Glossary: O

Definitions in this project whose names start with O. 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

OBase structure: Represent an orthonormal basis with full index set α. · OAI.PowerSaving.Binary.OBase
obj abbrev: The chosen colimit of the canonical-extension tower. · OAI.Thomason.Construction.InfiniteExtension.obj
obj def: Only finite powers of the simplicial interval are needed for homotopy groups. · OAI.Thomason.Construction.RealizedCube.obj
Object structure: The objects used by the construction retain the real margin together with an actual cancellative conical monoid; no saturation property is built in. · OAI.PartitionConsistency.MonoidConstruction.Object
object abbrev: The already constructed genuine derived iterate, viewed in integer degrees. · OAI.DerivedBar.IterateSupport.object
object abbrev: Genuine bar-derived iteration of the actual X over the actual D. · OAI.LittleFinitistic.SelectedBaseIterates.object
object abbrev: The actual derived object represented by the full B-valued fixed tensor complex. · OAI.LittleFinitistic.SelectedTensorSplitting.object
object def: The actual finitely generated projective right module eB. · OAI.LittleFinitistic.VertexProjectives.object
object def: The free monoid on one generator, with the source's concrete base heights. · OAI.PartitionConsistency.MonoidConstruction.Base.object
object_zero_isTerminal def: Its terminality uses the full strict functor laws, not just its objects. · OAI.Thomason.Construction.NormalizedSimplexIdentities.object_zero_isTerminal
objectAdjunction def: The object/codiscrete adjunction, proved from the explicit endpoint formula. · OAI.Thomason.Construction.StrictOmegaCategory.objectAdjunction
objectFunctor def: The actual zero-dimensional objects of a strict category. · OAI.Thomason.Construction.StrictOmegaCategory.objectFunctor
objectIndex def: The vertex index of either object boundary of an oriental cell. · OAI.Thomason.Construction.SimplexComplex.objectIndex
objective def: The finite-target variational objective as a function of affine intercepts. · OAI.Problem358.MaxPotential.objective
objective def: The objective is the final coordinate, retained through every projection. · OAI.UniformKServer.FourierMotzkin.objective
objectiveIntegrand def: Variational objective: the logarithmic endpoint (k=1), or the linear (k=2). · OAI.PettyProjection.Spherical.objectiveIntegrand
objectLabelsEquiv def: The explicit object/codiscrete bijection, without bundling functions as arrows. · OAI.Thomason.Construction.FiniteReflectionInput.objectLabelsEquiv
objectsEquiv def: The objects of the concrete oriental are canonically its ordered vertices. · OAI.Thomason.Construction.SimplexComplex.objectsEquiv
ObjectwiseWeak def: Weak equivalence at the three objects of this span. · OAI.Thomason.Construction.SimplicialSpan.ObjectwiseWeak
observable def: The integer form of the manuscript's observable coefficient. · OAI.ContingencyTables.CountingPhaseSchedule.observable
observableConstant def: The manuscript's observable constant. · OAI.ContingencyTables.ObservableBounds.observableConstant
observation def: Observation time zero is the true initial spatial row. Subsequent times are the true state-dependent spatial switch rows. · OAI.ActualLabels.observation
observation def: Observation time zero is the true initial spatial row. Subsequent times are the true state-dependent spatial switch rows. · OAI.SignedDisk.ActualLabels.observation
observation def: The source endpoint observation in the ambient coordinate space. Its finite index consists ONLY of the starting coarse bin and endpoint coarse bin; an internal frame witness is not part of this index. · OAI.SmoothObstruction.EndpointObservation.observation
observationBits def: Separate current-parameter and next-parameter oracle words are reserved at every observed state, including states at which they are unused. · OAI.ContingencyTables.CompanionPhaseWords.observationBits
ObservationCode structure: The exact three-copy observations at the special carrier. · OAI.PureTypeSystem.Labelled.ObservationCode
observationEquiv def: Full matrix space is a product of attainable visible rows and private kernel-valued maps. A section of A is not required. · OAI.MaxCutGames.Decoder.AdviceFibers.observationEquiv
observationEquiv def: Full matrix space is a product of attainable visible rows and private kernel-valued maps. A section of A is not required. · OAI.UniqueGamesTheorem.Decoder.AdviceFibers.observationEquiv
observationFibreEquiv def: The actual observation fibre, not an assumed rectangular probability law. · OAI.MaxCutGames.Soundness.ConditionalIncidences.observationFibreEquiv
observationFibreEquiv def: The actual observation fibre, not an assumed rectangular probability law. · OAI.UniqueGamesTheorem.Soundness.ConditionalIncidences.observationFibreEquiv
ObservationLayers def: Positive closure of the sort-profile seeds. · OAI.PureTypeSystem.ObservationLayers
observedAgreement def: The literal product of the two observation-local stochastic responses. · OAI.MaxCutGames.Decoder.VisiblePolicies.observedAgreement
observedAgreement def: The literal product of the two observation-local stochastic responses. · OAI.UniqueGamesTheorem.Decoder.VisiblePolicies.observedAgreement
observedBase def: A canonical base depends only on observed advice, never on the hidden matrix. · OAI.MaxCutGames.Decoder.AdviceFibers.observedBase
observedBase def: A canonical base depends only on observed advice, never on the hidden matrix. · OAI.UniqueGamesTheorem.Decoder.AdviceFibers.observedBase
ObservedRow abbrev: Only attainable rows are observation values. · OAI.MaxCutGames.Decoder.AdviceFibers.ObservedRow
ObservedRow abbrev: Only attainable rows are observation values. · OAI.UniqueGamesTheorem.Decoder.AdviceFibers.ObservedRow
observer def: The cube used by the balanced Euclidean main result. · OAI.ForcedComputation.BalancedStretch.observer
obsI def: Evaluation of individual matrices on Xpair · OAI.HadwigerCounterexample.obsI
obstruction def: Obstruction for a face joining the products of its opposite pairs. · OAI.HyperbolicObstruction.HSpacePathObstruction.obstruction
Occ def: Each original block has its own actual finite cyclic boundary. Empty blocks contribute no occurrences, rather than an artificial marked vertex. · OAI.TorsionFreeZeroDivisors.BlockBoundary.Occ
occupation def: The occupation itself has order one. · OAI.Problem335.FactorialExpansion.occupation
occupationAmplitude def: The nonnegative coefficient of one normalized creation/annihilation step. · OAI.Problem335.occupationAmplitude
occupationAverage def: Uniform average over all weak compositions of t in the coordinates ι. · OAI.Problem335.occupationAverage
occupationRatio def: The finite product appearing when a composition factorial moment is normalized by the independent geometric factorial moment. · OAI.Problem335.occupationRatio
occupationShift def: The signed shift of a product of variable operators. · OAI.Problem335.occupationShift
occupied def: Occupied shapes have at least one selected leaf below every vertex. · OAI.TraceReconstruction.OccupiedParityTree.occupied
OccupiedOverlapEndpoint abbrev: Exact occupied-board main. Carriers are in orthonormal coordinates; arbitrary unitary irreducibles of the two actual line groups are quantified over (equivalently, the tuples of line partitions in the manuscript). Labelling e chooses the … · OAI.RowColumn.OccupiedOverlapEndpoint
OccupiedParityTree inductive: A term in the parity expansion retains the occupied ordered shape, all roles, and all extra children, rather than assuming it is binary. · OAI.TraceReconstruction.OccupiedParityTree
occupiedRootWeight def: The factor at the current vertex of a fixed actual occupied role choice. It only uses that vertex's displacement, never any descendant mechanism. · OAI.TraceReconstruction.occupiedRootWeight
occupiedTermEncoding def: Unrestricted shape/label space really covers the coefficient index without multiplicity. · OAI.TraceReconstruction.occupiedTermEncoding
occupiedTermLabels def: The real numeric labels of the selected occupied configuration. · OAI.TraceReconstruction.occupiedTermLabels
occupiedTileAnchor def: Empty image tiles use the original neighboring sheet as anchor. An occupied tile retains its exact geometric anchor. · OAI.PolycyclicRecognition.occupiedTileAnchor
OccupiedWindowChoice def: The role choices depend only on the selected subset, not on any hidden orientation. This common index type permits averaging before norm estimates. · OAI.TraceReconstruction.OccupiedWindowChoice
occupiedWindowShape def: Occupied shape of an actual subset of a complete window tree. Only occupied children are enumerated, so its leaves are exactly the selected monomial variables, not the other complete-tree leaves. · OAI.TraceReconstruction.occupiedWindowShape
occupiedWindowTerm def: The parity tree of a role choice is independent of all displacement bits. · OAI.TraceReconstruction.occupiedWindowTerm
Occurrence abbrev: All recorded occurrences, retaining the actual original layer as their owner. · OAI.Superstring.Hierarchical.Family.Occurrence
occurrence def: An occurrence of a target value in f[B_i]. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.SliceData.occurrence
occurrenceCoefficient def: The total encoding also handles an empty formula with one dummy equation. · OAI.MaxCutGames.Foundations.Hastad.SourceBounds.occurrenceCoefficient
occurrenceCoefficient def: The total encoding also handles an empty formula with one dummy equation. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceBounds.occurrenceCoefficient
occurrenceEquiv def: Reordering a phase and a stable layer ID does not alter the recorded words. · OAI.Superstring.Hierarchical.Threading.occurrenceEquiv
occurrenceEquiv def: The occurrence index is retained separately from the common chain grid. · OAI.TorsionFreeZeroDivisors.RepeatedBlocks.occurrenceEquiv
OccurrenceHole def: An occurrence gap has present endpoints and an absent interior position. · OAI.TwoPointCorrelations.OccurrenceHole
OccurrenceIntervals def: Interval-shaped occurrence sets on a half-open range of positions. · OAI.TwoPointCorrelations.OccurrenceIntervals
occurrenceList def: Mapping the ordered enumeration preserves all sampled occurrences. · OAI.MaxCutGames.Foundations.Hastad.SourceOccurrences.occurrenceList
occurrenceList def: The full occurrence is the displayed equation together with a fresh index. · OAI.MaxCutGames.Reduction.SourceIncidence.occurrenceList
occurrenceList def: Mapping the ordered enumeration preserves all sampled occurrences. · OAI.UniqueGamesTheorem.Foundations.Hastad.SourceOccurrences.occurrenceList
occurrenceList def: The full occurrence is the displayed equation together with a fresh index. · OAI.UniqueGamesTheorem.Reduction.SourceIncidence.occurrenceList
occurrences def: Positions of calls of type i within a fixed sector program. · OAI.ExactFourier.Packing.occurrences
occurrences def: Each root-to-occurrence path is recorded in the order of the cleared ancestor recursion, so the newest reversal occurs first. · OAI.Ostmann.MovingSlotData.occurrences
occurrencesAt def: Count the syntactic occurrences of one variable in each three-slot clause. · OAI.MaxCutGames.Foundations.PCP.occurrencesAt
occurrencesAt def: Count the syntactic occurrences of one variable in each three-slot clause. · OAI.UniqueGamesTheorem.Foundations.PCP.occurrencesAt
occurrenceShape def: One index for every actual included trial, with all seven fan slots and no duplication of a top by its boundary cycle or by decoration coordinates. · OAI.HyperbolicObstruction.ConeSkeleton.occurrenceShape
occurrenceSuccess def: Acceptance probability directly on the uniform occurrence set. · OAI.MaxCutGames.Explicit.BipartiteGame.occurrenceSuccess
occurrenceSuccess def: Acceptance probability directly on the uniform occurrence set. · OAI.UniqueGamesTheorem.Explicit.BipartiteGame.occurrenceSuccess
occurrenceValues def: The literal signed values at original field positions. · OAI.HyperbolicObstruction.ConeSkeleton.occurrenceValues
