Glossary: T

Definitions in this project whose names start with T. 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

T def

The two other explicit affine opens of the original plane. · OAI.KummerAffineExtra.T

T def

The actual open D((st+1)(s+1)(t−1)) of the blowup chart. · OAI.KummerShifted.T

T def

The actual open D((st+s+1)(s+1)(t+1)) of the blowup chart. · OAI.KummerShiftedDiagonal.T

T def

The actual open D((s−1)(st−1)(1−t)) of the blowup chart. · OAI.KummerTriple.T

T def

The actual open D((s−1)(st−1)(1−t)) of the blowup chart. · OAI.KummerTripleDiagonal.T

T def

The fully executed, omega-one-length saturation. · OAI.PartitionConsistency.ConcreteSystem.T

T def

Threshold for a large weighted degree. · OAI.TalagrandDiscreteConvexity.Covering.NumericalBounds.T

t7 def

The first positive root t₇ of the normalized profile. · OAI.Bernoulli.DeSilvaJerison.t7

tab def

Build Tape len. As for sow we can first allocate then perform writes. · OAI.PowerSaving.RAM.Bill.tab

tab def

Materialized array. · OAI.PowerSaving.Tape.tab

Table def

Unrestricted nonnegative integer tables with the specified margins. · OAI.ContingencyTables.Table

Table structure

Each state designates one reading side. A cell test has at most one result; boundary tests only change the state. A failed (or absent-boundary) test has no result. This is exactly the source's disjoint-case convention. · OAI.FiniteAlgebraicEnvelopes.Compiler.Table

Table structure

A finite prefix-replacement table, with one input-group twist on each brick. The uniqueness clauses say that the source and target bricks are partitions. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Table

Table structure

A finite lookup table, without any assumption about addresses visited by a run. · OAI.ForcedComputation.Lattice.Table

Table structure

Stored reverse-dart indices, with a proof of the finite involution law. · OAI.MaxCutGames.Foundations.PCP.ExpanderTables.Table

Table structure

Finite executable input data, with the two graph laws certified on lookup. · OAI.MaxCutGames.Foundations.PCP.PortTables.Table

Table structure

Only a nonempty finite list of binary vectors is required as static noise data. The correcting function is a separate witness used in the proof. · OAI.MaxCutGames.Integration.NoiseTables.Table

Table def

A set-indexed table, bounded by the order type of the ground code set. · OAI.PartitionConsistency.OrderAssignment.Table

Table structure

Stored reverse-dart indices, with a proof of the finite involution law. · OAI.UniqueGamesTheorem.Foundations.PCP.ExpanderTables.Table

Table structure

Finite executable input data, with the two graph laws certified on lookup. · OAI.UniqueGamesTheorem.Foundations.PCP.PortTables.Table

Table structure

Only a nonempty finite list of binary vectors is required as static noise data. The correcting function is a separate witness used in the proof. · OAI.UniqueGamesTheorem.Integration.NoiseTables.Table

table def

A literal, dimension-recursive fallback for the sampling algorithm. · OAI.ContingencyTables.GreedyFeasibleTable.table

table def

The literal finite union of validation and simulation equations. · OAI.FiniteAlgebraicEnvelopes.Compiler.Control.table

table def

The displayed validator equations as an executable finite scan table. · OAI.FiniteAlgebraicEnvelopes.Compiler.Validator.table

table def

Independently sample one response for each possible private question. · OAI.MaxCutGames.Foundations.Games.FiniteDistribution.table

table def

A bit table specified by two bounded-arithmetic expressions. These are syntax, not assumed computable functions or unit-time arithmetic oracles. · OAI.OptimalMaxCut.CounterMachine.table

table def

The open table; the complex plane carries its ordinary real Euclidean metric. · OAI.TriangularBilliards.Triangle.table

table def

Independently sample one response for each possible private question. · OAI.UniqueGamesTheorem.Foundations.Games.FiniteDistribution.table

table def

Both the forward and inverse arrays have one entry for every alphabet label. · OAI.UniqueGamesTheorem.MachineTranslationTable.table

table def

Literal finite Boolean relation iteration. · OAI.WLIdentification.FiniteRefinement.table

Tableau abbrev

A tableau is an enumeration of all cells by the position set. · OAI.Saxl.Tableau

tableauInclusion def

Position embedding induced by inclusion of the actual Young diagrams. · OAI.Saxl.tableauInclusion

tableauMove def

A change of tableau enumeration only applies a position permutation. · OAI.Saxl.tableauMove

tableCanonical def

Redundant rectangular-table entries and harmless dummy primes are ignored unless they divide m; duplicates impose exactly the same congruence. · OAI.ExactQuantumFactoring.tableCanonical

tableCertificate def

The initial graph and entire repeated table transformation have concrete finite-machine executions with the same formula and table encodings. · OAI.MaxCutGames.Foundations.Complexity.MachineTableIteration.tableCertificate

tableCertificate def

The initial graph and entire repeated table transformation have concrete finite-machine executions with the same formula and table encodings. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineTableIteration.tableCertificate

tableComputer def

A genuine polynomial-time finite Boolean-tape Turing machine for any closed bounded-arithmetic bit table. This is an operational theorem, not a closure axiom or a unit-cost machine convention. · OAI.OptimalMaxCut.CounterMachine.tableComputer

tableConstant def

Walsh coefficients of the log of a positive two-spin table. · OAI.BufferedIsing.tableConstant

tableCovariance def

Covariance of the two canonical spins of a positive two-spin table. · OAI.BufferedIsing.tableCovariance

tableFactor def

A zero/unused-row word is replaced by the harmless prime 2. · OAI.ExactQuantumFactoring.BitArithmetic.tableFactor

tableFactors def

A fixed rectangular table of prime candidates. Unused rows and zero padding become 2. This requires no support enumeration or factor search. · OAI.ExactQuantumFactoring.PhysicalTree.tableFactors

tableFavorable def

Pairwise comparison avoids searching for a least component. The supplied rectangular table may contain duplicates and harmless primes not dividing m. · OAI.ExactQuantumFactoring.tableFavorable

tableFavorableCountNet def

The exact source F, computed from supplied prime rows using polynomial integer circuits. Distinctness tests remove repeated table entries. · OAI.ExactQuantumFactoring.BitArithmetic.tableFavorableCountNet

tableFintypeTransport def

Enumeration instances affect the proofs of the margins, never the entries. · OAI.ContingencyTables.tableFintypeTransport

tableFlag def

The flag on an orientation of A. Test all entries on the table spaces with column in the B pins. Test the other entries by the reciprocal flag. Together these are §5 admissibility (when units are in the pin leaves). · OAI.HadwigerCounterexample.tableFlag

tableGraph def

Interpret the numeric table over its explicitly decoded Boolean labels. · OAI.MaxCutGames.Foundations.Complexity.FinalCNFPattern.tableGraph

tableGraph def

Interpret the numeric table over its explicitly decoded Boolean labels. · OAI.UniqueGamesTheorem.Foundations.Complexity.FinalCNFPattern.tableGraph

tableHeaderInTime def

Specialization to the entire input of the actual numbered graph codec. · OAI.MaxCutGames.Foundations.Complexity.FinalCNFMachine.Program.tableHeaderInTime

tableHeaderInTime def

Specialization to the entire input of the actual numbered graph codec. · OAI.UniqueGamesTheorem.Foundations.Complexity.FinalCNFMachine.Program.tableHeaderInTime

tableInj def

small table injecting condition, at this role. Row directions are in a.sp and wholly in the indicated primal block; output is viewed modulo hPin at the same sign, not the sign of the tested columns. · OAI.HadwigerCounterexample.tableInj

tableInTime def

Timed witness for the actual controller, with a fixed alphabet-dependent bound on its full forward table. Other row fields and unread input are preserved. · OAI.MaxCutGames.Explicit.MachineSubdivisionTableRead.tableInTime

tableInTime def

Construct the entire table execution from its actual initialized inputs. No row trace, loop trace, or abstract computational witness is a hypothesis. · OAI.MaxCutGames.Foundations.Complexity.MachineExpanderTable.tableInTime

tableInTime def

Timed witness for the actual controller, with a fixed alphabet-dependent bound on its full forward table. Other row fields and unread input are preserved. · OAI.UniqueGamesTheorem.Explicit.MachineSubdivisionTableRead.tableInTime

tableInTime def

Construct the entire table execution from its actual initialized inputs. No row trace, loop trace, or abstract computational witness is a hypothesis. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineExpanderTable.tableInTime

tableNormal def

The actual partial table-first procedure used at the fixed point. · OAI.FiniteAlgebraicEnvelopes.Bootstrap.tableNormal

tableOrder def

A literal order circuit using only retained verified table entries and modular powering. No mathematical factorization/support enumeration appears in its definition, even on malformed inputs. · OAI.ExactQuantumFactoring.BitArithmetic.tableOrder

tableOrderBound def

This bound is polynomial in n, table length and the individual stored-word circuit cost. Resizing is itself a bounded circuit, not a word-RAM primitive. · OAI.ExactQuantumFactoring.BitArithmetic.tableOrderBound

tableOrderWidth def

The extra n bits make zero-extension unconditional, including empty malformed tables. On verified input this holds the entire repeated product. · OAI.ExactQuantumFactoring.BitArithmetic.tableOrderWidth

tableOutput def

Fixed finite noise data suffice for executable output. · OAI.MaxCutGames.Decoder.TableKeysAddressGame.tableOutput

tableOutput def

The concrete vector-table interface contains no correcting-map witness. · OAI.MaxCutGames.Reduction.AddressGame.tableOutput

tableOutput def

Fixed finite noise data suffice for executable output. · OAI.UniqueGamesTheorem.Decoder.TableKeysAddressGame.tableOutput

tableOutput def

The concrete vector-table interface contains no correcting-map witness. · OAI.UniqueGamesTheorem.Reduction.AddressGame.tableOutput

tableOutputsInTime def

The concrete driver outputs exactly the checked table encoding. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Runtime.tableOutputsInTime

tableOutputsInTime def

The concrete driver outputs exactly the checked table encoding. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Runtime.tableOutputsInTime

tablePolynomial def

A two-parameter bound for the complete unary graph codec. · OAI.MaxCutGames.Foundations.PCP.PreprocessingMachineBounds.tablePolynomial

tablePolynomial def

A two-parameter bound for the complete unary graph codec. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingMachineBounds.tablePolynomial

tablePolynomialTime def

Fixed q determines one finite machine and one natural polynomial. The input and output alphabets use the exact shared Bool serialization interfaces. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Runtime.tablePolynomialTime

tablePolynomialTime def

Powering and alphabet reduction now use their constructed finite machines. The only remaining input is an execution certificate for preprocessing. · OAI.MaxCutGames.Foundations.PCP.RoundComputation.tablePolynomialTime

tablePolynomialTime def

Fixed q determines one finite machine and one natural polynomial. The input and output alphabets use the exact shared Bool serialization interfaces. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Runtime.tablePolynomialTime

tablePolynomialTime def

Powering and alphabet reduction now use their constructed finite machines. The only remaining input is an execution certificate for preprocessing. · OAI.UniqueGamesTheorem.Foundations.PCP.RoundComputation.tablePolynomialTime

tableRealizer def

Every loop is over an input row or column; binary margins are never counted down one unit at a time. · OAI.ContingencyTables.GreedyFeasibleTable.tableRealizer

tableRowMachineInTime def

With the actual computed data on its input tape, the existing finite row machine emits precisely the serialized relation of the actual output table. The unread input suffix and prior output tape are preserved as specified. · OAI.MaxCutGames.Foundations.PCP.PoweringRowData.tableRowMachineInTime

tableRowMachineInTime def

With the actual computed data on its input tape, the existing finite row machine emits precisely the serialized relation of the actual output table. The unread input suffix and prior output tape are preserved as specified. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringRowData.tableRowMachineInTime

tableRowsBits def

The unread graph rows in the actual graph codec. · OAI.MaxCutGames.Foundations.Complexity.FinalCNFMachine.Program.tableRowsBits

tableRowsBits def

The unread graph rows in the actual graph codec. · OAI.UniqueGamesTheorem.Foundations.Complexity.FinalCNFMachine.Program.tableRowsBits

tablesEquiv def

The terminal feasible-profile count is the original bounded-table count, with the instance's necessarily nonnegative margins. · OAI.ContingencyTables.CompanionPhysicalTables.tablesEquiv

tableSetupInTime def

Specialization to the checked graph-table codec. The bound follows from the actual copy and delimiter-scan traces above. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Setup.tableSetupInTime

tableSetupInTime def

Specialization to the checked graph-table codec. The bound follows from the actual copy and delimiter-scan traces above. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Setup.tableSetupInTime

tableSign def

The private sign function, formed from the table on one visible row fiber. · OAI.MaxCutGames.Decoder.PrivateFourier.tableSign

tableSign def

The private sign function, formed from the table on one visible row fiber. · OAI.UniqueGamesTheorem.Decoder.PrivateFourier.tableSign

tableSkeleton def

The projection is used only to fill an erased witness field. It is never used when generating constraints or evaluating their permutation tables. · OAI.MaxCutGames.Integration.TableReduction.tableSkeleton

tableSkeleton def

The projection is used only to fill an erased witness field. It is never used when generating constraints or evaluating their permutation tables. · OAI.UniqueGamesTheorem.Integration.TableReduction.tableSkeleton

tableStep def

A complete uniform table-building step: evaluation of a compiled callback, indexed counter update, and append of an escaped dense record. · OAI.RationalHitting.StackLanguage.tableStep

tableTotient def

At most n repetitions of each rectangular prime entry suffice. Unlike order stripping, the totient accumulator does not need a wide product. · OAI.ExactQuantumFactoring.BitArithmetic.tableTotient

tableVerifier abbrev

The concrete Boolean verifier associated with an already indexed table. · OAI.MaxCutGames.Foundations.Complexity.FinalCNFPattern.tableVerifier

tableVerifier abbrev

The concrete Boolean verifier associated with an already indexed table. · OAI.UniqueGamesTheorem.Foundations.Complexity.FinalCNFPattern.tableVerifier

tableWords def

Existing graph codec, with header [n,n*d] and 4098 words in every row. · OAI.MaxCutGames.Foundations.PCP.PortTables.tableWords

tableWords def

`tableWords table` lists the natural values of `table.images` in label order. · OAI.UniqueGamesTheorem.Foundations.Complexity.tableWords

tableWords def

Existing graph codec, with header [n,n*d] and 4098 words in every row. · OAI.UniqueGamesTheorem.Foundations.PCP.PortTables.tableWords

tabloidOrbitMap def

The actual orbit sum giving the map out of the tabloid permutation module. It is unnormalized, which is harmless for testing nonzero Young symmetrizers. · OAI.RowColumn.tabloidOrbitMap

tabloidSub def

The actual row-tabloid permutation module, embedded in the word space. · OAI.Saxl.tabloidSub

tag def

Code for a move: a leading tag, plus zero-padded first and second frame encodings. · OAI.Fifths.Flow.tag

tag def

The phase coordinate of a transition in the shared analytic family. · OAI.LipschitzHilbert.ShiftedBlocks.tag

taggedAssignment def

Padded query tables are evaluated at one coordinate per context. Left and right tables have disjoint tags, and the unused dummy bit is fixed to false. · OAI.MaxCutGames.Foundations.Hastad.SourceHonest.taggedAssignment

taggedAssignment def

Padded query tables are evaluated at one coordinate per context. Left and right tables have disjoint tags, and the unused dummy bit is fixed to false. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceHonest.taggedAssignment

taggedTranslations def

Complete interface from exact-projection coordinate families to the actual uncolored graphs, with every private leaf and its owner included. · OAI.WLIdentification.AffineBlocks.taggedTranslations

Tags structure

Live structure prices and hyperparameters selected before running the program. · OAI.ExactDerandomization.Helix.Moon.Tags

TagStable def

Testing degree intervals converts stable uncolored tuples back into their named base classes, without assuming color information as a new WL hypothesis. · OAI.WLIdentification.TagStable

Tail inductive

The right suffix tested from N: clean workspace, D, then clean anchors. · OAI.FiniteAlgebraicEnvelopes.Compiler.Validator.Tail

tail def

Partial identities on rational tails, in the ambient partial-map monoid. · OAI.FiniteAlgebraicEnvelopes.PartialAffine.tail

tail def

Discarding finitely many matrix stages preserves every geometric field of the tower and its dense union. · OAI.KadisonSimilarity.TracialMatrixTower.tail

tail def

Ambient actions restrict to the same cofinal sequence of matrix stages. · OAI.KadisonSimilarity.WordLevelMatrixActions.tail

tail def

The exact tail of the density in eq. tier-radius-density. The clipped formula includes both support endpoints (which have zero atom). · OAI.KServer.PartitionProbabilities.tail

tail def

Taylor remainder of the n-th power after its quadratic term. · OAI.PettyProjection.Scalar.tail

tail def

Survival function of the manuscript's truncated exponential radius, normalized by r. It includes support endpoints with their exact values. · OAI.UniformKServer.RadiusTail.tail

tailAt def

The tail from the unique segment through g, or the empty segment if absent. · OAI.DiamondDistortion.Ancestry.tailAt

tailBlock def

The remaining coordinates after a bounded prefix. · OAI.Problem358.FinitePrefix.tailBlock

tailCoefficient def

The common decay coefficient for the two probability sheets in the tail. · OAI.BoltzmannNonuniqueness.SheetMeasures.tailCoefficient

tailCoefficient def

Coefficients conditionally independent below the exit forest. · OAI.IndependentProducts.FiniteHead.tailCoefficient

tailCollisionDefect def

The finite defect for the two summands, with the second one reflected. · OAI.Ostmann.tailCollisionDefect

TailData structure

The tails and the actual normal subgroup of eventual pointwise fixers. For components, these data are supplied by Chain.tailFixer_normal. No bijections or selector are included among the data. · OAI.PartitionConsistency.TailSelector.TailData

tailData def

The abstract tail data are supplied by the actual component stabilizer and the union of its pointwise tail fixers previously constructed. · OAI.PartitionConsistency.LocalDiagramSystem.Vertex.Chain.tailData

tailDual def

Complementary coordinate projection, never assumed contractive without proof. · OAI.DiamondDistortion.SegmentSystem.tailDual

tailFamily def

Tail maps retain their actual outer resolvent and generated inner map. · OAI.ReflexiveFixedPoints.tailFamily

tailFixer def

The increasing union of pointwise tail fixers, regarded inside the actual component stabilizer. · OAI.PartitionConsistency.LocalDiagramSystem.Vertex.Chain.tailFixer

tailGroup def

The old tail fixer viewed faithfully as coordinate permutations. · OAI.PartitionConsistency.ProductForcing.ChainCode.tailGroup

tailJoin def

Regroup the next scalar coordinate with the remaining tail. · OAI.Problem358.FinitePrefix.tailJoin

tailMean def

On one real kernel path, retain its prefix and compute the true mean of its descendant shape. Omitted sampled branches have integrated to one. · OAI.DilutedSpinGlass.PrescribedTree.tailMean

tailPairBound def

A deliberately coarse but absolute bound on unseparated pair dimensions. It replaces the manuscript's unbounded-subsequence argument by a finite energy propagation along a certified spanning tree of the exact complex. · OAI.SidorenkoCounterexample.tailPairBound

tailPorts def

The actual ports before a selected time, with no vertex-dependent choice. · OAI.MaxCutGames.Foundations.PCP.PoweringFieldPlan.tailPorts

tailPorts def

The actual ports before a selected time, with no vertex-dependent choice. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringFieldPlan.tailPorts

tailProjection def

Dropping one coordinate does not increase Euclidean distance. · OAI.PositiveEinsteinFour.Angular.tailProjection

tailResidues def

Residues of a summand after deleting the initial interval through N + p. · OAI.Ostmann.tailResidues

tailSpan def

Algebraic finite blocks in the specified tail, with no completion added. · OAI.SeparableQuotient.ActualSpace.BlockSequence.tailSpan

TailSpectatorFamily structure

Every field refers to the same selected prime family and the original tail masks. The tolerance is available both for Fourier factors and for all translated nonprincipal characters. · OAI.Ostmann.TailSpectatorFamily

tailSupport def

All residues of the first summand's infinite tail, labeled in [0,p). · OAI.Ostmann.tailSupport

TailVanishing def

Uniform pure-tail vanishing at the unit sphere, the qualitative mass-removal input. No power law or quantitative modulus is inserted into this hypothesis. · OAI.DiamondDistortion.Midpoint.Heads.TailVanishing

take def

Restrict a coordinate tuple to an initial prefix. · OAI.Problem358.TriangularRecursion.take

takeWitnessPacks def

Preserve every chunk position while truncating the concatenated witness list. · OAI.GammaFPP.CrossingBridge.takeWitnessPacks

tallyH def

halving and log. All-natural iterations. · OAI.PowerSaving.RAM.tallyH

Tangent abbrev

Tangential directions to the normal v. · OAI.Problem358.TangentNormalCoordinates.Tangent

tangent def

The real tangent subspace at a unitary, represented in the ambient algebra. · OAI.KervaireUnitaryTangent.tangent

tangent def

A tangent vector to side i, with no choice of inward or outward normal. · OAI.TriangularBilliards.Triangle.tangent

TangentCone abbrev

The completed cone of represented directions, built from radial CAT(0) comparison limits rather than postulated to exist. · OAI.HyperbolicObstruction.TangentCone

tangentConstraint def

The tangent space is the kernel of two independent complex linear forms. · OAI.ArnoldCounterexample.Quadric.tangentConstraint

tangentCoordinates def

In coordinates (u,v,a,b) on the tangent the Hessian is va'+av'. · OAI.ArnoldCounterexample.ThreePoints.tangentCoordinates

TangentCritical def

dh vanishes on the full constrained tangent of S² × S². · OAI.ArnoldCounterexample.ThreePoints.TangentCritical

tangentCurve def

A genuine curve in the unit cone, not an assumed tangent parametrization. · OAI.ArnoldCounterexample.Quadric.tangentCurve

tangentDist def

Distance between formal nonnegative multiples of two radial directions. · OAI.HyperbolicObstruction.CATZero.tangentDist

tangentH def

The two tangent Ricci eigenvalues as continuous functions of the six scalar profile jets. The harmless positive clamp is inactive wherever q≥1/2. · OAI.HarmonicCounterexample.Berger.tangentH

tangentHeight def

The tangent-plane height used in all source sections. · OAI.AffineBernstein.tangentHeight

tangentImaginaryOperator def

The imaginary derivation with the original model-space norm. · OAI.PolycyclicRecognition.tangentImaginaryOperator

tangentMap def

Tangent-first affine coordinates, convenient for iterated strip integrals. · OAI.Problem358.TangentNormalCoordinates.tangentMap

tangentMass def

Mass in the increasing family of tangent disks, away from its tangency cap. · OAI.DiskGeometry.tangentMass

tangentMass def

Mass in the increasing family of tangent disks, away from its tangency cap. · OAI.SignedDisk.DiskGeometry.tangentMass

tangentMatrix def

A literal real skew matrix lifting any horizontal vector. · OAI.ArnoldCounterexample.CriticalBound.tangentMatrix

tangentModelEquiv def

The manuscript's epigraph; the domain is not silently replaced by all of space. Literal tangent affine coordinates, ordered as one base direction and n transverse directions. · OAI.AffineBernstein.tangentModelEquiv

tangentPairing def

Definition by bounded supremum, so no completeness of the ambient CAT(0) space is required to form comparison products in its tangent cone. · OAI.HyperbolicObstruction.CATZero.tangentPairing

tangentProjection def

Orthogonal tangential projection when `e` is a unit vector. · OAI.AffineBernstein.tangentProjection

tangentProjection def

Fixed tangential coordinates relative to a chosen point of the reference plane. · OAI.Problem358.AffinePlaneGraph.tangentProjection

TangentRay structure

Represented directions with an arbitrary nonnegative radial length factor. Zero-distance representatives are identified by the Hausdorff completion. · OAI.HyperbolicObstruction.TangentRay

tangentSection def

The manuscript's open tangent section. · OAI.AffineBernstein.tangentSection

tangentShift def

Subtract the tangent plane and a positive section height; this does not alter the Hessian, hence preserves the original classical PDE. · OAI.AffineBernstein.tangentShift

tangentTorusUnitAction def

This version uses the original model-space norm and scalar structures. · OAI.PolycyclicRecognition.tangentTorusUnitAction

tangentVolume def

A constant ambient form restricting to the determinant volume on the unitary tangent. · OAI.KervaireIncidenceCurrent.tangentVolume

taoExponent def

The five-cycle exponent matrix in equation (cubic-matrix). The entries 2 represent the exponents -1 modulo three. · OAI.MUB6.taoExponent

Tape structure

A literal bounded workspace. The left list is in reverse tape order, and an empty right list is outside the supplied quota, not a fresh blank cell. · OAI.FiniteAlgebraicEnvelopes.Compiler.Simulator.Tape

Tape abbrev

Shared original table, query, scan and copy scratch, then input coordinate fields and the corresponding gathered output fields. · OAI.MaxCutGames.Explicit.MachineProductGather.Tape

Tape structure

Register arrays. · OAI.PowerSaving.Tape

Tape abbrev

Shared original table, query, scan and copy scratch, then input coordinate fields and the corresponding gathered output fields. · OAI.UniqueGamesTheorem.Explicit.MachineProductGather.Tape

tape def

Every input experiment extended to full spatial resolution · OAI.FourierLLogL.Dyadic.Procedure.Wire.tape

TapeConfig structure

The same head/tape representation on any cyclic position alphabet. · OAI.WLIdentification.Machine.TapeConfig

tapeCount abbrev

Cleanup's concrete tape count is the size of the global tape enumeration. · OAI.MaxCutGames.Foundations.Complexity.PoweringRuntimeBudget.tapeCount

tapeCount abbrev

Cleanup's concrete tape count is the size of the global tape enumeration. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringRuntimeBudget.tapeCount

TapeProgram structure

A finite tape machine with distinguished literal input/output states. · OAI.FiniteAlgebraicEnvelopes.MachineRealization.CodeMachine.TapeProgram

tapeQueries def

The three actual query tables read from a uniform finite tape. · OAI.MaxCutGames.Foundations.Hastad.SourceTape.tapeQueries

tapeQueries def

The three actual query tables read from a uniform finite tape. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceTape.tapeQueries

tapes def

The frame records all tapes, including untouched header and output tapes. At every loop guard the four fields are empty and scratch is empty. · OAI.MaxCutGames.Explicit.MachineClone100Loop.tapes

tapes abbrev

Three designated tapes, with the complete surrounding tape family retained. · OAI.MaxCutGames.Foundations.Complexity.MachineFixedDivMod.tapes

tapes def

Three caller tapes with every other tape retained verbatim. · OAI.MaxCutGames.Foundations.Complexity.MachineLookup.tapes

tapes def

Tape 0 is protected input, 1 is the working row stream, 2 is scratch, and 3 and 4 hold the extracted vertex and dart counts. · OAI.MaxCutGames.Foundations.Complexity.MachineTableSplit.tapes

tapes def

Slots 0 and 1 are the operands; slots 2 and 3 are their saved prefixes. · OAI.MaxCutGames.Foundations.Complexity.MachineUnaryLessAt.tapes

tapes def

The frame records all tapes, including untouched header and output tapes. At every loop guard the four fields are empty and scratch is empty. · OAI.UniqueGamesTheorem.Explicit.MachineClone100Loop.tapes

tapes abbrev

Three designated tapes, with the complete surrounding tape family retained. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineFixedDivMod.tapes

tapes def

Three caller tapes with every other tape retained verbatim. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineLookup.tapes

tapes def

Tape 0 is protected input, 1 is the working row stream, 2 is scratch, and 3 and 4 hold the extracted vertex and dart counts. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineTableSplit.tapes

tapes def

Slots 0 and 1 are the operands; slots 2 and 3 are their saved prefixes. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineUnaryLessAt.tapes

