonsingularAction.prod

prod def

Independent actual outcomes. A failure value in either coordinate does not prevent the other coordinate from being drawn. · OAI.LowSpaceSubsetSum.FiniteLaw.prod

prod def

The second component is the low-order block coordinate. · OAI.MaxCutGames.Foundations.Hastad.SourceOccurrences.Encoding.prod

prod def

The second component is the low-order block coordinate. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceOccurrences.Encoding.prod

prodSubmoduleKernelEquiv def

The kernel of the last-block projection is the first-block intersection. · OAI.Kaplansky.prodSubmoduleKernelEquiv

producedCutSeed def

The old generator coordinates after the necessary based-circle gauge. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.producedCutSeed

ProducedMarkedFirstShortEquation def

The framed flag equation at the first produced short seam. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.ProducedMarkedFirstShortEquation

ProducedMarkedSeamEquations def

The framed flag equations at every produced marked seam. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.ProducedMarkedSeamEquations

ProducedMarkedSurfaceEquation def

The surface equation of the produced marked diagram. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.ProducedMarkedSurfaceEquation

ProducedMarkedVertexEquation def

The vertex equations of the produced marked diagram. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.ProducedMarkedVertexEquation

product def

Independent tuples of edges and labels. The two endpoint types keep the two parts separate, even when the base sets happen to have the same type. · OAI.MaxCutGames.Explicit.BipartiteGame.product

product def

Independent conjunction product. Each answer may depend on the full local pair of questions; locality is inherited from Game and Strategy. · OAI.MaxCutGames.Repetition.product

product def

Independent product of the outer and conditional laws. · OAI.MaxCutGames.Repetition.ProjectionKernel.product

product def

Concatenation, absorbing the leading old coefficient of the second normal form into the last letter of the first. · OAI.PartitionConsistency.MergerAlgebra.Word.product

product def

Independent tuples of edges and labels. The two endpoint types keep the two parts separate, even when the base sets happen to have the same type. · OAI.UniqueGamesTheorem.Explicit.BipartiteGame.product

product def

Independent conjunction product. Each answer may depend on the full local pair of questions; locality is inherited from Game and Strategy. · OAI.UniqueGamesTheorem.Repetition.product

product def

Independent product of the outer and conditional laws. · OAI.UniqueGamesTheorem.Repetition.ProjectionKernel.product

productBaseChangeIso def

Pulling back a product projection gives the actual product over the new base. All maps are canonical and preserve the new base coordinate. · OAI.Thomason.Construction.productBaseChangeIso

productBody def

The genuine Cartesian product T₁₀ × T₁₀ in Euclidean R²⁰. · OAI.ProjectionCounterexample.productBody

productBoundaryClass def

The actual bounded family of unitary classes on the product boundary. · OAI.KadisonKastler.productBoundaryClass

productBoundaryConstantFamily def

The corresponding representation as a family of constant operator classes. · OAI.KadisonKastler.productBoundaryConstantFamily

productBoundaryRepresentation def

The commuting left and right copies form an actual representation. · OAI.KadisonKastler.productBoundaryRepresentation

productBoundaryRepresentative def

A uniformly unitary representative of the product field. · OAI.KadisonKastler.productBoundaryRepresentative

productBoundaryValue def

The pointwise product of the left cocycle and its tracial right copy. · OAI.KadisonKastler.productBoundaryValue

productCharacters def

Product of two characters, viewed at their common product modulus. · OAI.TwoPointCorrelations.productCharacters

productChosenVertex def

The product inclusion of that same chosen vertex. · OAI.Thomason.Construction.productChosenVertex

productCoordinates def

First coordinate is signed distance to the chosen affine plane; the others parameterize its orthogonal coordinates. · OAI.Problem358.HouseholderCoordinates.productCoordinates

productCost def

Total cost of a product expression. · OAI.PolycyclicRecognition.productCost

ProductEntries abbrev

Literal children suffice to name the Cartesian product of two sets, at arbitrary ranks. The activation test retains both membership values. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.ProductEntries

productEquiv def

Left component first, with the right component varying fastest. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Enumeration.productEquiv

productEquiv def

Left component first, with the right component varying fastest. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Enumeration.productEquiv

productFacetDomain def

The product-simplex parameter domain for a facet in the selected block. · OAI.ProjectionCounterexample.productFacetDomain

productFacetLinear def

Linear part of the affine product facet chart. · OAI.ProjectionCounterexample.productFacetLinear

productFiberCoefficient def

The selected coefficient at one ordinary product-compressed frequency. · OAI.MaxCutGames.Inverse.KMSAnalyticHybridEnergy.productFiberCoefficient

productFiberCoefficient def

The selected coefficient at one ordinary product-compressed frequency. · OAI.UniqueGamesTheorem.Inverse.KMSAnalyticHybridEnergy.productFiberCoefficient

productFrequencyEquiv def

The exact change of frequency coordinates for a product restriction. · OAI.MaxCutGames.Inverse.KMSAnalyticHybridEnergy.productFrequencyEquiv

productFrequencyEquiv def

The exact change of frequency coordinates for a product restriction. · OAI.UniqueGamesTheorem.Inverse.KMSAnalyticHybridEnergy.productFrequencyEquiv

productFunctional def

The twenty-two bounding inequalities of the actual Cartesian product. · OAI.ProjectionCounterexample.productFunctional

ProductGaps def

Centered coefficients may separate arbitrary signs; unit coefficients may only separate equal signs. This excludes cancellation through an internal unit. · OAI.Generator.Tracial.ProductGaps

productGraphLineUnit def

The graph evaluation unit also covers zero-dimensional factors. · OAI.Kaplansky.productGraphLineUnit

productInitialVertex def

The initial-vertex inclusion with an arbitrary simplicial parameter. · OAI.Thomason.Construction.productInitialVertex

ProductInstance structure

A product's exact stored formation data and its raw environment. · OAI.PureTypeSystem.ObservationValues.ProductInstance

productLabel def

Arbitrary bounded coordinates give a genuine finite measurable partition, with exactly this finite alphabet, rather than an assumed cardinal bound. · OAI.SmoothObstruction.GridMargin.productLabel

productLetters def

Concatenate the explicitly alternating expansions of all corner blocks. · OAI.KadisonSimilarity.DihedralCornerWords.productLetters

productLineCore def

L₀ is trivial; Lᵢ is the actual hyperplane line pulled back from factor i. · OAI.Kaplansky.ComplexGrassmann.productLineCore

ProductMap structure

A componentwise injection on a nonempty Boolean product domain. Values of mapping off the domain are immaterial. · OAI.WLIdentification.ProductMap

productMass def

Product mass of independent particle data. · OAI.CoordinateSweeps.Coverage.productMass

productMass def

Product of genuine root-core probabilities at every vertex. · OAI.FKCRT.Inventory.CoreTree.productMass

ProductMatrixStages structure

An increasing sequence of finite product matrix algebras, in its cyclic separating product representation. This describes the construction data rather than any of its centralizer or factor consequences. · OAI.KadisonKastler.ProductMatrixStages

productMean def

The product of the two marginal statistics. · OAI.JointDickman.productMean

productMeasure def

The product law of independent copies of the multiplier. · OAI.IndependentProducts.productMeasure

ProductMixedSliceBound def

The actual mixed-slice hypothesis in all coordinates adapted to the compressed image. It is a bound on smallComponent, with the original function and translate transported by the explicit product equivalence. · OAI.MaxCutGames.Inverse.KMSAnalyticHybridEnergy.ProductMixedSliceBound

ProductMixedSliceBound def

The actual mixed-slice hypothesis in all coordinates adapted to the compressed image. It is a bound on smallComponent, with the original function and translate transported by the explicit product equivalence. · OAI.UniqueGamesTheorem.Inverse.KMSAnalyticHybridEnergy.ProductMixedSliceBound

productNet def

Multiplication is modulo the sufficiently wide product register; no repetition of earlier expression DAGs occurs in this linear fold. · OAI.ExactQuantumFactoring.BitArithmetic.productNet

productOccupationLaw def

The law of a measurable chart sampled independently in its two factors. · OAI.PolycyclicRecognition.productOccupationLaw

productPairBaseSubsetIso def

Relative product coordinates over an arbitrary subset of the base. · OAI.Kaplansky.productPairBaseSubsetIso

productPairChartFiberMap def

A fiber inside a restricted product chart. · OAI.Kaplansky.productPairChartFiberMap

productPairCoordinateMap def

A map from a product pair to its fiber pair. · OAI.Kaplansky.productPairCoordinateMap

productPairFiberMap def

Inclusion of one genuine fiber of a product pair. · OAI.Kaplansky.productPairFiberMap

productPath def

A coefficient multiplied by a continuous unknown jet path. · OAI.ForcedComputation.PlaneScalarMild.productPath

ProductPureStateSeparation def

The product-pure-state separation condition in Tomiyama's Theorem 5. · OAI.WeakPureInfiniteness.ProductPureStateSeparation

productQuestionEquiv def

Regroup independently sampled question pairs into the two local questions. · OAI.MaxCutGames.Repetition.productQuestionEquiv

productQuestionEquiv def

Regroup independently sampled question pairs into the two local questions. · OAI.UniqueGamesTheorem.Repetition.productQuestionEquiv

productScore def

The nonnegative subset-product score used by the rounding argument. · OAI.TalagrandThreshold.productScore

productShift def

The product-shift map before the final quotient. · OAI.MaxCutGames.Gadget.Enlargement.productShift

productShift def

The product-shift map before the final quotient. · OAI.UniqueGamesTheorem.Gadget.Enlargement.productShift

productSpan def

The closed real L1 span of the path products. · OAI.IndependentProducts.productSpan

productStage def

The product before the quotient. Its finite noise sampler first chooses an injection uniformly, then independently chooses the original base noise. · OAI.MaxCutGames.Gadget.EnlargementConstruction.productStage

productStage def

The product before the quotient. Its finite noise sampler first chooses an injection uniformly, then independently chooses the original base noise. · OAI.UniqueGamesTheorem.Gadget.EnlargementConstruction.productStage

productState def

Independent, not necessarily identically distributed, retained registers. · OAI.ExactQuantumFactoring.productState

ProductStateCentralizes def

The state centralizer identity used by a finite-site swap. · OAI.KadisonKastler.ProductStateCentralizes

productStemIncrementSq def

The actual change of a product of proper-child conditional means, when its common evaluation depth advances by one on the SAME full sampled path. · OAI.DilutedSpinGlass.PrescribedTree.productStemIncrementSq

productStemMeanAt def

Actual product of the proper-child conditional means on a single path. · OAI.DilutedSpinGlass.PrescribedTree.productStemMeanAt

productSubcomplexIso def

A product with a subcomplex is exactly the inverse image under the second simplicial projection, including all degenerate simplices. · OAI.Thomason.Construction.productSubcomplexIso

productSubspaceHomeomorph def

The punctured trivial-bundle subspace really is the product of the base and the punctured model. · OAI.Kaplansky.productSubspaceHomeomorph

productSubspaceHomeomorph def

The subspace selected by the second coordinate is the ordinary product with the selected subspace. · OAI.Thomason.Construction.productSubspaceHomeomorph