occurrenceWeight def: The list retains repeated index occurrences in a raw tuple. · OAI.LowSpaceSubsetSum.occurrenceWeight
Occurs def: A substring occurrence retains its starting position, so overlapping occurrences are counted separately. · OAI.Superstring.Counts.Occurs
Occurs def: Literal realization of an enlarged fixed pattern and its bin profile. The prescribed-side embeddings are into the actual two independent matchings. The comparisons refer to normalized traversals of the same suppressed graph. · OAI.TorsionFreeZeroDivisors.ActualPatternEvent.Occurs
occurs def: All indices and loop counters below are explicitly represented unary data. · OAI.Superstring.Computable.occurs
occursLater def: Equality is tested on runtime word values, not on network syntax. · OAI.ExactQuantumFactoring.BitArithmetic.occursLater
oddDoubleAction def: The selecting action on the odd double. · OAI.LittleFinitistic.FormalSummandAction.oddDoubleAction
oddEndpoint def: Common vertex of the two odd branches, using the fourth coordinate plane. · OAI.HarmonicCounterexample.Control.oddEndpoint
oddFrame def: Canonical frames choose the odd-slot orientation. Different choices conjugate by a diagonal sign; the signed action is exactly reordering the odd slots. · OAI.RowColumn.Signed.oddFrame
oddPartNet def: Compute the odd part by a bounded gcd and exact integer division. · OAI.ExactQuantumFactoring.BitArithmetic.oddPartNet
oddPrimorial def: The product of the first r odd primes. · OAI.Problem337.oddPrimorial
OddSlots abbrev: Odd slots of a word, genuinely a subset of the board slots. · OAI.RowColumn.Signed.OddSlots
oddSubsetProduct def: Denominator before introducing any inverse, available in a coefficient ring. · OAI.CirculantHadamard.oddSubsetProduct
oddWord def: Odd alternating word completing one Gram trace to an even moment. · OAI.CoordinateSweeps.TraceHolder.oddWord
of def: Canonical maps into the directed union, represented as a direct limit. No ambient monoid or model-existence hypothesis is needed. · OAI.PartitionConsistency.DiagramOrder.ConservativeLimit.of
of_adjoint_orthogonal def: Construct the geometric decomposition from the operator form of orthogonality of the inclusions. · OAI.KadisonSimilarity.OrthogonalSummandDecomposition.of_adjoint_orthogonal
of_matrix_reduction def: Compose the proved matrix reduction with the explicit three-stage finish. All parameters, machines, alphabets, and time polynomials precede the input. The original raw binary input codec is retained by sequential composition. · OAI.MaxCutGames.FinishReduction.of_matrix_reduction
of_matrix_reduction def: Compose the proved matrix reduction with the explicit three-stage finish. All parameters, machines, alphabets, and time polynomials precede the input. The original raw binary input codec is retained by sequential composition. · OAI.UniqueGamesTheorem.FinishReduction.of_matrix_reduction
of_off_diagonal def: Loops have no effect on separator closure. · OAI.TorsionFreeZeroDivisors.RecursiveSeparator.Split.of_off_diagonal
of_pointed_approximation def: Build the approximation data directly from pointed metric approximations and uniform closeness of the normalized coordinate functions. · OAI.RegularPointCoordinates.OpenCoordinateData.of_pointed_approximation
of_relators def: Rooted potentials are provided by the actual cycle relators on fixed routes. · OAI.HyperbolicObstruction.MasterGraph.Potential.of_relators
ofAccepted def: Express a positive table in the Boolean-face presentation. · OAI.Thomason.Construction.DirectedComplex.Cell.ofAccepted
ofAccepted def: Positive chain maps also admit the bundled-group presentation. · OAI.Thomason.Construction.DirectedComplex.Hom.ofAccepted
ofAccepted def: Bundle the chain groups of a carrier-and-instance directed complex. · OAI.Thomason.Construction.DirectedComplex.ofAccepted
ofAction def: Evaluate the quadratic presentation in any actual algebra action. · OAI.LittleFinitistic.QuadraticEncoding.Representation.ofAction
ofAlgEquiv def: An algebra isomorphism supplies polynomial coordinates for both directions. · OAI.StableCoordinate.AmbientMaps.PolynomialEquiv.ofAlgEquiv
ofBlock def: Regard a block for one fixed ambient group as a member of the class of all finite-group blocks. · OAI.Donovan.FiniteGroupBlock.ofBlock
ofBlocks def: Mathlib's increasing finite-set enumeration is implemented by sorting; the coordinate map contains no choice of a solution subset. · OAI.LowSpaceSubsetSum.TopCoordinates.ofBlocks
ofBody def: Recover the relative extended gauge from an actual relative convex body. · OAI.PettyProjection.RelativeGauge.ofBody
ofBool def: A truth table on one Boolean has two literal finite output words. This performs no run-time computation in its Boolean branch labels. · OAI.ExactQuantumFactoring.BitStackProgram.Procedure.ofBool
ofBoundary def: Insert the literal root break in a lifted genus-zero full matching. This retains every graph dart and all endpoints; no graph path is replaced by its rose label. The initial existence of this full matching must still be proved. · OAI.TorsionFreeZeroDivisors.FinitePicture.ofBoundary
ofBounded def: Construct a sequence from an actual uniform norm bound. · OAI.Tingley.UltraSequence.ofBounded
ofCardBound def: Construct all encodings and representative maps from the fixed label family and its realized-cardinality bounds. The construction has no measure or coefficient-vector parameter. Nonempty X supplies a representative even at codes outside … · OAI.MetricEntropyDuality.FixedLabels.ofCardBound
ofCell def: Globular cells are the augmentation-one positive tables. · OAI.Thomason.Construction.DirectedComplex.Cell.PositiveTable.ofCell
ofCells def: Finite sums of atoms are positive tables. · OAI.Thomason.Construction.DirectedComplex.Cell.PositiveTable.ofCells
ofChart def: Taking coordinates of the explicitly constructed linear chart. · OAI.IntegralCharacterVarieties.ExactSequenceChart.MatrixSplitData.ofChart
ofChart def: Left translation of radial contraction makes the same constants work about every center of a left-invariant metric. · OAI.PolycyclicRecognition.ControlledContraction.ofChart
ofChord def: Conversion of an actual simple chord (with its unweighted boundary ports) to the ordered-center model used in the tensor noncancellation proof. · OAI.CriticalHoneycomb.Transfer.ChordOrientations.ofChord
ofCloudTables def: Materialize every reverse index and every 4096-bit constraint row. · OAI.MaxCutGames.Foundations.PCP.PreprocessingRegularTables.ofCloudTables
ofCloudTables def: Materialize every reverse index and every 4096-bit constraint row. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingRegularTables.ofCloudTables
ofCoeffLaw def: Construct an additive algebraic action from the coefficient identities of the actual two-parameter additive group law. · OAI.StableCoordinate.CoactionLND.PolynomialAdditiveAction.ofCoeffLaw
ofContactCandidate def: A contact candidate calibrates the common marginal potential integral. This constructs the abstract certificate without assuming the candidate is optimal. · OAI.Problem356.ShiftedCoulombCertificate.ofContactCandidate
ofCycle def: The actual vertex-simple macro polygon obtained from a library cycle. · OAI.HyperbolicObstruction.MacroGraph.ofCycle
ofEquivalentProjectionFamily def: A countable strong partition into equivalent projections supplies actual full matrix units with the prescribed initial Hilbert corner. · OAI.KadisonKastler.FullMatrixCorner.ofEquivalentProjectionFamily
ofF def: The real-algebraic inclusion into the algebraic plane. · OAI.PlaneFiveColor.Spectral.ofF
off def: K translated by -z. · OAI.GeneralMahler.Body.off
ofFamily def: Reindex an explicitly finite orthogonal family by Fin. · OAI.Generator.Tracial.CornerPartition.ofFamily
ofFamily def: Family tables are computed once, then captured by the lookup functions. The runtime interval loop only indexes these stored arrays. · OAI.LowSpaceSubsetSum.SeparatorBlocks.ofFamily
offBranch def: The literal affine complement of the five branch lines in A². This subalgebra model keeps the original fraction field C(x,y) definitionally. · OAI.KummerLines.offBranch
offBranchRestrictionEquiv def: Restricting the normalization of the actual original affine plane to its branch complement gives the same literal finite etale cover. This is an isomorphism of algebras, not only a generic-field identification. · OAI.KummerLines.offBranchRestrictionEquiv
OffDiagonal def: The off-diagonal ordinary bimodule O ⊕ T over L × R. · OAI.LittleFinitistic.OrdinarySimulation.OffDiagonal
offDiagonalLinearMap def: The signed off-diagonal linear embedding used for paired representations. · OAI.KadisonKastler.offDiagonalLinearMap
offDiagonalSeminorm def: The off-diagonal distance seminorm from Corollary cor:hyperreflexivity. · OAI.KadisonSimilarity.offDiagonalSeminorm
ofFinite def: The concrete finite-seed information model is a special case, with exactly the same decisions rather than an unrelated replacement rule. · OAI.MatroidProphet.General.OnlineRule.ofFinite
ofFinite def: Extend only beyond the finite window by its terminal value. This extension is algebraically invertible at every index and never affects an actual derivative equation within the window. · OAI.SmoothObstruction.ShortBin.Gauge.ofFinite
ofFinset def: A canonical representation for the semantic correspondence. Enumeration can instead retain its already generated sublist, with no sorting step. · OAI.LowSpaceSubsetSum.IndexedMask.ofFinset
ofFinset def: Increasing enumeration of the actual finite knot set. · OAI.ThompsonNonamenability.SortedSubdivision.ofFinset
offlineCost def: All finite serving-label choices, knowing the entire request word. · OAI.UniformKServer.offlineCost
ofFormula def: Any fixed formula with two distinguished free variables gives a regular binary forcing relation. Its other free variables remain actual parameters. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.BinaryRelation.ofFormula
offset def: Basing a lift at the first boundary vertex fixes every corner lift. · OAI.HyperbolicObstruction.IntegralIncidence.Polygons.offset
offset def: The coordinate within the size-2^h class of a raw host position. · OAI.Problem348.Construction.HostSampling.offset
offset def: The number of entries in groups preceding p. · OAI.ReflexiveFixedPoints.OrderedGroupConcat.offset
offsetComputableInPolyTime def: The same machine, specialized to the actual offset and full-table encodings. · OAI.UniqueGamesTheorem.MachineTranslationTable.offsetComputableInPolyTime
offsetFamily def: A proved producer of the exact-projection interface, not a hypothesis asserting that local translations happen to extend. · OAI.WLIdentification.ComparisonSchemas.SpecialImages.offsetFamily
OffsetInput abbrev: Ambient dimension and requested relative accuracy. · OAI.ContingencyTables.CountingOracleScheduleProgram.OffsetInput
Offsets abbrev: Actual offset set for nonwrapping source D-windows. · OAI.SmoothObstruction.WordStep.Offsets
ofFunction def: A C^k function whose first k derivatives are uniformly bounded defines a jet. · OAI.ForcedComputation.BoundedSpatialJets.ofFunction
ofFunctionCurve def: Bounded derivatives on a compact time interval, with a uniform time modulus. · OAI.ForcedComputation.BoundedSpatialJets.ofFunctionCurve
ofFunctionFamily def: Bundle the actual derivatives of a function with bounded jets. · OAI.ForcedComputation.BoundedSpatialJets.ofFunctionFamily
ofGeometric def: The velocity supplied by the geometric construction, expressed in the analytic second-jet coordinates. All its identities survive this pullback. · OAI.ClosedSurfaceR4.SurfaceVelocityFamily.Loop.ofGeometric
ofGroups def: Assemble the actual extension from the positive, normalized child groups. · OAI.ReflexiveFixedPoints.FrontierExtension.ofGroups
ofHalving def: Two equivalent orthogonal halves, as a family with a common initial corner. · OAI.Generator.Tracial.EquivalentPartition.ofHalving
ofInstance def: Retain every original list occurrence as its own edge identifier. · OAI.MaxCutGames.Explicit.SubdivisionTarget.ofInstance