TapeStep inductive

The intended finite-quota operations of the finite compiler. These are defined from the given machine table, not postulated accepting transitions. · OAI.FiniteAlgebraicEnvelopes.Compiler.Simulator.TapeStep

tapeSymbol def

Tape lookup includes two distinct endmarkers, even on empty input. · OAI.TwoWayComplementation.tapeSymbol

Target def

The normal target family used in the no-normal-proof obstruction. · OAI.PureTypeSystem.Labelled.Expr.Target

target def

Fully specified indexed oriented boundaries for actual included rows. This is a concrete instantiation of normal-picture cancellation, not an abstract diagram carrying the desired fan bound or isoperimetry. · OAI.HyperbolicObstruction.ConeSkeleton.target

target def

The exact indexed oriented cell target constructed from ordinary polygon boundaries, suitable for CellPicture.normalize. · OAI.HyperbolicObstruction.PictureSwitch.PolygonTarget.Boundary.target

target abbrev

The actual primary H(Y), concentrated in degree zero, and its shift by three. · OAI.LittleFinitistic.SelectedNativeEvaluation.target

target def

The map k([x,i]) = x c_i; compatibility uses the terminal-factor identity, not an incorrectly presumed injective transition map. · OAI.PartitionConsistency.ProbeAlgebra.Path.target

target def

The image position of a prescribed comparison. · OAI.TorsionFreeZeroDivisors.BlockConstraints.Translation.target

targetConcatenation def

The outside and successive endpoint charges, including the final jump. · OAI.EditApproximation.targetConcatenation

targetContractionChangeSq def

Squared norm of the change of the two-leaf contraction in its actual joint target law, not a product of its marginals. · OAI.DilutedSpinGlass.PrescribedTree.targetContractionChangeSq

targetDual def

A Borel extension of the target term from the finite set of sites. · OAI.Problem358.FiniteAffine.targetDual

targetDualAt def

The target term at a specified site. · OAI.Problem358.FiniteAffine.targetDualAt

targetError def

The strict correction bound 2.216 × 10⁻⁴. · OAI.StationaryNS.targetError

targetEuler def

Euler operator of the second coordinate block. · OAI.Problem346.PolynomialShift.targetEuler

targetEvent def

The finite event that this signature becomes monochromatic in bit b. · OAI.QuantitativeVanDerWaerden.SparsePerturbation.targetEvent

targetGraphConormal def

The actual graph conormal expressed in an arbitrary fixed target space. · OAI.AffineBernstein.targetGraphConormal

targetIntercept def

This intercept is determined once the visible base and affine witness are fixed; it is not silently erased when decoding from a row fiber. · OAI.MaxCutGames.Decoder.TableKeysPrivateRow.targetIntercept

targetIntercept def

This intercept is determined once the visible base and affine witness are fixed; it is not silently erased when decoding from a row fiber. · OAI.UniqueGamesTheorem.Decoder.TableKeysPrivateRow.targetIntercept

targetIso def

The previous component target is exactly X in degree zero, with both D-actions. · OAI.LittleFinitistic.OrdinarySimulation.XResolution.targetIso

targetKey def

Every target control specifies the stack whose top identifies the incoming row. · OAI.ForcedComputation.Balanced.targetKey

targetTheory def

ZF with the Partition Principle, ordinal-indexed Choice and failure of Choice. · OAI.PartitionConsistency.SetLanguage.targetTheory

targetTheory def

The exact theory in target thm:main-consistency. · OAI.PartitionConsistency.SetLanguage.targetTheory

taskBound def

The source's whole-tree task cap, expressed without real logarithms. · OAI.ExactQuantumFactoring.AuxiliaryTree.taskBound

tau def

The squared Euclidean norm of a finite holomorphic map. · OAI.Mahler.tau

tauCoeff def

The τ coefficient (not the reduced σ coefficient) in CD(K,N). · OAI.WeakHessian.tauCoeff

tauPower abbrev

Positive real powers embedded into the actual complex multiplier field. · OAI.Nagata.Workers.W10.tauPower

Tear abbrev

Enter a transparent Bernoulli coin (pure rational expression). · OAI.ExactDerandomization.Arena.Amber.Tear

Tells def

Computes a value from a fixed finite type. · OAI.Fifths.Entry.Tells

tempered def

Require each batch's total own-energy below temp; at success the available past temp is expanded. · OAI.FourierLLogL.Dyadic.Procedure.Road.tempered

Template structure

A fixed admissible pilot template, independent of the input law and horizon. · OAI.UniformKServer.PilotEdits.Template

template def

This fixed token list is directly accepted by the actual template machine. · OAI.MaxCutGames.Reduction.CanonicalBodyMachine.template

template def

Coefficient equality alone selects the record tag and the input field. · OAI.MaxCutGames.Reduction.CanonicalWordTemplate.template

template def

This fixed token list is directly accepted by the actual template machine. · OAI.UniqueGamesTheorem.Reduction.CanonicalBodyMachine.template

template def

Coefficient equality alone selects the record tag and the input field. · OAI.UniqueGamesTheorem.Reduction.CanonicalWordTemplate.template

templateSupportConstant def

A fixed Lipschitz constant for the full template support. · OAI.PolycyclicRecognition.SupportDatum.templateSupportConstant

templateSupportExponent def

Each generator uses its own prescribed supporting Cartan position. · OAI.PolycyclicRecognition.CartanWordModel.templateSupportExponent

templateSymmetryGroup def

The group in equation (support-symmetry-group) of boxes.tex. · OAI.PolycyclicRecognition.templateSymmetryGroup

templateType def

Zero-based notation for the source complementary template; true is minus. The motif itself is fixed independently of the hypothesis. · OAI.TraceReconstruction.templateType

templateVisitPath def

The specified support visit, retained for measurable path sampling. · OAI.PolycyclicRecognition.CartanWordModel.templateVisitPath

Tensor abbrev

Tensor over the possibly noncommutative algebra A. · OAI.LittleFinitistic.BalancedTensor.Tensor

Tensor structure

Symmetric three-by-three matrix with four-vector entries. off i is the entry in positions (next i, previous i) and its transpose. · OAI.PositiveEinsteinFour.Bianchi.Tensor

tensor def

The actual tensor sheaf: sheafification of the objectwise tensor product of presheaves of modules. It will provide the Cartier line-bundle twists. · OAI.ActualSheafTensor.tensor

tensor abbrev

Noncommutative balanced tensor D ⊗_A V, presented using tensors over the field. · OAI.DerivedBar.Balanced.tensor

tensor abbrev

Genuine balanced X ⊗_D V, with its exterior left D-action. · OAI.DerivedBar.BimoduleTensor.tensor

tensor def

Cartesian coefficient tensor of dr²+r²G(a,q,J), with u=x/r, v=Jx/r and c=a²q^(-1/m). The radial eigenvalue remains exactly one. · OAI.HarmonicCounterexample.Berger.tensor

tensor abbrev

The actual left B-module complex P tensor_B N. · OAI.LittleFinitistic.FullTensorRectification.tensor

tensor def

The global right interval tensor in the separate-boundary presentation. · OAI.Thomason.ConstructedInterval.tensor

tensorAcross def

Tensor a bimodule map and an input map, retaining the outside B-action. · OAI.LittleFinitistic.BimoduleTotal.tensorAcross

tensorAction def

The commuting actions give the actual tensor-algebra representation. · OAI.LittleFinitistic.OrdinarySimulation.Compression.tensorAction

tensorAncestorMarkKernel def

The ancestor-dependent mark transition, with density c_i(z,a)^b_i. · OAI.InvariantIsing.tensorAncestorMarkKernel

TensorAncestorPairLaw def

The exact finite tensor specialization of the published joint marked path law. The unperturbed cascade supplies the shared-prefix depth and the normalized ancestor kernels supply the conditional marks. · OAI.InvariantIsing.TensorAncestorPairLaw

tensorAncestorPairPathMean def

Iterated conditional expectation of two Gaussian increment paths. Before their branching depth the two paths receive the same tilted increment; afterwards their next increments are conditionally independent. · OAI.InvariantIsing.tensorAncestorPairPathMean

tensorAngular def

Angular split, first slow split, and the final centered slow average. · OAI.ThreeTorus.tensorAngular

tensorAnti def

A sign intertwiner in the second factor twists the diagonal product. · OAI.Saxl.tensorAnti

tensorAugmentation def

The augmentation induces the concrete finite-total transformation. · OAI.LittleFinitistic.BimoduleTotal.AugmentedProjective.tensorAugmentation

tensorBarBicomplexIso def

The termwise balanced/free comparison is an isomorphism of bicomplexes. · OAI.DerivedBar.BimoduleTensor.tensorBarBicomplexIso

tensorBarComplex def

The actual balanced tensor of D with every term and differential of the verified bar resolution. · OAI.DerivedBar.tensorBarComplex

tensorBarComputesLeftDerived def

The left-derived balanced tensor functor is computed by exactly the original tensorBarComplex, via the proved free bar projective resolution. · OAI.DerivedBar.Balanced.tensorBarComputesLeftDerived

tensorBarDerivedIso def

Exact scalar/action-preserving identification with the source of the bar worker's actual derived splitting. · OAI.LittleFinitistic.SelectedPDDetection.tensorBarDerivedIso

tensorBarIso def

The explicit two-sided bar is isomorphic to balanced tensor applied to a genuine projective resolution. · OAI.DerivedBar.tensorBarIso

TensorClass abbrev

Finite complex tensors, with two presentations identified exactly when each is a linear restriction of the other. · OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.TensorClass

tensorClass def

The class of a finite tensor presentation. · OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.tensorClass

tensorClassHomEquiv def

Concrete characters are exactly monotone real-valued tensor semiring homomorphisms. · OAI.MatrixMultiplication.AuxiliarySeparation.Character.tensorClassHomEquiv

tensorCoefficientMatrix def

The transposed matrix-unit coefficients give the trace-dual matrix. · OAI.KadisonSimilarity.tensorCoefficientMatrix

tensorCompatibleCyclicRepresentation def

The particular cyclic representation for which the tensor extension has just been constructed. · OAI.KadisonKastler.tensorCompatibleCyclicRepresentation

tensorComplement def

The algebraic tensor polynomial whose norm expresses the required lower bound. · OAI.NuclearUltrapower.tensorComplement

tensorCongr def

Balanced tensor respects actual right-module equivalences. · OAI.LittleFinitistic.ConcreteTensorEvaluation.tensorCongr

tensorCoordinateDerivative def

The derivative of the coordinate map in the real pair coordinates used by the three-component tensor representation. · OAI.ClosedSurfaceR4.JetPolynomial.tensorCoordinateDerivative

tensorCoordinateJacobian def

The three Jacobian coefficients of one pulled-back tensor component. · OAI.ClosedSurfaceR4.JetPolynomial.tensorCoordinateJacobian

tensorCoordinatePermutation def

Reordering the tensor coordinates. · OAI.KadisonSimilarity.tensorCoordinatePermutation

tensorCoordinatePullbackValue def

Covariant pullback of a symmetric tensor by the actual derivative. · OAI.ClosedSurfaceR4.JetPolynomial.tensorCoordinatePullbackValue

tensorCoordinates def

A fixed coordinate relabeling of the ordinary Kronecker tensor power. · OAI.ExactFourier.tensorCoordinates

tensorCountSplitting def

The tensor of the actual free resolution splits by X-count as a D-module chain complex. The identification of count r with the shifted r-fold derived tensor remains separate. · OAI.DerivedBar.SquareZero.tensorCountSplitting

tensorCurry def

Currying is a morphism of genuine presheaves of modules. · OAI.CoherentDual.tensorCurry

tensorCurryAt def

A bilinear map of actual presheaves defines local linear maps by restricting its second argument. · OAI.CoherentDual.tensorCurryAt

tensorCurryLocal def

The sectionwise tensor-Hom adjunction on a slice. · OAI.CoherentDual.tensorCurryLocal

tensorDiagonalUnitary def

A scalar diagonal whose entries lie on the unit circle. · OAI.KadisonSimilarity.tensorDiagonalUnitary

tensorDisorderPressure def

The normalized spin/leaf log partition retaining all three disorders. · OAI.InvariantIsing.tensorDisorderPressure

tensorEmbedding def

The map to the joint rational-function field. · OAI.EuclideanRamsey.IndependentTensor.tensorEmbedding

tensorEnergy def

The coordinate-invariant expression for the regularized energy density. · OAI.CircleDomainRigidity.FiniteTransfer.tensorEnergy

tensorEquiv def

The two independently defined balanced quotients are naturally B-linearly isomorphic. · OAI.LittleFinitistic.BimoduleBarComparison.tensorEquiv

tensorEquiv def

Evaluation on a vertex projective is its actual vertex space. · OAI.LittleFinitistic.IdempotentTensor.tensorEquiv

tensorEquiv def

Restriction through a ring isomorphism commutes with tensor product. The formula on pure tensors is literally the identity on both factors. · OAI.TensorChangeScalars.tensorEquiv

tensorEval def

The natural map eA tensor_A N → eN. · OAI.LittleFinitistic.IdempotentTensor.tensorEval

tensorEvaluation def

Evaluation into the exact scalar-extended affine cone. · OAI.SourceConeMorphism.tensorEvaluation

tensorFlux def

The flux of the angular tensor acting on the projected vector field. Written in a fixed ambient basis, so no unspecified moving-frame derivatives appear. · OAI.AffineBernstein.tensorFlux

tensorFunctor def

Actual balanced tensor by the arbitrary bimodule X, as an additive endofunctor. · OAI.DerivedBar.BimoduleTensor.tensorFunctor

tensorFunctor def

The actual tensor evaluation functor on all right A-modules. · OAI.LittleFinitistic.BalancedTensor.tensorFunctor

tensorFunctor abbrev

Tensor with the actual right-projective Q_X on arbitrary input complexes of D-modules. · OAI.LittleFinitistic.OrdinarySimulation.XResolution.tensorFunctor

tensorFunctor def

Tensor with a fixed right module, on actual left modules over A. · OAI.LittleFinitistic.PDTensor.tensorFunctor

tensorFunctor def

The genuine additive tensor-with-D functor required by the path exact sequence. · OAI.LittleFinitistic.TensorComplex.tensorFunctor

tensorFunctorCongr def

Naturality in every input module retains the exterior D-action. · OAI.DerivedBar.BimoduleTensor.tensorFunctorCongr

tensorFunctorEquiv def

Compare the functor's canonical scalar restriction with an existing compatible k-action. · OAI.LittleFinitistic.BalancedTensor.tensorFunctorEquiv

tensorFunctorIso def

Agreement with the exact ordinary bimodule functor used by DerivedBar. · OAI.LittleFinitistic.BimoduleBarComparison.tensorFunctorIso

tensorFunctorObjIso def

Coherence between the categorical tensor functor and the literal balancing quotient. · OAI.DerivedBar.BimoduleTensor.tensorFunctorObjIso

tensorGaussianCoordinatePressure def

The coordinate pressure is the actual root/cascade pressure averaged over the rotation, with precisely one Gaussian amplitude replaced. · OAI.InvariantIsing.tensorGaussianCoordinatePressure

tensorHeadAction def

Act on the head factor while leaving the completed tail unchanged. · OAI.KadisonSimilarity.tensorHeadAction

tensorHomAdjunction def

Actual tensor sheaves are a left adjoint, so tensoring is right exact. · OAI.CoherentDual.tensorHomAdjunction

tensorHomEquiv def

Actual tensor-Hom equivalence for presheaves. · OAI.CoherentDual.tensorHomEquiv

tensorHorizontal def

Actual tensor horizontal maps satisfy every requirement of the finite total construction. · OAI.LittleFinitistic.ComplexTensor.tensorHorizontal

tensorIsometry def

Isometries remain isometries upon completion, proved using density. · OAI.NuclearUltrapower.tensorIsometry

tensorJacobianCoefficient def

Store a Jacobian factor as an actual smooth positional coefficient. · OAI.ClosedSurfaceR4.JetPolynomial.tensorJacobianCoefficient

tensorKernel def

Independent tensor of two kernels. · OAI.MaxCutGames.Repetition.tensorKernel

tensorKernel def

Independent tensor of two kernels. · OAI.UniqueGamesTheorem.Repetition.tensorKernel

tensorKraus def

Tensor products of the single-site Kraus operators, with the full Kraus index. · OAI.GAD.tensorKraus

tensorLabelCount def

The multiplicity of a label in an elementary tensor. · OAI.KadisonSimilarity.tensorLabelCount

tensorLeafCoefficients def

Level zero is the common root mark. Positive levels encode the corresponding ancestor prefix of the labeled leaf. · OAI.InvariantIsing.tensorLeafCoefficients

tensorLeafState def

The actual sum of the root and all Gaussian marks on a sampled leaf. · OAI.InvariantIsing.tensorLeafState

tensorLeft def

Tensoring with a fixed sheaf, on both objects and morphisms. · OAI.ActualSheafTensor.tensorLeft

tensorLeftCompOverIso def

The restriction isomorphism for tensor sheaves is natural in the second variable. This is the exactness-descent bridge for Cartier twisting. · OAI.ActualSheafTensor.tensorLeftCompOverIso

tensorLeftHom def

The two commuting unital coordinate tensor embeddings. · OAI.NuclearUltrapower.tensorLeftHom

tensorLeftIso def

An isomorphism of sheaves induces a natural tensor isomorphism. · OAI.ActualSheafTensor.tensorLeftIso

tensorLeftMap def

Tensor a morphism of module sheaves on the left. · OAI.ActualSheafTensor.tensorLeftMap

tensorLeftSequence def

First tensor factor, represented uniformly at every coordinate. · OAI.NuclearUltrapower.tensorLeftSequence

tensorLineCore def

Tensor product of two actual line cores. Its transition functions are the products of the two line transition functions. · OAI.Kaplansky.tensorLineCore

tensorLinePair def

Multiplication in core fibers is the tensor pairing. This formula is compatible with every tensor-product chart. · OAI.Kaplansky.tensorLinePair

tensorMap abbrev

Tensor an actual map of the underlying right-module complexes. · OAI.LittleFinitistic.SelectedUnderlyingHomology.tensorMap

tensorMapBetween def

Heterogeneous completion of the Hilbert tensor functor. · OAI.NuclearUltrapower.tensorMapBetween

tensorMatrix def

Tensor product in the computational basis. · OAI.QAC.tensorMatrix

tensorMatrixUnit def

A matrix unit with row a and column b. · OAI.KadisonSimilarity.tensorMatrixUnit

tensorModel def

Explicit identification of this coordinate model with the source tensor product. · OAI.LittleFinitistic.OrdinarySimulation.TerminalTensor.tensorModel

tensorMoment def

The moment functional of independent line variables, defined algebraically so only the finitely many moments actually used need be integrable. · OAI.CoordinateSweeps.MomentExpansion.tensorMoment

tensorNamespacedPlaneCoefficients def

The actual derivative coefficient map uses precisely the same Gaussian namespaces and ancestor tags as the Hamiltonian. · OAI.InvariantIsing.tensorNamespacedPlaneCoefficients

tensorNamespaceTag def

Namespace zero contains the linear field; namespace j+1 contains precisely the jth spectral tensor at all ancestor levels. · OAI.InvariantIsing.tensorNamespaceTag

tensorNatIso def

The residue projection comparison is natural on all actual left A-modules. · OAI.LittleFinitistic.PDResidueProjection.tensorNatIso

tensorNetwork def

Direct concatenation prepares K fixed blocks. No branch-indexed truth table is allocated; the count is the sum of the supplied K finite networks. · OAI.ExactQuantumFactoring.tensorNetwork

tensorOpenRestrictIso def

The tensor sheaf commutes with open-scheme restriction. · OAI.ActualSheafTensor.tensorOpenRestrictIso

tensorOpenRestrictNatIso def

Restriction of tensoring is natural, with the coefficient changes. · OAI.ActualSheafTensor.tensorOpenRestrictNatIso

tensorOperator def

Algebraic finite-rank operators represented by Hilbert tensors. · OAI.NuclearUltrapower.tensorOperator

tensorOperatorMatrix def

This matrix has scalar coefficients in the fixed factor; the first factor may be infinite-dimensional. · OAI.NuclearUltrapower.tensorOperatorMatrix

tensorOrdinary def

The resolved ordinary tensor object, in the native derived category retaining both outside actions. The resolution, its quasi-isomorphism, finite support and right projectivity are fields of Q. · OAI.LittleFinitistic.BoundedBimoduleResolution.tensorOrdinary

tensorOverIso def

The tensor sheaf commutes with restriction to every slice site. · OAI.ActualSheafTensor.tensorOverIso

tensorPacket def

One completed multiplicity word and one completed endpoint word. · OAI.KadisonSimilarity.tensorPacket

tensorPathProfile def

The root has variance h 0; each child adds the next field increment. For a tree monomial of degree zero the entire variance is at the root. · OAI.InvariantIsing.tensorPathProfile

tensorPermutationIsometry def

Tensor-position permutations preserve the Hilbert inner product. · OAI.KadisonSimilarity.tensorPermutationIsometry

tensorPermutationLinear def

The algebraic map permuting tensor positions. · OAI.KadisonSimilarity.tensorPermutationLinear

tensorPermutationMatrix def

Its matrix uses the same successor convention as permutationContraction. · OAI.KadisonSimilarity.tensorPermutationMatrix

tensorPiLinearEquiv def

The fixed-factor tensor/direct-sum identification, with its actual Hilbert norm. · OAI.NuclearUltrapower.tensorPiLinearEquiv

tensorPolynomialCoordinatePullback def

Apply the tensor coordinate transformation coefficient by coefficient to a finite polynomial in the perturbation parameter. · OAI.ClosedSurfaceR4.JetPolynomial.Perturbation.tensorPolynomialCoordinatePullback

tensorProjection def

The canonical projection morphism for the tensor sheaves. · OAI.ActualSheafTensor.tensorProjection

tensorProjectionIso def

The canonical projection-formula isomorphism for any line bundle. · OAI.ActualSheafTensor.tensorProjectionIso

tensorProjectionUnitTargetIso def

Target trivialization for the projection formula with unit coefficients. · OAI.ActualSheafTensor.tensorProjectionUnitTargetIso

tensorReadBallConstant def

The atlas-only enlargement needed when a global encoded tensor ball is read in any one chart. It is independent of reference tensors and all orders. · OAI.ClosedSurfaceR4.FiniteOrderSmoothing.SmoothingAtlas.tensorReadBallConstant

tensorResolution def

Balanced X tensor of the actual free projective resolution of N. · OAI.DerivedBar.BimoduleTensor.tensorResolution

tensorResolutionIso def

The explicit two-sided bar is isomorphic to genuine balanced tensor of the free resolution. · OAI.DerivedBar.BimoduleTensor.tensorResolutionIso

tensorRestoreTransition def

The three-coordinate form of the genuine tensor-bundle transition, including the single cutoff used by restoration. · OAI.ClosedSurfaceR4.FiniteOrderSmoothing.SmoothingAtlas.tensorRestoreTransition

tensorRestrictionEquiv def

The module-theoretic pullback of a dual to a principal divisor is the canonical quotient of that dual. · OAI.DualCartierRestriction.tensorRestrictionEquiv

tensorRestrictionQuotientEquiv def

Quotient-linearity of the affine pullback/quotient identification. · OAI.DualCartierRestriction.tensorRestrictionQuotientEquiv

tensorRootMark def

Reindex independent standard Euclidean coordinates, then apply the coordinate square roots of the prescribed root variances. · OAI.InvariantIsing.tensorRootMark

tensorRootTreeLaw def

Independent common Gaussian root and marked cascade. · OAI.InvariantIsing.tensorRootTreeLaw

TensorScalars abbrev

Scalars at the same universe as the Hilbert space. · OAI.KadisonSimilarity.TensorScalars

tensorSchemeRestrictIso def

The tensor sheaf commutes with open-scheme restriction. · OAI.ActualSheafTensor.tensorSchemeRestrictIso

tensorSectionsEquiv def

Sections over every vertex subset commute with flat extension of scalars for a finite moment graph. No freeness or BMP character input is used. · OAI.KLInvariance.MomentGraph.Sheaf.tensorSectionsEquiv

tensorSeries def

The tensor product of finitely supported one-dimensional coefficient series, represented on its literal vector-valued exponents. · OAI.TraceReconstruction.tensorSeries

tensorSharedCoordinateProduct def

Only the shared ancestor increments remain after the independent suffixes have been integrated. At the branching level the integrand is the product of actual backward gradients. · OAI.InvariantIsing.tensorSharedCoordinateProduct

tensorSlice def

The slice extends uniquely to the completed Hilbert tensor product. · OAI.KadisonSimilarity.tensorSlice

tensorSlice def

Apply the second game to a fixed first question and first answer. The square-root factor changes its weighted output space into counting measure. · OAI.MaxCutGames.Repetition.ProjectionKernel.tensorSlice

tensorSlice def

Apply the second game to a fixed first question and first answer. The square-root factor changes its weighted output space into counting measure. · OAI.UniqueGamesTheorem.Repetition.ProjectionKernel.tensorSlice

tensorSlotAction def

The induced action on all covariant slots. · OAI.PositiveEinsteinFour.Geometry.tensorSlotAction

tensorSymmetrizer def

Intrinsic symmetrization of a bilinear form. · OAI.ClosedSurfaceR4.FiniteOrderSmoothing.tensorSymmetrizer

tensorTiltedPairPathMean def

The actual tilted two-leaf Gaussian path expectation, with the spin residual integrated in the finite Ising terminal. · OAI.InvariantIsing.tensorTiltedPairPathMean

tensorToSymPow def

Multiplication of tensors, with codomain restricted to degree n. · OAI.Problem346.tensorToSymPow

tensorToSymPowRaw def

The tensor-power presentation of the homogeneous symmetric power. · OAI.Problem346.tensorToSymPowRaw

tensorTotal def

Actual signed tensor with the manuscript's Q_T on an arbitrary complex of BC-modules. · OAI.LittleFinitistic.OrdinarySimulation.TensorResolution.tensorTotal

tensorTotalExplicitIso def

Resolving the whole input and applying balanced tensor gives precisely the explicit two-sided bar total. Finite diagonal sums suffice; X need not be flat. · OAI.DerivedBar.BimoduleTensor.tensorTotalExplicitIso

tensorTotalFunctor def

Resolve the entire input complex before applying the actual balanced tensor functor. · OAI.DerivedBar.tensorTotalFunctor

tensorTotalIterate def

Repeated resolution and balanced tensor, proceeding from the right as in Section 6. The identification with the independently signed literal multibar is a separate obligation. · OAI.DerivedBar.tensorTotalIterate

tensorTranspose def

Transpose a covariant tensor by interchanging its two arguments. · OAI.ClosedSurfaceR4.FiniteOrderSmoothing.tensorTranspose

tensorTrialRadius def

A fixed global trial radius for a prescribed local geometric margin. · OAI.ClosedSurfaceR4.FiniteOrderSmoothing.SmoothingAtlas.tensorTrialRadius

tensorUncurry def

Uncurrying is compatible with restriction maps. · OAI.CoherentDual.tensorUncurry

tensorUncurryApp def

Evaluation gives the inverse to currying on each object. · OAI.CoherentDual.tensorUncurryApp

tensorUnitaryAction def

Exact completed tensor action used in the source uniform spectral bound. · OAI.NuclearUltrapower.tensorUnitaryAction

tensorUnitLeftIso def

The tensor-with-unit functor is naturally the identity. · OAI.ActualSheafTensor.tensorUnitLeftIso

tensorVarianceProfile def

Variances shared by the coordinates of each spectral monomial. · OAI.InvariantIsing.tensorVarianceProfile

tensorVector def

The map inserting a fixed vector in the second tensor leg. · OAI.KadisonKastler.tensorVector

tensorVectorSlice def

The vector slice is concrete compression by the tensor-vector map. · OAI.KadisonKastler.tensorVectorSlice

tensorVectorSliceCP def

The concrete vector slice is completely positive. · OAI.KadisonKastler.tensorVectorSliceCP

tensorVertexEquiv def

Evaluation uses the actual balanced tensor over BC, not only over C. · OAI.LittleFinitistic.OrdinarySimulation.BimoduleO.tensorVertexEquiv

tensorVertexSpaceEquiv def

The BC vertex space is the same actual subspace of O as the C vertex space. · OAI.LittleFinitistic.OrdinarySimulation.BimoduleO.tensorVertexSpaceEquiv

tensorWord def

The elementary word determined by its letters. · OAI.KadisonSimilarity.tensorWord

tensorWordLiftLinear def

The multilinear universal property of the iterated algebraic tensor word. · OAI.KadisonSimilarity.tensorWordLiftLinear

tensorWordMultilinear def

The elementary word map, with its multilinearity bundled. · OAI.KadisonSimilarity.tensorWordMultilinear