ProductTest def

Arbitrary indexed product tests. · OAI.PureTypeSystem.Candidates.ProductTest

productVectorLinear def

The linear vector functional used by the finite product calculation. · OAI.KadisonKastler.productVectorLinear

productVectorState def

The ambient vector functional, without any trace assumption. · OAI.KadisonKastler.productVectorState

productWeight def

Literal independent product tape; independence is proved by multiplication, not postulated as an assumption about adaptive states. · OAI.KServer.PartitionProbabilities.productWeight

productWeight def

Real-valued finite probabilities; used only to select a deterministic finite table. · OAI.SquareRootDegree.FiniteRounding.productWeight

Profession structure

mating on the pre-cap main ports. Captures baseline order by specifying only its needed invariants. · OAI.ExactDerandomization.SiteX.Line.Fabric.Profession

Profile inductive

Literal ancestor offspring/slot list of a preorder vertex. Slot counts siblings to its right. · OAI.FKCRT.TreeCoding.PlaneTree.Profile

Profile abbrev

Cut profiles in the direct sum over unordered components. · OAI.HadwigerCounterexample.Profile

Profile def

Sorts assigned to the full normal form. · OAI.PureTypeSystem.Profile

Profile abbrev

The full profile retains both final and original colors. · OAI.TalagrandThreshold.Selector.Profile

profile def

The initial axial velocity at scale s. · OAI.BoltzmannNonuniqueness.AxialFlow.profile

profile def

A Lipschitz tent profile suffices for the pressure argument; no derivative of the profile is used in the layer or interval integration. · OAI.HenonLaneEmden.CapWeights.profile

profile def

Finite minimum with a default value 1, also for the empty center set. · OAI.KServer.HeavyCenters.profile

profile def

Finite minimum with a default value 1, also for the empty center set. · OAI.KServer.HeavyEdit.profile

profile def

Expected ramp under independent survival probabilities. · OAI.KServer.RosterExpectation.profile

profile def

McShane cones over the finite ancestral chain. The recursive minimum is used instead of linear interpolation, so zero-length incidences need no division. · OAI.PlanarL1.HeightTree.profile

profile def

The source profile on syntax with fixed global variable names. · OAI.PureTypeSystem.ContextTower.profile

profile def

The positive, unnormalized bound state. · OAI.SharpLiebThirring.ExtremizerProof.profile

profile def

Canonical finite vertex level profile; its mass is not an extra assumption. · OAI.TorsionFreeZeroDivisors.SeparatorLevels.profile

profileAlongMap def

Evaluate a profile and a coordinate map jointly on fixed profile and parameter sets. · OAI.PolycyclicRecognition.profileAlongMap

profileBoxStep def

Invalid vectors get harmless stationary rows. Every actual profile row is exactly the computed sampler transition transported to the box. · OAI.ContingencyTables.SmallGraphProfiles.profileBoxStep

profileChart def

A Cartan chart, in the coordinates used for the profiles. · OAI.PolycyclicRecognition.CartanFactorModel.profileChart

profileCoefficients def

The coordinates are S, D, N, and z L. The xx entry in the input has already been multiplied by the fast scale. · OAI.ClosedSurfaceR4.GeometryPreservation.profileCoefficients

profileFactor def

Each factor in the finite product, including its manuscript normalization. · OAI.DefocusingNLS.ProfileCertificate.profileFactor

ProfileFeasible structure

Each field is a finite integer test on the candidate profile. · OAI.ContingencyTables.SmallGraphProfiles.ProfileFeasible

profileIncrementMap def

A normalized increment of a profile between two coordinate maps. · OAI.PolycyclicRecognition.profileIncrementMap

profileInTime def

Compute every matrix entry while preserving all original field stacks. · OAI.MaxCutGames.Foundations.Complexity.MachineFieldProfile.profileInTime

profileInTime def

The actual profile execution, placed in the loaded context arena. · OAI.MaxCutGames.Foundations.Hastad.SourceSignaturePrepare.profileInTime

profileInTime def

Compute every matrix entry while preserving all original field stacks. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineFieldProfile.profileInTime

profileInTime def

The actual profile execution, placed in the loaded context arena. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceSignaturePrepare.profileInTime

profileInTime_bounded def

A linear bound in the maximum original field length, for fixed q. The actual machine and execution witness are retained. · OAI.MaxCutGames.Foundations.Complexity.MachineFieldProfile.profileInTime_bounded

profileInTime_bounded def

A linear bound in the maximum original field length, for fixed q. The actual machine and execution witness are retained. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineFieldProfile.profileInTime_bounded

profileMachine def

A concrete finite machine for the full field equality matrix. · OAI.MaxCutGames.Foundations.Complexity.MachineFieldProfile.profileMachine

profileMachine def

A concrete finite machine for the full field equality matrix. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineFieldProfile.profileMachine

profileMixture def

The actual nonnegative column mixture, before imposing any sign or mass condition on its coefficients. · OAI.MetricEntropyDuality.ProfileMixtures.profileMixture

profileProgram def

The fixed list of pairs determines all control labels and original fields. The program compares only copied work-stack heads, then stores one finite bit. · OAI.MaxCutGames.Foundations.Complexity.MachineFieldProfile.profileProgram

profileProgram def

The fixed list of pairs determines all control labels and original fields. The program compares only copied work-stack heads, then stores one finite bit. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineFieldProfile.profileProgram

profileRowEncoding def

The literal row record with its proved size and histogram constraints. · OAI.TalagrandThreshold.Selector.profileRowEncoding

profileRows def

Padded residual margins can be computed before establishing feasibility. · OAI.ContingencyTables.SmallGraphProfiles.profileRows

profileSum def

Sum of the genuine ancestral record types of a finite preorder prefix. · OAI.FKCRT.TreeCoding.profileSum

ProfileWordSchedule abbrev

The schedule contains the number of steps and the two step widths. · OAI.ContingencyTables.ProfilePrograms.ProfileWordSchedule

Program inductive

A topologically ordered scalar DAG; available values are never consumed. · OAI.ExactFourier.Program

Program inductive

A directed adjacent sorting gallery. Labels here are occurrences, not facet names. Every crossing removes an inversion in the target order. · OAI.IntegralCharacterVarieties.DirectedSort.Program

Program structure

The homogeneous parts of a finite machine, without dependent alphabet casts. · OAI.MaxCutGames.Foundations.Complexity.MachineCanonicalOutput.Program

Program inductive

A bounded-loop register language. Every repeat uses the ORIGINAL input length, not a mutable register. Nesting is static in the finite program. Copy is assignment, not concatenation; the only operation increasing maximum word length is a … · OAI.Superstring.BitCode.Program

Program structure

The homogeneous parts of a finite machine, without dependent alphabet casts. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineCanonicalOutput.Program

Program inductive

Word-register programs with bounded, captured-count loops. No unbounded natural register, whole-input transition or host-language oracle is available. · OAI.WLIdentification.BoundedWords.Program

program def

A single finite control graph, independent of the number of retries. · OAI.ContingencyTables.OnlineIntegerRetry.program

program def

Both physical charges, including rational scaling and integer rounding, are computed by a finite Boolean-stack program with a polynomial runtime. · OAI.ContinuumCoulomb.ChargePrograms.program

program def

Literal fixed-gate program for the raw extended record; no oracle for the future data, success probability, or distinguished guess occurs in its code. · OAI.ExactQuantumFactoring.Completion.program

program def

One actual fixed-gate circuit: the completed random list is generated first; only that same list supplies the bases for the ensuing order slots. · OAI.ExactQuantumFactoring.FixedSplit.program

program def

No probability or canonical factor list occurs in the raw unitary. · OAI.ExactQuantumFactoring.Quarter.program

program def

A fixed finite program: the table reader counts only q delimiters in finite control, while all unbounded integers are physical unary tape contents. · OAI.MaxCutGames.Explicit.MachineSubdivisionProgram.program

program def

The actual finite program. All graph-dependent naturals remain on unary stacks. The only buffered graph data is one fixed 4096-bit relation table. · OAI.MaxCutGames.Foundations.Complexity.FinalCNFMachine.Program.program

program def

A single fixed program copies the input once, skips two headers, then compares the first field of each 4098-field row. · OAI.MaxCutGames.Foundations.Complexity.MachineCloudCount.program

program def

No input-dependent natural number occurs in the finite program. · OAI.MaxCutGames.Foundations.Complexity.MachineDummyRows.program

program def

The complete finite program. The vertex count is supplied on a tape, while the small table and all finite control types depend only on fixed d. · OAI.MaxCutGames.Foundations.Complexity.MachineExpanderTable.program

program def

Six fixed labels and a single optional-bit register, independent of input. · OAI.MaxCutGames.Foundations.Complexity.MachineLookup.program

program def

The checked splitter is placed directly in five homogeneous Boolean stacks, with its actual halt replaced by the next preparation label. · OAI.MaxCutGames.Foundations.Complexity.MachineOverlayTable.program

program def

Actual composition on shared Boolean tapes. No runtime bridge operation is required, because both programs use the same physical level tape. · OAI.MaxCutGames.Foundations.Complexity.MachinePaddedExpanderFamily.program

program def

The three labels are transfer, restore-and-copy, and fixed-offset emission. · OAI.MaxCutGames.Foundations.Complexity.MachineRegisterEmit.program

program def

Every size-dependent operation is performed by the checked native program. The entry load only resets the finite control state. · OAI.MaxCutGames.Foundations.Complexity.MachineRegularFamily.program

program def

Fixed finite code. Original graph/global-index inputs are never replaced by specification values; owner and rank queries are produced on the tapes. · OAI.MaxCutGames.Foundations.Complexity.MachineRegularMetadata.program

program def

All graph-dependent loop counts remain on tapes. The code contains only the fixed expander base and the finite subprogram labels. · OAI.MaxCutGames.Foundations.Complexity.MachineRegularTable.Top.program

program def

Six finite control labels, independent of all input sizes and contents. · OAI.MaxCutGames.Foundations.Complexity.MachineTableSplit.program

program def

A fixed finite controller. Its only unspecified instruction is the emitter body, whose actual leaf traces are supplied to NestedCycle. · OAI.MaxCutGames.Foundations.Complexity.MachineTupleOdometer.Control.program

program def

A concrete guard and halt are added to the supplied body statements. The body may return to Sum.inl false; it starts at Sum.inr entry. · OAI.MaxCutGames.Foundations.Complexity.PCPIterationMachine.program