ofInstance def: Retain every original list occurrence as its own edge identifier. · OAI.UniqueGamesTheorem.Explicit.SubdivisionTarget.ofInstance
ofInterior def: Convert a coherent map on the genuine interior faces into a persistent optional index on the entire diagram. · OAI.HyperbolicObstruction.WordPictures.WordPicture.Indexing.ofInterior
ofIso def: A module-sheaf isomorphism transports a line atlas. · OAI.ActualSheafTensor.LineTrivialization.ofIso
ofIso def: The same data transported across a commuting square of isomorphisms. · OAI.Thomason.Construction.TopologicalDeformation.Data.ofIso
ofLawAt def: The zero row-sum is obtained by differentiating the actual stochastic partition of unity, not postulated for the row derivatives. · OAI.SignedAssembly.RowJet.ofLawAt
ofLawAt def: The zero row-sum is obtained by differentiating the actual stochastic partition of unity, not postulated for the row derivatives. · OAI.SignedDisk.SignedAssembly.RowJet.ofLawAt
ofList def: Retain every list position as its own occurrence identifier. · OAI.MaxCutGames.Reduction.ActualSource.Source.ofList
ofList def: Retain every list position as its own occurrence identifier. · OAI.UniqueGamesTheorem.Reduction.ActualSource.Source.ofList
ofLocal def: Restrict a locally smooth family to the admissible domain. · OAI.ClosedSurfaceR4.LocalPeriodicExpansion.Family.ofLocal
ofMachine def: Regard a Machine as a table (no reindexing). · OAI.ExactDerandomization.Table.TM.ofMachine
ofMarks def: The literal source cut construction has at least one cut per boundary, using one chosen indexing mark there. · OAI.TorsionFreeZeroDivisors.PlanarRuns.CutSystem.ofMarks
ofModule def: In particular, every R-module supplies the required representation. · OAI.LittleFinitistic.QuadraticEncoding.Representation.ofModule
ofMultilinearMap def: Read coefficients from actual basis evaluations. Symmetry will show that this does not depend on the representative selected for the multiset. · OAI.MetricEntropyDuality.SymmetricForm.ofMultilinearMap
ofNat def: The executable little-endian digit map; no search through 2^d addresses. · OAI.WLIdentification.BinaryAddress.ofNat
ofNonneg def: Construct the level chain by sorting the actual finite image. · OAI.ThompsonNonamenability.FiniteLevelChain.ofNonneg
ofNonpositiveResolution def: Package exactly the ordinary worker's actual bounded bimodule resolution API. Augmentation, quasi-isomorphism, projectivity and support are all explicit. · OAI.LittleFinitistic.BimoduleTotal.ofNonpositiveResolution
ofParallelFlows def: Normalized Ricci curvature supplies the contraction field of the parallel-frame calculus. · OAI.PositiveEinsteinFour.Geometry.ParallelMetricData.ofParallelFlows
ofPartial def: A partial sewing, with every unused port left exposed exactly once. Injectivity prevents accidental multiple use of a repeated occurrence. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.PortPatch.ofPartial
ofPortGraph def: Materialize a reversible port graph and its predicates into the fixed rows. · OAI.MaxCutGames.Foundations.PCP.PortTables.ofPortGraph
ofPortGraph def: Materialize a reversible port graph and its predicates into the fixed rows. · OAI.UniqueGamesTheorem.Foundations.PCP.PortTables.ofPortGraph
ofPrimal def: Explicit conversions keep the chosen norm distinct from unrelated Finsupp structures. · OAI.DiamondDistortion.SegmentSystem.ofPrimal
ofProjection def: Total projection maps give ordinary finite classical games. Question weights may be arbitrary nonnegative real weights summing to one. · OAI.MaxCutGames.Repetition.ofProjection
ofProjection def: Total projection maps give ordinary finite classical games. Question weights may be arbitrary nonnegative real weights summing to one. · OAI.UniqueGamesTheorem.Repetition.ofProjection
ofQuasiIsometry def: Choose a coarse inverse from the coarse-surjectivity part of a quasi-isometry, and prove its affine bound using the lower inequality. · OAI.PolycyclicRecognition.MetricQuasiEquivalence.ofQuasiIsometry
ofRational def: A genuine rational-error certificate supplies every weaker real threshold. The input and output encodings and finite transition program remain identical. · OAI.MaxCutGames.Integration.RealTarget.ofRational
ofRational def: A genuine rational-error certificate supplies every weaker real threshold. The input and output encodings and finite transition program remain identical. · OAI.UniqueGamesTheorem.Integration.RealTarget.ofRational
ofRel def: Regard a binary relation as a path-ordered endomorphism. · OAI.TwoWayComplementation.RelationMonoid.ofRel
ofRepresentatives def: Equality of represented scalar functions supplies compatibility automatically. · OAI.ForcedComputation.PlaneScalarMild.CompatibleData.ofRepresentatives
ofRetract def: An arrow retract of a stationary deformation inclusion is again one. · OAI.Thomason.Construction.TopologicalDeformation.Data.ofRetract
ofRows def: The actual reverse and predicate vectors extracted from validated flat rows. · OAI.MaxCutGames.Foundations.PCP.PortTables.ofRows
ofRows def: The actual reverse and predicate vectors extracted from validated flat rows. · OAI.UniqueGamesTheorem.Foundations.PCP.PortTables.ofRows
ofTensor def: Replace arbitrary finite coordinate sets by standard finite sets. · OAI.MatrixMultiplication.AuxiliarySeparation.TensorSemiring.FiniteTensor.ofTensor
ofTensorClassHom def: A monotone semiring homomorphism on tensor classes gives a character on all finite coefficient tensors. Normalization follows from preservation of 0 and 1. · OAI.MatrixMultiplication.AuxiliarySeparation.Character.ofTensorClassHom
ofUnits def: A full matrix frame is obtained from a specified matrix-unit range. · OAI.KadisonKastler.FullMatrixFrame.ofUnits
ofVertices def: Group-vertex labels determine the labelled directions: the nine positive letters and their inverses are distinct in the actual triangle Artin group. · OAI.ArtinCAT0.TriangleMesh.SimplicialLabeling.ofVertices
ofWeights def: The probability whose mass at each point is its prescribed weight. · OAI.QuantitativeVanDerWaerden.FiniteLocalLemma.Probability.ofWeights
ofZarankiewiczParametrization def: Transfer a genuine affine parametrization, retaining the identical path and knots. · OAI.Paper170.ofZarankiewiczParametrization
OK structure: Evaluation guarantee. · OAI.PowerSaving.RAM.Bill.OK
Old def: The only possible old left-side shapes at a selector stage. · OAI.FiniteAlgebraicEnvelopes.Bootstrap.Selectors.Old
old def: The canonical inclusion of all old vertices through finitely many refinements. · OAI.ForestSpace.Diamond.Network.old
oldAction def: An action on an old object in the completion is an actual old action. · OAI.LittleFinitistic.FormalSummandAction.oldAction
oldCloudIndex def: An original global dart carries its own owner and executable local rank. · OAI.MaxCutGames.Foundations.PCP.PreprocessingCloudIndex.oldCloudIndex
oldCloudIndex def: An original global dart carries its own owner and executable local rank. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingCloudIndex.oldCloudIndex
oldCollapse def: The old-valued endpoint k in the local zigzag. · OAI.Thomason.Construction.SieveRetraction.oldCollapse
oldCollapseRetraction def: The endpoint k factors through the old subposet as a retraction. · OAI.Thomason.Construction.SieveRetraction.oldCollapseRetraction
oldData def: Removing the ghost radial alphabet from a reduced picture changes NEITHER its permutation genus nor its boundary representatives. · OAI.TorsionFreeZeroDivisors.ConePicture.oldData
olderCorrection def: The signed contribution of letters strictly older than the first survivor. · OAI.FKCRT.Inventory.olderCorrection
OldGate structure: An old gate controls both old right multiples and intersections with old cones not containing the new element. In probe normal forms this is exactly the first-letter exit rule. · OAI.PartitionConsistency.DiagramOrder.OldGate
OldGateStep def: The least-old-multiple alternative at a successor stage, including the old-input implication used for descent. The new word must belong to the successor stage; arbitrary elements of the final monoid are not called new at this stage. · OAI.PartitionConsistency.DiagramOrder.OldGateStep
oldGauge def: The gauge at an old apex is its literal radial edge to the fixed fan base. The bouquet vertex has gauge one. · OAI.HyperbolicObstruction.ConeSkeleton.oldGauge
oldHom def: The old monoid embeds by words with no new letters. · OAI.PartitionConsistency.MergerAlgebra.Word.oldHom
oldHom def: A literal conservative homomorphism of old elements into probe normal forms. · OAI.PartitionConsistency.ProbeAlgebra.Word.oldHom
oldInclusion def: The monotone inclusion of the old subposet. · OAI.Thomason.Construction.RelativePrismData.oldInclusion
OldIntersections def: Only old cone intersections are required for the recursive gate calculation. The path alphabet supplies this law from old conditional joins. · OAI.PartitionConsistency.ProbeAlgebra.Word.OldIntersections
oldLeaf def: The old paths keep their separate labels when a genuine fresh branch is added. · OAI.DilutedSpinGlass.PrescribedTree.oldLeaf
OldMinimum def: A least old cone containing a word's old right multiples. It is used only to choose a cost cap; the gate theorem proves its existence when needed. · OAI.PartitionConsistency.ProbeAlgebra.Word.OldMinimum
oldPatternLaw def: Empirical pattern law of the old spin array at independently drawn sites. · OAI.DilutedSpinGlass.oldPatternLaw
oldPredicate def: The old predicate is read in the same row-major order as a graph row. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Relations.oldPredicate
oldPredicate def: The old predicate is read in the same row-major order as a graph row. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Relations.oldPredicate
oldSample def: The projection forgets the new branch but preserves every old value. · OAI.DilutedSpinGlass.PrescribedTree.oldSample
oldThenMovable def: One legal interleaving: all old occurrences, followed by all movable occurrences. · OAI.MatroidProphet.MarkedPivots.oldThenMovable
omega def: The restricted Fubini--Study form, evaluated on horizontal lifts. · OAI.ArnoldCounterexample.omega
omega def: The ordinary Euclidean unit ball volume, in Cartesian Lebesgue measure. · OAI.HarmonicCounterexample.omega
omega abbrev: The arithmetic matrix multiplication exponent over the complex numbers, in the repository's existing scalar-operation model. · OAI.MatrixMultiplication.AuxiliarySeparation.omega
omegaObs def: Observation law weighted by the local subpiece. · OAI.FourierLLogL.Dyadic.Thermal.omegaObs
OmegaOne abbrev: The coordinate set of the negative construction is really ω₁. · OAI.SeparableQuotient.CoherentClosures.OmegaOne
omissionEnvelope def: The old-prime choices can be enumerated before drawing the other remainder. This is essential when applying the inclusion bound to a fixed numeric addition. · OAI.JointDickman.omissionEnvelope
omittedIndex def: The deterministic omitted-index choice used for both complementary masks. · OAI.FiniteFactor.omittedIndex
omitting def: Words generated without using the index k. · OAI.FiniteAlgebraicEnvelopes.Steinberg.omitting
onCells def: Change only the current pair of cells; the retained descriptor set shrinks. · OAI.Falconer.PolicyJetData.onCells
OnChosenLine def: A line through a mark, chosen by its projective direction. · OAI.Release075.OnChosenLine
onDomain def: Only the domain in which the polynomial coefficients are read changes. The phase chart, reconstruction bounds and every numeric profile are retained. · OAI.ClosedSurfaceR4.JetPolynomial.Perturbation.PolynomialSolveData.onDomain
one def: The identity frame lifts the identity change in a quaternion basis. · OAI.PositiveEinsteinFour.QuaternionFrame.QuaternionLift.one
OneCardUniform def: The source's one-card marginal property, stated for a line law. This is proved for both U and the actual binary law below, not a main-proof assumption. · OAI.CoordinateSweeps.Grid.Holes.OneCardUniform