TensorWordSpace structure

Bundling keeps the vector-space structures available during the recursion. · OAI.KadisonSimilarity.TensorWordSpace

tensorWordSpace def

Right-associated algebraic Hilbert tensor powers, including degree zero. · OAI.KadisonSimilarity.tensorWordSpace

tent def

A continuous probability of passing a rare interval test. It vanishes outside the interval, and is rounded to deterministic tests on each finite input alphabet before constructing any final observation. · OAI.SquareRootDegree.SmoothGaussian.tent

Term inductive

Concrete words used at seams, corners, coarsened swaps, split vertices and surface relations. A block is an direct sum, not an abstract map between unnamed quotient modules. All inverses are the preserved GL inverse variables. · OAI.IntegralCharacterVarieties.MatrixExpression.Term

Term inductive

Finite expressions over the cells of the small diagram. · OAI.Thomason.Construction.StrictOmegaCategory.ColimitConstruction.Term

term def

The augmented bar terms: term 0 is N; term (n+1) is A ⊗ term n. · OAI.DerivedBar.term

term def

The exact Peano--Baker iterates, with oriented integrals on the entire real line. · OAI.HarmonicCounterexample.LinearODE.term

termCast def

Explicit degree transport; it changes no tensor factor or action. · OAI.DerivedBar.ParityBridge.termCast

termD def

Actual parameter-jet Peano–Baker terms. · OAI.HarmonicCounterexample.LinearODE.termD

termEquiv def

The canonical balanced quotient comparison retains the outside B action. · OAI.LittleFinitistic.BimoduleTotalPresentation.termEquiv

termEquiv def

No extra support assumptions: P is required to vanish only above a+l here. · OAI.LittleFinitistic.OrdinarySimulation.FullTensorTerm.termEquiv

Terminal def

An internal terminal configuration has no forward transition. · OAI.FiniteAlgebraicEnvelopes.Compiler.Terminal

Terminal structure

The endpoint projection has a specified bounded inverse on the actual fast space. It need not annihilate the slow space. · OAI.SmoothObstruction.FiniteGreen.Splitting.Terminal

terminal def

The unique terminal factor from a node to a. · OAI.PartitionConsistency.ProbeAlgebra.Path.terminal

terminal def

The single actual terminal sequence fixed for this constructed tree. · OAI.ReflexiveFixedPoints.SwitchingTreeData.terminal

TerminalAddress abbrev

Terminal addresses include the leaf name, preventing collisions between tables. · OAI.Problem310.FiniteTableModel.TerminalAddress

terminalAddress def

The terminal coordinate read along the selected leaf. · OAI.Problem310.FiniteTableModel.terminalAddress

terminalAugmentation def

The terminal character extended along the B-factor. · OAI.LittleFinitistic.OrdinarySimulation.terminalAugmentation

terminalAugmentationMap def

A genuine chain map to the terminal simple placed in degree l. · OAI.LittleFinitistic.OrdinarySimulation.ChainAlgebra.terminalAugmentationMap

TerminalBound def

Every terminal continuation is within B of the same closed-prefix value. · OAI.SignedAssembly.TerminalBound

terminalCharacter def

The terminal simple right module is the character at vertex l. · OAI.LittleFinitistic.OrdinarySimulation.ChainAlgebra.terminalCharacter

terminalCoefficient def

The fixed coefficient in the constructive terminal length estimate. · OAI.Problem337.TerminalAssembly.terminalCoefficient

TerminalConditions def

The first coordinate is s; the equation is d*s=t*e. · OAI.TotientAsymptotic.TerminalConditions

terminalDefect def

The unweighted displacement of all maps strictly above terminal depth. · OAI.ReflexiveFixedPoints.TreeEvaluation.terminalDefect

terminalEndpoints def

Each terminal table uses its leaf's incoming-edge endpoint. · OAI.Problem310.FiniteTableModel.terminalEndpoints

TerminalFlow structure

A finite path flow with a unique final edge for each terminal top. The balance law includes both continuing and terminating mass. · OAI.LipschitzHilbert.TerminalFlow

terminalLaw def

A finite table of independent Bernoulli terminal entries. · OAI.Problem310.ConcreteLabels.terminalLaw

terminalLeave def

A sector cannot have any omega except its last letter. · OAI.FiniteAlgebraicEnvelopes.Compiler.Path.terminalLeave

TerminalProfile def

The fixed terminals are exactly the codomains of the internal domain edges of the logic SCC. This includes the chosen logical bottom. · OAI.PureTypeSystem.Labelled.TerminalProfile

terminalProjection def

Fixed endpoint-bin orthogonal projection, together with its genuine inverse on the moving fast space. Empty endpoint spaces are included. · OAI.SmoothObstruction.FiniteGreen.Splitting.terminalProjection

terminalProjectiveComplex def

The source's projective complex, extended by zero after degree l. · OAI.LittleFinitistic.OrdinarySimulation.ChainAlgebra.terminalProjectiveComplex

terminalRange def

The terminal numerators not covered by the binary factor. · OAI.Problem337.GeometricRandomRanges.terminalRange

TerminalRun structure

An actual raw terminal execution, with arbitrary remaining state and tapes. · OAI.MaxCutGames.Foundations.Complexity.MachineCanonicalOutput.TerminalRun

TerminalRun structure

An actual raw terminal execution, with arbitrary remaining state and tapes. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineCanonicalOutput.TerminalRun

terminalRun def

The terminal witness is the checked full lazy-table execution. The wrapper then drains the retained original input rather than assuming it disappeared. · OAI.MaxCutGames.Foundations.Complexity.MachineLazyTableRuntime.terminalRun

terminalRun def

The raw terminal witness is the actual three-phase execution. No phase trace, output-size bound, or complexity assertion is an input hypothesis. · OAI.MaxCutGames.Foundations.Complexity.MachinePaddedOverlayRuntime.terminalRun

terminalRun def

The terminal witness is the checked full lazy-table execution. The wrapper then drains the retained original input rather than assuming it disappeared. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineLazyTableRuntime.terminalRun

terminalRun def

The raw terminal witness is the actual three-phase execution. No phase trace, output-size bound, or complexity assertion is an input hypothesis. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachinePaddedOverlayRuntime.terminalRun

TerminalSimple def

W is the genuine terminal right simple, via the terminal character. · OAI.LittleFinitistic.OrdinarySimulation.TerminalSimple

TerminalTensor def

Coordinate model for B tensor_k W, with its specified terminal right action. · OAI.LittleFinitistic.OrdinarySimulation.TerminalTensor

terminalTower def

Auxiliary levels contain no physical spin randomness. The last level is exactly the Gibbs sum under its uniform spin prior. · OAI.DilutedSpinGlass.KernelTower.terminalTower

termIso def

Additive functors preserve the zero components added by integer extension. · OAI.LittleFinitistic.IntegerExtensionFunctor.termIso

termIso def

The total complex has exactly the three finite columns. · OAI.LittleFinitistic.RectificationTotal.termIso

termMap def

Functorial map on every augmented bar term. · OAI.DerivedBar.termMap

termMem def

Membership is independent of representatives because first-order equality has its actual substitution rule. · OAI.PartitionConsistency.Completeness.termMem

termPatternIso def

Exterior actions agree as well as the underlying field-tensor modules. · OAI.DerivedBar.Multibar.termPatternIso

termSetoid def

Quotient of the natural-number constant codes by their syntactically provable equality. · OAI.PartitionConsistency.Completeness.termSetoid

ternaryAsPolynomialInY def

Regard the actual ternary ring as polynomials in Y with binary U,Z coefficients. · OAI.Nagata.Workers.W24.ternaryAsPolynomialInY

ternaryEulerFactor def

The standard ternary local Euler factor, extended by one at nonprimes. · OAI.Problem337.ternaryEulerFactor

TernaryMangoldtAsymptotic def

The classical ternary Mangoldt asymptotic, stated as a proposition to keep the genuinely unproved analytic input explicit. Its main term is the singular series times the area of the positive fixed-sum simplex, namely u²/2. · OAI.Problem337.TernaryMangoldtAsymptotic

ternarySign def

Literal signs: 0 is negative, 1 is zero, and 2 is positive. · OAI.QuantitativeVanDerWaerden.ternarySign

terr def

error < 1e-24 through terms n=0:24. · OAI.GeneralMahler.Cert.IV.terr

Test def

Compactly supported smooth real tests in the open set. · OAI.Bernoulli.Test

test def

Segment testing as a Euclidean-valued linear map. · OAI.DiamondDistortion.SegmentSystem.Family.test

test def

Equivalent form of dyadic completion for a retrospective, data-based test of the guessed string. No canonical output is used during preparation. · OAI.ExactQuantumFactoring.Completion.test

TestBound def

Squared bound on every disjoint family of segment tests. · OAI.ForestSpace.TestBound

testBundleZeroTotalHomeomorph def

A global linear bundle chart for E₀. This proves topological triviality, not just that its fibers have the desired dimension. · OAI.Kaplansky.testBundleZeroTotalHomeomorph

testCoefficient def

The slot tuples, two queries, and noise tapes per clause tuple. The empty-context branch uses the global dummy and adds no random tape. · OAI.MaxCutGames.Foundations.Hastad.SourceBounds.testCoefficient

testCoefficient def

The slot tuples, two queries, and noise tapes per clause tuple. The empty-context branch uses the global dummy and adds no random tape. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceBounds.testCoefficient

TestCompletion abbrev

The completion used in the test-space constructions. · OAI.BoundedTreePotentials.TestCompletion

testCut def

Value of the usual correlated long-code Max-Cut test, before aggregating its finite rational outcomes into graph weights. · OAI.OptimalMaxCut.LongCode.Game.testCut

testedClass def

A computable, uniquely comparable coding of the tested class for the auxiliary monoid. No rational-number coding is used as an oracle. · OAI.FiniteAlgebraicEnvelopes.PartialAffine.testedClass

TestedStack inductive

Tests follow the exact normal successive types, not other typings. · OAI.PureTypeSystem.Candidates.TestedStack

testEmbedding def

Real continuous test integrals embed probability laws into a locally convex product space. This permits direct reuse of Mathlib's Krein--Milman lemma, without an unproved ergodic-decomposition input. · OAI.SmoothObstruction.CompactInvariant.testEmbedding

testEndpoint def

The test endpoint map is conjugate-linear because matrix star is. · OAI.KadisonSimilarity.MatrixHilbertStage.testEndpoint

testers def

The tester set uses the domain Read precisely at this primary component. · OAI.PureTypeSystem.ObservationValues.ProductInstance.testers

testEval def

Evaluation on the actual hypersurface algebra along the test line. · OAI.StableCoordinate.SourceFiltration.testEval

TestFamily structure

A coefficient test family, before imposing any particular tree budget. · OAI.BoundedTreePotentials.TestFamily

testFunctional def

A genuine continuous linear scalar functional in the test-vector slot. · OAI.KadisonKastler.ScalarInvariantMean.testFunctional

testGradients def

Actual gradients of compactly supported smooth tests, in L2. · OAI.Bernoulli.HarmonicReplacement.testGradients

TestGraph def

A distributional graph described by test identities. Instantiating it with weak differentiation requires constructing its actual test functionals. · OAI.TriangularBilliards.Analytic.TestGraph

TestHyperplaneLine abbrev

The actual dual hyperplane-line candidate on the projection product. · OAI.Kaplansky.TestHyperplaneLine

TestIndex abbrev

Each event keeps its actual period, rational step and recorded word. · OAI.QuantitativeVanDerWaerden.EligiblePatternTests.TestIndex

testJetSubmodule def

Compactly supported interior test jets, with the actual support condition. · OAI.Elasticity.testJetSubmodule

testLineChartIndex def

The zero label denotes L₀; positive labels denote hyperplane pullbacks. · OAI.Kaplansky.testLineChartIndex

testLineCoreHyperplaneHomeomorph def

The positive-labelled line is topologically the preserved hyperplane pullback, including its total-space topology. · OAI.Kaplansky.testLineCoreHyperplaneHomeomorph

testMap def

The reverse-Taylor test on bounded-degree polynomials multiplies each Taylor coefficient by t^a and places it in output degree M-1-a. · OAI.RationalHitting.SmallExpansion.testMap

testModule abbrev

Actual finite-dimensional modules over the one fixed A; their projective dimensions are the remaining homological conclusion. · OAI.LittleFinitistic.SelectedOrdinaryModules.testModule

testNextStageTotalHomeomorph def

The regrouped target and E_{j+1} are isomorphic as actual topological bundles, not merely vector spaces of the same rank. · OAI.Kaplansky.testNextStageTotalHomeomorph

testNextStageUnitary def

The replacement-plus-fresh target is the standard next stage as an ordinary Hermitian bundle, so normalization survives this regrouping. · OAI.Kaplansky.testNextStageUnitary

testPairingTest def

Independent label choices on test-test pairs. · OAI.KadisonSimilarity.testPairingTest

testPairLabels def

The analogous tuple on the test side. · OAI.KadisonSimilarity.testPairLabels

testPartial def

The i-th coordinate derivative of a smooth test function. · OAI.CenteredDiskEndpoint.testPartial

testPartial def

The i-th coordinate derivative of a smooth test function. · OAI.SignedDisk.CenteredDiskEndpoint.testPartial

testPn def

finite arithmetic at upper cells · OAI.GeneralMahler.SCal.testPn

testPoint def

A line of points on H = 0, varying only the coordinate F. · OAI.StableCoordinate.SourceFiltration.testPoint

testPrefixDimension def

Exact prefix dimensions, retaining every later occurrence through R. · OAI.Kaplansky.testPrefixDimension

testProgram def

A three-instruction output test after a structured command, followed by a literal accepting or rejecting stack-machine halt. · OAI.WLIdentification.StackModel.Command.testProgram

testSequenceStep def

All 24 normalized bundle maps for all levels, constructed solely from m>0. Both forward maps and their genuine adjoints are continuous over the single full projective product Y. · OAI.Kaplansky.testSequenceStep

TestTape abbrev

Two full Boolean query tables and one finite noise entry per right index. · OAI.MaxCutGames.Foundations.Hastad.SourceTape.TestTape

TestTape abbrev

Two full Boolean query tables and one finite noise entry per right index. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceTape.TestTape

testValues def

All squared disjoint-segment tests. Quantifying over finite index types is equivalent to families S₁,...,Sₘ; a zero-sized family adds only zero to the set of test values. · OAI.ForestSpace.testValues

TestVector def

c00(I) with its actual test norm, not the ambient Finsupp topology. · OAI.BoundedTreePotentials.TestVector

TestVector def

Finite coordinate vectors with the segment norm (a type synonym prevents clashes with any other norm on finitely supported functions). · OAI.ForestSpace.TestVector

tetrahedron def

The regular edge-2 tetrahedron in physical coordinates. · OAI.RuledApproximation.tetrahedron

tetraVertices def

The four vertices of the regular tetrahedron. · OAI.TriangularCovering.tetraVertices

TExpr inductive

Finite tree expressions are uniformly linear in the size of their argument; no arithmetic or lookup oracle is concealed in an expression. · OAI.ParityWL.Complexity.TExpr

text def

The letter removed by the down step at each integer start coordinate. · OAI.Superstring.Hierarchical.Threading.text

TextOccurs def

An occurrence in a two-sided text, retaining its integer origin. · OAI.Superstring.Hierarchical.TextOccurs

textWord def

The finite integer window of a two-sided text. · OAI.Superstring.Hierarchical.textWord

thermalDefect def

The L1 thermal defect for the literal score of all selected marks. · OAI.DilutedSpinGlass.KernelTower.thermalDefect

theta def

A concrete permissible choice in source04/06. R is still chosen last. · OAI.CoordinateSweeps.SparseScalar.theta

theta def

The actual theta on E0 in the primary's source Verdier quotient. · OAI.LittleFinitistic.SourceQuotientAction.theta

theta def

The one-draw success probability for the prescribed left count. · OAI.LowSpaceSubsetSum.SeparatorCheck.theta

theta def

The generator indexed by i. · OAI.PixtonChow.theta

thetaCosetEquiv def

The eight left cosets are precisely the three translation residues. · OAI.NuclearUltrapower.Elementary.thetaCosetEquiv

thetaCosetEquiv def

The eight left cosets are precisely the three translation residues. · OAI.NuclearUltrapower.thetaCosetEquiv

thetaDeriv def

The derivative profile of theta. · OAI.ThompsonNonamenability.Analytic.thetaDeriv

thetaDiskMajorant def

An explicit summable majorant for all integer indices on a fixed disk. · OAI.Nagata.W22.thetaDiskMajorant

thetaExponent def

Exponent of the theta parameter, with a = K - U. · OAI.Nagata.W07.thetaExponent

thetaFrameCenter def

Frame center forcing the source interval's left endpoint to be one half. · OAI.Nagata.W02.thetaFrameCenter

thetaIndices def

The consecutive integers used to index a theta basis. · OAI.Nagata.W09.thetaIndices

thetaJacobiArgument def

The affine first argument of Jacobi theta for the source normalized series. · OAI.Nagata.W08.thetaJacobiArgument

thetaJacobiParameter def

The upper-half-plane period parameter corresponding to the real nome τ. · OAI.Nagata.W08.thetaJacobiParameter

thetaJetMatrix def

The actual mixed-derivative matrix of all three scalar coefficient branches. The marks are offsets aᵢ = x₀ - xᵢ in (0,1), as in §03's product limit; they are not the original point exponents xᵢ. · OAI.Nagata.Workers.W11.thetaJetMatrix

thetaMargin def

Distance of the selected exponent from the two endpoints. · OAI.Nagata.W07.thetaMargin

thetaProduct def

The negative-index product is written with index n+1, starting at zero. · OAI.Nagata.W21.thetaProduct

thetaProductSection def

A translate of the source theta product defines a section with multiplier -a. · OAI.Nagata.W08.thetaProductSection

thetaScalarCombination def

The literal finite scalar expression for a coefficient vector. · OAI.Nagata.Workers.W11.thetaScalarCombination

thetaScalarCombinationLinear def

The actual finite scalar expansion is linear in its coefficients. · OAI.Nagata.Workers.W11.thetaScalarCombinationLinear

Thick inductive

The actual thick closure: zero, isomorphisms, shifts, extensions, and retracts. · OAI.LittleFinitistic.ThickEvaluation.Thick

thicken def

A genuine full-dimensional approximation of an arbitrary relative gauge. The sum regularization avoids any constant-rank assumption. · OAI.PettyProjection.RelativeGauge.thicken

thickening def

An open old-space neighborhood thickened by the outer half of each disk. Membership is tested by radial retraction of the disk to its attaching sphere. · OAI.GroupRingTopology.Attachment.thickening

Thicket structure

Given number of beats and axes, tape of trees (exact-width layer schedules + metadata). · OAI.PowerSaving.RAM.Bench.Thicket

ThielDiffuseCornerHaarInput def

The local specialization of H. Thiel, Diffuse traces and Haar unitaries, arXiv:2009.06940v3, Proposition 7.2 and the following paragraph (printed p.20). A nonzero corner of a diffuse von Neumann algebra is diffuse; its normalized normal … · OAI.KadisonSimilarity.ThielDiffuseCornerHaarInput

ThinnedWindows structure

Successively thinned windows. The threshold indexed by n is used strictly after n. · OAI.SeparableQuotient.ActualSpace.ThinnedWindows

thinParameter def

Strictly subunit parameters approaching arbitrary permitted parameters. · OAI.HyperedgeComparison.thinParameter

ThinPaths def

Every path component is simply connected; no connectedness is imposed. · OAI.EilenbergGanea.ForestCover.ThinPaths

thinRescalingEquiv def

The rescaling is genuinely an invertible linear image, not an abstract limit operation. · OAI.AffineBernstein.thinRescalingEquiv

thinSlice def

Thin slices and rescalings before reordering the k+1 base coordinates. · OAI.AffineBernstein.thinSlice

thinSupplyBase def

Prime-prefix supply with an arbitrary logarithmic coefficient. · OAI.Problem337.thinSupplyBase

ThinTree def

`ThinTree C k G T` requires `T` to be a spanning tree and to use at most a $C/k$ fraction of the whole graph's edges across every nontrivial vertex cut. · OAI.AlgorithmicThinTrees.ThinTree

ThinTriangles def

A uniform thinness constant for geodesic triangles in the Cayley graph. · OAI.CannonRelease.CayleyData.ThinTriangles

ThinTriangles def

Literal vertex thin triangles. · OAI.HyperbolicObstruction.VertexPath.ThinTriangles

thirdCoordinate def

The final free coordinate is the common residual in the three off-diagonal quaternion coordinates. · OAI.PositiveEinsteinFour.Bianchi.thirdCoordinate

thirdProjectionMap def

The third local map carries the single Fourier normalization factor. · OAI.MatrixMultiplication.AuxiliarySeparation.thirdProjectionMap

thirdQuery def

The third queried Boolean table is g * (f ∘ π) * μ in sign notation. · OAI.MaxCutGames.Foundations.Hastad.thirdQuery

thirdQuery def

The third queried Boolean table is g * (f ∘ π) * μ in sign notation. · OAI.UniqueGamesTheorem.Foundations.Hastad.thirdQuery

thirdSeparationMap def

The third local Fourier map followed by its possibly negative weight. · OAI.MatrixMultiplication.AuxiliarySeparation.thirdSeparationMap

thirdWeight def

Weight minus one on the middle third-leg interval, zero elsewhere. · OAI.MatrixMultiplication.AuxiliarySeparation.Sector.thirdWeight

thirdWeight def

Weight on the third tensor leg. · OAI.MatrixMultiplication.AuxiliarySeparation.thirdWeight

thomasonAdjunction def

Twice subdivision followed by categorification is genuinely adjoint to Ex² of the constructed Street nerve. No strict-category model structure is used. · OAI.Thomason.Construction.NormalizedSimplexIdentities.thomasonAdjunction

thomasonCofibrations def

The prescribed cofibrations, using the lifting definition of the paper. · OAI.Thomason.Construction.NormalizedSimplexIdentities.thomasonCofibrations

thomasonFibrations def

The prescribed fibrations are detected by actual horn lifting. · OAI.Thomason.Construction.NormalizedSimplexIdentities.thomasonFibrations

thomasonGeneratingAcyclicCofibrations def

Twice-subdivided horn generators, categorified by the actual left adjoint. · OAI.Thomason.Construction.NormalizedSimplexIdentities.thomasonGeneratingAcyclicCofibrations

thomasonGeneratingCofibrations def

Twice-subdivided boundary generators, categorified by the actual left adjoint. · OAI.Thomason.Construction.NormalizedSimplexIdentities.thomasonGeneratingCofibrations

thomasonLeft def

The left adjoint in the stated Thomason adjunction. · OAI.Thomason.Construction.NormalizedSimplexIdentities.thomasonLeft

thomasonLeftLocalizer def

Brown preservation makes the actual left adjoint a morphism of localizers. · OAI.Thomason.Construction.thomasonLeftLocalizer

ThomasonModelResult structure

The main conclusion in the usual derived formulation of Quillen equivalence. Weak equivalences, fibrations, and both generators are fixed by the actual Street nerve and twice-subdivided left adjoint. · OAI.Thomason.Construction.ThomasonModelResult

thomasonRight def

The right adjoint candidate Ex² N on the actual strict category. · OAI.Thomason.Construction.NormalizedSimplexIdentities.thomasonRight

thomasonRightLocalizer def

The morphism of localizers underlying the actual Thomason right adjoint. · OAI.Thomason.Construction.thomasonRightLocalizer

thomQuotientZeroSection def

The actual zero section into the disk/sphere quotient. · OAI.Kaplansky.thomQuotientZeroSection

thomZeroHomologyMap def

The integral homology morphism of the genuine Thom zero section. Its definition supplies no nonvanishing, Thom class, or Chern-class axiom. · OAI.Kaplansky.thomZeroHomologyMap

Threading structure

Down-edge tokens threaded through the acyclic up-edge runs. · OAI.Superstring.Hierarchical.Threading

threading def

Realize the base down multiset as a periodic circulation on actual vertices. · OAI.Superstring.Counts.Data.threading

ThreeCopyCode structure

Five odd segments and the original source channel for tagged copies. · OAI.PureTypeSystem.Labelled.ThreeCopyCode

threeMapSystem def

The three named translations, with all positivity and normalization supplied explicitly. · OAI.EntropyRateDimension.Extensions.threeMapSystem

threePolarRadii def

Source, reverse anchor, and within-tile radii, in that order. · OAI.PolycyclicRecognition.threePolarRadii

threePrimeArithmeticFactor def

The normalized actual local unit-triple count at prime moduli, with one at nonprime indices. The inactive branch makes this a total natural-indexed function without imposing a finiteness instance on ZMod 0. · OAI.Problem337.threePrimeArithmeticFactor

threePrimeLocalFactor def

The local density relative to the heuristic density of three unit classes. The prime hypothesis is used when identifying this with a singular-series factor; the finite expression is defined for every nonzero modulus. · OAI.Problem337.threePrimeLocalFactor

threePrimeLocalPairs def

Two-coordinate parametrization of nonzero residue triples with sum u: the third coordinate is u - x - y. · OAI.Problem337.threePrimeLocalPairs

threePrimeLocalTriples def

The literal ordered triples underlying the local density. · OAI.Problem337.threePrimeLocalTriples

threePrimeNonzeroResidues def

Nonzero residue classes modulo p. For prime p, these are exactly its unit classes. · OAI.Problem337.threePrimeNonzeroResidues

threePrimeOddPerturbation def

The odd-prime local perturbation in the ternary Goldbach singular series. Index n corresponds to the integer n + 3; nonprimes contribute zero. · OAI.Problem337.threePrimeOddPerturbation

threePrimeSingularPartial def

The finite canonical singular product, including the factor at two. · OAI.Problem337.threePrimeSingularPartial

threePrimeSingularSeries def

The local factor at two, with all the odd-prime factors following it. · OAI.Problem337.threePrimeSingularSeries

ThreeQuota def

The three-edge fan quota, expressed without choosing an orientation for its fields. With distinct successive fields this says at most two per fan. · OAI.HyperbolicObstruction.CoreGeometry.ThreeQuota

threeSpaceToHyperplane def

The hyperplane perpendicular to a nonzero vector in four-space is isometrically three-dimensional. · OAI.ClosedSurfaceR4.SphericalStart.threeSpaceToHyperplane

ThreeVertexAlgebra abbrev

Coordinates by the six possible path endpoints: three vertices, 0-to-1 arrows, 1-to-2 arrows, and length-two paths modulo relations. · OAI.LittleFinitistic.ThreeVertexAlgebra