program def

Actual finite control, independent of both runtime header values. · OAI.MaxCutGames.Foundations.Complexity.PoweringMachineFinish.program

program def

The complete program, fixed solely by the reduction parameters. · OAI.MaxCutGames.Foundations.Complexity.PoweringMachineGlobal.program

program def

Ten binary stacks and fixed finite control. Every loop scans one actual bit or tests one actual counter bit; there is no arithmetic oracle in the state. · OAI.MaxCutGames.Foundations.Hastad.SourceClauseLookup.program

program def

Every label either seeds one field, scans that field, or executes the final halt. · OAI.MaxCutGames.Foundations.Hastad.SourceFieldArray.program

program def

Concrete n+1-label program, terminating after resetting the finite state. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Cleanup.program

program def

Header, digit initialization, physical tuple body, carry loops, and complete finalization are placed in a single actual finite transition program. · OAI.MaxCutGames.Reduction.AddressMachineProgram.program

program def

A standalone local program, also usable with a local continuation. · OAI.MaxCutGames.Reduction.MachineFieldTemplate.program

program def

Eight binary stacks and fixed finite control. Every loop scans one actual bit or tests one actual counter bit. The arbitrary ambient state is preserved. · OAI.MaxCutGames.Reduction.SourceEquationLookup.program

program def

A fixed finite program: the table reader counts only q delimiters in finite control, while all unbounded integers are physical unary tape contents. · OAI.UniqueGamesTheorem.Explicit.MachineSubdivisionProgram.program

program def

The actual finite program. All graph-dependent naturals remain on unary stacks. The only buffered graph data is one fixed 4096-bit relation table. · OAI.UniqueGamesTheorem.Foundations.Complexity.FinalCNFMachine.Program.program

program def

A single fixed program copies the input once, skips two headers, then compares the first field of each 4098-field row. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineCloudCount.program

program def

No input-dependent natural number occurs in the finite program. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineDummyRows.program

program def

The complete finite program. The vertex count is supplied on a tape, while the small table and all finite control types depend only on fixed d. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineExpanderTable.program

program def

Six fixed labels and a single optional-bit register, independent of input. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineLookup.program

program def

The checked splitter is placed directly in five homogeneous Boolean stacks, with its actual halt replaced by the next preparation label. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineOverlayTable.program

program def

Actual composition on shared Boolean tapes. No runtime bridge operation is required, because both programs use the same physical level tape. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachinePaddedExpanderFamily.program

program def

The three labels are transfer, restore-and-copy, and fixed-offset emission. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRegisterEmit.program

program def

Every size-dependent operation is performed by the checked native program. The entry load only resets the finite control state. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRegularFamily.program

program def

Fixed finite code. Original graph/global-index inputs are never replaced by specification values; owner and rank queries are produced on the tapes. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRegularMetadata.program

program def

All graph-dependent loop counts remain on tapes. The code contains only the fixed expander base and the finite subprogram labels. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRegularTable.Top.program

program def

Six finite control labels, independent of all input sizes and contents. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineTableSplit.program

program def

A fixed finite controller. Its only unspecified instruction is the emitter body, whose actual leaf traces are supplied to NestedCycle. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineTupleOdometer.Control.program

program def

A concrete guard and halt are added to the supplied body statements. The body may return to Sum.inl false; it starts at Sum.inr entry. · OAI.UniqueGamesTheorem.Foundations.Complexity.PCPIterationMachine.program

program def

Actual finite control, independent of both runtime header values. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringMachineFinish.program

program def

The complete program, fixed solely by the reduction parameters. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringMachineGlobal.program

program def

Ten binary stacks and fixed finite control. Every loop scans one actual bit or tests one actual counter bit; there is no arithmetic oracle in the state. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceClauseLookup.program

program def

Every label either seeds one field, scans that field, or executes the final halt. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceFieldArray.program

program def

Concrete n+1-label program, terminating after resetting the finite state. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Cleanup.program

program def

Header, digit initialization, physical tuple body, carry loops, and complete finalization are placed in a single actual finite transition program. · OAI.UniqueGamesTheorem.Reduction.AddressMachineProgram.program

program def

A standalone local program, also usable with a local continuation. · OAI.UniqueGamesTheorem.Reduction.MachineFieldTemplate.program

program def

Eight binary stacks and fixed finite control. Every loop scans one actual bit or tests one actual counter bit. The arbitrary ambient state is preserved. · OAI.UniqueGamesTheorem.Reduction.SourceEquationLookup.program

programInPolynomialTime def

The fixed program's genuine execution fits a quadratic polynomial in the combined lengths of its two encoded operand fields. · OAI.MaxCutGames.Foundations.Complexity.MachineUnaryMultiply.programInPolynomialTime

programInPolynomialTime def

The fixed program's genuine execution fits a quadratic polynomial in the combined lengths of its two encoded operand fields. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineUnaryMultiply.programInPolynomialTime

programMatrix def

Matrix of executing a list of instructions, in chronological order. · OAI.ExactQuantumFactoring.programMatrix

programPushBound def

The finite maximum is a proof-bound constant, not an operation of the machine being analyzed. · OAI.MaxCutGames.Foundations.Complexity.Runtime.programPushBound

programPushBound def

The finite maximum is a proof-bound constant, not an operation of the machine being analyzed. · OAI.UniqueGamesTheorem.Foundations.Complexity.Runtime.programPushBound

programPushes def

A finite program has a fixed bound on pushes in one machine step. · OAI.ContinuumCoulomb.TM2Size.programPushes

progressionCorrelationWeight def

The actual correlation weight on one interval/residue piece. Values outside the piece are set to zero only to state finite variation on ℕ. · OAI.Ostmann.progressionCorrelationWeight

progressionSequence def

Applying a progression mask to the original values preserves their unit bound. · OAI.TwoPointCorrelations.progressionSequence

progressionSum def

The progression sum includes every residue class, including zero. · OAI.TwoPointCorrelations.progressionSum

ProhibitedInterval def

A concrete local prohibition, including the complete occurrence pattern and an interior suffix divisibility relation. · OAI.TwoPointCorrelations.ProhibitedInterval

ProhibitedPrimeFamily structure

The arithmetic data used by the prohibited-word argument. · OAI.TwoPointCorrelations.ProhibitedPrimeFamily

Proj abbrev

The additive category of finitely generated projective right B-modules. · OAI.LittleFinitistic.ProjectiveComplexes.Proj

proj def

Given p : α ×ₗ (Fin n) := (f i, i) with p ∈ graph f, graph.proj p is defined to be f i. · OAI.Nagata.Workers.W10.LocalTupleSort.graph.proj

proj def

Projection action. · OAI.PowerSaving.Binary.OBase.proj

proj def

The projective scheme of the section-coefficient algebra. · OAI.ReesProj.proj

proj def

The projective scheme of the actual section-coefficient algebra. · OAI.SectionCompletion.proj

proj abbrev

Projection to one homogeneous component, regarded as an endomorphism. · OAI.StableCoordinate.LeadingComponent.proj

projBaseChartEquiv def

The actual degree-zero homogeneous localization in Proj of the WHOLE normalized cone, not a surrogate affine model, is the literal smooth H-chart. · OAI.ExplicitCone.projBaseChartEquiv

projBaseOpenIso def

A genuine scheme isomorphism from the distinguished basic open of the actual Proj of normalized coefficients to the explicit Kummer H-chart. · OAI.ExplicitCone.projBaseOpenIso

ProjData structure

Onto projection in coordinates, killing a specified disjoint subspace. · OAI.HadwigerCounterexample.ProjData

project def

Projection onto an actual vertex subspace. · OAI.LittleFinitistic.VertexDecomposition.project

project def

Ambient coordinate form of the actual retained-coordinate projection. · OAI.MaxCutGames.Soundness.RepeatedGameBounds.ActualProjection.project

project def

The projection onto the specified finite collection of Fourier characters. · OAI.Problem313.BiasedFourier.project

project def

Orthogonal projection onto the hyperplane normal to u. · OAI.ProjectionCounterexample.project

project def

Invalid exit tags receive exactly the distinguished target. · OAI.PureTypeSystem.ObservationValues.project

project def

Ambient coordinate form of the actual retained-coordinate projection. · OAI.UniqueGamesTheorem.Soundness.RepeatedGameBounds.ActualProjection.project

projectD def

Real or imaginary projection of a complex first derivative. · OAI.SymmetricMahler.projectD

projectD2 def

Explicit restriction in both arguments of a nested complex derivative. · OAI.SymmetricMahler.projectD2

ProjectedAdvice structure

The second prover sees actual equation/variable tags, retaining ordered block positions but omitting every hidden singleton occurrence ID. · OAI.MaxCutGames.Decoder.AdviceExperiment.ProjectedAdvice

ProjectedAdvice structure

The second prover sees actual equation/variable tags, retaining ordered block positions but omitting every hidden singleton occurrence ID. · OAI.UniqueGamesTheorem.Decoder.AdviceExperiment.ProjectedAdvice

projectedBranchPolynomial def

A projected branch, recorded as an ordinary polynomial after Fourier averaging. · OAI.MatrixMultiplication.AuxiliarySeparation.projectedBranchPolynomial

projectedCylinder def

A genuine cylinder in its perpendicular plane, not an oblique intercept patch. · OAI.RuledApproximation.projectedCylinder

projectedInput def

Decode only the displayed question and the visible row/complement maps. · OAI.MaxCutGames.Decoder.ActualAdviceUpper.projectedInput

projectedInput def

Decode only the displayed question and the visible row/complement maps. · OAI.UniqueGamesTheorem.Decoder.ActualAdviceUpper.projectedInput

projectedKey def

The reduced table attached to visible data is independent of the compatible full question used to realize it. This is equality of the entire key. · OAI.MaxCutGames.Decoder.TableKeys.projectedKey

projectedKey def

The reduced table attached to visible data is independent of the compatible full question used to realize it. This is equality of the entire key. · OAI.UniqueGamesTheorem.Decoder.TableKeys.projectedKey

projectedMinor def

Actual projected maximal minor in fixed orthonormal endpoint coordinates. The rows use the selected projection coordinates and the columns are arbitrary starting vectors; starting bases need not be orthogonal. · OAI.SmoothObstruction.ExteriorStability.projectedMinor

projectedOverlap def

Overlap restricted to a deterministic spectral coordinate group. · OAI.InvariantIsing.projectedOverlap

projectedPolynomial def

Apply the finite Fourier projection and square weights branch by branch. · OAI.MatrixMultiplication.AuxiliarySeparation.projectedPolynomial

projectedPreferredNormal def

Normalize the actual global normal projection wherever it is nonzero. · OAI.ClosedSurfaceR4.FiniteOrderSmoothing.SmoothingAtlas.projectedPreferredNormal