oneEigenvalueMoment def: Retain the largest weight of a single negative eigenvalue of the actual operator. The supremum of the empty family is zero. · OAI.SharpLiebThirring.OperatorProof.oneEigenvalueMoment
oneEquation def: The constant equation 1 = 0, of multidegree zero. · OAI.Nagata.ProjectiveGeometry.oneEquation
oneFiberComparison def: The canonical map from the reflected original fibre to the reflected label's fibre. · OAI.Thomason.Construction.FiniteReflectionInput.oneFiberComparison
oneFiberHomEquiv def: The universal bijection of the reflected one fibre. · OAI.Thomason.Construction.FiniteReflectionInput.oneFiberHomEquiv
oneFiberOverEquiv def: Reflection descends maps to the initial cone while retaining the label. · OAI.Thomason.Construction.FiniteReflectionInput.oneFiberOverEquiv
oneFiberUnit def: Restriction of the original reflection unit to its one fibre. · OAI.Thomason.Construction.FiniteReflectionInput.oneFiberUnit
oneForm def: A continuous linear one-form regarded as a continuous alternating one-form. · OAI.Mahler.oneForm
oneModeBeamCoefficient def: One-mode passive beam-splitter coefficient, in number bases. It is the coefficient of (c†)^k (d†)^e in (√η c† − √(1−η) d†)^a (√(1−η) c† + √η d†)^b, with the normalized number-vector factorials. · OAI.EntropyPhotonNumber.oneModeBeamCoefficient
OnePositiveDirection def: A symmetric form is nonpositive on the form-orthogonal complement of every vector of positive square. This is the quadratic formulation of positive inertia at most one. · OAI.ContingencyTables.OnePositiveDirection
oneQuasitraceOfConstraints def: All coordinate requirements reconstruct the original bundled object, including complex linearity on each closed nonunital abelian algebra. · OAI.Kaplansky.oneQuasitraceOfConstraints
oneSidedModulus def: The usual one-sided modulus, whose directions have norm exactly 1. · OAI.IndependentProducts.oneSidedModulus
oneStateMoment def: The moment retaining at most one negative eigenvalue. · OAI.SharpLiebThirring.oneStateMoment
oneStepOffset def: An interior period slot admits the three literal shifts. · OAI.TraceReconstruction.oneStepOffset
oneVector def: The constant-one vector, so the initial stage is literally span{1}. · OAI.ExactModuli.KW.oneVector
onL2 def: The same average of the actual unitary operators. · OAI.PolycyclicRecognition.FiniteOrbitAverage.onL2
OnlineRule structure: A decision uses only the entire initial seed, the initially observed vector, and the label/value history through the current arrival. · OAI.MatroidProphet.General.OnlineRule
OnlineRule structure: A rule on physical labels receives only the same sample and history data. · OAI.MatroidProphet.LabeledTransport.OnlineRule
onlyLength def: order environments · OAI.PowerSaving.RAM.Bench.onlyLength
onModeSupport def: The function is unchanged; only its compact-support index is transferred. · OAI.ClosedSurfaceR4.JetPolynomial.Perturbation.onModeSupport
onProjection def: The identity partial isometry on a projection. · OAI.Generator.Tracial.PartialIsometry.onProjection
OnshuusPostRealizationInput def: The part of the published triangular realization needed by this proof. Unipotence is required only for exponentials of the actual canonical ideal. · OAI.PolycyclicRecognition.OnshuusPostRealizationInput
onSide def: Contraction of the complement of S to a new vertex. Edges retain identities; only edges with both endpoints outside S are discarded. · OAI.MatchingEntropy.LooplessGraph.onSide
onSlice def: Membership in the slice, for a tree of any total depth. · OAI.OptimalMaxCut.TreeCode.onSlice
onSphere def: The boundary lift for attaching an n-cell. · OAI.GroupRingTopology.Cellular.PartialLift.onSphere
OntoClock def: A clock visits every nonnegative internal time and never visits a negative one. · OAI.ForcedComputation.OntoClock
opair def: The equivariant name for the actual Kuratowski pair. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.Name.opair
openAffineIso def: An open affine chart of any scheme computes its cotangent sheaf. · OAI.ActualCotangent.openAffineIso
openAffineIso def: The comparison of the top exterior sheaf on an affine open. · OAI.ActualExterior.openAffineIso
openAt def: Open one cyclic passage into two interval endpoints. · OAI.TorsionFreeZeroDivisors.PathIncidences.Opened.openAt
openCell def: Open disks are inserted as genuine open embeddings. · OAI.GroupRingTopology.Attachment.openCell
openChartCoordinateLocalHomologyIso def: Local homology in a full open vector-space chart, at an arbitrary point. · OAI.Kaplansky.openChartCoordinateLocalHomologyIso
openChartLocalExcisionIso def: Excision at any point of an open chart, using the preserved chain-level comparison and the proved open-cover small-chain theorem. · OAI.Kaplansky.openChartLocalExcisionIso
OpenCoordinateData structure: Raw approximation data: small normalized targets have approximate preimages, and changing the center and going down one quarter-scale costs at most two in the residual norm. All conditions are local to the open domain. · OAI.RegularPointCoordinates.OpenCoordinateData
openCoverMayerVietorisBoundaryIso def: Full singular Mayer–Vietoris boundary isomorphism for a cover by two contractible open subspaces, in degrees at least two. · OAI.Kaplansky.openCoverMayerVietorisBoundaryIso
openCoverMayerVietorisOneIso def: Full integral H₁ with the sign fixed by the ordered two intersection components. · OAI.Kaplansky.openCoverMayerVietorisOneIso
openCoverSmallHomologyIso def: Comparison from the homology of small chains to full integral singular homology for a genuine two-set open cover. · OAI.Kaplansky.openCoverSmallHomologyIso
openDerivationEquiv def: Derivations are local for open immersions. · OAI.ActualCotangent.openDerivationEquiv
OpenDiagram structure: A planar partially exposed lifetime diagram. The anchor has zero weight and merely keeps a distinguished boundary rotation available at the empty word. · OAI.KaplanskyCounterexample.Planar.OpenDiagram
openerEmbedding def: Transport a core matching through its strictly increasing position map. · OAI.FKCRT.Chords.openerEmbedding
openFaceContraction def: The explicit contraction to the barycenter keeps every prescribed star coordinate strictly positive at every time, including endpoints. · OAI.Thomason.Construction.PosetGeometry.openFaceContraction
openGraph def: The graph of retained bonds. fromEdgeSet discards loops. · OAI.CriticalPercolation.BondGraph.openGraph
openIntegerInterval def: Every integer strictly between two real endpoints, including integral endpoints. · OAI.Nagata.FiniteExponents.openIntegerInterval
OpenInterval inductive: The four kinds of nonempty open real intervals. Valid excludes reversed finite endpoints; no endpoints at infinity are ever evaluated. · OAI.HenonLaneEmden.IntervalGeometry.OpenInterval
openInverseHomEquiv def: The Hom-set version of the open cotangent comparison. · OAI.ActualCotangent.openInverseHomEquiv
openIso def: The intrinsic cotangent sheaf is local for open immersions. · OAI.ActualCotangent.openIso
openIso def: Sheafified exterior powers commute with open restriction without rank or freeness hypotheses. · OAI.ActualExterior.openIso
openPresheafRestrict def: The concrete presheaf restriction underlying open-scheme restriction. · OAI.ActualSheafTensor.openPresheafRestrict
openPresheafTensorRestrictIso def: Tensor presheaves commute with restriction to an open subscheme. · OAI.ActualSheafTensor.openPresheafTensorRestrictIso
openPullbackEIso def: The global double-dual model is the actual double-dual model on EACH open affine chart. It is not an independently chosen local replacement. · OAI.CoherentModelActual.openPullbackEIso
openPullbackEIsoOfComp def: The same comparison with a specified structural map on the chart. · OAI.CoherentModelActual.openPullbackEIsoOfComp
openRingRestrictHom def: The coefficient-ring map for open-scheme restriction. · OAI.ActualSheafTensor.openRingRestrictHom
openSheafifyRestrictMap def: Canonical open restriction comparison for module sheafification. · OAI.ActualSheafTensor.openSheafifyRestrictMap
openStar def: The open star of a vertex in the actual barycentric realization. · OAI.HyperbolicObstruction.FinitePolyhedron.openStar
operandLength def: The actual total bit length of the isolated radix and digit fields. · OAI.MaxCutGames.Foundations.Complexity.MachineHorner.operandLength
operandLength def: The actual total bit length of the isolated radix and digit fields. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineHorner.operandLength
Operator def: The operator type with its normed-space structure fixed before taking another operator space. · OAI.ForcedComputation.BoundedSpatialJets.Operator
operator def: Actual bounded operators on the finite-dimensional Hilbert fibers. · OAI.Kaplansky.NormalizedHermitianTuple.operator
operator def: The mismatch operator ∑ᵢ (I - |vᵢ⟩⟨vᵢ|)ᵢ. · OAI.QAC.Count.operator
operator def: The self-adjoint realization of -d²/dx²-W, obtained by inverting the coercively shifted H¹ form. The graph identity below characterizes it without reference to the choice of potential decomposition or shift. · OAI.SharpLiebThirring.OperatorProof.operator
operatorAction def: The inverse pullbacks form a group action. · OAI.KadisonKastler.NonsingularAction.operatorAction
operatorBimoduleMean def: The constructed scalar-coefficient mean is bimodular conditionally only on the indicated published projection theorem. · OAI.KadisonKastler.ScalarBoundaryMean.operatorBimoduleMean
operatorClosedRangeMap def: The same operator, with target restricted to the closure of its range. · OAI.KadisonKastler.operatorClosedRangeMap
operatorCompressionCP def: Compression by any bounded operator is completely positive. · OAI.KadisonKastler.operatorCompressionCP
operatorEquiv def: Inverse pullback gives the boundary action on actual operator classes. · OAI.KadisonKastler.NonsingularAction.operatorEquiv
operatorFactorial def: Integral coefficient supplied by a mixed derivative on a monomial. · OAI.Problem335.operatorFactorial
operatorFieldAERel def: Equality almost everywhere is an algebra congruence. · OAI.KadisonKastler.operatorFieldAERel
operatorFieldAlgebra def: The closed field algebra commuting with the constant commutant. · OAI.KadisonKastler.operatorFieldAlgebra
operatorFieldDiagonal abbrev: Constant fields form the target C-star subalgebra of the expectation. · OAI.KadisonKastler.operatorFieldDiagonal
operatorFieldFamilyInclusion def: The inclusion of bounded M-valued field families into all field families. · OAI.KadisonKastler.operatorFieldFamilyInclusion
operatorFieldZeroIdeal def: The zero fields form a closed two-sided star ideal. · OAI.KadisonKastler.operatorFieldZeroIdeal
operatorInverseTranspose def: The inverse-transpose operation on all bounded operators. Only its behavior at an invertible operator is used. · OAI.RegularPointCoordinates.operatorInverseTranspose
operatorLift def: Conjugation/compression by an isometric Hilbert-space embedding. · OAI.NuclearUltrapower.operatorLift
operatorMatrix def: Matrix entries for the counting orthonormal basis or any chosen orthonormal basis. · OAI.LeanBlast.GotsmanLinial.operatorMatrix
operatorMatrixAction def: The actual left action of an operator-valued matrix on a Hilbert direct sum. · OAI.NuclearUltrapower.operatorMatrixAction
operatorMatrixActionHom def: Matrix amplification is a genuine unital star homomorphism. · OAI.NuclearUltrapower.operatorMatrixActionHom
operatorMatrixEquiv def: Matrix representation of a continuous operator in an orthonormal basis, bundled with its algebraic compatibility. · OAI.LeanBlast.GotsmanLinial.operatorMatrixEquiv
operatorMatrixRepresentation def: The algebraic operator-matrix action, before changing the matrix norm. · OAI.KadisonSimilarity.operatorMatrixRepresentation
operatorMatrixVectorFunctional def: The quadratic form of a finite operator matrix, as a positive functional. · OAI.BoundedHochschild.KadisonRingrose.operatorMatrixVectorFunctional
operatorMean def: The scalar boundary mean induces a contractive complex-linear map on actual essentially bounded measurable operator fields. · OAI.KadisonKastler.ScalarBoundaryMean.operatorMean