ThreeVertexConnected def

Deleting any set of at most two vertices leaves a connected graph of order at least two. · OAI.Barnette.ThreeVertexConnected

threshold def

No selected map or convex coefficient is an argument to this threshold. · OAI.ReflexiveFixedPoints.CandidateContext.threshold

thresholdContribution def

A threshold contribution of the coarse estimator. · OAI.EditApproximation.thresholdContribution

thresholdExpr def

The natural floor implementation of the exact dyadic acceptance threshold. · OAI.ExactQuantumFactoring.thresholdExpr

thresholdFamily def

All strict homogeneous threshold supports of the fixed coefficient vectors. · OAI.QuantitativeVanDerWaerden.Arrangement.thresholdFamily

thresholdLabel def

Labels use only the local vertex and the shared coordinate. · OAI.MaxCutGames.Repetition.thresholdLabel

thresholdLabel def

Labels use only the local vertex and the shared coordinate. · OAI.UniqueGamesTheorem.Repetition.thresholdLabel

thresholdLevels def

All squared amplitudes, including repetitions, determine a single shared finite threshold partition. · OAI.MaxCutGames.Repetition.thresholdLevels

thresholdLevels def

All squared amplitudes, including repetitions, determine a single shared finite threshold partition. · OAI.UniqueGamesTheorem.Repetition.thresholdLevels

thresholdPoints def

The only additional discontinuities of the truncated label are the two threshold points in each period. Equality at either threshold is retained. · OAI.QuantitativeVanDerWaerden.AdaptiveMesh.thresholdPoints

thresholdSample def

Finite integer-threshold rejection sampling. A mass table can be sampled using shared proposals (label, height) drawn from a finite rectangle. · OAI.MaxCutGames.Foundations.CorrelatedSampling.thresholdSample

thresholdSample def

Finite integer-threshold rejection sampling. A mass table can be sampled using shared proposals (label, height) drawn from a finite rectangle. · OAI.UniqueGamesTheorem.Foundations.CorrelatedSampling.thresholdSample

ThresholdSeed abbrev

The seed is a coordinate and a finite threshold interval. · OAI.MaxCutGames.Repetition.ThresholdSeed

ThresholdSeed abbrev

The seed is a coordinate and a finite threshold interval. · OAI.UniqueGamesTheorem.Repetition.ThresholdSeed

thresholdSign def

The threshold convention assigns 1 at zero. · OAI.LeanBlast.GotsmanLinial.thresholdSign

thresholdWeight def

The manuscript's thresholded and renormalized disk average (equation 22). · OAI.EuclideanFiveColor.thresholdWeight

through def

First draw from P, then apply a channel. · OAI.FourierLLogL.Law.through

through def

Incidences with one fixed literal full label. · OAI.QuantitativeVanDerWaerden.AnchoredPatterns.through

through def

true denotes the positive sign; false denotes the negative sign. · OAI.TwoWayComplementation.Diagram.through

ThroughLabel abbrev

Multisets containing a prescribed label. · OAI.KadisonKastler.ThroughLabel

throughLabelEquiv def

Erasing one occurrence of the prescribed label. · OAI.KadisonKastler.throughLabelEquiv

tick def

A halted configuration is absorbing. Each nonhalting tick is exactly one write or one move, or the detection of halt. · OAI.MatchingFPRAS.tick

tierDrift def

The constant is shared by all tiers; the overlap integer is fixed. · OAI.KServer.TierProcess.tierDrift

tierRegion def

The finite region whose mass is charged by one real tier insertion. · OAI.KServer.Pilot.tierRegion

tieSpace def

The equality hyperplane of two affine scores. · OAI.Problem358.TripleTieAvoidance.tieSpace

tighten def

The actual adjacent reverse-edge deletion with the forced partner splice. In contrast to the paired band move, p and q need not be partners. · OAI.TorsionFreeZeroDivisors.FinitePicture.Data.tighten

tilde def

tightening used by compression and evidence posterior bounds. · OAI.FourierLLogL.Dyadic.Thermal.tilde

tilde def

The normalized quarter-turn spectral vector. · OAI.PositiveEinsteinFour.Bianchi.Shape.tilde

tildeIso def

The base-change isomorphism for quasicoherent sheaves on spectra. · OAI.AffineClosedBaseChange.tildeIso

tildeScalar def

Scalar map on an associated sheaf. · OAI.CartierAffineLattice.tildeScalar

tile def

Dirichlet factor normalized to one. · OAI.FourierLLogL.Dyadic.Procedure.Capture.tile

tile def

Blocks left i≤280 step8, right j≤720 step8 · OAI.GeneralMahler.SCal.Mid.tile

TiledProbeBoxData structure

Fixed source box data; packaging it keeps later three-chain applications small when the ambient algebra is an actual tangent Lie algebra. · OAI.PolycyclicRecognition.CartanWordModel.TiledProbeBoxData

TiledProbeChainData structure

The joint sampler data, packaged before specializing the templates. This keeps dependent elimination small at the polar-template application. · OAI.PolycyclicRecognition.CartanWordModel.TiledProbeChainData

TileNetwork structure

A finite gluing of the exact physical lozenges. A wire label shared by tiles is summed only once; open boundary values are imposed separately. · OAI.CriticalHoneycomb.Transfer.PhysicalTiles.TileNetwork

tilt def

One tilted path kernel (trees.tex, original-tilt). · OAI.DilutedSpinGlass.FiniteLaw.tilt

tiltMean def

At a regular attempt membership uses the specified old-channel conditional mean, never a sample. · OAI.FourierLLogL.Dyadic.Thermal.Update.tiltMean

tiltTable def

Multiplying by terminal fields does not change the effective interaction. · OAI.BufferedIsing.tiltTable

time def

The explicit final polynomial. All coefficients depend only on the fixed degree/radius and the chosen finite machine's push bound and tape count. · OAI.MaxCutGames.Foundations.Complexity.PoweringPolynomialBudget.time

time def

The explicit final polynomial. All coefficients depend only on the fixed degree/radius and the chosen finite machine's push bound and tape count. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringPolynomialBudget.time

TimeBound inductive

A concrete syntax for nonnegative polynomial bounds. This is only used to bound actual stack transitions, never as an additional machine instruction. · OAI.RationalHitting.TimeBound

timeBound def

A fixed-degree polynomial in input bits, ε⁻¹, and log δ⁻¹. Natural ceilings and binary ceiling-log change this bound only by constants. · OAI.MatchingFPRAS.timeBound

timeBound def

For each fixed degree, the exact cost is linear in index and prior output. · OAI.MaxCutGames.Foundations.Complexity.MachineRegularDummyRow.timeBound

timeBound def

For each fixed degree, the exact cost is linear in index and prior output. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRegularDummyRow.timeBound

TimeBounds def

Thm 1 (intro main) and decimal corollary estimates. · OAI.PowerSaving.TimeBounds

timeCell def

Equal-Lebesgue-mass time cells. The endpoint one is assigned to the last cell; this retains a total, surjective coding without changing any mass. · OAI.FKCRT.timeCell

TimeInput abbrev

The natural mixing factor and the natural height-plus-grid allowance, followed by the rational per-call accuracy. · OAI.ContingencyTables.CountingOracleScheduleProgram.TimeInput

timeIntegral def

Integration in time, as a genuine bounded linear operator. No operator-norm continuity of the unitary group is presupposed. · OAI.TriangularBilliards.Analysis.HilbertFlow.timeIntegral

timeless def

An ambient representative independent of the time coordinate. · OAI.KervaireIncidenceCurrent.timeless

timeMap def

Rokhlin's multiple-mixing problem asks whether ordinary mixing of one invertible probability-preserving transformation implies mixing of every finite order. The formalization proves this implication. For each $k\ge3$ and every $k$ … · OAI.Rokhlin.timeMap

timeOrdinal def

The ordinal map recording which side of the prism cut a vertex lies on. · OAI.Thomason.Construction.PrismBarycentric.timeOrdinal

timePartial def

Repeated time differentiation, retaining space as a parameter. · OAI.ForcedComputation.timePartial

timePolynomial def

The bound is one fixed quadratic polynomial in the actual unary input length. Its coefficient may depend on the fixed growth factor. · OAI.MaxCutGames.Foundations.Complexity.MachineCeilingPower.timePolynomial

timePolynomial def

A linear polynomial for each fixed port count d. · OAI.MaxCutGames.Foundations.Complexity.MachineDummyRows.timePolynomial

timePolynomial def

A single polynomial in the combined two input lengths. · OAI.MaxCutGames.Foundations.Complexity.MachineLookup.timePolynomial

timePolynomial def

A cubic polynomial for fixed degree, including the final zero-fuel guard. The one-time relation initializer can be accounted for separately. · OAI.MaxCutGames.Foundations.Complexity.MachinePaddingBounds.timePolynomial

timePolynomial def

A polynomial in initial semantic size; two extra steps include the final zero guard and actual halt. The encoding polynomial is applied once. · OAI.MaxCutGames.Foundations.Complexity.PCPIterationMachine.timePolynomial

timePolynomial def

Fixed program dimensions determine this explicit polynomial in input bits. · OAI.MaxCutGames.Integration.AddressTupleBudget.timePolynomial

timePolynomial def

The exponent and coefficients depend only on fixed reduction parameters. · OAI.MaxCutGames.Reduction.AddressMachineLoop.timePolynomial

timePolynomial def

A fixed polynomial in a common magnitude bound for the input fields, serialized words, radix, capacity, and digits. · OAI.MaxCutGames.Reduction.MachineAddressEdge.timePolynomial

timePolynomial def

A fixed polynomial in the serialized input length bounds actual steps. · OAI.MaxCutGames.Reduction.MachineAddressHeaders.Arithmetic.timePolynomial

timePolynomial def

Sum of fixed row polynomials; the additional constant is the dispatcher. This proof-side polynomial contains no runtime input field values. · OAI.MaxCutGames.Reduction.MachineOutcomeRows.timePolynomial

timePolynomial def

A fixed polynomial in a magnitude bounding input fields, the serialized word list, the radix, and the digits. Width and token count are static. · OAI.MaxCutGames.Reduction.MachineTemplateAddress.timePolynomial

timePolynomial def

The bound is one fixed quadratic polynomial in the actual unary input length. Its coefficient may depend on the fixed growth factor. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineCeilingPower.timePolynomial

timePolynomial def

A linear polynomial for each fixed port count d. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineDummyRows.timePolynomial

timePolynomial def

A single polynomial in the combined two input lengths. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineLookup.timePolynomial

timePolynomial def

A cubic polynomial for fixed degree, including the final zero-fuel guard. The one-time relation initializer can be accounted for separately. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachinePaddingBounds.timePolynomial

timePolynomial def

A polynomial in initial semantic size; two extra steps include the final zero guard and actual halt. The encoding polynomial is applied once. · OAI.UniqueGamesTheorem.Foundations.Complexity.PCPIterationMachine.timePolynomial

timePolynomial def

Fixed program dimensions determine this explicit polynomial in input bits. · OAI.UniqueGamesTheorem.Integration.AddressTupleBudget.timePolynomial

timePolynomial def

The exponent and coefficients depend only on fixed reduction parameters. · OAI.UniqueGamesTheorem.Reduction.AddressMachineLoop.timePolynomial

timePolynomial def

A fixed polynomial in a common magnitude bound for the input fields, serialized words, radix, capacity, and digits. · OAI.UniqueGamesTheorem.Reduction.MachineAddressEdge.timePolynomial

timePolynomial def

A fixed polynomial in the serialized input length bounds actual steps. · OAI.UniqueGamesTheorem.Reduction.MachineAddressHeaders.Arithmetic.timePolynomial

timePolynomial def

Sum of fixed row polynomials; the additional constant is the dispatcher. This proof-side polynomial contains no runtime input field values. · OAI.UniqueGamesTheorem.Reduction.MachineOutcomeRows.timePolynomial

timePolynomial def

A fixed polynomial in a magnitude bounding input fields, the serialized word list, the radix, and the digits. Width and token count are static. · OAI.UniqueGamesTheorem.Reduction.MachineTemplateAddress.timePolynomial

timeVertex def

The exact degree-mass coding, using every nonroot preorder pole. · OAI.FKCRT.Inventory.CoreTree.timeVertex

tipRole def

A lane's absolute label changes its incoming/outgoing role between opposite sides of a cell. · OAI.OneWayLiveness.CellNetwork.tipRole

TipSeparates def

Strict separation of representatives by the chord, with an arbitrary linear cut outside its first open arc. · OAI.PlanarL1.Columns.Model.Field.TipSeparates

TM structure

Finiteness of X, J and Y supplied in main theorem. Finite functions in the table present no uniformity issue; all are fixed for the machine. · OAI.Fifths.TM

tmin def

Exact finite decimals of the paper as rationals cast to ℝ · OAI.GeneralMahler.tmin

toAccepted def

Transpose the degree and boundary indices of a positive globular table. · OAI.Thomason.Construction.DirectedComplex.Cell.toAccepted

toAccepted def

A positive chain map is unchanged on each chain group by the conversion. · OAI.Thomason.Construction.DirectedComplex.Hom.toAccepted

toAccepted def

Express a directed complex using its chain carriers and group instances. · OAI.Thomason.Construction.DirectedComplex.toAccepted

toAlternatingCreationCoordinates def

Normalize the elementary creation data using the proved norms of the actual symmetry and corner Haar vectors. · OAI.KadisonSimilarity.TracialCreationFormulas.toAlternatingCreationCoordinates

toAnalyticIterationData def

Assemble the analytic iteration inputs from genuine finite patch fields. The four field bounds and every correction jet are conclusions of the patch calculus, rather than fields of the geometric input. · OAI.RegularPointCoordinates.PatchIterationGeometry.toAnalyticIterationData

toArtin def

The kernel presentation is compared with the signed trace group, not with an assumed normal-form model. · OAI.EilenbergGanea.TraceWords.toArtin

toBadFamily def

Every small successful index for a bad numerator lands at a bad numerator at the next level. In particular a zero residue cannot remain bad. · OAI.Problem337.DensityResidueFamily.toBadFamily

toBar def

This comparison retains the actual outside B-action. · OAI.LittleFinitistic.BimoduleBarComparison.toBar

toBarLinear def

The field-linear balanced universal map to the bar worker's B-linear quotient. · OAI.LittleFinitistic.BimoduleBarComparison.toBarLinear

toBasic def

Translation to the finite Boolean/existential basis used in the quotient truth proof. This changes no set-theoretic predicate or quantifier domain. · OAI.PartitionConsistency.Formula.toBasic

toBipartiteGame def

The list presentation yields exactly the bipartite occurrence abstraction used by the v2 subdivision proof, without quotienting or discarding occurrences. · OAI.MaxCutGames.Explicit.MachineOutputContract.SimpleBipartite.toBipartiteGame

toBipartiteGame def

The list presentation yields exactly the bipartite occurrence abstraction used by the subdivision proof, without quotienting or discarding occurrences. · OAI.UniqueGamesTheorem.Explicit.MachineOutputContract.SimpleBipartite.toBipartiteGame

toBiPolynomial def

The coefficient-preserving bridge to the existing bivariate interpolation library. · OAI.Paper170.toBiPolynomial

toBlocks def

Re-express all maps in the chosen domain/codomain coordinates at once. · OAI.MaxCutGames.Appendix.CoordinateTransport.toBlocks

toBlocks def

Re-express all maps in the chosen domain/codomain coordinates at once. · OAI.UniqueGamesTheorem.Appendix.CoordinateTransport.toBlocks

toBounded def

Forgetting the projective subcategory gives actual bounded projective module complexes. · OAI.LittleFinitistic.SourceModelBridge.toBounded

toCanonicalAnalyticIterationData def

The explicit patch constants supply a canonical field coefficient and jet base, with no dependence on a coordinate iterate. · OAI.RegularPointCoordinates.PatchIterationGeometry.toCanonicalAnalyticIterationData

toCompletion def

The coefficient ring maps canonically to the vertex-adic completion. · OAI.SectionCompletion.toCompletion

toConcrete def

Conversion of a produced genuine oracle into the manuscript invariant. The scalar bounds here are exactly those to be discharged by scale margins. · OAI.TraceReconstruction.OracleTracePair.toConcrete

toConditional def

Forgetting the auxiliary injectivity retains an actual conditional star of master matches. Its common tuple is named by I. · OAI.HyperbolicObstruction.StarModel.toConditional

toConstant def

A map from a diagram to a constant target, induced by its colimit. · OAI.Thomason.Construction.CobaseChangeCellularClosure.toConstant

toContinuous def

The L-infinity coefficient map is jointly continuous. · OAI.KadisonKastler.ScalarOperatorKernel.toContinuous

toContinuousBilinear def

No continuity premise is required in the published correction theorem: it follows directly from the scalar part of its Haagerup estimate. · OAI.KadisonKastler.HaagerupBound.toContinuousBilinear

toContinuousLinearMap def

Turn a completely bounded algebraic map into a continuous linear map. · OAI.KadisonKastler.CompletelyBoundedBy.toContinuousLinearMap

toContinuousLinearMap def

The continuous functional furnished automatically by a positive normalized mean. · OAI.ThompsonNonamenability.InvariantMean.toContinuousLinearMap

toData def

The normalized stage's actual data, with its original finite noise choices. Passing the inclusion explicitly avoids dependent casts of the shift carrier. · OAI.MaxCutGames.Gadget.StageData.toData

toData def

Forgetting the schedule gives ordinary stationary deformation data. · OAI.Thomason.Construction.TopologicalDeformation.ScheduledData.toData

toData def

The normalized stage's actual data, with its original finite noise choices. Passing the inclusion explicitly avoids dependent casts of the shift carrier. · OAI.UniqueGamesTheorem.Gadget.StageData.toData

toDepth5Circuit def

Relabel each finite gate type by Fin without changing any wire or degree. · OAI.Problem335.IndexedDepth5Circuit.toDepth5Circuit

toDMachine def

A deterministic machine with exactly the same states and finite accepting runs. No halting or totality condition is introduced. · OAI.TwoWayComplementation.CompanionBridge.toDMachine

toDyadicPLWitness def

Uniform affine data is an ordinary finite dyadic PL subdivision. · OAI.ThompsonNonamenability.UniformDyadicPLWitness.toDyadicPLWitness

toeplitzRowCompression def

Compression in the ambient algebra, with relative unit q. · OAI.WeakPureInfiniteness.toeplitzRowCompression

toeplitzRowVacuum def

The defect of the finite isometry row before compression. · OAI.WeakPureInfiniteness.toeplitzRowVacuum

toEquiv def

The underlying point equivalence, retaining its polynomial witnesses in e. · OAI.StableCoordinate.AmbientMaps.PolynomialEquiv.toEquiv

toEvaluation def

The final nonlinear evaluator uses exactly the maps stored in the frontiers. · OAI.ReflexiveFixedPoints.FrontierTower.toEvaluation

toFamily def

Construct the exact raw family from ceiling residue information at the canonical levels. This definition discharges every ceiling, divisibility, terminal-expansion, and geometric-transition field. · OAI.Problem337.GeometricDensity.ResidueData.toFamily

toffoliCount def

The product projection in the exact Toffoli identity I-2 C⊗|-><-|. · OAI.QAC.toffoliCount

toffoliWord def

A Toffoli flips its target iff all controls are one. · OAI.QAC.Gate.toffoliWord

toFin def

Transport to the already proved finite-coordinate fan inequality. · OAI.HyperbolicObstruction.FanLink.IndexedFanData.toFin

toFiniteComplex def

The exact finite-complex representation in the release reference. · OAI.HyperbolicObstruction.FinitePolyhedron.toFiniteComplex

toFiniteOmega def

Conversion restricted to categories with a uniform finite dimension bound. · OAI.Thomason.Construction.toFiniteOmega

toFinsupp def

The finitely supported coordinate function of an actual cylinder chain. · OAI.Thomason.Construction.PastingGlobeCylinderBasis.toFinsupp

toFraction def

The same localization embeds into the fraction field used for the alternating products. · OAI.CirculantHadamard.BinaryLocalRing.toFraction

toFree def

The map out of the algebra defined by all entries of GS−I. · OAI.IntegralCharacterVarieties.ExactSequenceChart.MatrixSplitData.toFree

toFrontier def

The next frontier evaluation is built from the preceding one. · OAI.ReflexiveFixedPoints.FrontierExtension.toFrontier

toGame def

Each actual edge has equal weight. Simplicity makes the acceptance predicate at its endpoint pair exactly its permutation constraint. · OAI.MaxCutGames.Explicit.BipartiteGame.toGame

toGame def

Each actual edge has equal weight. Simplicity makes the acceptance predicate at its endpoint pair exactly its permutation constraint. · OAI.UniqueGamesTheorem.Explicit.BipartiteGame.toGame

toggle def

Insert or remove the join of the pivot with the fixed element. · OAI.FiniteChain.toggle

toGraph def

The precise reverse Tietze map, keeping each new radial generator's value. · OAI.TorsionFreeZeroDivisors.GraphConePresentation.toGraph

toHomotopy def

On a whole finite-dimensional complex the final partial homotopy is an ordinary continuous homotopy, not merely cellwise data. · OAI.GroupRingTopology.Cellular.PartialHomotopy.toHomotopy

toImage def

Send an old flag vector to its class in the quotient subsystem, retaining its membership in the induced image flag. · OAI.IntegralCharacterVarieties.IntrinsicInterval.Data.toImage

toImage def

The canonical map onto the finite image, allowing state-to-word collisions. · OAI.TreeEdit.BinaryLower.ParameterImage.toImage

toInvariantMultilinear def

The concrete coefficient parametrization is linear into actual invariant multilinear maps. · OAI.MetricEntropyDuality.SymmetricForm.toInvariantMultilinear

Token inductive

A literal word or a reference to one of the four saved input fields. · OAI.MaxCutGames.Reduction.CanonicalWordTemplate.Token

Token inductive

A literal word or a reference to one of the four saved input fields. · OAI.UniqueGamesTheorem.Reduction.CanonicalWordTemplate.Token

tokenComputation def

The token-stream phase has a displayed finite machine and actual execution bound; its polynomial argument is the token stream's physical bit length. · OAI.MaxCutGames.BinaryRenameMachine.tokenComputation

tokenComputation def

The token-stream phase has a displayed finite machine and actual execution bound; its polynomial argument is the token stream's physical bit length. · OAI.UniqueGamesTheorem.BinaryRenameMachine.tokenComputation

tokens def

The program copies saved endpoint words and emits complete forward tables. · OAI.MaxCutGames.Explicit.MachineSubdivisionRows.tokens

tokens def

The program copies saved endpoint words and emits complete forward tables. · OAI.UniqueGamesTheorem.Explicit.MachineSubdivisionRows.tokens

toKernel def

The determinant map with codomain restricted to the diagonal kernel. · OAI.MatrixMultiplication.AuxiliarySeparation.DeterminantBounds.toKernel

toLaurent abbrev

Alias for the canonical map from G to B[τ,τ⁻¹]. · OAI.StableCoordinate.ModificationEmbedding.toLaurent

tolerance def

Per-phase logarithmic tolerance. · OAI.ContingencyTables.CountingPhaseSchedule.tolerance

tolerance def

§03: the literal absolute reference tolerance p*/8, p*=1/24. · OAI.KServer.CoreFlags.tolerance

tolerance def

The dyadic tolerance obtained from the actual cumulative budgets. · OAI.ReflexiveFixedPoints.OrderedTree.tolerance

toLinear def

Bundle the coefficient identities as a sesquilinear map. · OAI.KadisonKastler.ScalarOperatorKernel.toLinear

toMetricGreenData def

Metric differential identities give the tensor-identity structure. · OAI.PositiveEinsteinFour.Geometry.ParallelMetricData.toMetricGreenData

tomitaClosedLinearOperator def

The actual closed operator, expressed using a fixed conjugate copy of H. · OAI.KadisonKastler.tomitaClosedLinearOperator

tomitaGraphAntiunitary def

Adjoint restricts to an antiunitary involution on the closed graph. · OAI.KadisonKastler.tomitaGraphAntiunitary

tomitaGraphFirst def

First coordinate on the graph Hilbert space. · OAI.KadisonKastler.tomitaGraphFirst

tomitaGraphFlip def

Adjoint on graph vectors, with the fixed second-coordinate identification. · OAI.KadisonKastler.tomitaGraphFlip

TomitaGraphHilbert abbrev

The fixed Hilbert space in which the two rescaled graphs are compared. · OAI.KadisonKastler.TomitaGraphHilbert

tomitaGraphRoot def

The positive square root of the bounded graph resolvent. · OAI.KadisonKastler.tomitaGraphRoot

tomitaLinearPolarUnitary def

The linear unitary part of the Tomita operator in the chosen conjugate coordinates. · OAI.KadisonKastler.tomitaLinearPolarUnitary

tomitaPolarConjugation def

The polar conjugation constructed on the original represented Hilbert space. · OAI.KadisonKastler.tomitaPolarConjugation

tomitaPolarCoordinates def

Polar identification of H with the graph Hilbert space. · OAI.KadisonKastler.tomitaPolarCoordinates

tomitaProductCoordinates def

Changing only the finite product norm does not change the graph topology. · OAI.KadisonKastler.tomitaProductCoordinates

tomitaProductGraph def

The closed graph in the ordinary product used by LinearPMap. · OAI.KadisonKastler.tomitaProductGraph

toModel def

The homomorphism from the supplied infinite presentation to the finite model. · OAI.LittleFinitistic.FinitePresentation.toModel

toMultilinearMap def

The coefficient formula is an actual multilinear map, not an assumed law. · OAI.MetricEntropyDuality.SymmetricForm.toMultilinearMap

toNuclei def

Integer positive charges satisfy the real-charge library interface. · OAI.ContinuumCoulomb.NuclearData.toNuclei

toObstruction def

Assemble the source's actual graph configuration, rather than assuming common centers or tangent-cone distances, into the proved obstruction. · OAI.HyperbolicObstruction.CoreGeometry.MetricCore.toObstruction

toOmega def

The cell-preserving conversion to separate source and target. · OAI.Thomason.Construction.toOmega

toOmegaCategory def

Interpret Boolean faces as separate source and target operations. · OAI.Thomason.Construction.StrictOmegaCategory.toOmegaCategory

toOmegaFunctor def

The same cell function is a functor in the separate-boundary presentation. · OAI.Thomason.Construction.StrictOmegaCategory.Hom.toOmegaFunctor

toOneFunctor def

The object and 1-cell action of a strict functor is an ordinary functor. · OAI.Thomason.Construction.StrictOmegaCategory.Hom.toOneFunctor

toOperator def

The bounded complex-linear operator reconstructed from the kernel. · OAI.KadisonKastler.ScalarOperatorKernel.CoherentCore.toOperator

toOption def

A precise normal-form identification with the set in the manuscript. · OAI.FiniteAlgebraicEnvelopes.ZeroProduct.toOption

toOrderedTree def

The single infinite ordered tree extracted from all concrete levels. · OAI.ReflexiveFixedPoints.FrontierTower.toOrderedTree

toOuterStrategy def

Convert a pair of genuine local point strategies into the ordinary answer type used by the clean-coordinate local simulation. · OAI.MaxCutGames.Clean.AnswerBridge.toOuterStrategy

toOuterStrategy def

Convert a pair of genuine local point strategies into the ordinary answer type used by the clean-coordinate local simulation. · OAI.UniqueGamesTheorem.Clean.AnswerBridge.toOuterStrategy