projectedTensor def

The exact coefficient tensor after Fourier projection, before weighting. · OAI.MatrixMultiplication.AuxiliarySeparation.projectedTensor

projection def

The actual linear projection onto words with a prescribed X-count. · OAI.DerivedBar.SquareZero.projection

projection def

Bounded linear extraction of any spatial jet. · OAI.ForcedComputation.BoundedSpatialJets.projection

projection def

Endpoint projection of the genuine path-class extension. · OAI.GroupRingTopology.PathExtension.projection

projection def

Left multiplication by e_i on the actual induced-column first factor B e_j. · OAI.LittleFinitistic.CornerProjection.projection

projection def

The genuine projection onto the vertex component. · OAI.LittleFinitistic.VertexFunctor.projection

projection def

Projection to the original quotient. · OAI.PolycyclicRecognition.FactorExtension.projection

projection def

Projection back to the fixed reference plane along the common transverse direction. · OAI.Problem358.MovingPlaneEnvelope.projection

projection def

Every ordinal/color rectangle projection is defined on the full completion. · OAI.SeparableQuotient.ActualSpace.projection

projection def

Contractive interval/rectangle projection on the actual Banach completion. · OAI.SeparableQuotient.NormConstruction.NormingSet.projection

projection def

Collapse of the transported cylinder to its source. · OAI.Thomason.ConstructedInterval.projection

projection def

Collapsing the right interval is a positive augmented chain map. · OAI.Thomason.Construction.DirectedComplex.RightCylinder.projection

projection def

The associated idempotent endomorphism of the ambient poset. · OAI.Thomason.Construction.SieveRetraction.projection

projection def

The global strict cylinder collapse is the unique extension of these maps. · OAI.Thomason.Construction.StrictOmegaCategory.GrayChainExtension.projection

projection def

The quotient projection from the bounded representative category. · OAI.Thomason.Construction.StrictOmegaCategory.IntelligentTruncation.projection

projection def

Projection to an original diagram component. · OAI.Thomason.Construction.StrictOmegaCategory.LimitConstruction.projection

projection def

The cylinder projection. · OAI.Thomason.Construction.TopologicalInterval.projection

projection def

Orthogonal projection onto the plane normal to a unit vector. · OAI.TriangularCovering.projection

projectionBody def

The body with support function equal to the volumes of orthogonal shadows, expressed as its defining intersection of supporting half-spaces. · OAI.PettyProjection.projectionBody

projectionBody def

The projection body, defined entirely by its directional projection volumes. The bounds are the normalized Hausdorff measures of orthogonal projections. · OAI.ProjectionCounterexample.projectionBody

projectionBound def

The upper bound for the projection coefficient after introducing the nonnegative spectral defects V = x-v and W = y-b. · OAI.PositiveEinsteinFour.Quartic.projectionBound

projectionCentralData def

Construct all the direct-integral data needed for the separable commutator estimate, before the separate factor-fiber theorem. The only geometric inputs are genuine conull representations and actual L² action identities for the isometric … · OAI.KadisonSimilarity.projectionCentralData

projectionCoefficient def

The coefficient g in the weighted Hessian projection bound. · OAI.PositiveEinsteinFour.QuarticEnergy.projectionCoefficient

projectionColumn def

The projected coordinate vector supplies a local frame wherever nonzero. · OAI.Kaplansky.projectionColumn

ProjectionCorner abbrev

The closed supported corner, equipped below with unit q. · OAI.WeakPureInfiniteness.ProjectionCorner

projectionCornerElement def

A supported element, regarded as an element of the unital corner. · OAI.WeakPureInfiniteness.projectionCornerElement

projectionCornerInclusion def

The algebra inclusion sends the corner unit to q. · OAI.WeakPureInfiniteness.projectionCornerInclusion

projectionCornerUnitary def

Extending by the complementary identity gives an ambient unitary. · OAI.WeakPureInfiniteness.projectionCornerUnitary

ProjectionDiffuse def

The diffuseness condition, independent of a finite or semifinite factor structure. · OAI.KadisonKastler.ProjectionDiffuse

projectionDimensionIndex def

A fixed countable index type for each dimension stratum. · OAI.KadisonKastler.projectionDimensionIndex

projectionDualLineCore def

The dual (hyperplane) line core, with inverse unitary transitions. · OAI.Kaplansky.projectionDualLineCore

projectionDualLineFiberEquiv def

The constructed dual fiber is the full algebraic dual of the actual line. · OAI.Kaplansky.projectionDualLineFiberEquiv

projectionEndpoint def

The normalized initial endpoint used by the dihedral word columns. · OAI.KadisonSimilarity.TracialCornerGeometry.projectionEndpoint

projectionFamilyEntry def

The matrix entries associated to partial isometries with a common initial projection. · OAI.KadisonKastler.projectionFamilyEntry

projectionFiber def

The actual closed range of an orthogonal projection. · OAI.KadisonSimilarity.projectionFiber

projectionFiberAction def

A commuting operator acts on the closed range by ordinary restriction. · OAI.KadisonSimilarity.projectionFiberAction

projectionFiberCompression def

Compression is defined on every operator, even before support is proved. · OAI.KadisonSimilarity.projectionFiberCompression

projectionFiberEquivOfEq def

Equal projections have the identity isometry between their ranges. · OAI.KadisonSimilarity.projectionFiberEquivOfEq

projectionFiberRepresentation def

A genuine representation commuting with a projection restricts to a genuine unital representation on its actual Hilbert range. · OAI.KadisonSimilarity.projectionFiberRepresentation

projectionJoin def

Join formed from the orthogonal part of the second projection. · OAI.KadisonSimilarity.projectionJoin

projectionLine def

The actual one-dimensional subspace selected by a rank-one projection. · OAI.Kaplansky.projectionLine

projectionLineCore def

A genuine complex line-bundle core on the compact space of orthogonal rank-one projections. Its transition functions are constructed, not assumed. · OAI.Kaplansky.projectionLineCore

projectionLineFiberEquiv def

Identify each constructed fiber with the actual projected line in Euclidean space. · OAI.Kaplansky.projectionLineFiberEquiv

projectionLineTotalHomeomorph def

The constructed line bundle is homeomorphic to the actual incidence subspace of the compact projection base times Euclidean space. · OAI.Kaplansky.projectionLineTotalHomeomorph

projectionNorm def

The exact orthogonal residual norm of the actual two-center target. · OAI.RegularPointCoordinates.ActualProjectionConfiguration.projectionNorm

projectionPhaseUnitary def

Put one unit-modulus scalar on a projection and one on its complement. · OAI.KadisonKastler.projectionPhaseUnitary

projectionRangeFrame def

The nonzero projected frame, as vectors of the genuine closed range. · OAI.KadisonKastler.projectionRangeFrame

projectionRangeHilbertBasis def

Every closed projection range has the basis given by its surviving frame. · OAI.KadisonKastler.projectionRangeHilbertBasis

projectionRangeRestriction def

Restriction to an invariant projection range, retaining the original norm. · OAI.KadisonKastler.projectionRangeRestriction

projectionReflection def

The reflection associated to a projection, written without scalar coercions so that its algebraic identities apply in any star ring. · OAI.KadisonSimilarity.projectionReflection

projectionReflectionLetter def

The actual centered letter associated to a half-trace projection. · OAI.KadisonSimilarity.TracialCornerGeometry.projectionReflectionLetter

ProjectionSeed abbrev

The actual two-stage sampler retains both its projection choices and its uniform map on that projected target. Thus target advice is not forgotten. · OAI.MaxCutGames.Decoder.AdviceLaw.ProjectionSeed

ProjectionSeed abbrev

The actual two-stage sampler retains both its projection choices and its uniform map on that projected target. Thus target advice is not forgotten. · OAI.UniqueGamesTheorem.Decoder.AdviceLaw.ProjectionSeed

projectionSmoothing def

Product scalar smoothing of the joint matrix--label observation law. · OAI.SubsphereRegression.projectionSmoothing

projectionSpace abbrev

Orthogonal rank-s projections, represented by equations with continuous left-hand sides; the trace equation is equivalent to the rank condition. · OAI.Kaplansky.projectionSpace

projectionStratumBasis def

A basis with a fixed index set on the whole dimension stratum. · OAI.KadisonKastler.projectionStratumBasis

projectionStratumCoordinates def

The coordinate unitary uses the same target Hilbert space throughout a stratum. · OAI.KadisonKastler.projectionStratumCoordinates

projectionStratumOperator def

An invariant operator field in a fixed Hilbert space on each rank stratum. · OAI.KadisonKastler.projectionStratumOperator

ProjectionTraceIndependent def

The degree-two consequence of freeness used for the tail estimate. · OAI.KadisonKastler.ProjectionTraceIndependent

projectionVolume def

Actual (d−1)-dimensional Euclidean volume of the orthogonal projection. · OAI.ProjectionCounterexample.projectionVolume

projectionWordWeight def

The two projection-word estimates, written with a square-root denominator. · OAI.KadisonKastler.projectionWordWeight

projectiveCoordinatePuncture def

The coordinate puncture is open in the actual compact projective product. · OAI.Kaplansky.ComplexGrassmann.projectiveCoordinatePuncture

projectiveHyperplaneHomeomorph def

The complementary hyperplane carries exactly the topology of CP^k. · OAI.Kaplansky.ComplexGrassmann.projectiveHyperplaneHomeomorph

projectiveLinearImage def

The actual image line, when the linear map is injective on that line. · OAI.Kaplansky.ComplexGrassmann.projectiveLinearImage

projectiveLinePunctureHomeomorph def

The projective line minus any point is the entire affine complex model. · OAI.Kaplansky.ComplexGrassmann.projectiveLinePunctureHomeomorph

projectiveNormalization def

Actual morphism X→P² from the precise normalized section cone. No claim identifying all of X with the source's smooth Kummer surface is included, and no finite W→V conclusion is assumed. · OAI.ExplicitCone.projectiveNormalization

projectiveOneFactorFundamentalClass def

An actual global lift of the preserved generator for an arbitrary-dimensional one-factor product. The multi-factor case is not assumed. · OAI.Kaplansky.projectiveOneFactorFundamentalClass

projectiveOneFactorPunctureHomeomorph def

Puncturing commutes with the actual one-factor product homeomorphism. · OAI.Kaplansky.projectiveOneFactorPunctureHomeomorph

projectiveOneFactorTopHomologyIso def

Global top homology is ℤ, normalized by the preserved canonical local class. · OAI.Kaplansky.projectiveOneFactorTopHomologyIso

projectiveOneFactorTopLocalizationIso def

The actual absolute-to-relative map, proved bijective in top degree. · OAI.Kaplansky.projectiveOneFactorTopLocalizationIso