operatorMean def: The actual contractive complex-linear operator average. · OAI.KadisonKastler.ScalarInvariantMean.operatorMean
operatorMeanCP def: Complete positivity follows by applying scalar positivity to the finite sum of all the matrix coefficients of a positive amplified family. · OAI.KadisonKastler.ScalarInvariantMean.operatorMeanCP
operatorMeanValue def: Reconstruct the operator-valued mean from the averaged scalar kernel. · OAI.KadisonKastler.ScalarBoundaryMean.operatorMeanValue
operatorOfForm def: Reconstruct an operator from a bounded sesquilinear matrix coefficient. · OAI.KadisonSimilarity.operatorOfForm
operatorOfPredual def: Riesz reconstruction from a bounded functional on the vector predual. · OAI.KadisonSimilarity.operatorOfPredual
operatorPositiveTrace def: The real part of the ordinary trace on finite-dimensional operators. · OAI.NuclearUltrapower.operatorPositiveTrace
operatorRangeClosure def: Closed range used to construct the support projection of an operator. · OAI.KadisonSimilarity.operatorRangeClosure
operatorRep def: Bounded operator products and adjoints descend to the Hilbert ultraproduct. · OAI.NuclearUltrapower.HilbertUltraproduct.operatorRep
operatorRepresentativeConst def: Constant operators as measurable fields. · OAI.KadisonKastler.operatorRepresentativeConst
operatorTracePairing def: The trace pairing is kept as a bounded functional so that its norm is the ordinary dual norm, with no additional trace-class definitions. · OAI.KadisonKastler.operatorTracePairing
operatorVectorState def: The ambient vector functional, without an algebra-membership wrapper. · OAI.KadisonKastler.operatorVectorState
operatorWeight def: The weighted-degree shift assigned to an operator variable: differentiation has negative weight, while multiplication has positive weight. · OAI.Problem335.operatorWeight
opinionAt def: Outside the full-radius neighborhood a fixed fallback makes the opinion function total. Actual tested path vertices will be certified inside it. · OAI.MaxCutGames.Foundations.PCP.PoweringOpinions.opinionAt
opinionAt def: Outside the full-radius neighborhood a fixed fallback makes the opinion function total. Actual tested path vertices will be certified inside it. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringOpinions.opinionAt
opPair def: Realized paper expressions via clm evaluations. · OAI.GeneralMahler.opPair
Opposite def: Equal-length opposite points of a cone. The zero pair is permitted. · OAI.HyperbolicObstruction.ConeData.Opposite
opposite def: Reversing the vertices transfers exactly the hypotheses to right modules. · OAI.LittleFinitistic.DirectedBound.DirectedSystem.opposite
opposite def: Reverse both boundaries and the order of every strict composition. · OAI.Thomason.Construction.StrictOmegaCategory.opposite
OppositeColoring abbrev: A binary assignment with opposite values on each marked matching edge. · OAI.CubeShuffle.CycleColoring.OppositeColoring
oppositeDerivative def: First derivative of the opposite stationary branch for K = 20. · OAI.Problem356.LocalHessian.oppositeDerivative
oppositeEquiv def: The opposite of the actual product ring is the product of the two opposite rings. · OAI.LittleFinitistic.ProductModules.oppositeEquiv
oppositeEquivalence def: Reversal is a genuine equivalence, allowing colimits to be transported. · OAI.Thomason.Construction.StrictOmegaCategory.oppositeEquivalence
OppositeFaces structure: Two oppositely oriented face boundaries, including their particular cyclic parametrizations. The two occurrences are disjoint, even if the target cell or some target boundary vertices coincide. · OAI.HyperbolicObstruction.PictureSwitch.PictureMap.OppositeFaces
oppositeFaces def: The two faces perpendicular to the first coordinate. · OAI.Thomason.Construction.RealizedCube.oppositeFaces
oppositeFacesContraction def: Contract the remaining coordinates without leaving either opposite face. · OAI.Thomason.Construction.RealizedCube.oppositeFacesContraction
oppositeFunctor def: Opposite is functorial without a change of universe. · OAI.Thomason.Construction.StrictOmegaCategory.oppositeFunctor
oppositeKeys def: The fixed keys that must flip for the target event. · OAI.QuantitativeVanDerWaerden.SparsePerturbation.oppositeKeys
oppositeLineFrame def: Interchange the two one-dimensional factors of a projective-line chart. · OAI.Kaplansky.ComplexGrassmann.oppositeLineFrame
oppositeMap def: Reversing all directions retains the underlying strict functor. · OAI.Thomason.Construction.StrictOmegaCategory.oppositeMap
oppositePositions def: Positions whose original bit differs from the target bit. · OAI.QuantitativeVanDerWaerden.SparsePerturbation.oppositePositions
oppositeProduct def: This is the unsigned product only on genuinely opposite signs. · OAI.BoundedTreePotentials.Segments.oppositeProduct
oppositeQuaternion def: The second quaternion triple is obtained by reversing the first frame axis. This exchanges the two Hodge-star eigenspaces. · OAI.PositiveEinsteinFour.Geometry.oppositeQuaternion
oppositeSign def: Opposite sign for the selected head sum; at zero either sign is harmless. · OAI.ForestSpace.oppositeSign
OppositeWeylData structure: The same classical first-order calculus for the opposite Weyl block, using the already specified actual parallel frame flow. · OAI.PositiveEinsteinFour.Geometry.OppositeWeylData
opt def: Min rent uses r rows. · OAI.FourierLLogL.Compress.opt
opt def: The optimum in symbols, not in bits. · OAI.Superstring.opt
optimalCost def: This declaration is an exact comparator-selected statement. Read the Lean statement below for its mathematical content. · OAI.KServer.optimalCost
optimalCost def: Offline optimum over the finite set of all label strings. For k ≥ 2 this range is nonempty and finite, hence its real infimum is the attained minimum. There is no restriction on the metric itself. · OAI.KServer.optimalCost
optimalDiagonal def: The diagonal coordinates attaining the constrained minimum. · OAI.Problem337.SelbergOptimal.optimalDiagonal
OptimalOrder def: All four extremal quantities have common absolute positive comparison constants. ENNReal bounds imply finiteness rather than mapping a potentially infinite density to zero through toReal. · OAI.CoveringOrder.OptimalOrder
OptimalOrder def: Every clause of the literal optimal-order endpoint. The single absolute C is in PHYSICAL time; convergence is to zero, not merely below a fixed cutoff. · OAI.CubeShuffle.OptimalOrder
optimalTrace def: A fixed minimizing labeled trajectory for each complete input. · OAI.KServer.FiniteExperiment.optimalTrace
optimalWeight def: Selberg weights obtained by upper Möbius inversion. · OAI.Problem337.SelbergOptimal.optimalWeight
optimisticHistory def: The accumulated optimistic objective before a time index. · OAI.EditApproximation.optimisticHistory
optimize def: Exact rational arithmetic and comparisons only. The nonnegative objective constraint is imposed by the base case even if not listed among the rows. · OAI.UniformKServer.FourierMotzkin.optimize
optimizedCutoff def: The conventional balanced natural cutoff for Vaughan's identity. · OAI.Problem337.Vaughan.optimizedCutoff
optimizerMixtureWeight def: Exact rational weight of the vertex selected at step j, after K steps. · OAI.EditApproximation.optimizerMixtureWeight
optimum def: The same offline maximum on an arbitrary finite label type. · OAI.MatroidProphet.LabeledTransport.optimum
optionAccept def: Exhaustion is a rejecting actual outcome, without conditioning the law. · OAI.LowSpaceSubsetSum.optionAccept
optionFamilyColor def: Total zero extension at a non-basic image as well as an inadmissible one. · OAI.ThompsonNonamenability.optionFamilyColor
optionFinsetEquiv def: Separate Fourier supports according to whether the distinguished coordinate occurs. · OAI.Problem313.BiasedFourier.optionFinsetEquiv
oracle def: Actual fixed-gate compute/copy/uncompute, with all trace wires restored. · OAI.ExactQuantumFactoring.BooleanNetwork.oracle
oracle def: Null-mass outcomes are assigned zero score. This proves, rather than assumes, the pointwise positivity needed by the finite simulation API. · OAI.TraceReconstruction.ConcreteLevelPair.oracle
OracleAlgorithm structure: Measurable adaptive queries and output, parametrized by private randomness. · OAI.LogConcaveSampling.OracleAlgorithm
oracleOutput def: The basis-state whose trace is zero and whose target is toggled by c(x). · OAI.ExactQuantumFactoring.BooleanNetwork.oracleOutput
oraclePlacement def: Place the oracle's retained trace after its output rather than between the input and output. A larger reservoir of work wires is permitted. · OAI.ExactQuantumFactoring.BooleanNetwork.oraclePlacement
oracleTime def: The source uses a linear reciprocal-error allowance inside the mixing time, avoiding any unimplemented logarithm computation. · OAI.ContingencyTables.CompanionOracleDensity.oracleTime
OracleTracePair structure: A finite genuine approximate oracle pair with a common complete-trace decoder. It does not include an overlap or any unproved scale conclusion. · OAI.TraceReconstruction.OracleTracePair
orbit def: A measure-preserving transformation gives precisely the stationarity needed below; no independence, ergodicity, or decay of correlations enters. · OAI.SmoothObstruction.DyadicChord.orbit
orbitAction def: The bounded action on the actual closed orbit subspace. · OAI.KadisonSimilarity.orbitAction
orbitAmplitude def: The signed amplitudes retain all determinant-correcting signs. · OAI.HarmonicCounterexample.Cycling.orbitAmplitude
orbitAverage def: The positive orbit-average functional. · OAI.PolycyclicRecognition.orbitAverage
orbitBusemann def: Busemann function based on the positive orbit of a minimizing point. · OAI.ArtinCAT0.orbitBusemann
orbitClosure def: The closed cyclic subspace generated by the realizing vector. · OAI.KadisonSimilarity.orbitClosure
OrbitCoherent inductive: A single coloured-interchange schema and its acting-group translates, closed under tree contexts. The action's two-transitivity is not hidden in the constructors: it is used explicitly in the completeness theorem below. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Tree.OrbitCoherent
OrbitCopy abbrev: An actual distinct copy, with no labeling multiplicity. · OAI.LeanBlast.SecondKahnKalai.OrbitCopy
orbitCopyEquiv def: The group acts bijectively on the set of actual orbit copies. · OAI.LeanBlast.SecondKahnKalai.orbitCopyEquiv
orbitCount def: The total number of distinct copies. · OAI.LeanBlast.SecondKahnKalai.orbitCount
orbitEquiv def: Subdivision gives a genuine equivalence of orbit quotients, including fixed points and the empty type. · OAI.HyperbolicObstruction.PictureSwitch.PermInsertion.orbitEquiv
orbitExtensionTree def: The actual extension tree. At each node its children are all distinct copies of the next set containing the current copy, with their uniform law. The dependent path type retains the entire sequence of choices. · OAI.LeanBlast.SecondKahnKalai.orbitExtensionTree
orbitFiberEquiv def: Exact sigma decomposition of a word orbit and its transporter fibers. · OAI.CoordinateSweeps.OrdinaryTensor.orbitFiberEquiv
orbitGraph def: The invariant graph of the partial name-orbit map in eq:partial-orbit-map. Each entry is activated by its actual moved diagram. · OAI.PartitionConsistency.ProductForcing.orbitGraph
orbitIntoClosure def: The orbit map with codomain restricted to the closed cyclic subspace. · OAI.KadisonSimilarity.orbitIntoClosure
orbitLinearMap def: The linear orbit map at the vector realizing the positive functional. · OAI.KadisonSimilarity.orbitLinearMap
orbitList def: Canonical cyclic list, retaining one-element fixed cycles. · OAI.FKCRT.MapCycles.orbitList
orbitList def: Unlike Perm.toList, this retains fixed-point boundary circles. · OAI.IntegralCharacterVarieties.BoundaryBand.orbitList
orbitMap def: An equivariant involution descends to orbit classes. · OAI.SharpLiebThirring.PLParity.orbitMap
orbitObservation def: All translates of a finite observation of the normal factor, simultaneously. · OAI.EilenbergGanea.orbitObservation