Top abbrev

Only the 32 actual top fields index a top occurrence. Decorations are NOT extra coordinates indexing redundant copies of a top cone. · OAI.HyperbolicObstruction.ConeSkeleton.Top

top def

The canonical positive top cell of either half. · OAI.Thomason.Construction.PastingGlobeBasis.top

top def

The top basis simplex on its own set of vertices. · OAI.Thomason.Construction.SimplexComplex.top

toPacket def

Equip a full lattice with the basis expected by the packet model. · OAI.DukePrimeDegree.FullLattice.toPacket

toPath def

Realize a schedule as the literal switching path of the fixed word. · OAI.PlanarL1.Columns.Model.Schedule.toPath

topBasis def

The distinguished top-dimensional simplex basis element. · OAI.Thomason.Construction.SimplexComplex.topBasis

topCell def

The universal n-cell is the positive table of its two faces in every lower degree and the single generator at its top degree. · OAI.Thomason.Construction.DirectedGlobeComplex.topCell

topChartSection def

Send a algebraic top differential to its intrinsic sheaf section. · OAI.ActualCotangent.topChartSection

topCode def

`topCode` returns `0` for an empty stack, `1` for a leading `false`, and `2` for a leading `true`. · OAI.CurrentKS.topCode

topDart def

The actual oriented macro edge in the indexed attaching polygon. · OAI.HyperbolicObstruction.ConeSkeleton.topDart

topDegree def

Number of top-dimensional faces incident to a vertex. · OAI.CoboundaryExpanders.Complex.topDegree

topDegree def

Number of top-dimensional faces incident to the given vertex. · OAI.CoboundaryExpanders.Complex.topDegree

topDifferentialMap def

The canonical map on the second exterior power of differentials. · OAI.CanonicalCoordinates.topDifferentialMap

topElement def

The value of a field, viewed inside the full operator algebra. · OAI.KadisonKastler.WeakOperatorRepresentative.topElement

toPeriodic def

A single lattice is a one-coset periodic covering. · OAI.CoveringOrder.LatticeCover.toPeriodic

topExteriorProduct def

The product of all generators, in the given increasing order. · OAI.PixtonChow.topExteriorProduct

topInclusion def

restriction along an inclusion of affine coordinate subrings. · OAI.CanonicalCoordinates.topInclusion

topLeftCorner def

The input corner used for the two relative isometries. · OAI.WeakPureInfiniteness.topLeftCorner

topMeetings def

The switches in the final layer used by two marked paths. A switch is indexed by the common tail at the end of the two child butterflies. · OAI.CubeShuffle.topMeetings

toPointMap def

The forward polynomial map. · OAI.StableCoordinate.AmbientMaps.PolynomialEquiv.toPointMap

toPolynomial def

Send the cyclic basis element at i to the i.valth power of the root. · OAI.CirculantHadamard.CyclicPolynomial.toPolynomial

toPolynomial def

Ascending coefficient lists interpreted as polynomials. · OAI.DefocusingNLS.SeparatorArithmetic.toPolynomial

toPolynomial def

The source quotient maps to the four-variable polynomial presentation. · OAI.StableCoordinate.Presentation.toPolynomial

topOpenAffineIso def

The intrinsic comparison on an affine open. · OAI.ActualCotangent.topOpenAffineIso

toPort def

Give a procedure a single external input/output port. The two transfers are real stack programs; scratch registers remain clean. · OAI.RationalHitting.StackLanguage.Procedure.toPort

toProperChainMetric def

Sampling minimizing paths provides the chain structure for the existing proper-homomorphism and canonical-height metric comparisons. · OAI.PolycyclicRecognition.ProperMinimizingMetric.toProperChainMetric

topSheaf def

The top exterior power of the cotangent sheaf. · OAI.ActualCotangent.topSheaf

topSpace def

Counting-measure top-level space. Equivalently the orthogonal complement of the span of proper-coordinate cylinder functions (conjugation preserves that span). · OAI.CoordinateSweeps.Placement.topSpace

toQuotientKernel def

The right-hand morphism in the elementary modification sequence. · OAI.CartierConnecting.Boundary.toQuotientKernel

Tor abbrev

Tor is the left derived functor of the actual balanced tensor functor. · OAI.LittleFinitistic.PDTensor.Tor

toRealization def

The continuous map from the geometric horn to the actual realization. · OAI.Thomason.Construction.HornBarycentric.toRealization

toRealOrder def

Convert the complex-embedding ordering to the real-embedding ordering used by the row-lattice packet model. Total reality makes this bijective. · OAI.DukePrimeDegree.EmbeddingOrder.toRealOrder

toReducedCuts def

Extract the actual shores and local vertex partitions from the occurrence subtrees; vertices are not double-counted even if they occur in many bags. · OAI.ParityWL.ReducedBagTree.toReducedCuts

toRel def

The underlying binary relation. · OAI.TwoWayComplementation.RelationMonoid.toRel

toRelaxed def

Forgetting the noncoincidence requirement at diagonal layers embeds each compatible quadruple into its relaxed class. · OAI.Problem335.MomentPairing.toRelaxed

toResidueData def

A normalized strict discrepancy conclusion implies the weaker ceiling residue interface, without demanding a strict inequality from later consumers. · OAI.Problem337.GeometricDensity.NormalizedResidueData.toResidueData

torIso def

Actual derived residue tensor is unchanged by the proved right-module identification. · OAI.LittleFinitistic.PDResidueProjection.torIso

torIsoBar def

No extra identification between the bar augmentation and M is assumed. · OAI.LittleFinitistic.PDBar.torIsoBar

torIsoBaseBar def

Actual Tor at the native bar model, using the canonical augmentation. · OAI.LittleFinitistic.SelectedPDDetection.torIsoBaseBar

torIsoModel def

The chosen minimal resolution computes this actual derived functor. · OAI.LittleFinitistic.PDTensor.torIsoModel

torIsoResolution def

Any actual projective resolution computes the same Tor, including the constructed absolute bar resolution. · OAI.LittleFinitistic.PDBar.torIsoResolution

torIsoTerm def

Homology of the tensor of a minimal resolution is its term, because all its differentials have been proved zero. · OAI.LittleFinitistic.PDTensor.torIsoTerm

torIsoTwoSided def

Tor computed by the literal two-sided bar, with no assumed quotient comparison. · OAI.LittleFinitistic.PDBarQuotient.torIsoTwoSided

TorsionFree def

No nonidentity element has finite order. This does not assert unique roots. · OAI.Release075.TorsionFree

TorsionFree def

A group is torsion-free when its only finite-order element is the identity. This is the usual group-theoretic condition, not the stronger unique-roots condition. · OAI.TorsionFreeBass.TorsionFree

TorsionFree def

Ordinary group torsion-freeness, without requiring uniqueness of all roots. · OAI.TorsionFreeZeroDivisors.TorsionFree

torusAutomorphism def

Each element of the compact coefficient group is a Lie automorphism. · OAI.PolycyclicRecognition.ComplexCartan.torusAutomorphism

torusCharacter def

The periodic character exp(i n·x). · OAI.DefocusingNLS.torusCharacter

torusChart def

The common chart into the genuine set quotient of nonzero complex points. · OAI.Nagata.Workers.W05.torusChart

torusCoordinateLine def

The real parameter t moves along the jth coordinate circle. · OAI.DefocusingNLS.torusCoordinateLine

torusDistanceSq def

Squared Euclidean distance to the nearest lattice point, using the centered representative in each coordinate. · OAI.ForcedComputation.VelocityDetector.torusDistanceSq

torusFourierLaplacian def

The spectral Laplacian, represented as a continuous function for k>8. · OAI.DefocusingNLS.torusFourierLaplacian

torusHeatDerivative def

The joint derivative, expressed entirely by evolutions of smooth initial data. · OAI.ForcedComputation.VelocityDetector.torusHeatDerivative

TorusHeatInput structure

The flat-torus specialization of the heat representation and Gaussian bound. · OAI.ForcedComputation.VelocityDetector.TorusHeatInput

torusLocalChart def

Every genuinely injective open piece supplies an actual local homeomorphism onto its image in the torus quotient. The chart injectivity must be proved by the caller. · OAI.Nagata.W21.torusLocalChart

torusMap def

The produced physical action in the fixed torus coordinate convention. · OAI.ThreeTorus.SmoothDynamics.torusMap

torusOperator def

A unit diagonal coefficient vector acts on the entire algebra. · OAI.PolycyclicRecognition.ComplexCartan.torusOperator

torusOrbitSetoid def

Two covering-space points represent the same point exactly when they differ by an integer power of the period. · OAI.Nagata.W21.torusOrbitSetoid

torusPartial def

First coordinate derivative, represented by its absolutely convergent Fourier series. · OAI.DefocusingNLS.torusPartial

TorusPoint def

The multiplicative quotient; for the source torus use G = ℂˣ. · OAI.Nagata.W21.TorusPoint

torusRepresentation def

The torus field is real-linear in its plane velocities. · OAI.HarmonicCounterexample.Control.torusRepresentation

torusRepresentative def

A representative in the fundamental half-open unit cube, used only to evaluate periodic derivatives. Periodicity makes those derivatives independent of this choice. · OAI.BoltzmannNonuniqueness.torusRepresentative

TorusScalarExistence def

Classical existence for smooth periodic scalar advection-diffusion on a finite time cylinder. · OAI.ForcedComputation.VelocityDetector.TorusScalarExistence

TorusScalarSolution structure

Smoothness and the actual scalar equation on a finite closed cylinder. The time derivative is one-sided at its two endpoints. · OAI.ForcedComputation.VelocityDetector.TorusScalarSolution

torusSecondPartial def

Second coordinate derivative, represented by its absolutely convergent Fourier series. · OAI.DefocusingNLS.torusSecondPartial

torusTuple def

The corresponding actual torus tuple. · OAI.Nagata.W06.TorusConfiguration.torusTuple

torusVolume def

The Riemannian volume measure of the unit flat torus. · OAI.CubicTorus.torusVolume

toScalarProfile def

The polynomial profile satisfies every hypothesis of the scalar growth theorem. · OAI.MatrixMultiplication.AuxiliarySeparation.Character.toScalarProfile

toSelection def

All infinite relators follow from the finite zero-argument patterns. · OAI.LittleFinitistic.FinitePresentation.CovariantData.toSelection

toSimplexApp def

Evaluation with explicit unbundled chain-group types. · OAI.Thomason.Construction.SimplexAtomsInput.toSimplexApp

toSimplexAtomsInput def

The signed-part formulas and composition generation give the simplex-atom data. · OAI.Thomason.Construction.PublishedSimplexGenerationInput.toSimplexAtomsInput

toSimplexChains def

Evaluate the additive classes of strict cells in their original chains. · OAI.Thomason.Construction.SimplexAtomsInput.toSimplexChains

toSlice def

The induced U map retains its inclusion in the old graded module. · OAI.IntegralCharacterVarieties.OldContactSlice.toSlice

toSmooth def

The genuine embedding of Schwartz functions into the smooth differential algebra. · OAI.ElasticityPhysicalAlgebra.toSmooth

toSource def

Decode valid finite data into the existing actual spin Hamiltonian. · OAI.ContinuumCoulomb.BinaryHeisenberg.toSource

TossUp structure

Generic membership fork, same spatial node both children. · OAI.FourierLLogL.Dyadic.Procedure.TossUp

toStage def

The quotient is again a finite stage; its logical map is now the identity. · OAI.MaxCutGames.Gadget.StageQuotient.toStage

toStage def

The quotient is again a finite stage; its logical map is now the identity. · OAI.UniqueGamesTheorem.Gadget.StageQuotient.toStage

toStrictHom def

Strict functors preserve the combined Boolean face operation. · OAI.Thomason.OmegaCategory.Functor.toStrictHom

toStrictOmegaCategory def

Combine source and target into a Boolean-indexed boundary operation. · OAI.Thomason.OmegaCategory.toStrictOmegaCategory

toSubdivision def

Construct all subdivision data from the chain equalities. · OAI.ThompsonNonamenability.GapChain.toSubdivision

toSubdivision def

No changes to endpoints or to the original multigraph are made by subdividing. · OAI.TorsionFreeZeroDivisors.PortSubdivision.PortGraph.Walk.toSubdivision

tot def

Total weight. · OAI.HadwigerCounterexample.tot

Total def

The total symbolic component over all ordinary Cohen filters. There is no assertion here that any particular such filter is a ground generic. · OAI.PartitionConsistency.ProductForcing.ChainCode.Total

total def

All valid cells of degree k, without a height restriction. · OAI.CubicalChains.HeightFiltration.total

total def

Actual signed finite tensor total in modules over the outside algebra. · OAI.LittleFinitistic.BimoduleComplexTensor.total

total def

The same signed finite tensor total, retaining the full left B action. · OAI.LittleFinitistic.BimoduleTotal.total

total def

The actual three-column induced bimodule complex of Section 4. · OAI.LittleFinitistic.InducedRectification.total

total def

A genuine cochain complex rectifying a three-column homotopy diagram. The cone convention fixes all differential signs; the three-column term isomorphism is given below. · OAI.LittleFinitistic.RectificationTotal.total

total def

The number of entries after concatenating all groups. · OAI.ReflexiveFixedPoints.OrderedGroupConcat.total

totalAddDesc def

Additive maps out of a module totalization, using its preserved coproduct. This permits unit insertion, which need not be A-linear. · OAI.DerivedBar.totalAddDesc

totalAssociativityIso def

Full signed total associativity in S-modules, on every input complex Z. · OAI.LittleFinitistic.BimoduleComplexTensor.totalAssociativityIso

TotalAt def

Representation in the total space, retaining its actual filter. · OAI.PartitionConsistency.ProductForcing.ChainCode.TotalAt

totalCalls def

Total calls of a type before scheduling. · OAI.ExactFourier.Packing.totalCalls

totalDegree def

The polynomial degree attached to a surviving monomial. · OAI.MatrixMultiplication.AuxiliarySeparation.totalDegree

totalDifferential def

The full differential is the sum of the signed vertical and horizontal differentials. · OAI.LittleFinitistic.FiniteColumnComplex.totalDifferential

TotalFunctionalF def

The antecedent of a Replacement instance: totality with local unique witnesses, exactly as supplied by the existential forcing clause. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.TotalFunctionalF

totalFunctor def

The actual tensor total is a functor retaining its outside action on every complex. · OAI.LittleFinitistic.BimoduleComplexTensor.totalFunctor

totalFunctorIso def

Natural comparison on all input complexes, with the complete outside action. · OAI.LittleFinitistic.BimoduleTotalPresentation.totalFunctorIso

totalInterchange def

Arbitrary additive functors commute with the actual iterated-cone total. · OAI.LittleFinitistic.RectificationTotal.totalInterchange

totalIso def

The two signed totals coincide through their actual tensor-summand equivalences. · OAI.LittleFinitistic.BimoduleTotalPresentation.totalIso

totalIso def

Removing the proved zero shift identifies the ordinary tensor complex exactly. · OAI.LittleFinitistic.SingleBimoduleTensor.totalIso

totalMap def

Actual chain maps of the outside-action tensor total. · OAI.LittleFinitistic.BimoduleComplexTensor.totalMap

totalMap def

The original input tensor map is B-linear in every degree. · OAI.LittleFinitistic.BimoduleTotal.totalMap

totalMap def

The induced chain map of the native primary's actual iterated-cone total. · OAI.LittleFinitistic.RectificationTotal.MapData.totalMap

totalMapDesc def

Maps out of the image of a total term, using only a finite coproduct. · OAI.DerivedBar.totalMapDesc

totalMapIso def

Additive functors commute with first-quadrant totalization as chain complexes, with the actual horizontal and signed vertical differential. · OAI.DerivedBar.totalMapIso

totalMapIsoX def

Additive functors commute with each finite diagonal sum. · OAI.DerivedBar.totalMapIsoX

totalMove def

A lift of transport to the total space, before verifying the action laws. · OAI.PartitionConsistency.ProductForcing.ChainCode.totalMove

totalNumber def

Total number of quanta in a number-basis vector. · OAI.EntropyPhotonNumber.totalNumber

totalPaddingBinMass def

This is the literal S₀: the index restriction is on the left endpoint of each bin, exactly as in q:bins. · OAI.TwoPointCorrelations.totalPaddingBinMass

totalPlus def

Simultaneous plus rotation of both derivative slots and the matrix slots. · OAI.PositiveEinsteinFour.HighestHessian.totalPlus

totalRightIso def

Tensor total preserves actual isomorphisms of genuine bimodule complexes and their outside action. · OAI.LittleFinitistic.BimoduleComplexTensor.totalRightIso

TotalSpace def

The genuine quotient total space of the associated bundle. · OAI.Nagata.W16.AssociatedBundle.TotalSpace

totalSteps def

Exactly n supplied body blocks and n + 1 concrete guard transitions. · OAI.MaxCutGames.Foundations.Complexity.MachineCountedLoop.totalSteps

totalSteps def

Exactly n supplied body blocks and n + 1 concrete guard transitions. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineCountedLoop.totalSteps

totalTermIso def

The native total's term as the actual product of the three input modules. · OAI.LittleFinitistic.RectificationTotal.totalTermIso

totalUniformConditionalMass def

Null fibers are assigned a point mass. Their observation mass is zero, so this makes a probability kernel without changing any joint law. · OAI.JointDickman.totalUniformConditionalMass

totalVariation def

Half the unnormalized sum of absolute probability-mass differences. · OAI.BinaryCoordinateSweeps.totalVariation

totalVariation def

Total variation for real-valued finite mass functions. Distribution normalization is supplied explicitly in theorems that need it. · OAI.MaxCutGames.Foundations.CorrelatedSampling.totalVariation

totalVariation def

Total variation for real-valued finite mass functions. Distribution normalization is supplied explicitly in theorems that need it. · OAI.UniqueGamesTheorem.Foundations.CorrelatedSampling.totalVariation

totalVertexChain def

The full vertex set is the greatest vertex in a subdivided simplex. · OAI.Thomason.Construction.totalVertexChain

totalWeight def

Exponent of the degeneration parameter on one term. · OAI.MatrixMultiplication.AuxiliarySeparation.Sector.totalWeight

totalWeight def

Total weight of a target monomial after the three independent substitutions. · OAI.MatrixMultiplication.AuxiliarySeparation.totalWeight

toTensorClassHom def

A coefficient-level character descends to a normalized semiring homomorphism. · OAI.MatrixMultiplication.AuxiliarySeparation.Character.toTensorClassHom

totientFromList def

Run the polynomial accumulator scan from the queried word and one. · OAI.ExactQuantumFactoring.BitArithmetic.totientFromList

totientPair def

A division-based totient step. Redundant prime entries do nothing. · OAI.ExactQuantumFactoring.BitArithmetic.totientPair

totientScan def

Strip a given prime multiset, updating Euler's totient one prime factor at a time. The supplied list may contain redundant factors. · OAI.ExactQuantumFactoring.totientScan

totientStepBound def

A loose but explicit uniform polynomial bound; the residual, prime and accumulator subcircuits are evaluated a constant number of times per step. · OAI.ExactQuantumFactoring.BitArithmetic.totientStepBound

totientStepNat def

The finite-word accumulator semantics. Only the multiplicative accumulator is reduced; division of the residual never overflows. · OAI.ExactQuantumFactoring.totientStepNat

totientUpdate def

Carry the input unchanged and update two finite-word accumulators. · OAI.ExactQuantumFactoring.BitArithmetic.totientUpdate

toTopology def

The presented edge group maps into the actual topological pi1. · OAI.HyperbolicObstruction.FinitePolyhedron.EdgePresentation.toTopology

toTriple def

The coordinate convention used by the analytic packet calculations. · OAI.ThreeTorus.toTriple

Touch def

All actual processed groups that can affect this point through their narrow-band geometry. It is not just the groups having a prescribed ID. · OAI.PlanarL1.Columns.Model.RetainedSelection.Touch

Touch def

A real group meets the narrow-band radius-four-R ball through one of its actual retained endpoints. This test involves no fresh random choices. · OAI.PlanarL1.Columns.Model.SurvivingGroups.Touch

touchGraph def

Dual edges incident to a specified set of primal vertices. · OAI.TorsionFreeZeroDivisors.MapEuler.touchGraph

toUncompleted def

The canonical map from the completed cone to the uncompleted cone. · OAI.SourceConeMorphism.toUncompleted

toUnit def

Clamp affine coordinates into the unit interval. · OAI.Problem358.KernelQuantile.toUnit

Tour structure

A deterministic reference visit touches one selected block. All bounded state carried between visits, other than the blocks, is a raw c-bit controller. k>0, b=k*k; block tapings have at most b controller-width digits. · OAI.Fifths.Tour

tourNumbering def

Tour positions with initial position the original root dart. · OAI.FKCRT.tourNumbering

tourSelector def

Source exploration: start at the root and follow decided tree/non-tree rules until the first as-yet undecided edge. The fallback is only for states without a spanning-tree completion; it is never used on a valid branch. · OAI.FKCRT.tourSelector

toWeightedTree def

Extract the actual positive weights chosen by every finite extension. · OAI.ReflexiveFixedPoints.FrontierTower.toWeightedTree

tower def

A product-prior tower on the physical state and all realized marks. · OAI.DilutedSpinGlass.HeterogeneousMarks.tower

tower def

The infinite evaluated tree is extracted only after the finite recursion. · OAI.ReflexiveFixedPoints.AnchoredCandidateContext.tower

towerBand def

One finite matrix band, indexed by its domain cells. · OAI.WeakPureInfiniteness.towerBand

towerFunctor def

The whole sequence, before taking its colimit. · OAI.Thomason.Construction.InfiniteExtension.towerFunctor

towerInputStage def

The bottom bundle repeated once per matrix column. · OAI.Kaplansky.towerInputStage

towerRetainedStage def

The finite direct sum of the retained stages, for each matrix row. · OAI.Kaplansky.towerRetainedStage

toWiring def

The old assembly's actual whole-port wiring, not an abstract end matching. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.PortAssembly.toWiring

toYoung def

The column-recursive coordinates enumerate the actual Ferrers cells. · OAI.Saxl.Columns.toYoung

trace def

extend means continuing into the static system. · OAI.FourierLLogL.Dyadic.Procedure.Road.trace

trace def

The canonical faithful normal trace of the represented II₁ factor. · OAI.KadisonKastler.TracialUltrapower.trace

trace def

The vector sum restricted to the correct finite-trace algebra. · OAI.KervaireCentralizer.trace

trace def

The (unnormalized for one copy) vector trace. · OAI.KervaireRegular.trace

trace def

Analysis only: the finite output of a bounded number of advances. · OAI.LowSpaceSubsetSum.CyclicCursor.trace

trace def

Analysis view of repeated advances. The program can stop between them. · OAI.LowSpaceSubsetSum.RowMerge.trace

trace def

The actual set of representatives visited by a finite switching path, including every point of its vertical segments. · OAI.PlanarL1.Columns.Model.Path.trace

trace def

The finite geometric trace, including the degenerate chain's endpoint. · OAI.PlanarL1.Topology.SegmentChain.trace

trace def

Integral corner evaluation, with the outer face omitted. · OAI.Release075.FullFilling.trace

trace def

The literal one-epoch virtual routine. The deterministic filter continues independently through fallback; its discarded final state has no effect on service. Its full kept word is provided separately by keptTail. · OAI.UniformKServer.EpochShadow.trace

traceAverage def

Expectation over a length-n list of independent proposals with mass w. This is defined directly as a finite nested weighted sum. · OAI.MaxCutGames.Foundations.CorrelatedSampling.traceAverage

traceAverage def

Expectation over a length-n list of independent proposals with mass w. This is defined directly as a finite nested weighted sum. · OAI.UniqueGamesTheorem.Foundations.CorrelatedSampling.traceAverage

TraceBaseProjection structure

A trace projection is specified by precisely its range and pairing properties. · OAI.KadisonKastler.TraceBaseProjection

traceBinary abbrev

The field trace to the binary prime field. · OAI.MaxCutGames.Quadratic.traceBinary

traceBinary abbrev

The field trace to the binary prime field. · OAI.UniqueGamesTheorem.Quadratic.traceBinary

traceCenter def

Subtraction of the scalar trace is a complex-linear operation. · OAI.KadisonKastler.traceCenter

traceCentered def

Centering uses the normalized trace. · OAI.KadisonSimilarity.traceCentered

traceClosure def

The algebra of all two-norm limits of elements of a star subalgebra. · OAI.KadisonKastler.traceClosure

traceComponent def

The prescribed trace component of an actual Hessian. · OAI.PositiveEinsteinFour.TraceHessian.traceComponent

TraceConeEquiv structure

An equivalence of extended trace cones preserving addition, zero, positive scalar multiplication, and canonical convergence in both directions. · OAI.TraceConeClassification.TraceConeEquiv

traceCongruence def

Algebra congruence proved from actual trace-norm estimates. · OAI.UltrapowerBridge.traceCongruence

TraceCornerBound def

The original manuscript hypothesis is uniform over all projections of the selected trace in the other free factor. · OAI.KadisonSimilarity.TracialCornerGeometry.TraceCornerBound

traceCornerElements def

The test elements supported by a projection of the selected trace. · OAI.KadisonSimilarity.TracialCornerGeometry.traceCornerElements

traceDirection def

Barycentric direction, with value zero at the empty trace. · OAI.EilenbergGanea.TraceWords.traceDirection

traceDistance def

The normalized trace two-distance between genuine unitaries of the original algebra. · OAI.RelativeGeneration.traceDistance

traceDotForm def

The trace dot product, as an actual binary bilinear form. · OAI.MaxCutGames.Quadratic.traceDotForm

traceDotForm def

The trace dot product, as an actual binary bilinear form. · OAI.UniqueGamesTheorem.Quadratic.traceDotForm

traceDualEquiv def

The exact binary trace-pairing identification. · OAI.MaxCutGames.Quadratic.traceDualEquiv

traceDualEquiv def

The exact binary trace-pairing identification. · OAI.UniqueGamesTheorem.Quadratic.traceDualEquiv

traceExcursion def

The same specified heights, together with nonnegative segments and final first hit. · OAI.FKCRT.SimpleWalk.traceExcursion

traceFamily def

Ordered analytic product; factors may be noncommuting. · OAI.CoordinateSweeps.TraceHolder.traceFamily

traceFreeAmbientExtension def

Compress the ambient GNS representation to the canonical trace space. · OAI.KadisonKastler.traceFreeAmbientExtension

traceFreeBinaryPartition def

Choose the actual two-leg partition supplied by trace-free halving. · OAI.Generator.Tracial.traceFreeBinaryPartition

traceFreeCopy def

Algebraic identification only: it does not identify the two norms. · OAI.Generator.Tracial.traceFreeCopy

traceFreeDimension def

The same constructed lower cut, now normalized at the algebra identity. · OAI.Generator.Tracial.traceFreeDimension

traceFreeDyadicCut def

Membership contains an actual partial-isometry witness through Subequivalent. · OAI.Generator.Tracial.traceFreeDyadicCut

traceFreeDyadicMatrixUnits def

Actual matrix units of sizes 2^n, with coherent inclusions. · OAI.Generator.Tracial.traceFreeDyadicMatrixUnits

traceFreeDyadicPartition def

A coherent family: each successor halves the previous common initial projection once and transports that halving to all its equivalent pieces. · OAI.Generator.Tracial.traceFreeDyadicPartition

TraceFreeHilbert abbrev

The actual Hilbert completion; no representation is supplied. · OAI.Generator.Tracial.TraceFreeHilbert

traceFreeHilbertClosure def

The bicommutant is a von Neumann algebra; no image-closedness is assumed. · OAI.Generator.Tracial.traceFreeHilbertClosure

traceFreeHilbertClosureTrace def

Output tracial representation on the actual bicommutant. Its fields are proved above; no TracialRepresentation is an input to this construction. · OAI.Generator.Tracial.traceFreeHilbertClosureTrace

traceFreePositiveFunctional def

The constructed trace as a positive linear functional. · OAI.KadisonKastler.traceFreePositiveFunctional