projectivePointChart def

An actual full product graph chart containing any specified projective point. · OAI.Kaplansky.projectivePointChart

projectivePointLocalCoordinateIso def

Local homology at every point of the actual projective product, rather than only at the canonical section zero, has the genuine coordinate comparison. · OAI.Kaplansky.projectivePointLocalCoordinateIso

projectivePointLocalGenerator def

A primitive local class at every point, with no claim of a coherent orientation or a global lift when the total complex dimension exceeds one. · OAI.Kaplansky.projectivePointLocalGenerator

projectivePointLocalTopHomologyIso def

At every projective point the top local integral group is ℤ. Choices of charts and bases give separate generators; their orientation coherence is not asserted. · OAI.Kaplansky.projectivePointLocalTopHomologyIso

projectivePointPunctureLowerHomotopyEquiv def

The puncture retraction theorem holds at any actual projective point. · OAI.Kaplansky.ComplexGrassmann.projectivePointPunctureLowerHomotopyEquiv

projectiveProductFundamentalClass def

The unique actual global lift of the PRESERVED canonical local generator. This construction is unconditional apart from positive total dimension. · OAI.Kaplansky.projectiveProductFundamentalClass

projectiveProductGraphIntersectionHomeomorph def

Actual iterated graph-chart intersections, with their subspace topology, have the explicit affine-complement coordinates. · OAI.Kaplansky.ComplexGrassmann.projectiveProductGraphIntersectionHomeomorph

projectiveProductOneFactorHomeomorph def

Projection onto the unique positive-dimensional factor is a homeomorphism; all other actual Grassmann factors have one point. · OAI.Kaplansky.ComplexGrassmann.projectiveProductOneFactorHomeomorph

projectiveProductTopHomologyIso def

Global integral top homology is ℤ with the exact preserved local normalization. · OAI.Kaplansky.projectiveProductTopHomologyIso

projectiveProductTopLocalizationIso def

Top-degree localization is the actual absolute-to-relative map and is invertible at every point of every positive-dimensional projective product. · OAI.Kaplansky.projectiveProductTopLocalizationIso

projectiveProjectionEquiv def

Projection and graph-atlas models of complex projective space agree as sets of actual complex lines. · OAI.Kaplansky.ComplexGrassmann.projectiveProjectionEquiv

projectiveProjectionHomeomorph def

Identification of the compact orthogonal-projection model with the ordinary smooth graph-chart model of complex projective space. · OAI.Kaplansky.ComplexGrassmann.projectiveProjectionHomeomorph

projectivePunctureLowerHomotopyEquiv def

CP^(k+1) minus a chart center is homotopy equivalent to CP^k. · OAI.Kaplansky.ComplexGrassmann.projectivePunctureLowerHomotopyEquiv

projectiveResolution def

Explicit free absolute bar resolution, packaged for derived functor computations. · OAI.DerivedBar.projectiveResolution

projectiveSpaceLocalCoordinateIso def

Local homology of a single projective space in actual graph coordinates. · OAI.Kaplansky.projectiveSpaceLocalCoordinateIso

projectiveSpacePunctureHomologyIso def

The actual punctured-space homology reduces to the lower projective space. · OAI.Kaplansky.projectiveSpacePunctureHomologyIso

projectiveSpaceTopLocalizationIso def

Absolute-to-local homology is an isomorphism in the real top dimension of every positive-dimensional single complex projective space. · OAI.Kaplansky.projectiveSpaceTopLocalizationIso

projectiveSurface abbrev

Actual proper Proj of the entire normalized coefficient algebra. · OAI.ExplicitCone.projectiveSurface

projectiveTensorEquiv def

Canonical algebra equivalence used to interpret the paper's tensor-product cone as the already formalized degreewise completed ring. · OAI.ExplicitCone.projectiveTensorEquiv

projector def

The unnormalized character projector. Its identity coefficient is one, so normalization is unnecessary for constructing the nonzero test module. · OAI.LittleFinitistic.CharacterModule.projector

projector def

The matrix of the orthogonal rank-one operator associated with a vector. · OAI.MUB6.Weiner.projector

projectorChangeSq def

Single-copy spatial squared distance of two actual conditional means. · OAI.DilutedSpinGlass.ReducedTopology.projectorChangeSq

projectorVector def

Flatten the rank-one matrix associated with an arbitrary complex vector. · OAI.MUB6.Weiner.projectorVector

projectQ def

Projection to the successive quotient of the image flag on V/U. · OAI.IntegralCharacterVarieties.IntrinsicInterval.Data.projectQ

Projects def

Equality of the finite pushforward law, expressed without choosing a representation of fibers. · OAI.DilutedSpinGlass.FiniteLaw.Projects

Projects def

A genuine prior-law projection at every vertex, not a conclusion about weighted roots. · OAI.DilutedSpinGlass.KernelTower.Projects

projectWord def

Tree projection with untouched operand and clean finite work stacks. · OAI.ParityWL.Complexity.projectWord

projEval abbrev

The same evaluation at one used in constructing the section ring. · OAI.SourceSymmetry.projEval

projF def

In-place first or second projection. · OAI.Fifths.SM.projF

projJac def

Projection Jacobian, zero at undefined points. · OAI.GeneralMahler.projJac

projPairChartEquiv def

The homogeneous product chart is the common field-principal open. · OAI.SourceSymmetry.projPairChartEquiv

promised def

Deterministic realization promise at two orientations. Can be required only if both accept the specific parameter pair and their keys agree. · OAI.HadwigerCounterexample.Volley.promised

ProofContext def

Only proof variables are constrained. Each is a proof/proof converter whose first argument is one of the positive normal targets. This includes exactly the context shape used by the source, and is stable under data extensions. No converter … · OAI.PureTypeSystem.Labelled.ProofContext

ProofParity def

Uniform proof transport records open callbacks, not a meta-level implication between propositions. Every operation returns an actual typed term. · OAI.PureTypeSystem.Labelled.ProofParity

ProofWrapper inductive

Fixed hygienic wrapper syntax. The code is selected once from a path, never reselected when an input is substituted or its sorting profile grows. · OAI.PureTypeSystem.Labelled.ProofWrapper

propagate def

A matrix is stored by columns, so each new coordinate is one dot product. · OAI.ContingencyTables.IntegerMatrixPrograms.propagate

propagationData def

The full abstract propagation data are derived from the raw word realization and the actual corner Haar unitary. · OAI.KadisonSimilarity.TracialRepeatedWordRealization.propagationData

propagator def

Actual later-on-the-left product used in the manuscript. · OAI.SmoothObstruction.EuclideanSplitting.propagator

Proper structure

Exactly the source normalization: delete zero facets/children and sew identity seams BEFORE closed factors are separated. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.Proper

ProperChainMetric structure

A compatible proper left-invariant metric with large-scale-geodesic chains. · OAI.PolycyclicRecognition.ProperChainMetric

ProperChainMetric structure

A compatible proper left-invariant metric together with its explicit large-scale-geodesic chains. · OAI.PolycyclicRecognition.ProperChainMetric

ProperCocompactHom structure

A proper continuous homomorphism whose image has a compact coset cover, the cocompactness condition in the CT arrows. · OAI.PolycyclicRecognition.ProperCocompactHom

ProperColoring def

This declaration is an exact comparator-selected statement. Read the Lean statement below for its mathematical content. · OAI.EuclideanFiveColor.ProperColoring

ProperCombinatorialAxioms structure

The model axioms and proper/local-presentability conclusions needed from the published abstract transfer theorem. This is data to be proved or supplied by a precisely stated published implication, not a global axiom. · OAI.Thomason.Construction.ProperCombinatorialAxioms

ProperCrossing structure

A proper double crossing is locally homeomorphic to two coordinate axes. · OAI.Paper170.ProperCrossing

properFace def

Proper cell-face incidence, including vertices of closed triangles. · OAI.EilenbergGanea.properFace

ProperlyInfiniteIdentity structure

The ordinary two-isometry description of a nonzero properly infinite identity. The cross product states that the two range projections are orthogonal. · OAI.KadisonKastler.ProperlyInfiniteIdentity

ProperlyInfiniteProjection def

A projection admits two orthogonal copies of itself inside itself. · OAI.BoundedHochschild.KadisonRingrose.ProperlyInfiniteProjection

ProperMinimizingMetric structure

The metric consequences of a complete left-invariant Riemannian metric. · OAI.PolycyclicRecognition.ProperMinimizingMetric

property def

The seven source structural tags G,C0,Cl,Cr,El,Er,R, in that order. · OAI.PureTypeSystem.Labelled.SpecializedCode.property

proportional def

Change density proportionally to a nonnegative function. · OAI.FourierLLogL.Law.proportional

proposal def

The selection stage is the actual binary list-index program. · OAI.ContingencyTables.CompanionArithmeticProposal.proposal

proposal def

Each valid exchange neighbor has one code. Every unused code holds the current state; the full transition has another independent lazy bit. · OAI.ContingencyTables.CompanionBooleanPhase.proposal

proposalLaw def

Each feasible neighbor is assigned its prescribed dyadic mass; the remaining proposal words hold at the present state. · OAI.ContingencyTables.CompletionTranslation.proposalLaw

Protected def

Protected cells: at least one anchor/dummy position, or two root positions. · OAI.Problem348.Construction.HostSampling.Protected

protectedAt def

All leaves protected, at a general positive insertion, not just along a single direction. This retains the original finite-kernel law. · OAI.DilutedSpinGlass.PrescribedTree.protectedAt

protectedBy def

L is the set excluded at this projected space. · OAI.HadwigerCounterexample.protectedBy

protectedLog def

Exactly the denominator cancellation at the selected protected leaves, written in logarithmic form. · OAI.DilutedSpinGlass.PrescribedTree.protectedLog

provedCircleInput def

The complete published circle-index input, now a theorem. · OAI.CoarseAssembly.provedCircleInput

Proven def

Eventual derivability in the directed filter, with all free variables interpreted as natural-number constant codes. · OAI.PartitionConsistency.Completeness.Proven

Proves def

An actual labelled inhabitant, retaining its typing and proof mode. · OAI.PureTypeSystem.Labelled.Proves

prSmall def

prSmall a z consists of the pure-primal coordinates kept by the small table. · OAI.HadwigerCounterexample.prSmall

prune def

Stop at an attempted repeated or otherwise unavailable coordinate. · OAI.MatroidProphet.MarkedQueryProgram.prune

psdRank def

The least positive order of a real PSD factorization. · OAI.PerfectMatchingPSD.psdRank

pSieveRatio def

The actual cubic-sieve bound, divided by the dispersion benchmark. · OAI.CubicFirstMoment.pSieveRatio

PT abbrev

Pointwise semantic LF functions requiring only wire encodings. · OAI.ExactDerandomization.Relay.PT

publicPurifiedEntries def

Matrix-valued output entries after purification and publication of the transcript. · OAI.ZeroKey.publicPurifiedEntries