orbitProbability def: The regular Riesz measure associated with the invariant orbit average. · OAI.PolycyclicRecognition.orbitProbability
orbitProduct def: The product of the valuations of all six transforms of an object. · OAI.MatrixMultiplication.AuxiliarySeparation.LegPermutation.orbitProduct
orbitProductEquiv def: Reconstruct a point in a free orbit from its representative and a group element. · OAI.SharpLiebThirring.PLParity.orbitProductEquiv
orbitProjection def: The orbit map at the literal unit isotropic vector, extended linearly. · OAI.ArnoldCounterexample.CriticalBound.orbitProjection
orbitRepresentation def: Restriction to a cyclic reducing subspace is a unital star representation. · OAI.KadisonSimilarity.orbitRepresentation
orbitRows def: The row word arising from repeated translation in an additive parameter space. This is the row operation in both tree constructions. · OAI.TreeEdit.BinaryLower.CheapTranslations.orbitRows
orbitSecondMoment def: The four scalar statistics needed for the expected graph cost. · OAI.KadisonKastler.orbitSecondMoment
orbitSet def: The full cycle as an orbit, including one-element cycles. · OAI.CubeShuffle.orbitSet
orbitSubgraphEquiv def: Distinct edge orbits correspond bijectively to actual unlabelled subgraphs. The target has no isolated vertices, so its image subgraph is uniquely determined by its edges. This also covers the empty graph, with its empty vertex set. · OAI.LeanBlast.SecondKahnKalai.orbitSubgraphEquiv
orbitSynthesis def: Sum of orbit vectors against placement coordinates. The repeated stabilizer fibers are left in the sum, so no transversal choice is needed. · OAI.CoordinateSweeps.PlacementOccurrence.orbitSynthesis
OrbitV structure: Prism for diagonal, endpoint channel frozen nevertheless read. · OAI.ExactDerandomization.Vessel.OrbitV
orbitValue def: Candidate values along one residue class. The starting coefficient can be zero. · OAI.Nagata.W08.orbitValue
orbitWeight def: Circuit weights in their actual operator multiplication order. · OAI.KervaireCycle.orbitWeight
orbitWindowMeasure def: N + 1 orbit samples, so the probability law is defined also at N = 0. · OAI.DukePrimeDegree.orbitWindowMeasure
Order abbrev: The cyclotomic order as an actual subalgebra of the complex numbers. · OAI.CirculantHadamard.CyclotomicLocalBase.Order
Order abbrev: none is the flexible order; a typed order is some type. · OAI.FKCRT.Inventory.Order
order def: Largest order of a jet factor. Coefficients themselves use jets through order two. · OAI.ClosedSurfaceR4.JetPolynomial.Expression.order
orderAtLeast def: A polynomial vanishes to order at least m when it belongs to the mth power of the point's evaluation ideal. · OAI.Nagata.AffineMultiplicity.orderAtLeast
OrderBound def: A quantitative bound, with constants independent of concentration and of the half-space point. The polynomial spatial weight is integrable against our Gaussian/normal exponential. · OAI.ElasticityBoundaryOrderBound.OrderBound
orderBounds def: The finite real comparison scale, viewed in extended nonnegative reals. · OAI.CoveringOrder.orderBounds
Ordered def: Every pair of members of a finite set is comparable. · OAI.FiniteChain.Ordered
ordered def: A reusable explicit increasing-index enumeration. · OAI.MaxCutGames.Foundations.PCP.AlphabetTable.Enumeration.ordered
ordered def: A reusable explicit increasing-index enumeration. · OAI.UniqueGamesTheorem.Foundations.PCP.AlphabetTable.Enumeration.ordered
orderedAtomBasis def: The base ordered atoms are inherited from the explicit signed cylinder. · OAI.Thomason.Construction.PastingGlobeBasis.orderedAtomBasis
orderedAtomBasis def: The concrete ordered atom data of two equal globes pasted along a face. · OAI.Thomason.Construction.PastingGlobeCylinderBasis.orderedAtomBasis
orderedAtomBasis def: Every field is the corresponding literal normalized-chain construction. No generation assertion is required by this interface. · OAI.Thomason.Construction.SimplexComplex.orderedAtomBasis
OrderedBandGridCompatibility def: The manuscript's ordered compatibility: a new label meets no nontrivial component formed using only earlier labels at a free point. · OAI.CircleDomainRigidity.FiniteTransfer.OrderedBandGridCompatibility
OrderedBodyLocalizes def: Localization of bodies in the six actual host modes, using the two orders induced by the chosen enumerations. · OAI.Problem348.Construction.OrderedBodyLocalizes
orderedBox def: Closed truncations of the actual ordered positive-node chamber. · OAI.Honeycomb370.orderedBox
orderedConstantEdges def: Ordered neighboring pairs at which a real function has equal values. · OAI.LeanBlast.GotsmanLinial.orderedConstantEdges
OrderedData structure: The ordered-chart data. The base ordered field never changes during this identifier's lifetime. The auxiliary field is replaced by its actual signed increment history; its bound is 100, not an unbounded sum of the individual increments. · OAI.PlanarL1.Columns.Model.OrderedData
OrderedDistinctPoints abbrev: The genuine set of ordered tuples of pairwise distinct plane points. · OAI.Nagata.ProjectiveGeometry.OrderedDistinctPoints
orderedEval def: Exactly one O-corner, with its orientation sign. · OAI.Release075.BlockPresentation.orderedEval
OrderedFrom def: Monotonicity on the two arcs, expressed in the circle cut at a. The separate minimum condition records the wrap-around step to index zero. · OAI.PlanarL1.Columns.Model.Field.OrderedFrom
orderedGroupRationalCone def: Positive fractions in the rational hull. A positive numerator may be chosen after multiplying numerator and denominator; no unperforation of G is assumed. · OAI.Kaplansky.orderedGroupRationalCone
orderedInsertPermutation def: Insert the distinguished derivative slot at p, preserving the order of all remaining slots before applying their permutation. · OAI.Mahler.orderedInsertPermutation
orderedIntervalPairs def: The ordered separated pairs of actual interval occurrences. · OAI.ThompsonNonamenability.orderedIntervalPairs
orderedLabel def: An ordered object function extends uniquely to a strict poset label. · OAI.Thomason.Construction.StrictOmegaCategory.orderedLabel
orderedLineIndex def: The component containing an ordered mark; all additional marks are on zero. · OAI.Nagata.Workers.W14.orderedLineIndex
orderedList def: The list induced by an explicit numbering, in increasing numeric order. · OAI.MaxCutGames.Foundations.PCP.RawInitialRows.orderedList
orderedList def: The list induced by an explicit numbering, in increasing numeric order. · OAI.UniqueGamesTheorem.Foundations.PCP.RawInitialRows.orderedList
orderedMissingLabelMatrix def: The operator at the uniquely omitted label, with zero on intersecting pairs. · OAI.KadisonKastler.orderedMissingLabelMatrix
orderedNeighborPairs def: All ordered pairs of neighboring vertices. · OAI.LeanBlast.GotsmanLinial.orderedNeighborPairs
orderedPair def: Kuratowski ordered pair: p = {{x},{x,y}}. · OAI.PartitionConsistency.SetLanguage.orderedPair
orderedPermutationEquiv def: An order-preserving decomposition of every permutation into its first image and a permutation of the remaining slots. · OAI.Mahler.orderedPermutationEquiv
orderedPreservationMatrix def: The preservation matrix with ordered row and column labels. · OAI.KadisonKastler.orderedPreservationMatrix
OrderedPrimeSystem structure: The index order records elimination order; it is unrelated to the names or numerical values of the ambient prime labels. · OAI.TwoPointCorrelations.OrderedPrimeSystem
orderedProduct def: Coordinate zero acts first. · OAI.CoordinateSweeps.orderedProduct
orderedProduct def: Ordered composition: coordinate 0 is visited first. · OAI.CoordinateSweeps.orderedProduct
orderedProjectiveConfiguration def: An actual element of U_(k²+n), with all projective points explicitly constructed. · OAI.Nagata.W06.LineArrangement.orderedProjectiveConfiguration
orderedRealEmbedding def: The real ring embedding at one place of the ordering. · OAI.DukePrimeDegree.orderedRealEmbedding
orderedRealizationCylinderIso def: The actual extended cylinder agrees with the positive chain cylinder on every ordered atomic realization whose cylinder basis has the proved order. · OAI.Thomason.Construction.StrictOmegaCategory.GrayChainExtension.orderedRealizationCylinderIso
orderedSlotEquiv def: The first three positions are the first lexicographic block, etc. · OAI.PixtonChow.orderedSlotEquiv
OrderedStep structure: A step records the permitted vertex and all of its ports; the two lists only designate the main band through that vertex. Branch ports are not removed or identified. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.OrderedStep
orderedTupleEquiv def: Enumerate the subset from an ordered tuple known to lie over it. · OAI.KadisonKastler.orderedTupleEquiv
orderedTupleFiberEquiv def: An ordering fibre is the type of bijections from its positions onto its underlying subset. · OAI.KadisonKastler.orderedTupleFiberEquiv
orderedTupleList def: The ordered list used to form a noncommutative word. · OAI.KadisonKastler.orderedTupleList
orderedTupleMiddleEquiv def: The preservation-cut bijection recovers the omitted middle letter from the complementary singleton of the two retained subwords. · OAI.KadisonKastler.orderedTupleMiddleEquiv
orderedTupleOfEquiv def: Realize a bijective enumeration of a subset as an ordered tuple in W. · OAI.KadisonKastler.orderedTupleOfEquiv
orderedTupleSplitEquiv def: The exact word-to-pair bijection at a cut without preservation. · OAI.KadisonKastler.orderedTupleSplitEquiv
OrderedUnitSums def: Ordered positive denominator tuples with a prescribed rational sum. Repetitions are allowed. · OAI.Problem337.OrderedUnitSums
orderedVertexHomotopy def: A pointwise inequality on vertices gives a simplicial homotopy into a poset nerve, even when the source is not itself an ordinary nerve. · OAI.Thomason.Construction.PosetGeometry.orderedVertexHomotopy
OrderedWalk structure: Finite sequence of states and vertex tables. States do not impose extra conditions on solutions: they are the ordered facet names computed by the split/permutation/merge recipe. · OAI.IntegralCharacterVarieties.OccurrenceIncidence.VertexTable.OrderedWalk
orderedWeight def: Scalar weights for the exact ordered generator formula. Prefix correction uses η=1. The suffix uses the first t coefficients of ℓ/u^k. · OAI.ExactFourier.Boundary.orderedWeight
orderEquiv def: Bijective enumeration of complete arrival lists; the arrival index is unchanged. · OAI.MatroidProphet.Relabeling.orderEquiv
orderFromList def: Start at the complete product of a known prime list and scan it once. This is sufficient whenever the group order divides that product. · OAI.ExactQuantumFactoring.BitArithmetic.orderFromList
orderingGridMass def: The mass of the finite disjoint strip approximation. · OAI.JointDickman.orderingGridMass
orderInterpolate def: Interpolate between monotone maps along a monotone binary vertex label. · OAI.Thomason.Construction.orderInterpolate
orderKey def: column = false gives row order; true reverses the signs. · OAI.Problem348.TreePositions.orderKey
orderNerveHomotopy def: Every pointwise inequality induces a homotopy of the actual ordinary nerves. · OAI.Thomason.Construction.orderNerveHomotopy
OrderPCompared def: The conclusion of comparison, expressed by an actual unit and membership in the actual ramified maximal ideal. · OAI.CirculantHadamard.LocalComparison.OrderPCompared
orderPosition def: The position of an occurrence is the number of its strict predecessors. · OAI.MatroidProphet.Pivots.orderPosition
orderPrefix def: The first K labels of a complete order, used only to specify the experiment. · OAI.MatroidProphet.orderPrefix
orderPRoot def: The fixed primitive-order-p root used at every induction level. · OAI.CirculantHadamard.LocalComparison.orderPRoot
orderPValue def: Actual order-p evaluation of an element of the cyclic group ring of order p ^ (e + 1). · OAI.CirculantHadamard.LocalComparison.orderPValue