TraceFreePreHilbert def

Algebraic type copy indexed by the original factor hypotheses. · OAI.Generator.Tracial.TraceFreePreHilbert

tracefreeProjection def

Linear local SL chart: project away scalar matrices after translating the exact target to the identity. No abstract manifold chart is assumed. · OAI.HarmonicCounterexample.FiniteControl.Frobenius.tracefreeProjection

tracefreeProjection def

Linear local SL chart: project away scalar matrices after translating the exact target to the identity. No abstract manifold chart is assumed. · OAI.HarmonicCounterexample.FiniteControl.tracefreeProjection

traceFreeRelativeImage def

The actual WOT-closed image of the original given subalgebra. · OAI.Generator.Tracial.traceFreeRelativeImage

traceFreeSubalgebraClosure def

The double commutant of the represented intersection with Q. · OAI.Generator.Tracial.traceFreeSubalgebraClosure

traceFreeTuple def

Coordinate-free trace-zero tuples of actual endomorphisms. · OAI.HarmonicCounterexample.Control.traceFreeTuple

traceGNSPositive def

The represented image of a positive element in the GNS weak closure. · OAI.TraceConeClassification.ExtendedTrace.traceGNSPositive

traceHomeomorph def

The transported metric topology agrees with the trace-ball topology on genuine unitaries. · OAI.Generator.FactorTraceUnitary.traceHomeomorph

traceK0 def

Coefficient at the identity class of the Hattori--Stallings trace on K₀. · OAI.TorsionFreeBass.traceK0

traceLift def

One binary-linear lift on the trace-coordinate domain, obtained from the single common parent map. It is fixed before the outgoing line/orientation. · OAI.MaxCutGames.Gadget.BlockDescent.traceLift

traceLift def

The explicit trace lift 3/5 δij S + 1/10 Σ I_s,ij Σ_s S. · OAI.PositiveEinsteinFour.TraceHessian.traceLift

traceLift def

One binary-linear lift on the trace-coordinate domain, obtained from the single common parent map. It is fixed before the outgoing line/orientation. · OAI.UniqueGamesTheorem.Gadget.BlockDescent.traceLift

traceLog def

A regularized logarithm evaluated by a trace. · OAI.KervaireTraceCalculus.traceLog

traceMass def

Law of a single complete trace. · OAI.TraceReconstruction.traceMass

traceMatrix def

Supermatrix of the unnormalized trace map X ↦ trace(X) I. · OAI.MUB6.Weiner.traceMatrix

traceNorm def

The canonical trace two-norm is the norm of the identity-vector image. · OAI.UltrapowerBridge.traceNorm

traceNormalWeight def

The supremum of the finite-vector functionals over finite-trace strict contractions. · OAI.TraceConeClassification.ExtendedTrace.traceNormalWeight

traceOfMask def

true retains the bit. The observer gets only the resulting list. · OAI.TraceReconstruction.traceOfMask

traceOneNorm def

The tracial one-norm, using the C*-algebra absolute value. · OAI.KadisonKastler.traceOneNorm

TraceOrthogonal def

Orthogonality to the base algebra, equivalently zero conditional expectation onto it. · OAI.KadisonKastler.TraceOrthogonal

tracePair def

The binary trace pairing between oppositely oriented matrices. · OAI.MaxCutGames.Fourier.MatrixCharacters.tracePair

tracePair def

The binary trace pairing between oppositely oriented matrices. · OAI.UniqueGamesTheorem.Fourier.MatrixCharacters.tracePair

tracePairCLM def

Real trace pairing as a continuous linear functional. · OAI.SharpLiebThirring.MatrixProof.tracePairCLM

tracePairedMechanisms def

The paired experiment uses the actual unseparated complete deletion traces, and the actual normalized minimum of their mass functions. · OAI.TraceReconstruction.tracePairedMechanisms

TracePairingApproximation def

The norm-density property for arbitrary vector functionals. · OAI.KadisonKastler.TracePairingApproximation

tracePoints def

Successive specified heights, with free terminal suffix. · OAI.FKCRT.SimpleWalk.tracePoints

traceQuotient def

The trace quotient interface used by the constructed coordinate embedding into the original factor ultrapower. · OAI.KadisonKastler.FiniteMatrixUltraproduct.traceQuotient

traceQuotient def

The published ultrapower supplies every premise of the constructed expectation, with its canonical trace. · OAI.KadisonKastler.TracialUltrapower.traceQuotient

TraceQuotientMap structure

The trace formula and quotient map furnished by a tracial ultrapower. The target trace is an explicit parameter so its construction stays separate from the elementary expectation proofs. · OAI.KadisonKastler.TraceQuotientMap

TraceRel def

The trace-null equivalence relation, before descending algebra operations. · OAI.UltrapowerBridge.TraceRel

TraceSeparable def

Separability for the tracial two-norm, rather than the operator norm. · OAI.KadisonKastler.TraceSeparable

traceSpace def

Coordinates of a binary character subspace under the trace pairing. · OAI.MaxCutGames.Gadget.BlockDescent.traceSpace

traceSpace def

All possible complete outputs from length-n inputs, with lengths included. · OAI.TraceReconstruction.traceSpace

traceSpace def

Coordinates of a binary character subspace under the trace pairing. · OAI.UniqueGamesTheorem.Gadget.BlockDescent.traceSpace

traceStateEmbedding def

Completion of the trace-state isometry. · OAI.KadisonKastler.traceStateEmbedding

traceStatePreEmbedding def

The algebraic trace Hilbert space embeds in the extended state's GNS space. · OAI.KadisonKastler.traceStatePreEmbedding

traceStateRight def

Right multiplication, supplied by the same tracial GNS construction. · OAI.KadisonKastler.traceStateRight

TraceStrongContinuous def

Continuity from the tracial two-norm to the strong topology on each norm-bounded ball. This is the exact normal-representation property used by the clipping argument; no existence assertion is made here. · OAI.KadisonKastler.TraceStrongContinuous

traceTopology abbrev

The topology generated by trace two-distance balls, not the inherited operator norm topology. · OAI.RelativeGeneration.traceTopology

traceTV def

1/2 sum of absolute differences, on the full finite output space. · OAI.TraceReconstruction.traceTV

traceTwoNorm def

The tracial two-seminorm, on the faithful tracial algebra in use. · OAI.KadisonKastler.traceTwoNorm

traceUnitaryHomeomorph def

A metric presentation with the trace two-distance gives precisely the trace-ball topology on all genuine unitaries. · OAI.RelativeGeneration.traceUnitaryHomeomorph

TraceVectorApproximation def

Norm approximation of every vector functional by a trace functional. The order of the inner product makes the vector functional complex-linear. · OAI.KadisonKastler.TraceVectorApproximation

traceVectorForm def

A finite-trace positive element defines a vector functional. · OAI.TraceConeClassification.ExtendedTrace.traceVectorForm

traceWeakClosure def

The weak-operator closed star algebra of the square-integrable representation. · OAI.TraceConeClassification.ExtendedTrace.traceWeakClosure

tracialCenter def

Center a represented algebra vector by its actual orthogonal projection. · OAI.KadisonSimilarity.tracialCenter

TracialCommutationInput def

The finite-trace specialization of the tracial commutation theorem: Hiai, Theorem 2.2 and Example 2.4 (JMJ = M′); equivalently every bounded left-module endomorphism of L²(P,τ) is right multiplication by an element of P. This remains … · OAI.KadisonSimilarity.TracialCommutationInput

tracialConjugation def

The actual tracial conjugation is obtained by extending algebraic involution from the dense represented algebra. · OAI.KadisonSimilarity.tracialConjugation

tracialConjugationDefect def

The complex-linear comparison operator of two conjugations. · OAI.KadisonKastler.tracialConjugationDefect

TracialCornerCyclicityInput def

The local tracial module realization needed after comparison: a partial isometry from r into e realizes the corner representation on eL²(P)r as a cyclic module. In a concrete tracial model this is proved using the unitary ξ ↦ ξ v*, … · OAI.KadisonSimilarity.TracialCornerCyclicityInput

TracialCornerGeometry structure

The concrete data of a tracial Hilbert representation. · OAI.KadisonSimilarity.TracialCornerGeometry

TracialCreationFormulas abbrev

The standard free-product model supplies these elementary word identities. All letters and tensor factors are the actual centered tracial base spaces; no complete-prefix estimate is part of the data. · OAI.KadisonSimilarity.TracialCreationFormulas

tracialCreationFormulas def

The normalized Haar letters are among the centered algebra letters in the published model, so the raw propagation formulas need no new hypothesis. · OAI.KadisonSimilarity.ReducedWordCreationModel.tracialCreationFormulas

tracialDiagonalFockModel def

The faithful GNS action has the complete structural Fock model once the finite diagonal family generates its algebra. · OAI.KadisonKastler.tracialDiagonalFockModel

tracialDihedralColumns def

The actual finite dihedral column family on the tracial base. · OAI.KadisonSimilarity.TracialCornerGeometry.tracialDihedralColumns

tracialDihedralInclusion def

Actual dihedral word inclusion, with all base-space and compression data supplied by the ambient tracial representation. · OAI.KadisonSimilarity.TracialCornerGeometry.tracialDihedralInclusion

tracialEndpointWord def

The other-factor endpoint and centered internal letters. · OAI.KadisonSimilarity.tracialEndpointWord

TracialExpectation structure

The minimal trace-pairing data for expectation onto a tracial subalgebra. Existence is required only when a subalgebra is realized as a von Neumann subalgebra; no existence assertion for arbitrary star subalgebras is made. · OAI.KadisonSimilarity.TracialExpectation

TracialHeadCoordinates structure

Grouping reduced words by their possible initial D letter yields these coordinates. The sole action field is ordinary left multiplication on that initial tensor factor; no propagation estimate is included. · OAI.KadisonSimilarity.TracialHeadCoordinates

tracialHilbertConjugation def

The constructed involution, packaged as a conjugate-linear isometric equivalence. · OAI.KadisonKastler.tracialHilbertConjugation

tracialMatrixHaarProbability def

Actual normalized Haar measure of the finite matrix unitary group. · OAI.KadisonSimilarity.TracialCornerGeometry.tracialMatrixHaarProbability

tracialMatrixLeftRepresentation def

The matrix stage acts by actual ambient left multiplication. · OAI.KadisonSimilarity.TracialCornerGeometry.tracialMatrixLeftRepresentation

tracialMatrixStage def

The concrete normalized matrix stage associated to a trace-preserving matrix star homomorphism and the left tracial representation. · OAI.KadisonSimilarity.tracialMatrixStage

TracialMatrixTower structure

Local tracial realization data for the nested matrix stages. In particular, there is no invariant coefficient formula or intertwiner classification here. · OAI.KadisonSimilarity.TracialMatrixTower

tracialMatrixTowerOfEmbeddings def

Every scalar geometric field of the invariant-theory tower is now derived from the actual tracial representation and raw matrix embeddings. The only remaining sequence data is an algebraic real orthonormal sequence. · OAI.KadisonSimilarity.tracialMatrixTowerOfEmbeddings

tracialMatrixTowerOfHaar def

A Haar unitary removes the independent real-orthonormal-sequence input from the tracial matrix tower construction. · OAI.KadisonSimilarity.tracialMatrixTowerOfHaar

tracialMatrixUnitaryOperator def

Forgetting the unitary subtype gives the group homomorphism consumed by the weak Haar integral. · OAI.KadisonSimilarity.TracialCornerGeometry.tracialMatrixUnitaryOperator

tracialMatrixUnitaryRepresentation def

The finite matrix group acts by its constructed free-product automorphisms. · OAI.KadisonSimilarity.TracialCornerGeometry.tracialMatrixUnitaryRepresentation

TracialMatrixUnitColumnFamily structure

The frame together with the corner trace used in small-corner estimates. · OAI.KadisonSimilarity.TracialMatrixUnitColumnFamily

tracialMatrixUnitColumnFamilyOfEmbedding def

A trace-preserving unital matrix embedding supplies both frame and trace. · OAI.KadisonSimilarity.tracialMatrixUnitColumnFamilyOfEmbedding

tracialMatrixUnitColumnFamilyOfNonempty def

A nonempty finite matrix index type provides a distinguished corner without an additional choice of frame data. · OAI.KadisonSimilarity.tracialMatrixUnitColumnFamilyOfNonempty

tracialMatrixWordAction def

The genuine completed endpoint tensor action of a matrix unitary. · OAI.KadisonSimilarity.tracialMatrixWordAction

tracialMatrixWordCovariance def

Actual matrix-unitary intertwining supplies the word covariance used by the finite invariant theory. The source and test word formulas are proved here. · OAI.KadisonSimilarity.tracialMatrixWordCovariance

tracialMinimalMultiplication def

The finite tracial amenability representation, constructed from the expectation by the existing minimal tensor extension theorem. · OAI.WeakPureInfiniteness.tracialMinimalMultiplication

tracialModularConjugationData def

A tracial state gives modular conjugation data. · OAI.KadisonKastler.tracialModularConjugationData

tracialMultiplicityWord def

The base-algebra part of an elementary word. · OAI.KadisonSimilarity.tracialMultiplicityWord

TracialOperatorPartition structure

A central partition into two classes for which the primitive is proved. · OAI.BoundedHochschild.KadisonRingrose.TracialOperatorPartition

tracialOrthogonalWordLevels def

The completed orthogonal word decomposition, built from freeness. · OAI.KadisonSimilarity.tracialOrthogonalWordLevels

TracialQuotient abbrev

Underlying ring of the tracial ultrapower when l is a free ultrafilter. · OAI.UltrapowerBridge.TracialQuotient

tracialQuotientTrace def

The quotient trace is faithful because exactly the two-null ideal was removed. · OAI.KadisonKastler.tracialQuotientTrace

tracialRangeIsometry def

The completed trace-preserving map lands in the actual closed range space. · OAI.KadisonSimilarity.tracialRangeIsometry

tracialRangeUnitary def

The range completion is unitarily the original tracial GNS space. · OAI.KadisonSimilarity.tracialRangeUnitary

TracialRepeatedWordRealization structure

The remaining raw reduced-word geometry for a fixed positive degree. All operators acting on the ambient space are specified as left multipliers. There is no approximation estimate among these fields. · OAI.KadisonSimilarity.TracialRepeatedWordRealization

tracialRepeatedWordRealizationOfCreation def

Construct the propagation realization from elementary reduced-word creation and multiplication. Both complete-prefix fields are conclusions. · OAI.KadisonSimilarity.tracialRepeatedWordRealizationOfCreation

TracialRepresentation structure

A concrete finite tracial von Neumann algebra in a faithful vector-trace representation. These are operator-algebra data, without approximation assumptions. · OAI.FiniteFactor.TracialRepresentation

TracialRepresentation structure

A concrete finite tracial von Neumann algebra in a faithful vector-trace representation. These are operator-algebra data, without approximation assumptions. · OAI.UltrapowerBridge.TracialRepresentation

tracialRepresentationOfVectorState def

A faithful vector trace supplies precisely the concrete tracial representation data. · OAI.KadisonKastler.tracialRepresentationOfVectorState

TracialRightAction structure

The right multiplication convention, including faithfulness. Composition has the opposite order: R_b ∘ R_a = R_(a*b). · OAI.KadisonSimilarity.TracialRightAction

TracialRightSupportInput def

The application-specific consequence of support projections and polar decomposition (Blackadar, I.5.2.1--3): the right support of qv has trace at most that of its containing left projection q. · OAI.KadisonSimilarity.TracialRightSupportInput

tracialSmallColumn def

The actual small-column map is left multiplication by the projection, restricted to the completed base Hilbert space. · OAI.KadisonSimilarity.TracialCornerGeometry.tracialSmallColumn

tracialStarIsometry def

The adjoint map extends isometrically to the trace Hilbert space. · OAI.KadisonKastler.tracialStarIsometry

TracialTypeII def

Precisely the trace data needed by the proved tracial primitive theorem. · OAI.BoundedHochschild.KadisonRingrose.TracialTypeII

TracialUltrapower structure

A represented tracial ultrapower, with the published coordinate trace formula. The quotient is a concrete von Neumann algebra. · OAI.KadisonKastler.TracialUltrapower

tracialUltrapowerCompression def

Compression of the constant Hilbert-ultrapower operator to the tracial quotient's GNS space. No invariance of the original operator is needed. · OAI.KadisonSimilarity.tracialUltrapowerCompression

tracialUltrapowerGNSInclusion def

The closed GNS subspace of the Hilbert ultrapower, constructed by completion. · OAI.KadisonSimilarity.tracialUltrapowerGNSInclusion

tracialUltrapowerMap def

The induced star homomorphism of tracial ultrapowers. · OAI.KadisonSimilarity.tracialUltrapowerMap

tracialUnitaryConjugation def

Left multiplication followed by the conjugate right multiplication gives the actual trace-preserving matrix-conjugation isometry. · OAI.KadisonSimilarity.tracialUnitaryConjugation

tracialVacuumSubspace abbrev

Transport the actual vacuum subspace through tracial word coordinates. · OAI.KadisonSimilarity.tracialVacuumSubspace

tracialWordCoefficient def

Coefficients with the test letters conjugated by the tracial involution. · OAI.KadisonSimilarity.tracialWordCoefficient

TracialWordFormula def

Regrouped ordinary free-product words. The multiplicity space and the endpoint space are fixed concrete tensor products of the two base spaces. · OAI.KadisonSimilarity.TracialWordFormula

tracialWordHeadAction def

Left multiplication acts only on the first other-factor endpoint. · OAI.KadisonSimilarity.TracialCornerGeometry.tracialWordHeadAction

tracialWordInclusion def

The genuine isometric inclusion of one completed word level. · OAI.KadisonSimilarity.tracialWordInclusion

tracialWordPrefix def

The degree-m + 1 alternating algebra prefix. · OAI.KadisonSimilarity.tracialWordPrefix

tracialWordRepeatedOperator def

The canonical multiplicity compression for the actual matrix tower. · OAI.KadisonSimilarity.TracialCornerGeometry.tracialWordRepeatedOperator

tracialWordTensor def

The tensor product of the base-word and endpoint factors. · OAI.KadisonSimilarity.tracialWordTensor

track def

A homotopy with its time coordinate in the cylinder's second factor. · OAI.Thomason.Construction.TopologicalDeformation.Data.track

track def

Product with the interval preserves the colimit used to assemble the tracks. · OAI.Thomason.Construction.TopologicalDeformation.Sequential.track

trackCocone def

Tracks form a cocone after product with the interval. · OAI.Thomason.Construction.TopologicalDeformation.Sequential.trackCocone

trackLeg def

Finite tracks mapped into the actual colimit cocone. · OAI.Thomason.Construction.TopologicalDeformation.Sequential.trackLeg

trajectoryPlacement def

Position zero is the caller's starting vertex; later positions are the separate physical trajectory tapes. · OAI.MaxCutGames.Foundations.Complexity.PoweringMachineTapes.trajectoryPlacement

trajectoryPlacement def

Position zero is the caller's starting vertex; later positions are the separate physical trajectory tapes. · OAI.UniqueGamesTheorem.Foundations.Complexity.PoweringMachineTapes.trajectoryPlacement

trans def

Compose actual comparisons, including explicit bounds for the errors of their composite inverse maps. · OAI.PolycyclicRecognition.MetricQuasiEquivalence.trans

transactionBound def

A deterministic bound on the number of completed transactions. · OAI.MatroidProphet.MarkedQueryProgram.transactionBound

TransF def

The semantic transitivity clause, with unrestricted HS names. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.TransF

Transfer def

A term transformer under one temporary proof input. No product rule is postulated for its input/output pair; that rule generally does not exist. · OAI.PureTypeSystem.Labelled.Transfer

transfer def

The genuine elementary transfer id + T E_xy from the source's edge schedule, expressed on vectors rather than with a choice of enumeration. · OAI.KLInvariance.Comparison.transfer

transfer def

Transfer epsilon from donor i to receiver j, exactly as in §04. · OAI.KServer.SimplexTracker.transfer

transfer def

Transport a local monoid across a split pair. · OAI.OneWayLiveness.Corner.transfer

transfer def

Move one head symbol if present, popping the first register before updating the second, including when both registers coincide. · OAI.RationalHitting.StackLanguage.transfer

transferAtInTime def

A timed execution witness suitable for composing this bridge with other phases of the same actual transition system. · OAI.MaxCutGames.Reduction.MachineTransfer.transferAtInTime

transferAtInTime def

A timed execution witness suitable for composing this bridge with other phases of the same actual transition system. · OAI.UniqueGamesTheorem.Reduction.MachineTransfer.transferAtInTime

TransferClaim def

Full sample-to-rank contract for the actual rounded greedy survivor groups. · OAI.MatroidProphet.TransferClaim

transferConjugations def

true means complex conjugation, as in rationalTreeAmplitude. · OAI.Ostmann.transferConjugations

transferCopySign def

The true branch is the positive copy. · OAI.Ostmann.transferCopySign

transferFrequencyRange def

A fixed covering range for the signed output frequency. Zero is excluded in the summand, so the same interval also works when the range is empty. · OAI.Ostmann.transferFrequencyRange

TransferHistorySystem structure

A state records the current actual labels. Child states insert the reconstructed pivot and retain the corresponding branch. · OAI.Ostmann.TransferHistorySystem

transferInt def

A single unit transfer, allowing source and receiver to coincide. · OAI.ContingencyTables.transferInt

transferLeafList def

The flag records the conjugations accumulated from right branches. · OAI.Ostmann.transferLeafList

transferNextGap def

The gap for step j+1. · OAI.Ostmann.transferNextGap

transferOrbit def

Actual chronological product of endpoint transmission operators. · OAI.HarmonicCounterexample.Cycling.transferOrbit

transferredAgreement def

Literal uniform target agreement on the projected affine slice. The slice origin is an analysis witness, independent of the canonical visible row base. · OAI.MaxCutGames.Decoder.SelectedDecoding.transferredAgreement

transferredAgreement def

Literal uniform target agreement on the projected affine slice. The slice origin is an analysis witness, independent of the canonical visible row base. · OAI.UniqueGamesTheorem.Decoder.SelectedDecoding.transferredAgreement

transferredGraph def

none is the erased pivot; some (inl h) is a copied slot and some (inr y) is a shared outside slot. · OAI.Ostmann.transferredGraph

transferredMap def

The left translation by h, transferred using f and r. · OAI.PolycyclicRecognition.transferredMap

transferredMatrixRetraction def

A finite-stage retraction on the original ambient operators. · OAI.KadisonKastler.transferredMatrixRetraction

transferStep def

A finite-table transfer step. The source is popped; a prescribed subset of other stacks receives its top symbol. An empty source returns true. · OAI.ParityWL.Complexity.transferStep

transferWeight def

Add the weight at a to the parallel edge b, leaving a temporary zero. · OAI.KaplanskyCounterexample.Planar.transferWeight

transform def

Preparation, actual clean triangular gate program, inverse preparation. The success event is the all-zero phase and scratch register; it is not enforced. · OAI.ExactQuantumFactoring.Triangular.transform

transform def

The actual function on variable-size regular input tables. · OAI.MaxCutGames.Foundations.PCP.PoweringTables.transform

transform def

The elementary transform constructs another admissible extension and preserves its punctured framing. · OAI.SourceConeMorphism.ExtensionState.transform

transform def

The actual function on variable-size regular input tables. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringTables.transform

transformedColumn def

The coefficient map is the adjoint of the map acting on pointwise coordinate values. · OAI.RegularPointCoordinates.transformedColumn

transformedPolarCone def

The target functionals whose pullbacks lie in the source cone. · OAI.PolycyclicRecognition.transformedPolarCone

transformSequence def

Sequence G⊗J|X → E'|X → ker(P→G) of the elementary transformation formula. · OAI.ActualCartier.transformSequence

Transition def

One transition appends exactly one actual selected map and its detector. · OAI.ReflexiveFixedPoints.CandidateContext.Transition

Transition structure

A doubly stochastic transition rule, with no entry exceeding one half. · OAI.TorsionFreeZeroDivisors.WeightedMoore.Transition

transition def

The disjoint union of the neighborhood transition kernels. · OAI.CliqueFreeIndependence.WeightedGraph.LocalWalk.transition

transition def

Lebesgue density supported on strictly positive radii. · OAI.FKCRT.Radial.transition

transition def

Right multiplication by a transition factor. This map is deliberately not assumed injective: the construction has only left cancellation. · OAI.PartitionConsistency.ProbeAlgebra.Path.transition

transition def

Gauge-coordinate fundamental matrix; both time directions are encoded. · OAI.SmoothObstruction.EuclideanSplitting.transition

transitionAverage def

Normalized residual law on the true point stabilizer. The reference is any compatible transition; it is a unitary frame, not a new hypothesis on the law. · OAI.CoordinateSweeps.PlacementInduction.Chart.transitionAverage

TransitionCache abbrev

The scalar is the multiplicity of the ignored completion-bit suffix. · OAI.ContingencyTables.ProfilePrograms.TransitionCache

transitionCacheMeasure def

The cache is charged by the finite completion box and the requested reserved word length; no random-tape enumeration appears in this measure. · OAI.ContingencyTables.ProfilePrograms.transitionCacheMeasure

TransitiveCommutant def

Closed common invariant subspaces, including all closed complex subspaces. · OAI.Hyperinvariant.TransitiveCommutant

transitiveSet def

Transitivity of a set (not transitivity of the external structure). · OAI.PartitionConsistency.SetLanguage.transitiveSet

translate def

Translation on the actual Lebesgue L² space. · OAI.Bernoulli.Mollification.translate

translate def

Integer translations act on this pullback without changing the base. · OAI.CircleDomainRigidity.FiniteTransfer.CircleLiftCover.translate

translate def

Push a probability law forward by translation by c. · OAI.MetricMeasureLimits.DiameterCenter.translate

translate def

All requester windows and layer coordinates travel with their one record. · OAI.Superstring.Hierarchical.HostedGroup.Request.translate

translate def

Translate a hosted record by a whole number of host periods, transporting its entire child text, not only its interval endpoints. · OAI.Superstring.Hierarchical.HostedGroup.translate

translate def

Simultaneous coordinate transport of a child text and its layer. · OAI.Superstring.Hierarchical.Layer.translate

translatedGroupBoxProbability def

The normalized restriction to the actual translated unit box. · OAI.PolycyclicRecognition.CartanWordModel.translatedGroupBoxProbability

translatedPrefixAverage def

The average of the D translated positive prefixes. · OAI.TwoPointCorrelations.translatedPrefixAverage

translateFamily def

Translation invariance survives the coefficient construction. · OAI.KadisonKastler.ScalarInvariantMean.translateFamily

translatePair def

Normalize domain translation, on pairs of continuous maps. · OAI.PolycyclicRecognition.translatePair

Translation structure

A set of translated positions; the integral shift is positive. · OAI.TorsionFreeZeroDivisors.BlockConstraints.Translation

translation def

The fibre-valued left regular representation. · OAI.Dixmier.KernelObstruction.translation

translation def

A one-variable translation is obtained by fixing all but one input of an operation. · OAI.FiniteCongruence.translation

translation def

A fixed translation does not increase any derivative norm. · OAI.ForcedComputation.BoundedSpatialJets.translation

translationComputableInPolyTime def

Binary-translation producers discharge the only table-law condition. · OAI.MaxCutGames.Explicit.MachineSubdivisionCompose.translationComputableInPolyTime

translationComputableInPolyTime def

Binary-translation producers discharge the only table-law condition. · OAI.UniqueGamesTheorem.Explicit.MachineSubdivisionCompose.translationComputableInPolyTime

translationDefect def

Uniform defect bound for translations on the kernel of the linear action. · OAI.PolycyclicRecognition.HeightQuasiAction.translationDefect

translationIso def

Translation uses the same named coordinate offset in every incident block. · OAI.WLIdentification.AffineBlocks.translationIso

translationPulse def

The translation pulse used for both extraction and insertion. · OAI.ForcedComputation.PlanarHamiltonian.translationPulse

translationTable def

Both forward and inverse tables are explicitly generated from every alphabet index. Characteristic two makes these two tables equal. · OAI.MaxCutGames.Reduction.Encoding.translationTable

translationTable def

Both forward and inverse tables are explicitly generated from every alphabet index. Characteristic two makes these two tables equal. · OAI.UniqueGamesTheorem.Reduction.Encoding.translationTable

translationTube def

A filled tube containing every intermediate translated rectangle. · OAI.ForcedComputation.PlanarRouting.translationTube

transmission def

Actual center-regular value transmission, including the complete diagonal leg and with no pulse scalar normalization. · OAI.HarmonicCounterexample.LinearODE.HistoryGeometry.transmission