PublishedAcyclicKanLifting abbrev

The boundary-lifting consequence of Goerss–Jardine, Theorems I.11.3–I.11.4; Corollary II.8.6 records properness. · OAI.Thomason.Construction.PublishedAcyclicKanLifting

PublishedAdditiveLargeSieve def

The additive large sieve for separated frequencies and arbitrary complex coefficients. · OAI.Ostmann.PublishedAdditiveLargeSieve

PublishedAmalgamCriterion structure

Minasyan--Osin's general nontrivial-amalgam criterion, with the virtually cyclic alternative explicitly excluded. · OAI.HarmonicArtin.ParabolicIntersections.PublishedAmalgamCriterion

PublishedAmenableEmbedding def

The one-sided part of CCSSWW Theorem 3.3(i). The location of the unitary in the algebra generated by the two inputs is not needed here. · OAI.KadisonKastler.PublishedAmenableEmbedding

PublishedBlockCommutantApproximation def

The precise strict-approximation consequence of the published commutant-distance formula, restricted to two-by-two operator matrices. · OAI.KadisonKastler.PublishedBlockCommutantApproximation

PublishedBoundary def

The application-specific existence statement for a countable group: a probability boundary with scalar means for the original and separate product actions, and unitary double ergodicity for the diagonal action. · OAI.KadisonKastler.PublishedBoundary

PublishedBoundaryAssemblyInput structure

The exact published index/naturality specializations used here: Willett--Yu, Higher Index Theory (2020), §§5.1, 6.7, 7.1--7.2, with errata 30--31. The objects and both geometric/Roe maps are fixed above. There is no transfer-naturality … · OAI.CoarseAssembly.PublishedBoundaryAssemblyInput

PublishedC4FlowInput structure

Smooth dependence, Liouville's equation and a uniform finite-order variational estimate. · OAI.ContinuumCoulomb.PublishedC4FlowInput

PublishedCenterInjectivity def

The represented-center specialization; no perturbation conclusion is assumed. · OAI.KadisonKastler.PublishedCenterInjectivity

PublishedCentralPairDecomposition def

The central case of the cited decomposition and reconstruction results, for one pair and one chosen norm-separable common C-star algebra. · OAI.KadisonKastler.PublishedCentralPairDecomposition

PublishedCircleInput structure

Minimal published analytic data needed to test loops. Each pairing f is the index pairing with the scalar unitary circleUnitary f. · OAI.CoarseAssembly.PublishedCircleInput

PublishedCMPHardness def

Field-free signed square-lattice Heisenberg hardness of Cubitt–Montanaro–Piddock, arXiv:1701.05182v4, Theorem 48 and Section 10.1, with the polynomial-weight sparse construction of Lemma 47. · OAI.ContinuumCoulomb.PublishedCMPHardness

PublishedCommonTracialPosition def

Precise published input, specialized to the common irreducible amenable subalgebra needed by the finite-factor argument. · OAI.KadisonKastler.PublishedCommonTracialPosition

PublishedComplexZeroDensity def

The exact Montgomery–Inoue estimate, restricted to an arbitrary finite subfamily of the primitive nonprincipal characters of conductor at most Q. · OAI.Ostmann.PublishedComplexZeroDensity

PublishedComplexZeroRegion def

Bounded-height Landau–Page, with the entire possible exceptional character removed. The exception can be avoided by one prime deletion. · OAI.Ostmann.PublishedComplexZeroRegion

PublishedCyclicSimilarity def

Haagerup, Solution of the similarity problem for cyclic representations of C-algebras*, Ann. of Math. 118 (1983), Theorem 1.1 and its proof, pp. 217 and 225: a cyclic representation bounded by C is completely bounded by C³. This … · OAI.KadisonKastler.PublishedCyclicSimilarity

PublishedDerivationInputs structure

Approximate cyclic column factorization for every cyclic representation. · OAI.KadisonSimilarity.PublishedDerivationInputs

PublishedDiagonalFockRepresentation def

A precise specialization of the faithful reduced-free-product representation, with no quantitative word bounds in its conclusion. · OAI.KadisonKastler.PublishedDiagonalFockRepresentation

PublishedDixmierAveraging def

The published averaging statement used to identify centers. · OAI.KadisonKastler.PublishedDixmierAveraging

PublishedFactorCasesInput def

Precisely the factor alternatives needed downstream. The finite model is supplied by Popa's increasing-matrix subfactor theorem and the listed local inputs; the other two alternatives are projection/type classification. · OAI.KadisonSimilarity.PublishedFactorCasesInput

PublishedFactorialModularCore def

The factorial-core classification, specialized to a separably represented factor and a faithful normal vector state. · OAI.KadisonKastler.PublishedFactorialModularCore

PublishedFieldTomiyama def

The constant-field specialization of Tomiyama's Theorem 1(2). This is an explicit published-input hypothesis, not an axiom. · OAI.KadisonKastler.PublishedFieldTomiyama

PublishedFieldTomiyamaFamily def

Tomiyama's same published theorem, specialized to every scalar field base and group used on the fixed represented Hilbert space. · OAI.KadisonKastler.PublishedFieldTomiyamaFamily

PublishedFiniteMatrixUltraproduct def

Popa §1.6 specialized to the increasing finite matrix coordinates. · OAI.KadisonKastler.PublishedFiniteMatrixUltraproduct

PublishedFinitePosetUnitInput structure

Only the published unit comparison on finite posets, in the actual adjunctions. · OAI.Thomason.Construction.PublishedFinitePosetUnitInput

PublishedFiniteType def

The application-specific part of CSSW (2010), Lemma 3.5. No spatial equivalence or perturbation estimate is included in this published input. · OAI.KadisonKastler.PublishedFiniteType

PublishedFoldingGcd structure

The precise part of published gcd preservation needed below. The sphericity proof is retained in the type to identify the admissible finite-type folding to which the source theorem is applied. · OAI.HarmonicArtin.ParabolicIntersections.PublishedFoldingGcd

PublishedFrostmanAboveHalf def

Application-level specialization of Mattila (1995), Theorem 8.8. The upper bound on the exponent only fixes a convenient unit-size gap. · OAI.Falconer.PublishedFrostmanAboveHalf

PublishedGeometricTubeSupport def

A precise small consequence of the simple-curve inclusion in Tran–Yuan, Corollary 1.2, suited to the actual geometric continuation route. · OAI.SLEExactGauge.PublishedGeometricTubeSupport

PublishedHalfPlaneUnivalentCompactness def

The minimal nondegenerate univalent normal-family consequence used at the ribbon waypoint. · OAI.SLEExactGauge.PublishedHalfPlaneUnivalentCompactness

PublishedHardyMultiplierEvaluation def

The point-evaluation consequence of the published Hardy multiplier representation. The domain and the Poisson integral are concrete. · OAI.KadisonKastler.PublishedHardyMultiplierEvaluation

PublishedHydrogenBottom def

Exact published hydrogen bottom, specialized to the unperturbed well. · OAI.ContinuumCoulomb.PublishedHydrogenBottom

PublishedIncreasingMatrixInjectivity def

The AFD-to-injective implication specialized to the actual increasing full-matrix stages used in the irreducible subfactor construction. · OAI.KadisonKastler.PublishedIncreasingMatrixInjectivity

PublishedKaplanskyDensity def

Kaplansky's published unit-ball density theorem, retaining only finite vector tests needed in the nonseparable reduction. · OAI.KadisonKastler.PublishedKaplanskyDensity

PublishedLabelFiberInput structure

The small-colimit consequence of the precise published endpoint-fiber adjunction, on the actual functors defined above. · OAI.Thomason.Construction.PublishedLabelFiberInput

PublishedLocalizedAmenableEmbedding def

Precisely the inclusion, displacement and localization portions of the published one-sided embedding theorem. · OAI.KadisonKastler.PublishedLocalizedAmenableEmbedding

PublishedMeshInput def

The remaining finite-type input is positive-degree mesh nilpotence. · OAI.HarmonicArtin.ParabolicIntersections.PublishedMeshInput

PublishedModularSpectralResolution def

Tomita--Takesaki and the spectral theorem at a cyclic separating vector, restricted to the bounded data above. · OAI.KadisonKastler.PublishedModularSpectralResolution

PublishedMultiplicationCorrection def

The precise published input used for correction. No normality is required of the perturbed multiplication or of the correcting map. · OAI.KadisonKastler.PublishedMultiplicationCorrection

PublishedNegativePlanarGroundGap def

The negative-ground-energy specialization of Teschl's Theorems 10.2, 10.13 and 4.12. Its premises and conclusion concern the actual closed H¹ graph and the actual multiplication operator for a compact C¹ potential. · OAI.ContinuumCoulomb.PlanarSobolev.PublishedNegativePlanarGroundGap

PublishedNormalizerFactorization def

Precisely the quantitative normalizer factorization of CCSSWW (2014), Proposition 3.2. Distances are written in the symmetric order u₂ - u₁. · OAI.KadisonKastler.PublishedNormalizerFactorization

PublishedOmegaPosetSieveInput structure

The omega-category specialization of AM2014 Proposition 5.5, Lemma 5.12 and Proposition 5.13(c), for finite posets. · OAI.Thomason.Construction.PublishedOmegaPosetSieveInput

PublishedPlanarFurstenberg def

Exactly the range of the published planar Furstenberg estimate needed by the higher-dimensional slicing argument, in a bounded line chart. · OAI.Falconer.PublishedPlanarFurstenberg

PublishedPlanarRadialMoment def

The precise finite-energy consequence of the published radial theorem used in the planar application. The source and pin laws may be interchanged. · OAI.PlanarFalconer.PublishedPlanarRadialMoment

PublishedPlanarWienerUnitSupport def

A precise published input about ordinary planar Brownian motion, without any ribbon geometry or uniform-in-route probability conclusion. · OAI.SLEExactGauge.PublishedPlanarWienerUnitSupport

PublishedPointedMatrixOperations structure

The three elementary pointed-product facts used in the absorption argument. All coordinates carry the actual finite matrix-state moments. · OAI.KadisonKastler.PublishedPointedMatrixOperations

PublishedPopaFiniteBaseIndependence def

The published independence input for the two ultraproducts actually used in the controlled-lift construction: constant II1 factors and the increasing finite-matrix model. · OAI.KadisonKastler.PublishedPopaFiniteBaseIndependence

PublishedPosetSieveInput structure

The finite-poset specialization of published sieve preservation. · OAI.Thomason.Construction.PublishedPosetSieveInput

PublishedProgressionInput structure

The restricted classical analytic input. The family chooses the local exceptional primitive character, if one exists, for each modulus. · OAI.Ostmann.PublishedProgressionInput