orderRate def: The exact completed-order success expression. Its only runtime inputs are the actual unit order and its totient, both computed from retained rows. · OAI.ExactQuantumFactoring.Completion.Expressions.orderRate
orderResult def: The source order transition ignores empty trials and returns the least nonempty candidate, with zero for the entirely empty list. · OAI.ExactQuantumFactoring.OrderTrial.orderResult
orderResultNet def: Every ordinary raw trial uses its actual tensor block. Nonempty candidates are validated before a shared, linear-size least-positive-candidate fold. · OAI.ExactQuantumFactoring.OrderSlots.orderResultNet
orderSimplex def: The corresponding simplex of the ordinary nerve. · OAI.Thomason.Construction.NormalizedSimplexIdentities.orderSimplex
OrdF def: A transitive set of transitive sets; used to prove ordinal preservation, not as a replacement for the ordinal formula in the statement of ACwo. · OAI.PartitionConsistency.SetForcingNames.SymmetricSystem.OrdF
ordinal def: Left-to-right exact ordinal enumeration of the leaves. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Tree.ordinal
ordinal def: A von Neumann ordinal: a transitive set internally wellordered by membership. · OAI.PartitionConsistency.SetLanguage.ordinal
OrdinalCount def: The exact cardinal-count hypothesis used by the two density proofs. It is a theorem for the concrete system (not a construction premise). · OAI.PartitionConsistency.ProductForcing.ChainCode.OrdinalCount
ordinalLabel def: The canonical strict map from an oriental to its ordered vertices. · OAI.Thomason.Construction.NormalizedSimplexIdentities.ordinalLabel
ordinalMap def: Extend an ordinal map by the new final vertex. · OAI.Thomason.Construction.ConePrism.ordinalMap
ordinalRank def: The von Neumann rank of a selected code, not a check-name for its predetermined ground index. · OAI.PartitionConsistency.ProductForcing.ChainCode.ordinalRank
ordinalTable def: The bounded ground table depends on the component's fixed coding data, never on a condition or a moved generic computation. · OAI.PartitionConsistency.ProductForcing.ChainCode.ordinalTable
ordinalType def: The similarly supported total order-type name. · OAI.PartitionConsistency.ProductForcing.ChainCode.ordinalType
Ordinary inductive: The actual source equations, separate from the exceptional accepting exit for which only the complete raw-run evaluator is invariant. · OAI.FiniteAlgebraicEnvelopes.Compiler.Word.Ordinary
ordinary def: Payment from the already open plateau, without its children plateaus. · OAI.FourierLLogL.Dyadic.Procedure.Paths.CastShow.ordinary
ordinaryBlockEquiv def: The ordinary diagonal block (1,-A), without finite-rank assumptions. · OAI.EilenbergGanea.ordinaryBlockEquiv
ordinaryCoefficient def: Coefficients of the original physical packets, before selector masks. · OAI.LipschitzHilbert.PacketLatticeData.ordinaryCoefficient
ordinaryCoefficient def: Coefficients of the original physical packets, before selector masks. · OAI.LipschitzHilbert.SourceLabeledLattice.ordinaryCoefficient
ordinaryCrossingNumber def: The minimum spatial crossing count over admissible drawings. · OAI.Paper170.ordinaryCrossingNumber
OrdinaryData structure: The literal ordinary-chart scalar data. A nonconstant chart remembers its old band component and its one point source, rather than just a Lipschitz bound. The zero extension outside that component is never used on the chart's region. · OAI.PlanarL1.Columns.Model.OrdinaryData
ordinaryMultiplicity def: The least total degree of a nonzero Taylor coefficient. · OAI.Nagata.AffineMultiplicity.ordinaryMultiplicity
ordinaryMultiplicity def: The ordinary multiplicity is a finite minimum of genuine Taylor degrees. · OAI.Nagata.AffineMultiplicity.ordinaryMultiplicity
ordinaryPacketValue def: Original packet with its coefficient and original selector. · OAI.LipschitzHilbert.SourceLabeledLattice.ordinaryPacketValue
ordinaryPeriod def: The ordinary continuous periodic function associated with a test function. Continuous-map summation agrees pointwise by local finiteness. · OAI.PlaneFiveColor.InvariantMean.ordinaryPeriod
ordinaryRegularPresentation def: A fixed choice is used only to state the precise canonical-driver law on which the Rohde--Schramm trace-generation input is needed. · OAI.SLEExactGauge.ordinaryRegularPresentation
ordinarySectionClassMap def: The section's absolute class followed by localization away from the whole zero section, in the original total space. · OAI.Kaplansky.ordinarySectionClassMap
OrdinarySeed structure: The complete finite launch data, retaining both independent uniform variables and the random good-bump event required in the non-collapse argument. · OAI.BoundedTreewidthL1.LaunchNoise.OrdinarySeed
ordinarySize def: Binary digits, one sign per literal, and one terminal bit form a natural reference size. The concrete framing adds only a fixed multiplicative factor. · OAI.MaxCutGames.BinaryEncoding.ordinarySize
ordinarySize def: Binary digits, one sign per literal, and one terminal bit form a natural reference size. The concrete framing adds only a fixed multiplicative factor. · OAI.UniqueGamesTheorem.BinaryEncoding.ordinarySize
OrdinaryState structure: A finite ordinary chart partition for system A, with the actual globally nested sets and actual jointly distributed label variables. Chart regions may be disconnected; no artificial connected-component identifiers are introduced. · OAI.PlanarL1.Columns.Model.OrdinaryState
ordinaryTensorToDerived def: Ordinary balanced tensor on chains followed by the quasi-isomorphism localization. · OAI.DerivedBar.ordinaryTensorToDerived
OrdinaryTerms abbrev: Actual finite direct sum of terms; the finite product is canonically that sum. · OAI.LittleFinitistic.OrdinarySimulation.OrdinaryTerms
ordinaryXObject abbrev: The actual ordinary X, represented as a genuine D-bimodule. · OAI.LittleFinitistic.OrdinarySimulation.XResolution.ordinaryXObject
orient def: Orient a positive cube toward its unique word-length maximum. · OAI.EilenbergGanea.TraceWords.CubeDescents.orient
oriental def: The m-th oriental is the strict realization of the normalized directed simplex. · OAI.Thomason.Construction.SimplexComplex.oriental
orientalAbelianizationIso def: The actual orientals abelianize naturally to the normalized simplex complexes, without a separate published generation premise. · OAI.Thomason.Construction.NormalizedSimplexIdentities.orientalAbelianizationIso
orientalUnitIso def: On an actual oriental, the abelianization unit is a strict isomorphism. · OAI.Thomason.Construction.NormalizedSimplexIdentities.orientalUnitIso
orientationCoverCore def: The orientation cover is a bundle with the determinant-sign cocycle. · OAI.PositiveEinsteinFour.Geometry.orientationCoverCore
orientationCoverMetric def: In the preferred coordinates the derivative of the covering projection is identity, so the pullback inner product is the actual base inner product. · OAI.PositiveEinsteinFour.Geometry.orientationCoverMetric
orientationEquiv def: False and true are the adjacent orientation digits zero and one. · OAI.MaxCutGames.Foundations.PCP.PoweringEnumeration.orientationEquiv
orientationEquiv def: Choosing a reference isomorphism identifies all orientations with the automorphism group of the target. · OAI.MaxCutGames.Gadget.Orientation.orientationEquiv
orientationEquiv def: False and true are the adjacent orientation digits zero and one. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringEnumeration.orientationEquiv
orientationEquiv def: Choosing a reference isomorphism identifies all orientations with the automorphism group of the target. · OAI.UniqueGamesTheorem.Gadget.Orientation.orientationEquiv
OrientationMetricDoubleCover structure: Hatcher's connected orientation cover, equipped with the pullback metric. Preservation of Einstein and positive sectional curvature is the local-isometry specialization in Petersen §1.3.3. · OAI.PositiveEinsteinFour.OrientationMetricDoubleCover
orientationOracle def: Actual orientation-first, unconditioned augmented observation. · OAI.TraceReconstruction.PairedMechanisms.orientationOracle
orientationsGiven def: Exact posterior, with zero-mass metadata treated by the same canonical formula. In particular unseen orientations are independent fair bits. · OAI.TraceReconstruction.PairedMechanisms.orientationsGiven
orientationSum def: Literal finite orientation contraction of the actual path, with fixed source arrow and the final arrow summed (it will necessarily be the through arrow). This is not a new abstract kernel substituted for the center model. · OAI.CriticalHoneycomb.Transfer.ChordOrientations.CenterPath.orientationSum
orientationTransitionDet def: Determinant of the actual tangent transition, from chart i to chart j. · OAI.PositiveEinsteinFour.Geometry.orientationTransitionDet
orientBasis def: Reorient a row basis by changing one sign when necessary. · OAI.DukePrimeDegree.orientBasis
orientedBlockChange def: The nonlinear-change indicator for the actual oriented quadratic block. · OAI.MaxCutGames.Gadget.OrientedBlockKernel.orientedBlockChange
orientedBlockChange def: The nonlinear-change indicator for the actual oriented quadratic block. · OAI.UniqueGamesTheorem.Gadget.OrientedBlockKernel.orientedBlockChange
orientedBlockLinear def: The actual binary-linear embedding belonging to an oriented child. · OAI.MaxCutGames.Gadget.OrientedBlockKernel.orientedBlockLinear
orientedBlockLinear def: The actual binary-linear embedding belonging to an oriented child. · OAI.UniqueGamesTheorem.Gadget.OrientedBlockKernel.orientedBlockLinear
orientedBlockRestriction def: The actual child restriction after a choice of block orientation. · OAI.MaxCutGames.Quadratic.orientedBlockRestriction
orientedBlockRestriction def: The actual child restriction after a choice of block orientation. · OAI.UniqueGamesTheorem.Quadratic.orientedBlockRestriction
orientedCoefficient def: Actual coefficient of an n-form in the oriented omitted-coordinate basis. · OAI.MahlerStokes.orientedCoefficient
orientedDensity def: Outward surface coefficient in an ambient orthonormal frame. Division by Ω b fixes the orientation to the specified ambient volume form. At a unit normal x this is the cofactor formula for the restriction of B to x-perp. · OAI.Mahler.orientedDensity
OrientedInfix def: A contiguous subword in either orientation. · OAI.TwoPointCorrelations.OrientedInfix
orientedProd def: Orientation false uses P,Q in chronological order; true reverses them. · OAI.TraceReconstruction.FiniteLaw.orientedProd
OrientedPublishedInputs def: The classical hypotheses on a closed oriented metric, using its canonical Borel measurable structure. · OAI.PositiveEinsteinFour.OrientedPublishedInputs
orientedReachable def: The reachable set includes a possibly bad terminal site, exactly as in the manuscript's S. This is the finite set to which its contour argument applies. · OAI.PlanarFPP.orientedReachable
orientedSurvival def: n forward steps with all visited sites except possibly the last good. · OAI.PlanarFPP.orientedSurvival
orientPair def: The orientation records the conditioning endpoint first. · OAI.FiniteFactor.CouplingParameters.orientPair
origin def: The discrete launch location depends only on the rooted decomposition and active keys. In particular it does not consult any preparatory randomness. · OAI.BoundedTreewidthL1.AnchorTreeParticles.origin
origin def: The discrete launch location depends only on the rooted decomposition and active keys. In particular it does not consult any preparatory randomness. · OAI.BoundedTreewidthL1.TreeParticles.origin
originalBasis def: The original monomial bases in the s and w components. · OAI.MatrixMultiplication.AuxiliarySeparation.DeterminantFiltration.originalBasis
OriginalChildrenMatch def: The child identifications across every original internal seam. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.OriginalChildrenMatch
originalCoordinate def: Record the polynomial coordinate and the choice of s or w separately. · OAI.MatrixMultiplication.AuxiliarySeparation.DeterminantFiltration.originalCoordinate