Transport structure

Old parallel transport preserving the subsystem and old flag. · OAI.IntegralCharacterVarieties.IntrinsicInterval.Transport

Transport structure

Domain transport is computed from old/new held parameters and new rank inputs. For the manuscript it is the prepared update followed by deletion and zero insertion. It is not permitted to depend on the unrealized future. · OAI.KServer.AdaptiveMinimization.Transport

transport def

Relabeling outcomes does not change a finite conditional probability law. · OAI.DilutedSpinGlass.FiniteLaw.transport

transport def

Actual first-order matrix transport on compactly supported smooth tests. · OAI.ElasticityPlanar.transport

transport def

All concrete pointed-product data survive spatial transport. · OAI.KadisonKastler.PointedMatrixSites.transport

transport def

Half-trace completion and both support identities survive the factor embedding, as does the subsequently constructed centered reflection. · OAI.KadisonSimilarity.HalfTraceProjectionData.transport

transport def

A single embedding carries every piece of the small-corner data, with the original numerical cutoff unchanged. · OAI.KadisonSimilarity.SmallFreeCornerData.transport

transport def

Move all ambient vectors through a linear equivalence, retaining the same noise indices and their multiplicities. · OAI.MaxCutGames.Integration.SplitGadget.transport

transport def

Simulation on literal merger normal forms, by transporting each coefficient and each final-letter quotient class. · OAI.PartitionConsistency.MergerAlgebra.Word.transport

transport def

Simulation on literal strings. Reducedness will be proved, not postulated or hidden in a quotient of these words. · OAI.PartitionConsistency.ProbeAlgebra.Raw.transport

transport def

Smaller irreducible words remain irreducible in the larger presentation. · OAI.PartitionConsistency.ProbeAlgebra.Word.transport

transport def

Transport of the glued component to the moved Cohen filter, retaining its original chosen root. · OAI.PartitionConsistency.ProductForcing.ChainCode.transport

transport def

Literal transport carries the entire local-name collection to the moved diagram, including normalization and the full supporting fixer. · OAI.PartitionConsistency.ProductForcing.LocalName.transport

transport def

Transport of local symbolic names is between F and πF. This is not an action on some fixed evaluated universe. · OAI.PartitionConsistency.ProductForcing.LocalQuotient.transport

transport def

Change the linear coordinates on height, retaining the word multiplicities. · OAI.PolycyclicRecognition.SupportDatum.transport

transport def

Transport of a derivation through an algebra equivalence. · OAI.StableCoordinate.CoordinateDerivation.transport

transport def

Transport a derivation across an algebra equivalence. · OAI.StableCoordinate.LocalizationLND.transport

transport def

Move all ambient vectors through a linear equivalence, retaining the same noise indices and their multiplicities. · OAI.UniqueGamesTheorem.Integration.SplitGadget.transport

transportAction def

An isomorphism transports an automorphism action. · OAI.FiniteLattice.CommonCore.transportAction

transportAction def

Conjugating an action along an isomorphism. · OAI.LittleFinitistic.FormalSummandAction.transportAction

transportBoundedHom def

Transport a bounded homomorphism through an algebra equivalence and an isometric Hilbert-space equivalence. · OAI.KadisonSimilarity.SimilarityUniverseTransfer.transportBoundedHom

transportClosure def

The one finite set used to select a common partition level. · OAI.ThompsonNonamenability.transportClosure

transportCompactProfile def

Restriction to a changed compact chart, scalar normalization and the constant change of center. · OAI.PolycyclicRecognition.transportCompactProfile

transportCompletion def

Transport every restricted completion along one fixed basis change. · OAI.MaxCutGames.Inverse.KMSBasisComparisonPseudorandom.transportCompletion

transportCompletion def

Transport every restricted completion along one fixed basis change. · OAI.UniqueGamesTheorem.Inverse.KMSBasisComparisonPseudorandom.transportCompletion

transportCW def

Closed embeddings transport classical CW structures, including their weak topology, with precisely the same cells. · OAI.GroupRingTopology.Cellular.transportCW

transportDerivation def

Pull a derivation back through an actual algebra equivalence. · OAI.AbhyankarSathaye.CommutingDerivations.transportDerivation

transportDerivation def

Conjugation through the proved, concrete associated-graded equivalence. · OAI.StableCoordinate.ReesToDegeneration.transportDerivation

Transported def

The extra neutral invariant is exactly the G-prefix invariant in the manuscript's spine argument. · OAI.PureTypeSystem.ComponentInterface.Transported

transported def

Corner coefficients are conjugated, whereas the complementary isometry is moved by left multiplication, as in the finite Fock split. · OAI.WeakPureInfiniteness.BufferAbsorptionCorner.transported

transportedAcceptedThrough def

Irrevocable accepted labels in the original labeled stream. · OAI.MatroidProphet.Relabeling.transportedAcceptedThrough

transportedBilinear def

Transported multiplication as a bundled complex-bilinear product. · OAI.KadisonKastler.transportedBilinear

transportedDecision def

The transported decision uses only the seed, original samples, and history. · OAI.MatroidProphet.Relabeling.transportedDecision

transportedFiberEquiv def

Transfer the proved nonzero-fiber equivalence through a separately proved presentation equivalence that carries p to a specified polynomial. · OAI.StableCoordinate.NonzeroFiber.transportedFiberEquiv

transportedMatrix def

Both axes are reindexed by e. Columns encode the two coordinate indices. · OAI.EuclideanRamsey.NineRational.transportedMatrix

transportedMul def

Multiplication on the source transported through a linear equivalence. · OAI.KadisonKastler.transportedMul

transportedPositiveTrace def

Positive functional in the coordinates of the target algebra. · OAI.KadisonSimilarity.transportedPositiveTrace

TransportedProductBound def

The row-column defect of the transported multiplication. This is the concrete inequality used to invoke the published multiplication theorem. · OAI.KadisonKastler.TransportedProductBound

transportedReward def

Real reward, with no rounding or loss in the relabeling. · OAI.MatroidProphet.Relabeling.transportedReward

transportedSchedule def

Decreasing list, in the convention for the outermost-first matrix product. Its reverse is chronological increasing reflection order. · OAI.KLInvariance.BruhatGraph.transportedSchedule

transportedSecretaryDecision def

Exact decision on original labels, reading only the current available history. · OAI.MatroidProphet.Relabeling.transportedSecretaryDecision

transportEquiv def

Transport through the two faithful representations. · OAI.KadisonKastler.CommonTracialPositionData.transportEquiv

transportEquiv def

Literal name transport is a bijection of the whole ground collection, including the support and normalization requirements. · OAI.PartitionConsistency.ProductForcing.LocalName.transportEquiv

transportEquiv def

Source (symbolic-transport) is a genuine equivalence, not only a map. It does not assert an automorphism of one fixed interpreted model. · OAI.PartitionConsistency.ProductForcing.LocalQuotient.transportEquiv

transportEquiv def

The identity on underlying sections, with the scalar action transported along a coefficient-ring isomorphism. · OAI.TensorChangeScalars.transportEquiv

transporterEquiv def

A transporter fiber is a coset of the literal word stabilizer. · OAI.CoordinateSweeps.OrdinaryTensor.transporterEquiv

transporterEquiv def

All transporters between two words are exactly the independent bijections between corresponding color fibres. · OAI.RowColumn.WordOrbits.transporterEquiv

transportFlag def

Transport a degree flag through an invertible linear weight operator. · OAI.LeanBlast.GotsmanLinial.transportFlag

transportHom def

The literal normal-form embedding respects the defining reductions. · OAI.PartitionConsistency.ProbeAlgebra.Word.transportHom

transportInTime def

One-step simulations transport complete timed executions through the explicit finite register permutations. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.RuntimeModel.transportInTime

transportInTime def

One-step simulations transport complete timed executions through the explicit finite register permutations. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.RuntimeModel.transportInTime

transportKaroubiIso def

An isomorphism commuting with idempotents identifies their formal summands. · OAI.LittleFinitistic.QuadraticEncoding.transportKaroubiIso

transportMarked def

Target classes whose original traversals can use a rectangle from the fixed set J. · OAI.TwoWayComplementation.Diagram.transportMarked

transportPoly def

Rename the polynomial variables bijectively and include them in the fraction field. · OAI.EuclideanRamsey.NineRational.transportPoly

transportRealTracialSequence def

The real trace-orthonormal sequence is transported algebraically. · OAI.KadisonSimilarity.transportRealTracialSequence

transportResolvent def

The damped free-transport resolvent, with the incoming characteristic parametrization. · OAI.BoltzmannNonuniqueness.Published.transportResolvent

transportTracialMatrices def

A concrete matrix tower stays dense after a trace-preserving embedding. · OAI.KadisonSimilarity.transportTracialMatrices

transpose def

Transport the dual of a linear map along two perfect pairings. · OAI.Problem346.DualEmbedding.transpose

transposeField def

Reverse orientation exchanges the two mask fields and label arguments. · OAI.MaxCutGames.Foundations.PCP.PoweringRowData.transposeField

transposeField def

Reverse orientation exchanges the two mask fields and label arguments. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringRowData.transposeField

transposeSimplex def

Deletion plus zero insertion. At zero synthetic total choose a fresh canonical vector; this has exactly zero synthetic weighted movement. · OAI.KServer.ChangingDomains.transposeSimplex

transvection def

A matrix elementary operation on columns in an arbitrary left module. · OAI.FiniteAlgebraicEnvelopes.AffineAction.transvection

transversalEquiv def

The matching bijects its left and right transversal endpoints. · OAI.OneWayLiveness.Pairing.transversalEquiv

Transverse def

This is an actual genericity condition on linear functionals, not an assumed path or Hecke identity. · OAI.KLInvariance.TitsSpace.Transverse

transverse def

The transverse field. Its slope bound is inherited from the original field (and at least one), with the switch identity proved. · OAI.PlanarL1.Columns.Model.Field.transverse

transverseEuclideanIsometry def

Choose orthonormal transverse coordinates without changing the Euclidean norm used by the spherical fiber estimates. · OAI.AffineBernstein.transverseEuclideanIsometry

transverseExpression def

The local scalar expression of the b-th transverse derivative, written in a fixed finite family of actual holomorphic functions. · OAI.Nagata.W20.transverseExpression

transverseHilbertFlow def

The transverse geodesic flow is actual geodesic motion after a quarter turn and restoration of the direction, with no global commutation assumed. · OAI.TriangularBilliards.transverseHilbertFlow

TransversePolygons structure

Only proper interior meetings between the two polygonal walks. · OAI.Paper170.TransversePolygons

transverseVelocity def

The transverse velocity in the oriented sheet. · OAI.TriangularBilliards.transverseVelocity

traversal def

Actual complete tuple traversal, before the final output reversal. · OAI.MaxCutGames.Reduction.AddressMachineLoop.traversal

traversal def

Actual complete tuple traversal, before the final output reversal. · OAI.UniqueGamesTheorem.Reduction.AddressMachineLoop.traversal

Tree abbrev

The countably branching, finite-height, root-inclusive tree. · OAI.BoundedTreePotentials.Tree

Tree inductive

A finite coloured binary splitting forest with one root. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Tree

Tree inductive

Finite uniform branching includes each finite positive-refinement path forest by padding absent children with zero vectors. No triangle defect is assumed. · OAI.IntrinsicMarks.Tree

Tree inductive

Finite uniform branching includes each finite positive-refinement path forest by padding absent children with zero vectors. No triangle defect is assumed. · OAI.SignedDisk.IntrinsicMarks.Tree

tree def

An abstract height tree, including zero-length branches. · OAI.PlanarL1.Weighted.EnlargedTree.tree

treeAncestorScheme def

The selected ancestors are genuine prefixes. Fixed label products and which prefixes enter each coefficient may vary with the actual node. · OAI.Ostmann.treeAncestorScheme

treeBranch def

The side of an oriented tree edge towards its second endpoint. · OAI.HyperbolicObstruction.PlanarRecursion.treeBranch

treeBudget def

Number of nodes in the full k-ary tree of depth m. · OAI.UniformKServer.treeBudget

treeCell def

Horizontal Whitney strip crossed with one transverse cell in the reference coordinates of a tree. · OAI.LipschitzHilbert.SourceWhitney.treeCell

treeChronology def

Public chronological IDs. Each child carries its own local paired sides; the group's global sides and deterministic fillers surround that child. The order is fixed independently of all bits and both hypotheses. · OAI.TraceReconstruction.treeChronology

treeChronologyEquiv def

A genuine complete chronological bijection, not an assumed layout. · OAI.TraceReconstruction.treeChronologyEquiv

TreeCode inductive

Finite structured imperative code. Loops are data-dependent, not bounded by an external oracle. The cost counts the serialized registers traversed by the stack compiler. · OAI.ParityWL.Complexity.TreeCode

TreeContext inductive

A literal prescribed-tree context with one designated descendant hole; sibling branches are retained as genuine shapes, not erased from the test. · OAI.DilutedSpinGlass.PrescribedTree.TreeContext

TreeCoordinate abbrev

Coordinate type: the root is omitted only in the rootless sibling row. · OAI.BoundedTreePotentials.TreeCalculus.TreeCoordinate

TreeData structure

Data of the actual constructed ensemble. Retention and trace identities are verified by the particle recursion, not axioms about its endpoint map. · OAI.BoundedTreewidthL1.ExpandedConstruction.TreeData

treeDensityMajorant def

Parameter-free density majorant, including the zero convention. · OAI.Ostmann.treeDensityMajorant

TreeExec inductive

Work is the sum of one plus the size of all live data before each step. The compiler bound below is linear in this actual execution measure. · OAI.ParityWL.Complexity.TreeExec

treeExtensions def

Genuine multigraph spanning-tree completions of a partial exploration. · OAI.FKCRT.treeExtensions

TreeFrontier structure

The current maps and the evaluation strictly above a finite cutoff. · OAI.ReflexiveFixedPoints.TreeFrontier

treeIntegerResidues def

Equality of all leaf residues, expressed on the original binary tuple. · OAI.Ostmann.treeIntegerResidues

treeLeafCopyPath def

Left arithmetic children use the positive copy of H. · OAI.Ostmann.treeLeafCopyPath

treeLeafFiberSuccEquiv def

Exposing the product in the left half leaves independent product fibers below it. · OAI.Ostmann.treeLeafFiberSuccEquiv

treeLeafSplitEquiv def

Preorder split products are coordinates: root split first, followed by the left subtree and the right subtree. Each coordinate is an independent Haar unit even after the total leaf product is fixed. · OAI.Ostmann.treeLeafSplitEquiv

TreeMachine structure

Finite code over unbounded but explicitly serialized binary trees. · OAI.ParityWL.Complexity.TreeMachine

treeMean def

Recursive conditional mean of a descendant spin product. · OAI.DilutedSpinGlass.PrescribedTree.treeMean

TreeOutput def

`TreeOutput C x bits` means there exists a duplicate-free list of valid edge copies whose branch-specific serialization is exactly `bits` and whose selected edge set is `ThinTree C`. · OAI.AlgorithmicThinTrees.TreeOutput

treeOverlap def

The normalized product of all leaves, with the exact zero-size convention. · OAI.DilutedSpinGlass.PrescribedTree.treeOverlap

treePath def

A total extension of the path vector; the off-tree values are never used. · OAI.ForestSpace.Vertex.treePath

TreePathClosed def

Convexity with respect to the unique simple paths in a tree. · OAI.BoundedTreewidthL1.TreePathClosed

treePathHull def

Union of the root paths to a finite collection of terminals. · OAI.HyperbolicObstruction.PlanarRecursion.treePathHull

treeProductPolynomial def

A convenient product polynomial on the physical top coordinates. · OAI.LipschitzHilbert.treeProductPolynomial

TreeProgram structure

A verified program on a concrete tree encoding. time is a proved bound on both work and intermediate storage, never an oracle interpretation of code. · OAI.ParityWL.Complexity.TreeProgram

treeScan def

Sweep one prefix tree into dst in reverse order, tracking unvisited child slots on a unary stack. A null pending stack terminates; all transitions are constant-size finite-table operations. · OAI.ParityWL.Complexity.treeScan

treeScore def

The exact tree-score summand for one term of the full reservoir. · OAI.DilutedSpinGlass.HeterogeneousMarks.treeScore

treeSpace def

The quotient by d_e=0, its induced metric, and pushforward of uniform time. The equivalence closure is redundant for continuous e; the induced distance is independent of representatives, as required by the pseudometric proof. · OAI.FKCRT.treeSpace

treeSum def

Literal summation over all nodes up to height n. · OAI.BoundedTreePotentials.Segments.treeSum

treeTestFamily def

The four rows are specializations of this construction, not identified norms. · OAI.BoundedTreePotentials.TreeCalculus.treeTestFamily

trialAcceptOn def

Mode guards are outside the rational computations. In particular malformed labels and residue guesses cannot turn an irrelevant retention into acceptance. · OAI.ExactQuantumFactoring.OrderTrial.trialAcceptOn

TrialCountData abbrev

Cached walk vector, shared bit allowance, and the table/margin data with the requested precision. · OAI.ContingencyTables.DensePrograms.TrialCountData

trialLog def

Expectation over the root laws eta_0 of each independent message label. This also gives the r=0 convention (iid real messages, no power means). · OAI.DilutedSpinGlass.trialLog

trialResult def

Arithmetic validation is performed in every mode, before any returned candidate is used. The plain scan/guess never serves as an order certificate. · OAI.ExactQuantumFactoring.OrderTrial.trialResult

trialSmallValueNet def

Only VALID ordinary candidates are narrowed to n bits. This does not apply to the separately completed rare guess, whose whole transition word is kept. · OAI.ExactQuantumFactoring.OrderTrial.trialSmallValueNet

trialSplitEquiv def

Split a finite draw sequence into its first draw and remaining sequence. · OAI.LeanBlast.SecondKahnKalai.trialSplitEquiv

Triangle structure

Three affinely independent vertices in the Euclidean plane. · OAI.TriangularBilliards.Triangle

triangle def

Real triangular wave, period one; quotient norm makes continuity explicit. · OAI.ExactQuantumFactoring.OrderTrial.triangle

triangleAt def

Twice the triangle mass in the weighted neighborhood of u. · OAI.CliqueFreeIndependence.WeightedGraph.triangleAt

triangleBoundary def

The literal based integral boundary of the crossing three-cubes. · OAI.EilenbergGanea.LevelBasis.triangleBoundary

triangleDisplacement def

Triangle displacement pattern: zero at all nine original marks and one at each extra point. · OAI.Nagata.Workers.W14.triangleDisplacement

TriangleFilling structure

A finite genus-zero map with a prescribed exterior walk. Other components are allowed but ignored; only the component meeting the boundary must be filled. This handles components detached by folds, spurs, and empty boundary words. · OAI.Release075.TriangleFilling

TriangleMarking structure

The three roles are old side (0), incoming radial side (1), and outgoing radial side (2). Frozen polygons have only old sides. These are local incidence conditions on an existing side-pairing, not a genus or chain assumption. · OAI.HyperbolicObstruction.PictureSwitch.TriangleMarking

triangleMass def

Unordered triangle mass, written as one sixth of the ordered triangle mass. · OAI.CliqueFreeIndependence.WeightedGraph.triangleMass

TriangleMesh structure

Oriented finite triangular incidence data. Faces/edges may have repeated incidences; no topological assumption is built into this record. · OAI.ArtinCAT0.TriangleMesh

triangleOrderedConfiguration def

Every r≥9 has the concrete r-point triangle configuration in the genuine U_r. · OAI.Nagata.W06.LineArrangement.triangleOrderedConfiguration

TrianglePatch structure

A triangular disk may have unpaired boundary darts. No topological theorem is hidden in this interface: exactness is required and subsequently proved for the explicit grid disks used in the controlled-diagram construction. · OAI.ArtinCAT0.TrianglePatch

trianglePath def

The actual triangular closed boundary, with a root radial omitted exactly when it is collapsed in the finite CW model. No directed-edge duplication. · OAI.TorsionFreeZeroDivisors.ConePicture.trianglePath

TriangleRibbon structure

An oriented combinatorial triangle filling. Darts are incoming at the corner they carry; reverse * face rotates those corners about a vertex. This records finite incidence data, NOT existence of a filling for a nullword. Its single … · OAI.Release075.TriangleRibbon

triangleSet def

The whole two-skeleton of a finite standard simplex. · OAI.EilenbergGanea.AffineTransport.triangleSet

TriangleSphere structure

A finite labelled spherical map, possibly with several connected components. All faces are triangles; there is no distinguished outer face. · OAI.Release075.TriangleSphere

triangleWord def

The orientation of a face is L x R⁻¹, not a permuted product. · OAI.Release075.BlockPresentation.triangleWord

Triangular def

This is a support property of the decoded expressions, independent of all numerical values assigned to the prime labels. · OAI.TwoPointCorrelations.WitnessSystemData.Triangular

triangular def

Algebraic upper triangular operator product, without any norm assertion. · OAI.Dixmier.triangular

triangular def

The integer-valued quadratic appearing in a bi-infinite Laurent orbit. · OAI.Nagata.W08.triangular

TriangularAdjoint structure

A simultaneous real upper triangular basis for the adjoint representation. · OAI.PolycyclicRecognition.TriangularAdjoint

TriangularAdjoint structure

A simultaneous real upper triangular adjoint basis, including its actual matrix inequalities. This is the linear-algebra meaning of real triangulability. · OAI.PolycyclicRecognition.TriangularAdjoint

triangularBaseEquiv def

The induced invertible horizontal map of a triangular ambient affine map. · OAI.AffineBernstein.triangularBaseEquiv

triangularChoiceFiberEquiv def

Each triangular lookup fiber is exactly a finite interval of the claimed integer length. · OAI.EditApproximation.triangularChoiceFiberEquiv

triangularHilbertRepresentation def

The same triangular homomorphism on the actual Hilbert direct sum. · OAI.KadisonKastler.triangularHilbertRepresentation

triangularLattice def

Triangular lattice with nearest-neighbor spacing 4/5. · OAI.Problem160.triangularLattice

triangularLinearInverse def

Explicit inverse for the initial linearized endpoint map, apart from its nonzero scalar factor -1/ε. · OAI.SharpLiebThirring.PathProof.triangularLinearInverse

triangularMatrixBasis def

The ordered matrix units. · OAI.PolycyclicRecognition.triangularMatrixBasis

triangularMatrixCoordinates def

Matrix coordinates with respect to this ordering. · OAI.PolycyclicRecognition.triangularMatrixCoordinates

triangularMatrixOrder def

Rows increase and columns decrease in the ordering of matrix units. · OAI.PolycyclicRecognition.triangularMatrixOrder

triangularOperator def

The bounded upper triangular operator with equal diagonal blocks. · OAI.KadisonSimilarity.Derivation.triangularOperator

triangulation def

A triangular rotation system obtained by starring every face; old edge copies are preserved, and each old face has a distinct new center. · OAI.HyperbolicObstruction.PictureSwitch.PictureMap.triangulation

TriCylinder structure

A nondegenerate, closed triangular base in the perpendicular linear plane. · OAI.TriangularCovering.TriCylinder

trigPolys def

Trigonometric polynomials, as a subspace of continuous functions. · OAI.FourierLLogL.trigPolys

trim def

Retain the inclusive list if space remains; otherwise evict its oldest unprotected nonroot entry. The fallback is unreachable on admissible input. · OAI.BoundedTreewidthL1.AnchorCache.trim

trimMSD def

Removes the high zero bits of a most-significant-first word. · OAI.RationalHitting.StackLanguage.trimMSD

Trio structure

Uniform finite list of pre-incoming figures; product of independent scalar pieces. · OAI.ExactDerandomization.At.Trio

Trip def

deterministic integrity along a plateau list. · OAI.FourierLLogL.Dyadic.Procedure.Paths.Trip

Triple abbrev

A set of three distinct colors; sorting it gives its unique increasing triple. · OAI.PixtonChow.Triple

triple def

The manuscript's explicit three-plane complex structure: the disjoint coordinate swaps turn the standard pairs into (X_i,X_j), (Y_i,X_k), (Y_j,Y_k). It is an ACTUAL orthogonal complex structure, with no matrix equations assumed. · OAI.HarmonicCounterexample.Berger.ComplexStructure.triple

tripleCellLabel def

The product label map corresponding to the cell table. · OAI.Problem356.tripleCellLabel

tripleEnumeration def

Increasing enumeration of all three-color blocks in lexicographic order. · OAI.PixtonChow.tripleEnumeration

tripleLexOrder def

Lexicographic order on the sorted lists representing the triples. · OAI.PixtonChow.tripleLexOrder

tripleMatrixFlatten def

Flatten three matrix indices, retaining the outer-to-inner order. · OAI.KadisonSimilarity.tripleMatrixFlatten

tripleMoment def

The actual averaged fourth moment attached to a raw derivative triple. · OAI.MaxCutGames.Appendix.A13Reduction.tripleMoment

tripleMoment def

The actual averaged fourth moment attached to a raw derivative triple. · OAI.UniqueGamesTheorem.Appendix.A13Reduction.tripleMoment

triplePotential def

The sum of the marginal potential over the three coordinates. · OAI.Problem356.triplePotential

tripleRevealed def

Whether at least one of the three observation coins is set. · OAI.MatroidProphet.Secretary.tripleRevealed

triplesInTime def

The machine's bit-length bound includes every peek, literal run, and tally. · OAI.MaxCutGames.BinaryRenameLoop.triplesInTime

triplesInTime def

The machine's bit-length bound includes every peek, literal run, and tally. · OAI.UniqueGamesTheorem.BinaryRenameLoop.triplesInTime

tripleSpace def

The common equality locus of three affine scores. · OAI.Problem358.TripleTieAvoidance.tripleSpace

tripleSplit def

Regroup a prefix, its next scalar coordinate, and the final remainder. · OAI.Problem358.FinitePrefix.tripleSplit

tripleSumEquiv def

Literal coordinates for an ordered direct sum of three subspaces. · OAI.SidorenkoCounterexample.tripleSumEquiv

tripleTerm def

The weight after relabeling the actual raw derivative triple. · OAI.MaxCutGames.Appendix.A13Reduction.tripleTerm

tripleTerm def

The weight after relabeling the actual raw derivative triple. · OAI.UniqueGamesTheorem.Appendix.A13Reduction.tripleTerm

tripleWeight def

The independent H,D,C law at one label, retaining the thinning parameter. · OAI.MatroidProphet.Secretary.tripleWeight

trivialIntegral abbrev

The trivial integral representation used to compute integral finiteness. · OAI.PolycyclicRecognition.trivialIntegral

trivialLineCore def

A single global identity chart for the trivial complex line. · OAI.Kaplansky.trivialLineCore

trivialMapFrame def

Columns of a continuous injective map from a genuinely trivial stage are a continuous independent frame of its image. · OAI.Kaplansky.trivialMapFrame

trivialPowerIso def

The iterated tensor power trivialization, derived from the given line trivialization, retaining its compatibility with the ideal action. · OAI.CartierImageFiltration.trivialPowerIso

trivialTensorIso def

The tensor functor trivialization of a trivial line. · OAI.ActualSheafTensor.trivialTensorIso

trN def

Normalized trace over coordinates. · OAI.GeneralMahler.trN

trueBlocks def

The nonempty maximal contiguous blocks of positions satisfying p. · OAI.Problem335.WordRuns.trueBlocks

trueCost def

The source's c_i, extended by zero to unlisted true groups. · OAI.MatroidProphet.AccountingContracts.trueCost

trueCount def

The number of entries on which a Boolean function is true. · OAI.MaxCutGames.Fourier.Deletion.trueCount

trueCount def

The number of entries on which a Boolean function is true. · OAI.UniqueGamesTheorem.Fourier.Deletion.trueCount