PublishedProperInfiniteProjectionCopies def

Blackadar III.1.3.6 restricted to separably acting algebras. · OAI.KadisonKastler.PublishedProperInfiniteProjectionCopies

PublishedProperParabolicCriterion structure

CMMW (2026), Proposition 4.2, stated separately from its edge criterion. · OAI.HarmonicArtin.ParabolicIntersections.PublishedProperParabolicCriterion

PublishedQuadraticLargeSieve def

Heath-Brown's Theorem 1, specialized to integer bounds and finite coefficient arrays. This is an explicit hypothesis, not a global axiom. · OAI.Ostmann.PublishedQuadraticLargeSieve

PublishedQuadraticStationary def

The fixed-quadratic stationary-phase hypothesis, uniform over amplitudes and frequencies. · OAI.Falconer.PublishedQuadraticStationary

PublishedRealDiagonalDecomposition def

The diagonal-commutant decomposition theorem on a separable constant Hilbert field over Lebesgue measure. · OAI.KadisonKastler.PublishedRealDiagonalDecomposition

PublishedRealNestDistance def

The approximation consequence of Arveson's nest distance formula, with explicit slack instead of a nearest-point assertion. · OAI.KadisonKastler.PublishedRealNestDistance

PublishedRealSpectralCommutant def

Stone's spectral-commutant theorem, restricted to the concrete character multiplication group on the real line. · OAI.KadisonKastler.PublishedRealSpectralCommutant

PublishedRealZeroInput structure

Hadamard factorization and the full low-height zero-free region. Page uniqueness for the resulting actual real zeros is already in P. · OAI.Ostmann.PublishedRealZeroInput

PublishedRecognitionInputs structure

The cited theorem families, specialized by the proof to its chosen models. Gromov is retained solely as the polynomial-growth theorem. · OAI.PolycyclicRecognition.PublishedRecognitionInputs

PublishedRegularCommutant def

The real-group specialization of the published commutant-generator theorem. Strong continuity refers to the ordinary topology on ℝ. No fixed-point equality or perturbation assertion is assumed. · OAI.KadisonKastler.PublishedRegularCommutant

PublishedSeparableModel structure

A central direct-integral model with separable Hilbert fibers and the finite or properly infinite factor structure. · OAI.KadisonSimilarity.PublishedSeparableModel

publishedSeparableModel_of_central_factors def

A central decomposition into actual factors suffices; no factor model or free-position hypothesis is required from the decomposition theorem. · OAI.KadisonSimilarity.publishedSeparableModel_of_central_factors

PublishedSiegelBound def

The one-sided real-zero consequence of Siegel's theorem, for every fixed positive exponent. Its constant is allowed to be ineffective. · OAI.Ostmann.PublishedSiegelBound

PublishedSimpleDerivationInner def

The published theorem, with no separability assumption. · OAI.KadisonKastler.PublishedSimpleDerivationInner

PublishedSimpleTraceSupport def

The simple-curve inclusion in the published support theorem, restricted to finite deterministic capacity intervals. · OAI.SLEExactGauge.PublishedSimpleTraceSupport

PublishedSimplexGenerationInput structure

Steiner 2004, Theorem 5.6, restricted to generation by the simplex atoms explicitly constructed here. Atom existence, tables and endpoints are proved. · OAI.Thomason.Construction.PublishedSimplexGenerationInput

PublishedSimplicialGluingInput structure

Only the simplicial specialization of the classical gluing lemma. · OAI.Thomason.Construction.PublishedSimplicialGluingInput

PublishedSmoothPrincipalPNT def

The fixed-test, conductor-one consequence of the classical PNT. · OAI.Ostmann.PublishedSmoothPrincipalPNT

PublishedSobolevSmoothDensity def

The precise specialization of the published H=W theorem. · OAI.ContinuumCoulomb.PublishedSobolevSmoothDensity

PublishedSphericalAlgebraInputs structure

The two remaining published finite-type algebra inputs. · OAI.HarmonicArtin.ParabolicIntersections.PublishedSphericalAlgebraInputs

PublishedSphericalInputs structure

Only the published general mesh-algebra, admissible-folding and acyclic-evaluation facts; see the individual definitions for exact sources. · OAI.HarmonicArtin.ParabolicIntersections.PublishedSphericalInputs

PublishedStreetLocalizationInput structure

Gagna (2018), Theorem 5.6: the Street nerve induces an equivalence of homotopy categories for every positive finite dimension and omega. Original source: https://arxiv.org/pdf/1612.04450v2, p.12. · OAI.Thomason.Construction.PublishedStreetLocalizationInput

PublishedSummandSizeBound def

The only counting-function estimates needed from Elsholtz's Theorem 1.9. · OAI.Ostmann.PublishedSummandSizeBound

PublishedThomasonReduction structure

The precise published reduction, leaving both (d') and (e) as premises. No model conclusion is assumed without those premises. · OAI.Thomason.Construction.PublishedThomasonReduction

PublishedTraceGeneration def

The consequence of Rohde–Schramm Theorem 5.1 needed to pass from a Brownian driver to the continuous trace convention used in this development. · OAI.SLEExactGauge.PublishedTraceGeneration

PublishedTracialUltrapower def

Popa §1.6, specialized to the constant sequence of the given II₁ factor with its already constructed faithful normal trace. · OAI.KadisonKastler.PublishedTracialUltrapower

PublishedTwoRatioProduct def

Published pointed-product and modular tensor construction, restricted to the concrete data above. · OAI.KadisonKastler.PublishedTwoRatioProduct

PublishedTypeIInjectivity def

The factor specialization of Blackadar IV.2.2.10. · OAI.KadisonKastler.PublishedTypeIInjectivity

PublishedUniformInputs structure

Existence of central factor models for separable represented algebras. · OAI.KadisonSimilarity.PublishedUniformInputs

PublishedVisualSplitting structure

The exact two published implications used by the target corollary. · OAI.HarmonicArtin.ParabolicIntersections.PublishedVisualSplitting

pull def

monomial point move witnessing recovery · OAI.PowerSaving.Cluster.pull

pullAlphabet def

Pull an automaton back along a letter map, preserving its state set. · OAI.TwoWayComplementation.TwoNFA.pullAlphabet

pullback def

The pullback of a line bundle is a line bundle, via inverse-image trivializing opens and the pullback of the unit. · OAI.ActualSheafTensor.LineTrivialization.pullback

pullback def

Pull each finite averaging unitary back to the original representation. · OAI.KadisonKastler.FiniteInnerAverage.pullback

pullback def

Nonsingular pullback is a unital star-algebra homomorphism of the actual almost-everywhere quotient. · OAI.KadisonKastler.WeakOperatorField.pullback

pullback def

Pullback of bounded measurable representatives by a nonsingular map. · OAI.KadisonKastler.WeakOperatorRepresentative.pullback

pullback def

Pull back just the coordinates in an injective diagram. · OAI.PartitionConsistency.CohenForcing.Condition.pullback

pullback def

Change label coordinates only: every individual set and component graph is pulled back, and the upper invariant is transported by the actual measure. · OAI.PlanarL1.Columns.Model.SetSystem.pullback

pullback def

Graph isomorphisms, such as actual marked-chain subdivision, preserve all event data, including the root pin and exact directed labels. · OAI.TorsionFreeZeroDivisors.RealizationTransport.pullback

pullbackChartedSpace def

A pullback atlas whose preferred charts have explicit formulas. · OAI.PolycyclicRecognition.pullbackChartedSpace

pullbackChartIso def

Chartwise comparison with the built-in base-changed open cover. · OAI.SourcePullbackChart.pullbackChartIso

pullbackForm def

The usual pullback, defined using the actual derivative of the parametrization. · OAI.MahlerStokes.pullbackForm

pullbackIota def

The open chart of the literal source fiber product. · OAI.SourcePullbackChart.pullbackIota

pullbackIso def

The reflection unit identifies the descended cokernel with geometric pullback. · OAI.CartierUniversal.pullbackIso

pullbackIsoSpec def

Pullback comparison for the affine identifications of source and target, retaining the map of global section rings. · OAI.CoherentDual.pullbackIsoSpec

pullbackLineCore def

Pull a line core back along a continuous base map. · OAI.Kaplansky.pullbackLineCore

pullbackLineCoreTotalHomeomorph def

This is the ordinary topological pullback, not just a fiberwise copy. · OAI.Kaplansky.pullbackLineCoreTotalHomeomorph

pullbackMap def

Pulling back a map along a projection produces a map annihilating its kernel. · OAI.MaxCutGames.Soundness.LinearMapDensity.pullbackMap

pullbackMap def

Pulling back a map along a projection produces a map annihilating its kernel. · OAI.UniqueGamesTheorem.Soundness.LinearMapDensity.pullbackMap

pullbackPolynomialEquiv def

The source pullback is locally the affine line over the surface's own homogeneous chart, rather than a merely posited replacement scheme. · OAI.SourcePullbackChart.pullbackPolynomialEquiv

pullbackPunctureEquiv def

Localization of the actual line-bundle ring chart. · OAI.SourcePullbackChart.pullbackPunctureEquiv

pullbackRestrictNatIso def

Pullback commutes with open restriction through the canonical mate, compatibly with the pullback-pushforward unit. · OAI.CoherentBaseChange.pullbackRestrictNatIso

pullbackRow def

Raw row application after pulling back the retained coordinates along π. single records J, and omega is indexed by positions, never by names. · OAI.MaxCutGames.Soundness.ZeroInformation.pullbackRow

pullbackRow def

Raw row application after pulling back the retained coordinates along π. single records J, and omega is indexed by positions, never by names. · OAI.UniqueGamesTheorem.Soundness.ZeroInformation.pullbackRow

pullbackTildeIso def

Affine pullback of an associated module is the actual tensor-product extension of scalars, obtained by uniqueness of adjoints. · OAI.AffinePullback.pullbackTildeIso

pullbackUnit def

The original pullback maps weakly to the pullback of its constructed Kan replacements, with no properness hypothesis. · OAI.Thomason.Construction.InfiniteExtension.pullbackUnit

pullbackZeroQuotient def

The exact quotient at the zero section, with its S-algebra map. · OAI.SourcePullbackChart.pullbackZeroQuotient

pullCoeff def

Pull back an admissible coefficient function along the candidate-label equivalence. · OAI.ReflexiveFixedPoints.Prediction.Prefix.pullCoeff

pullCoeffEquiv def

Reindexing is an equivalence of the entire admissible coefficient domains. · OAI.ReflexiveFixedPoints.Prediction.Prefix.pullCoeffEquiv

pullDerivation def

Recover a derivation on an open scheme from one with values in its pushforward, using the restriction-pushforward counit. · OAI.ActualCotangent.pullDerivation