TrueLaw structure

Include balance even to a zero-mass destination. · OAI.FourierLLogL.Dyadic.Procedure.Fork.TrueLaw

trueRelation def

The actual 64-by-64 always-accepting relation. · OAI.MaxCutGames.Foundations.Complexity.MachineDummyRows.trueRelation

trueRelation def

The actual 64-by-64 always-accepting relation. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineDummyRows.trueRelation

truncate def

Forget higher regularity without changing the represented function. · OAI.ForcedComputation.BoundedSpatialJets.truncate

truncateAmplitude def

Truncating purification letters preserves positivity after partial trace. It is how the postselection mixture is constrained to even one-slot states. · OAI.RowColumn.Postselection.truncateAmplitude

truncateCLM def

The lossless bounded linear map forgetting higher spatial jets. · OAI.ForcedComputation.BoundedSpatialJets.truncateCLM

truncated def

Truncated point distance on EVERY column. Columns outside the metric component have the constant ceiling. · OAI.PlanarL1.Columns.Model.Point.truncated

truncatedDensity def

The radial Jacobian, set to zero once a conjugate point has occurred. · OAI.PositiveEinsteinFour.Jacobi.RegularJacobi.truncatedDensity

truncatedDivisorCount def

Number of positive divisors at most a real cutoff. · OAI.Problem337.truncatedDivisorCount

truncatedDivisors def

Positive divisors truncated at an integral threshold. · OAI.Problem337.DivisorMoment.truncatedDivisors

truncatedExcess def

The excess mass after every coordinate is truncated at the same cutoff. · OAI.TalagrandThreshold.Selector.truncatedExcess

truncatedField def

The symmetric cutoff is part of the definition, not replaced by a potentially non-absolutely-convergent whole-line Bochner integral. · OAI.SharpLiebThirring.truncatedField

truncatedInv def

A reciprocal truncated at Y, with its correct value at the singular point. The special value is needed for geometric exponential sums. · OAI.Problem337.RationalPhaseSpacing.truncatedInv

truncatedPoset def

A strict poset is an object of every positive finite dimension. · OAI.Thomason.Construction.FiniteReflectionInput.truncatedPoset

truncatePath def

Truncate every value of a continuous jet path. · OAI.ForcedComputation.PlaneScalarMild.truncatePath

truncation def

A finite polynomial truncation of a coefficient-compatible series. · OAI.SectionCompletion.truncation

truncationEmbedding def

Forget the finite level bound without changing a node or its candidate label. · OAI.ReflexiveFixedPoints.SwitchingTreeData.truncationEmbedding

TruncationNode abbrev

All nonroot nodes in a finite truncation, with their actual finite levels. · OAI.ReflexiveFixedPoints.OrderedTree.TruncationNode

TruncationSplit structure

A splitting of the actual truncation triangle, retaining its inclusion and projection. · OAI.LittleFinitistic.TwoDegreeFormality.TruncationSplit

truncHom def

Multiplication modulo z^t in the coefficient basis is a literal ring homomorphism. · OAI.ExactFourier.CoefficientTime.truncHom

Trust structure

Input ring geometry. · OAI.ExactDerandomization.Chime.Ember.Trust

Trusted def

f and J ambient original data. · OAI.FourierLLogL.Dyadic.Procedure.Road.Trusted

truth_second def

true stochastic row represented by the same building blocks. · OAI.ExactDerandomization.Net.Quest.truth_second

truthTable def

Explicit DNF with one possible conjunction for each of the 2^B rows. · OAI.TwoPointCorrelations.AC0Circuit.truthTable

truthVectors def

Truth fixes a coordinate to zero; false coordinates remain free. · OAI.WLIdentification.BufferAlgebra.truthVectors

tube def

The literal fixed-atlas coordinate tube, with chart sources checked. Only the base point chooses the chart; the second point need not be good. · OAI.SmoothObstruction.SelectedTube.tube

tube def

The actual open metric tube around the range of the curve. · OAI.ThompsonNonamenability.Analytic.RegularCurveData.tube

tubeAffineAreaJet def

The actual second-jet affine-area density in the base and orthonormal angular directions; the transverse column is (0,-e). · OAI.AffineBernstein.tubeAffineAreaJet

TubeIncident def

A point lies in the closed Euclidean δ-neighborhood of the line with slope l 0 and intercept l 1. · OAI.Falconer.TubeIncident

tubeIntegral def

The unrestricted product integral of a cutoff times a tube coefficient. · OAI.AffineBernstein.tubeIntegral

tubeMeasureCoefficient def

Literal invariant tube measure coefficient with respect to `ds dω`. · OAI.AffineBernstein.tubeMeasureCoefficient

tubeMeasureDensity def

The manuscript's invariant density `μ₀` on base times unit normals. · OAI.AffineBernstein.tubeMeasureDensity

tubeRadius def

The radius comes from proved uniform separation and the explicit speed bounds. · OAI.ThompsonNonamenability.Analytic.tubeRadius

tuple def

The normalized chain represented by a weakly increasing tuple. · OAI.Thomason.Construction.SimplexComplex.tuple

tupleAugmentation def

Augmentation of zero-chains. · OAI.Kaplansky.tupleAugmentation

tupleBitLength def

Total physical length of the saved occurrence-ID tapes. · OAI.MaxCutGames.Reduction.MachineSourceTuple.tupleBitLength

tupleBitLength def

Total physical length of the saved occurrence-ID tapes. · OAI.UniqueGamesTheorem.Reduction.MachineSourceTuple.tupleBitLength

tupleChainSpan def

Integral chains spanned by tuples satisfying a specified support condition. · OAI.Kaplansky.tupleChainSpan

tupleCode def

Explicit tuple address, with coordinate zero the least significant digit. · OAI.UniqueGamesTheorem.Explicit.ProductOutput.tupleCode

tupleCoefficient def

The convenient rational upper bound on the selector's coefficient. · OAI.TalagrandThreshold.tupleCoefficient

tupleColor def

Joint update: one multiset of whole replacement vectors, not independent marginals. · OAI.WLIdentification.tupleColor

tupleCone def

Prepending a fixed vertex, extended linearly on actual free chains. · OAI.Kaplansky.tupleCone

tupleDropMiddle def

Drop the distinguished middle position of a word. · OAI.KadisonKastler.tupleDropMiddle

tupleEmbed def

Inclusion of all child shifts before imposing the parent kernel condition. · OAI.MaxCutGames.Gadget.tupleEmbed

tupleEmbed def

Inclusion of all child shifts before imposing the parent kernel condition. · OAI.UniqueGamesTheorem.Gadget.tupleEmbed

tupleEnergy def

The quadratic form of the matching Laplacian. · OAI.NuclearUltrapower.tupleEnergy

tupleEvent def

Samples for which the specified tuple of sampled values satisfies bad. · OAI.MetricEntropyDuality.tupleEvent

tupleGapMap def

Maps abstract gap coordinates to their actual source slots. · OAI.TraceReconstruction.tupleGapMap

tupleHullMesh def

Support condition recording both the original hull and the new mesh. · OAI.Kaplansky.tupleHullMesh

tupleInsertMiddle def

Insert the missing label between the two subwords. · OAI.KadisonKastler.tupleInsertMiddle

tupleLower def

Bounds for the open gaps of an increasing tuple; slots and ranks are zero-based. · OAI.TraceReconstruction.tupleLower

tupleOfMask def

Canonical increasing tuple associated with a selected set of ranks. · OAI.TraceReconstruction.tupleOfMask

tuplePolynomial def

The actual symbolic evaluation on a tuple of vector blocks. · OAI.MetricEntropyDuality.tuplePolynomial

TuplePrimeAt def

A tuple prime occurs when it divides the whole tuple at that position. · OAI.TwoPointCorrelations.TuplePrimeAt

tupleRestriction def

The q-fold evaluation/restriction/quotient map used by the incidence locus. · OAI.Kaplansky.Incidence.tupleRestriction

tuples def

The product order is increasing radix address, with coordinate zero the least significant digit. The inverse computes each digit by division/modulo. · OAI.MaxCutGames.Explicit.ProductTarget.tuples

tuples def

The product order is increasing radix address, with coordinate zero the least significant digit. The inverse computes each digit by division/modulo. · OAI.UniqueGamesTheorem.Explicit.ProductTarget.tuples

tupleSimplicialHomotopy def

The usual prism is a genuine simplicial homotopy. · OAI.PolycyclicRecognition.tupleSimplicialHomotopy

tupleSourceOrder def

Clause, slot, left-query, noise, right-query loops, from outermost to innermost. Within each tuple loop coordinate zero changes fastest. · OAI.MaxCutGames.Foundations.Hastad.SourceLoopOrder.tupleSourceOrder

tupleSourceOrder def

Clause, slot, left-query, noise, right-query loops, from outermost to innermost. Within each tuple loop coordinate zero changes fastest. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceLoopOrder.tupleSourceOrder

tupleSSet def

Ordered tuples, with deletion and repetition of vertices. · OAI.Kaplansky.tupleSSet

tupleSSetMap def

Applying a function to each vertex defines a simplicial map. · OAI.Kaplansky.tupleSSetMap

tupleStabilizer def

Finite tuples have a common subgroup in the given normal filter. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.tupleStabilizer

tupleSubdivision def

A constructed chain endomorphism, for any supplied centers. · OAI.Kaplansky.tupleSubdivision

tupleSubdivisionHomotopy def

Subdivision is chain homotopic to the identity. · OAI.Kaplansky.tupleSubdivisionHomotopy

tupleSubdivisionMap def

A center is chosen for each positive-dimensional tuple. The chain identity does not require a metric; barycenters will supply mesh control. · OAI.Kaplansky.tupleSubdivisionMap

tupleSubdivisionPrism def

The prism is filled by a cone after subtracting its prescribed boundary. This recursion constructs the homotopy; it is not a homotopy premise. · OAI.Kaplansky.tupleSubdivisionPrism

tupleSubset def

Forget the ordering of the distinct labels. · OAI.KadisonKastler.tupleSubset

tupleSupport def

The set of indices appearing in an ordered tuple, including repeated-index tuples. · OAI.MetricEntropyDuality.tupleSupport

tupleSym def

The multiset of entries of a tuple, including every repetition. · OAI.MetricEntropyDuality.tupleSym

tupleTensorMatrix def

The matrix of a tensor product of finitely many matrix operators. · OAI.KadisonSimilarity.tupleTensorMatrix

tupleWeight def

Weight of the union of an injective tuple. · OAI.TalagrandDiscreteConvexity.Covering.tupleWeight

turnedExtend def

Injective continuation along fresh rows. · OAI.CriticalHoneycomb.FiniteContinuation.turnedExtend

turnEquiv def

Literal indexing equivalence from internal subdivision vertices to the half-integer-grid turn positions. · OAI.TorsionFreeZeroDivisors.ChainWeights.turnEquiv

turnGradient def

Passive rotation of all three slots of a matrix-valued covector. · OAI.PositiveEinsteinFour.QuaternionFrame.turnGradient

turnPoint def

A circular connector, with its actual normal ribbon coordinate a. The parameter runs from zero to pi, alternating above and below. · OAI.SLEExactGauge.Serpentine.turnPoint

turns def

Left transpositions in traversal order. · OAI.FKCRT.MapCycles.turns

tv def

Total variation distance of two finite probability vectors. · OAI.CliqueFreeIndependence.FiniteEntropy.tv

tv def

The finite total-variation convention of the source: half the l1 distance. · OAI.CoordinateSweeps.FiniteLaw.tv

tv def

Transvection carrying a chosen coordinate vector to a given norm-one t on disjoint support. · OAI.PowerSaving.Binary.OBase.tv

tVar def

Independent rational-function variables with arbitrary row labels. · OAI.EuclideanRamsey.NineRational.tVar

TWiseUniformDensity def

Every assignment to at most t selected coordinates has its uniform mass. · OAI.TwoPointCorrelations.TWiseUniformDensity

Twist structure

Restriction of the scalar action along a ring automorphism. · OAI.KLInvariance.ReflectionPairTranslation.Twist

twist def

Diagonal quantum-group weight, with q=exp(5πi/8). · OAI.CriticalHoneycomb.Transfer.FluxWard.twist

twist def

Coordinate permutation by the input group; precomposition is contravariant. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.twist

twist def

Remove the actual quotient homomorphism from every section action. · OAI.PolycyclicRecognition.NonabelianFactorSystem.twist

twistByCharacter def

Multiplication by a fixed Dirichlet character. · OAI.TwoPointCorrelations.twistByCharacter

twistedGraphBundleCoordinates def

Genuine linear coordinates for the original twisted bundle, over any subset of any product graph chart. · OAI.Kaplansky.twistedGraphBundleCoordinates

twistedHyperplaneLineCore def

The summands of (trivial(h) ⊕ H) tensored with the hyperplane line from the auxiliary factor. This is an explicit vector-bundle core. · OAI.Kaplansky.twistedHyperplaneLineCore

twistedHyperplaneSection def

A concrete section: auxiliary positive coordinates on the h trivial summands, and old positive coordinates multiplied by auxiliary coordinate 0 on the other summands. Its topological nonvanishing remains to be proved. · OAI.Kaplansky.twistedHyperplaneSection

twistedLayerQuotient def

The surjection from the jth conormal twist of the original restricted lattice onto its jth supported quotient (the ideal-image filtration). · OAI.ActualCartier.twistedLayerQuotient

twistedProjectiveDimensions def

The auxiliary factor for the stable obstruction has dimension h. · OAI.Kaplansky.twistedProjectiveDimensions

twistedQuotientCopy def

The shallow-to-deep edge operator, including its right holonomy twist. · OAI.CoarseAssembly.twistedQuotientCopy

twistedSectionFromVector def

Tensor an arbitrary old section with a homogeneous auxiliary coordinate, preserving the original bundle topologies. · OAI.Kaplansky.twistedSectionFromVector

twistedStandardSectionCoordinates def

Actual section coordinates in the proved tensor graph trivialization. · OAI.Kaplansky.twistedStandardSectionCoordinates

twistIter def

All integer powers of an intermediate twist in its coordinates. · OAI.ThreeTorus.twistIter

twistPrefix def

The induced coordinate permutation on finite prefixes. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.twistPrefix

two def

two independent retentions at rates u and t · OAI.ExactDerandomization.At.Ask.two

twoBlockNeutralGroup def

The block-diagonal determinant-one subgroup, expressed without a choice of bases in either block beyond the fixed coordinates. · OAI.DukePrimeDegree.twoBlockNeutralGroup

TwoBlockPartition structure

A chosen two-block partition. The fields contain only finite combinatorial partition data, no geometric or group-theoretic conclusion. · OAI.Release075.TwoBlockPartition

twoCenterProjectionSource def

The square root is taken only after the two summable energies are combined. Its square, rather than its square root, is summed. · OAI.RegularPointCoordinates.twoCenterProjectionSource

TwoCode structure

A deliberately generous finite codec. The last field stores k exceptional scalars rather than just one; the final numeric load still tends to zero. · OAI.KaplanskyCounterexample.FiniteGeometry.Candidate.TwoCode

twoColumnIso def

The ordinary two-column total identifies with the three-column API having zero relation column; its zero-column inclusion is unchanged. · OAI.ExternalTensor.Section4.SourceFiltration.twoColumnIso

TwoDegrees def

The exact cohomological support hypothesis in Section 4. · OAI.LittleFinitistic.DerivedTruncation.TwoDegrees

TwoDimensionalIndices structure

Explicit finite index data for vertices, edges and triangles. Every geometric field is an indexing identity, not an acyclicity assumption. · OAI.SimplicialChains.TwoDimensionalIndices

twoFormPairing def

The bilinear two-form contraction defining each Weyl block entry. · OAI.PositiveEinsteinFour.Geometry.twoFormPairing

twoFormWeitzenbockPotential def

The bounded potential in 1 + ∇*∇ = 2 R - 5 for harmonic two-forms when Ricci is normalized to 3g. · OAI.PositiveEinsteinFour.Geometry.twoFormWeitzenbockPotential

twoGroupCompositionExpansionMean def

Uniform averages in two independent weak-composition groups, with a shared term index. · OAI.Problem335.twoGroupCompositionExpansionMean

twoGroupGeometricExpansionValue def

The product-geometric expression for a two-group factorial expansion. · OAI.Problem335.twoGroupGeometricExpansionValue

twoGroupMean def

Independent uniform means, keeping the two occupation totals fixed separately. · OAI.Problem335.FactorialExpansion.twoGroupMean

TwoInfiniteSummandsImpossible def

The two infinite summands formulation, stated independently of its proof. · OAI.Ostmann.TwoInfiniteSummandsImpossible

twoJetCombination def

Literal real complete-jet M (N jets at0 and N atL²). · OAI.CriticalHoneycomb.FiniteFunctional.StripAsymptotic.twoJetCombination

TwoJetPolynomialApproximation def

Polynomial approximation of complete jets with uniform degree and exponential error. · OAI.CriticalHoneycomb.FiniteFunctional.StripAsymptotic.TwoJetPolynomialApproximation

TwoJetPolynomialCompression def

Fixed-parameter determinant compression for the complete jet family. · OAI.CriticalHoneycomb.FiniteFunctional.StripAsymptotic.TwoJetPolynomialCompression

twoKnotLabels def

Literal complete labels of M: N jets at 0, N at L^2. · OAI.CriticalHoneycomb.FiniteFunctional.StripAsymptotic.twoKnotLabels

twoLevelKernel def

The normalized two-removal kernel: line factors and the nested small coordinate factor are the only factors shared by two distinct indices. · OAI.ContingencyTables.twoLevelKernel

TwoNFA structure

A read-only 2NFA. The two endmarkers are Sum.inl 0 and Sum.inl 1. Move codes 0, 1, 2 mean left, stay, right. · OAI.TwoWayComplementation.TwoNFA

TwoPageDrawing structure

Equality with the actual circle/chord/inversion construction. · OAI.Paper170.TwoPageDrawing

TwoPath abbrev

One canonical orientation of each unoriented nonbacktracking two-edge path. · OAI.HyperbolicObstruction.IncidenceTemplate.TwoPath

twoPointThetaMultiply def

Restrict multiplication by the first theta factor to sections vanishing at the second point. · OAI.Nagata.W07.twoPointThetaMultiply

TwoQuasitraceConstraint inductive

Requirements on a function and its actual M₂ corner extension. The extension is deliberately not required to have unit value one. · OAI.Kaplansky.TwoQuasitraceConstraint

twoRatioConcreteCoordinates def

The original site coordinates generate the constructed stage closure. · OAI.KadisonKastler.twoRatioConcreteCoordinates

twoRatioConcreteStages def

Increasing full matrix stages with the required remote-copy moments. · OAI.KadisonKastler.twoRatioConcreteStages

twoRatioPointedProduct def

The two prescribed site families with their actual product-state representation. · OAI.KadisonKastler.twoRatioPointedProduct

TwoRatioProductModel structure

Concrete construction data for countably many sites of each of the two fixed matrix-state types. Finite complement moments, rather than irreducibility, are the retained input. · OAI.KadisonKastler.TwoRatioProductModel

TwoRays structure

The two genuine recursively generated rays, joined at an interior state. · OAI.TriangularBilliards.TwoRays

twoSampleReturn def

Probability of returning to A after a difference of two independent increments with law σ. · OAI.PolycyclicRecognition.twoSampleReturn

twoSidedBarBicomplex def

The literal two-sided bar bicomplex of an arbitrary nonnegative chain input. · OAI.DerivedBar.BimoduleTensor.twoSidedBarBicomplex

twoSidedBarFunctor def

A functor with both maps and differential defined on the explicit bar tensors. · OAI.DerivedBar.BimoduleTensor.twoSidedBarFunctor

twoSidedBarMap def

The two-sided bar map is defined on literal tensor coordinates. · OAI.DerivedBar.BimoduleTensor.twoSidedBarMap

twoSidedBarObjIso def

The same coherence for arbitrary X bimodule coefficients. · OAI.DerivedBar.Multibar.twoSidedBarObjIso

twoSidedBarPatternIso def

Coherence of a two-sided bar term with its literal X and D slots. · OAI.DerivedBar.Multibar.twoSidedBarPatternIso

twoSidedComplex def

Actual D-linear two-sided bar complex with degree-n term D ⊗_k A^⊗n ⊗_k N. · OAI.DerivedBar.twoSidedComplex

twoSidedDifferential def

The two-sided absolute bar differential after changing the free coefficient A to D. · OAI.DerivedBar.twoSidedDifferential

twoSidedIso def

Comparison to the literal two-sided bar complex used in the splitting. · OAI.LittleFinitistic.PDBarQuotient.twoSidedIso

twoSidedOrbit def

The finite two-sided orbit, indexed by independent choices. · OAI.KadisonKastler.twoSidedOrbit

twoSidedProjection def

Actual D-linear count projection on the two-sided bar term. · OAI.DerivedBar.SquareZero.twoSidedProjection

twoSidedSignedProjection def

The signed count operator preserves the exterior left D-action. · OAI.DerivedBar.SquareZero.twoSidedSignedProjection

TwoSidedTensor def

Balanced tensor with its two outside coefficient algebras recorded in the type. · OAI.LittleFinitistic.TwoSidedTensor

twoSidedTruncation def

The two hard-cutoff intervals in the directional Hilbert transform. · OAI.LipschitzHilbert.twoSidedTruncation

twoSidedTruncation def

The negative and positive parts of the original hard-truncated integral. · OAI.LipschitzHilbert.twoSidedTruncation

TwoSidedVertex abbrev

false is the left copy and true is the right copy. · OAI.MaxCutGames.Reduction.ActualOrbit.TwoSidedVertex

TwoSidedVertex abbrev

false is the left copy and true is the right copy. · OAI.UniqueGamesTheorem.Reduction.ActualOrbit.TwoSidedVertex

twoSidedWeight def

Count-r submodule of the two-sided bar term, with the leading D coefficient uncounted. · OAI.DerivedBar.SquareZero.twoSidedWeight

twoSidedWeightComplex def

A genuine D-module chain complex for each fixed number of square-zero entries. · OAI.DerivedBar.SquareZero.twoSidedWeightComplex

twoSidedWeightEquiv def

Canonical sum of homogeneous tensors, as a linear equivalence. · OAI.DerivedBar.SquareZero.twoSidedWeightEquiv

twoSiteCompressedInverseTerm def

Compression of the actual six-dimensional negative inverse term. · OAI.ContinuumCoulomb.twoSiteCompressedInverseTerm

twoSiteHopping def

Unit hopping amplitude, with the conventional minus sign. · OAI.ContinuumCoulomb.twoSiteHopping

twoSiteHoppingBlock def

The hopping block from the singly occupied spin space to virtual pairs. · OAI.ContinuumCoulomb.twoSiteHoppingBlock

twoSiteHoppingMatrix def

Matrix of the actual canonical hopping action on the two-electron sector. · OAI.ContinuumCoulomb.twoSiteHoppingMatrix

twoSiteInversePenaltyMatrix def

The inverse penalty vanishes on the singly occupied kernel. · OAI.ContinuumCoulomb.twoSiteInversePenaltyMatrix

twoSiteOccupancyPenalty def

The quadratic Coulomb occupancy penalty before the scalar -v shift. · OAI.ContinuumCoulomb.twoSiteOccupancyPenalty

twoSiteSecondOrder def

Exact negative second-order compression with excitation cost U-v. · OAI.ContinuumCoulomb.twoSiteSecondOrder

twoSiteWeight def

The individual diagonal weights of the two-dimensional state. · OAI.KadisonKastler.twoSiteWeight

twoSpinSwap def

Exchange of the two spin coordinates in the computational basis. · OAI.ContinuumCoulomb.twoSpinSwap

twoSpinTensor def

Tensor product written in the fixed four-state computational basis. · OAI.ContinuumCoulomb.twoSpinTensor

twoStateSummandProjection def

The projection onto one of the two original representation summands. · OAI.KadisonKastler.twoStateSummandProjection

TwoVectorKaplanskyInput def

Blackadar, I.9.1.3, specialized to two vectors, followed by quotient norm lifting. This precise published input contains no commutator estimate. · OAI.KadisonSimilarity.TwoVectorKaplanskyInput

type def

Specific source A0: two fixed odd-pair detours followed by the matched return. · OAI.PureTypeSystem.Labelled.ChannelReturn.type

typeContribution def

The character contribution of all words of one prescribed type. · OAI.MatrixMultiplication.AuxiliarySeparation.typeContribution

typedPartition def

Exact finite word normalizer for the map marginal. · OAI.FKCRT.typedPartition

TypedSubstitution def

A simultaneous substitution respects each declared variable's exact type. This is a premise only of the substitution metatheorem, not of the main target. · OAI.PureTypeSystem.Labelled.TypedSubstitution

TypedSubstitution def

A simultaneous substitution respects each declared variable's exact type. This is a premise only of the substitution metatheorem, not of the main target. · OAI.PureTypeSystem.TypedSubstitution

typeFamily def

The type-k family is the finite union over coordinate injections. Indexing by all injections avoids dependent coordinate casts; every k-element support is represented. · OAI.Problem326.Affine.typeFamily

TypeFP def

Type FP_n(ℤ): a projective resolution finite over ℤ[G] through degree n. · OAI.PolycyclicRecognition.TypeFP

TypeFPInfinity def

Type FP_∞(ℤ) means type FP_n(ℤ) in every finite degree. · OAI.PolycyclicRecognition.TypeFPInfinity

TypeI structure

A specified Type I expression. Child norming membership is imposed at its construction stage, outside this data structure. · OAI.SeparableQuotient.Norming.TypeI

typeICoefficient def

The short convolution coefficient appearing in Type I sums. · OAI.Problem337.Vaughan.typeICoefficient

typeICyclicEquiv def

The algebraic identification with all operators on the minimal cyclic space, constructed in the original representation. · OAI.KadisonKastler.typeICyclicEquiv

typeICyclicGroup def

Transfer a generating unitary group from the minimal cyclic space. · OAI.KadisonKastler.typeICyclicGroup

typeIIBlock def

The dyadic block with the actual arithmetic coefficients in Vaughan's identity. · OAI.Problem337.Vaughan.typeIIBlock

typeIICoefficient def

The coefficient appearing in the long-long Type II sum. · OAI.Problem337.Vaughan.typeIICoefficient

TypeIII1Factor structure

Type III₁ expressed by the published continuous-core characterization. The ordinary type III projection conditions are retained explicitly. · OAI.KadisonKastler.TypeIII1Factor

TypeIIIFactor structure

The ordinary nonzero type III factor conditions, with type III expressed as absence of nonzero finite projections. · OAI.KadisonKastler.TypeIIIFactor

typeIIIInitialSpace def

The closed initial Hilbert space associated to a concrete projection. · OAI.KadisonKastler.typeIIIInitialSpace

TypeIIInfinityFactor structure

The projection-theoretic description of a type II∞ factor. · OAI.KadisonKastler.TypeIIInfinityFactor

typeWeight def

The individual multinomial type of a word, evaluated at any integer weights. · OAI.CoordinateSweeps.OrdinaryTensor.typeWeight

typeWordCoordinate def

A coordinate on a retained split leg remembers its type word. · OAI.MatrixMultiplication.AuxiliarySeparation.typeWordCoordinate