pulledBase def

Pull the base action back along the actual quadric-to-determinant-ring map. · OAI.StableCoordinate.NormalizedDeterminantLift.pulledBase

pulledCurvatureEndomorphism def

Pull back the two tangent arguments of the curvature endomorphism. · OAI.PositiveEinsteinFour.Geometry.pulledCurvatureEndomorphism

pulledPlaneIdealChartIso def

Pulled-back plane ideal, on the completed source chart. · OAI.SourcePullbackChart.pulledPlaneIdealChartIso

pullOverHom def

Pull a morphism on a target open to a chart open whose image it contains. The scalar restriction uses the chart ring isomorphism. · OAI.ActualOpenComparison.pullOverHom

pullOverIso def

A puncture isomorphism restricts functorially to the puncture on every affine chart. · OAI.ActualOpenComparison.pullOverIso

pullSample def

Delete all unselected branches of an actual history. Unary vertices and all ancestor coordinates are retained. · OAI.DilutedSpinGlass.PrescribedTree.SplitMap.pullSample

pullSelector def

Use the target selector, including its concrete enumeration choices. · OAI.MaxCutGames.Foundations.PCP.PoweringPortReindex.pullSelector

pullSelector def

Use the target selector, including its concrete enumeration choices. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringPortReindex.pullSelector

pullState def

Pull back a function-indexed cyclic state to the list-path presentation. This is just relabeling coordinates, used to connect the closed-form vectors to the list recursion. · OAI.TreeEdit.BinaryLower.FiniteTree.pullState

pullStateHom def

Relabeling the product factors is additive, not merely a map of sets. · OAI.TreeEdit.BinaryLower.FiniteTree.pullStateHom

pulse def

Source period j is indexed here by j-1; no period lengths are changed. · OAI.HarmonicCounterexample.Schedule.pulse

PulseHistory structure

Raw center-regular history plus the quantitative hypotheses used on the current pulse. This structure contains no transmission conclusion. · OAI.HarmonicCounterexample.LinearODE.PulseHistory

pulseHistory def

This is a constructor from the actual stream. All nontrivial input facts are supplied by the single history-independent threshold theorem. · OAI.HarmonicCounterexample.LinearODE.HistoryGeometry.pulseHistory

PulseSystem structure

Fixed finite angular and radial constants. · OAI.HarmonicCounterexample.LinearODE.PulseSystem

punctureCollapse def

Collapse within the punctured projective space, including at c=0. · OAI.Kaplansky.ComplexGrassmann.punctureCollapse

punctureCollapseLinear def

Scale the removed-line coordinate and leave the complementary coordinates fixed. · OAI.Kaplansky.ComplexGrassmann.punctureCollapseLinear

puncturedChartIntersectionHomeomorph def

The punctured subspace and its ambient intersection have the same inherited topology, with the same underlying points. · OAI.Kaplansky.puncturedChartIntersectionHomeomorph

punctureDeformation def

The deformation scales [a:v] to [(1-t)a:v]; it stays outside the removed point and fixes the entire complementary hyperplane. · OAI.Kaplansky.ComplexGrassmann.punctureDeformation

PuncturedIncidence def

Only the chosen line's own mark is punctured. · OAI.Release075.PuncturedIncidence

puncturedLine def

The geometric affine line punctured only at its own marked point. · OAI.Release075.puncturedLine

puncturedVectorSphereHomotopyEquiv def

Radial normalization gives a genuine homotopy equivalence of the punctured vector space with its unit sphere. · OAI.Kaplansky.puncturedVectorSphereHomotopyEquiv

punctureEquiv def

The actual cone puncture agrees with the actual punctured line chart. This proof makes no degree-one generation assumption. · OAI.SectionCompletion.punctureEquiv

PunctureHyperplane def

The actual complementary projective hyperplane, as a subspace of the puncture. · OAI.Kaplansky.ComplexGrassmann.PunctureHyperplane

punctureHyperplaneHomotopyEquiv def

A proved homotopy equivalence, not a deformation-retraction premise. · OAI.Kaplansky.ComplexGrassmann.punctureHyperplaneHomotopyEquiv

Punctures structure

Scalar punctures, as prescribed full-ring units. Quasi-unipotence will be imposed at the density stage; the representing scheme itself needs only units. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.Punctures

punctureSectionsIso def

Restriction to a principal open supplies the localized-module isomorphism used for clearing denominators. · OAI.CartierAffineLattice.punctureSectionsIso

Pure def

Exact purity: top faces exist and every face lies in a top face. · OAI.CoboundaryExpanders.Complex.Pure

Pure def

Nonempty top face set and every face contained in an s-face: exact purity. · OAI.CoboundaryExpanders.Complex.Pure

pure def

The canonical pure-tensor section of the tensor sheaf. · OAI.ActualSheafTensor.pure

pureDistribution def

A pure choice, also used to fill unreachable decision histories. · OAI.KServer.pureDistribution

pureNormal def

Coordinates in an orthonormal frame of the two-dimensional normal plane. The complex plane is used only as a real inner product space. · OAI.ClosedSurfaceR4.RadialMetricAlgebra.pureNormal

pureWord def

Ordinary pure position tensor pairing, with no conjugation. · OAI.Saxl.Path.pureWord

push def

Exact aggregation of rational masses; all collisions are added, not silently treated as different edges. · OAI.OptimalMaxCut.LongCode.Law.push

push def

Apply the transpose of the strip matrix to a vector of coefficients. · OAI.SymmetricMahler.push

pushAll def

Tail before head: the recursive network occurs just ONCE in the syntax. All original factor fields remain read-only while the bounded queue is updated. · OAI.ExactQuantumFactoring.BitArithmetic.ChildQueue.pushAll

pushCone def

Linear image, using the inverse so the cone is obviously closed. · OAI.GeneralMahler.pushCone

pushDerivation def

Push a derivation back through the map of section rings. · OAI.ActualCotangent.pushDerivation

pushed def

Transport the character projector along an actual subgroup embedding. · OAI.KaplanskyCounterexample.CharacterIdempotents.pushed

pushes def

Maximum number of pushes along one finite statement branch. · OAI.ContinuumCoulomb.TM2Size.pushes

pushforward def

Pushforward counts all preimages, including collisions. · OAI.MaxCutGames.Foundations.Games.FiniteDistribution.pushforward

pushforward def

Pushforward counts all preimages, including collisions. · OAI.UniqueGamesTheorem.Foundations.Games.FiniteDistribution.pushforward

pushforward_fullyFaithful def

Actual reindexing plus coefficient isomorphism is fully faithful. · OAI.ModuleReindex.pushforward_fullyFaithful

pushforwardGammaIso def

Global sections of pushforward along an affine map are scalar restriction. · OAI.AffinePullback.pushforwardGammaIso

pushforwardIsoRestrict def

Pushforward under a scheme isomorphism is restriction along its inverse. · OAI.CoherentGlobal.pushforwardIsoRestrict

pushforwardIsoRoundtrip def

Pushforward around a scheme isomorphism. · OAI.CoherentGlobal.pushforwardIsoRoundtrip

pushforwardOverBaseIso def

Pushforward along the commuting parameter square. · OAI.SourceConeMorphism.pushforwardOverBaseIso

pushforwardRestrictAppIso def

Finite-open restriction of pushforward, on sections. · OAI.CoherentBaseChange.pushforwardRestrictAppIso

pushforwardRestrictIso def

Open base change for sheaves of modules; no assumption of flat base change or quasicoherence is needed for restriction to an open. · OAI.CoherentBaseChange.pushforwardRestrictIso

pushforwardRestrictNatIso def

The open basechange comparison is natural in the module sheaf. · OAI.CoherentBaseChange.pushforwardRestrictNatIso

pushforwardTildeIso def

The actual pushforward of an associated sheaf along an affine morphism is the associated sheaf of scalar restriction. · OAI.AffinePullback.pushforwardTildeIso

pushMass def

Push a nonnegative mass along an actual finite map. · OAI.HyperbolicObstruction.PlanarRecursion.pushMass

pushMass def

Pushing a finite entropy vector through a deterministic map cannot increase entropy. There is no measurability or independence assumption. · OAI.SmoothObstruction.FiniteEntropy.pushMass

pushoutData def

The pushout of a stationary deformation is itself a stationary deformation. · OAI.Thomason.Construction.TopologicalDeformation.pushoutData

PushoutGluing structure

Atomic data for gluing two pieces. The overlap condition is an actual common-source representative, rather than an equality assumed in the unknown target pushout. · OAI.Thomason.Construction.DirectedComplex.OrderedAtomBasis.PushoutGluing

pushoutRetraction def

Retraction on the actual pushout. · OAI.Thomason.Construction.TopologicalDeformation.pushoutRetraction

pushoutTrack def

Gluing the relative track uses preservation of pushouts by product with I. · OAI.Thomason.Construction.TopologicalDeformation.pushoutTrack

pushSymbol def

A fixed one-cell insertion. · OAI.ParityWL.Complexity.pushSymbol

pushTrue def

A syntactic chain of a fixed number of true-bit pushes. · OAI.MaxCutGames.Foundations.Complexity.PoweringMachineLoop.pushTrue

pushTrue def

A syntactic chain of a fixed number of true-bit pushes. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringMachineLoop.pushTrue

pushWord def

A fixed finite list is part of the program, not a run-time oracle. · OAI.ExactQuantumFactoring.BitStackProgram.pushWord

pushWord def

Push a fixed word in its listed order, then execute the continuation. · OAI.MaxCutGames.Reduction.MachineSubstitution.pushWord

pushWord def

A finite literal is written by a finite sequence; it is not a primitive which can materialize a word of unbounded length in one step. · OAI.RationalHitting.StackLanguage.pushWord

pushWord def

Write a fixed finite bit word, in the correct order. · OAI.Superstring.BitCode.Block.pushWord

pushWord def

Push a fixed word in its listed order, then execute the continuation. · OAI.UniqueGamesTheorem.Reduction.MachineSubstitution.pushWord

putLab def

Embedding of the affine coordinate space at a fixed label. · OAI.HadwigerCounterexample.putLab

PX def

Finite ancestral heads preserve the coordinate completion by their explicit finite-coordinate formula; tails then do so by subtraction. · OAI.ForestSpace.PX

pX def

The exponent on dot products whose first leg has dimension one. · OAI.MatrixMultiplication.AuxiliarySeparation.Character.pX

pY def

The exponent on dot products whose second leg has dimension one. · OAI.MatrixMultiplication.AuxiliarySeparation.Character.pY

pZ def

The exponent on dot products whose third leg has dimension one. · OAI.MatrixMultiplication.AuxiliarySeparation.Character.pZ

p₀ def

A conservative effective prescription coefficient, for the diameter union bound only. The empirical letter density remains 129/16513. · OAI.TorsionFreeZeroDivisors.TypedDiameter.p₀