originalCurvatureForm def: The complete original curvature form on an actual four-slot tensor. · OAI.PositiveEinsteinFour.HighestHessian.originalCurvatureForm
originalDart def: Literal darts at all original path positions. · OAI.TorsionFreeZeroDivisors.ChainChunkTransport.originalDart
originalEndpoint def: Fixed endpoint projection, still based on the old input's endpoint chart. · OAI.SmoothObstruction.ManifoldCoordinates.originalEndpoint
originalExecution def: Exact execution witness. Its stored count is derived from the four actual phases; a polynomial estimate is supplied separately. · OAI.MaxCutGames.Foundations.Complexity.MachineRegularOriginalBody.originalExecution
originalExecution def: Exact execution witness. Its stored count is derived from the four actual phases; a polynomial estimate is supplied separately. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRegularOriginalBody.originalExecution
OriginalFrameMatch def: Compatibility of the transported frames across an original internal seam. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.OriginalFrameMatch
OriginalFrameValuesAgree def: Equality of the two frame values on an original internal seam. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.OriginalFrameValuesAgree
OriginalInternalFlagsHold def: The framed flag equations on every original internal seam. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.OriginalInternalFlagsHold
OriginalInternalSideValuesTrivial def: The side values on every original internal seam are trivial. · OAI.IntegralCharacterVarieties.SurfacePresentation.Diagram.OriginalInternalSideValuesTrivial
originalInTime def: Concrete timed execution certificate, with no runtime premise. · OAI.MaxCutGames.Foundations.Complexity.MachineRegularMetadata.originalInTime
originalInTime def: Concrete timed execution certificate, with no runtime premise. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRegularMetadata.originalInTime
originalIota def: The direct original insertion in the native P and native fixed-submodule functor. · OAI.ExternalTensor.Section4.ArrowVertex.NativeBridge.originalIota
originalLayoutMargins def: Dummy leading margins reuse the rectangular free-array layout only. · OAI.ContingencyTables.ProfilePrograms.originalLayoutMargins
originalLetterAtBit def: The original letter whose block contains a specified encoded position. · OAI.TreeEdit.BinaryLower.Code.originalLetterAtBit
originalMachine def: A concrete finite program for original vertex blocks. · OAI.MaxCutGames.Foundations.Complexity.MachineRegularVertexBlock.originalMachine
originalMachine def: A concrete finite program for original vertex blocks. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRegularVertexBlock.originalMachine
originalMember def: Membership is a statement about the actual stored tail, used only to identify the numeric answer of the physical rank routine. · OAI.MaxCutGames.Foundations.Complexity.MachineRegularMetadata.originalMember
originalMember def: Membership is a statement about the actual stored tail, used only to identify the numeric answer of the physical rank routine. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRegularMetadata.originalMember
originalNeighborEquiv def: The neighbor fiber at an original vertex is exactly its port fiber. · OAI.TorsionFreeZeroDivisors.PortSubdivision.PortGraph.originalNeighborEquiv
originalPlusCurvatureForm def: The original plus part 4 − (5/6)T_D − (5/3)B_F(T). · OAI.PositiveEinsteinFour.HighestHessian.originalPlusCurvatureForm
originalPolarPairing def: Polarization of the original real quadratic curvature contraction. · OAI.PositiveEinsteinFour.HighestHessian.originalPolarPairing
originalPresentation abbrev: The unstabilized polynomial presentation. · OAI.StableCoordinate.Presentation.originalPresentation
originalSeeds def: The original presentation satisfies each of the finite seed equations. · OAI.LittleFinitistic.FinitePresentation.originalSeeds
originalStateSuccessEquiv def: Every accepted actual state/completion pair unpads to one original table, and every original table has one such pair. · OAI.ContingencyTables.SmallGraphProfiles.originalStateSuccessEquiv
originalTableTest def: Applied to the fixed rectangular shape generated by the enumerator. · OAI.ContingencyTables.ProfilePrograms.originalTableTest
originalToRotationalExtension def: The original group is properly and cocompactly included in the compact extension. · OAI.PolycyclicRecognition.ComplexWeightFactorModel.originalToRotationalExtension
originalTube def: An old tube tests the candidate in the original input's selected charts. The candidate is not required to belong to the good set. · OAI.SmoothObstruction.ManifoldCoordinates.originalTube
originFiber def: An origin fibre is exactly the prescribed outgoing type. · OAI.TorsionFreeZeroDivisors.TypedGraphs.originFiber
originTensorEquiv def: Closed fiber chart with its actual structural maps, not just an existential abstract isomorphism of rings. · OAI.SourcePullbackChart.originTensorEquiv
orthogonalConjugate def: Orthogonal coordinate conjugation, including both orientation components. · OAI.HarmonicCounterexample.Berger.ComplexStructure.orthogonalConjugate
orthogonalMatrixLinear def: An actual orthonormal coordinate map for continuous real endomorphisms. · OAI.HarmonicCounterexample.LinearODE.orthogonalMatrixLinear
orthogonalPairBasis def: A basis adapted to a subspace and its orthogonal complement. · OAI.KervaireSpectral.orthogonalPairBasis
orthogonalPairONB def: The normalized basis is orthonormal. · OAI.KervaireSpectral.orthogonalPairONB
OrthogonalProjectionFamily structure: A finite orthogonal decomposition of the identity by Hermitian projections. · OAI.LeanBlast.GotsmanLinial.OrthogonalProjectionFamily
orthogonalSum def: The sum of two orthogonal projections, in the original algebra. · OAI.Generator.Tracial.orthogonalSum
OrthogonalSummandDecomposition structure: Isometric summand inclusions with mutually orthogonal ranges and dense linear span. The summands may have different Hilbert spaces. · OAI.KadisonSimilarity.OrthogonalSummandDecomposition
orthogonalVector def: The canonical orthogonal unit vector in a qubit. · OAI.QAC.orthogonalVector
OrthogonalWordLevels structure: Actual isometric word inclusions with orthogonal ranges and dense linear span. No operator invariance is built into these geometric data. · OAI.KadisonSimilarity.OrthogonalWordLevels
OrthonormalFrame structure: Columns are the coordinates of an orthonormal tangent frame in the same chart as the concrete metric and curvature used by ClassificationOn. · OAI.PositiveEinsteinFour.Geometry.OrthonormalFrame
orthonormalMatrixUnits def: The rank-one matrix units associated with an orthonormal basis. · OAI.Generator.orthonormalMatrixUnits
otherMediatorPauli def: The second mediator spoke can meet the same or the opposite member. · OAI.ContinuumCoulomb.otherMediatorPauli
others def: The vertices other than the horn's cone vertex. · OAI.Thomason.Construction.HornBarycentric.others
Outcome abbrev: Each fiber records the actual observed total, not a conditional mean. · OAI.KServer.Rounding.Outcome
Outcome abbrev: All four sampling choices remain separate occurrence indices. · OAI.MaxCutGames.Decoder.TableKeysGame.Outcome
Outcome abbrev: Each product element is a separate edge occurrence. · OAI.MaxCutGames.Reduction.ActualGame.Outcome
Outcome abbrev: All four sampling choices remain separate occurrence indices. · OAI.UniqueGamesTheorem.Decoder.TableKeysGame.Outcome
Outcome abbrev: Each product element is a separate edge occurrence. · OAI.UniqueGamesTheorem.Reduction.ActualGame.Outcome
outcomeEquiv def: The reassociation changes only parentheses, not enumeration order. · OAI.MaxCutGames.Reduction.FixedOutcomes.outcomeEquiv
outcomeEquiv def: The reassociation changes only parentheses, not enumeration order. · OAI.UniqueGamesTheorem.Reduction.FixedOutcomes.outcomeEquiv
outcomeLaw def: Complete finite selector/terminal outcome law, with independent tables. · OAI.Problem310.ConcreteLabels.outcomeLaw
outer def: Genuine rank-one endomorphisms for a bilinear form. · OAI.HarmonicCounterexample.Control.outer
outer def: Reparametrization of the second outer branch by the central coordinate. · OAI.Problem356.CommonBranches.outer
OuterBalanced structure: The two balancing properties of the selected outer label coloring. · OAI.QuantitativeVanDerWaerden.OuterBalanced
outerBarSuffix def: The exterior free D bar uses its actual categorical coordinates. · OAI.DerivedBar.ParityBridge.outerBarSuffix
outerCast def: Retain the exterior D coefficient while applying an exact degree transport. · OAI.DerivedBar.ParityBridge.outerCast
OuterChoice structure: A concrete outer coloring, with its counted family of actual cyclic words. The family is retained so that its membership proof can be used directly by the subsequent progression dichotomy. · OAI.QuantitativeVanDerWaerden.ConstructionModel.OuterChoice
outerEmbedding def: The actual outer Hilbert sum of the component embeddings. · OAI.DiamondDistortion.RecursiveModel.outerEmbedding
OuterHybrid def: The hybrid selector for Q_A₀ Y|_B₀ at (A/A₀, B). · OAI.MaxCutGames.Appendix.Identities.OuterHybrid
OuterHybrid def: The hybrid selector for Q_A₀ Y|_B₀ at (A/A₀, B). · OAI.UniqueGamesTheorem.Appendix.Identities.OuterHybrid
outerMatrix def: The rank-one matrix associated with a vector. · OAI.ContingencyTables.outerMatrix
outerNeighbors def: Distinct tests are adjacent precisely when they share a label variable. · OAI.QuantitativeVanDerWaerden.outerNeighbors
OuterRibbon structure: A finite genus-zero relator ribbon with a marked outside vertex. Isolated zero-sign darts are retained, so cap surgery never changes the finite universe. · OAI.KervaireRibbon.OuterRibbon
outerSlope def: A fixed measurable choice of the outer slope. · OAI.Problem358.Sharpness.outerSlope
outerStatement def: Actual statements for the table loop. No unbounded natural number is stored in the state or evaluated by a state-update function. · OAI.MaxCutGames.Foundations.Complexity.MachineExpanderTable.outerStatement
outerStatement def: Actual statements for the table loop. No unbounded natural number is stored in the state or evaluated by a state-update function. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineExpanderTable.outerStatement
outerSuffixEmbed def: Retain the exterior D coefficient while inserting an arbitrary suffix. · OAI.DerivedBar.ParityBridge.outerSuffixEmbed
outerValues def: The literal outer ℓ² norm of a family of finite component seminorms. · OAI.DiamondDistortion.RecursiveModel.outerValues
OuterVectorInput abbrev: Walk data and margins; trial markers, denominator, and fallback matrix. · OAI.ContingencyTables.ProfilePrograms.OuterVectorInput
OuterVectorInput abbrev: Walk data and margins; trial markers, denominator, and fallback matrix. · OAI.ContingencyTables.UnitCountPrograms.OuterVectorInput
outerWordFamily def: Actual fixed outer coefficients needed for one-patch terms. · OAI.Generator.Tracial.outerWordFamily
Output structure: All output tuples have one common positive dimension. · OAI.RationalHitting.Output
Output def: Output is the binary word beginning at origin and ending at its first blank. · OAI.WLIdentification.Output
output def: Coordinate output, including the absolutely summable root-start paths and the child contribution. No finite-support restriction is made on a Rep. · OAI.DiamondDistortion.StartModel.output
output def: The output view is never a field of the executable state. · OAI.LowSpaceSubsetSum.MergedReader.output
output def: The actual output constructor depends only on finite tables, not on the correcting map or its equivariance proof. · OAI.MaxCutGames.Decoder.MatrixGap.Parameters.output
output def: A total executable map with finite vector data and no nonlinear-map input. · OAI.MaxCutGames.Integration.TableReduction.output
output def: The semantic output, with the control state updated after each input bit. · OAI.MaxCutGames.Reduction.MachineTransducer.output
output def: All proof-only numerical parameters and the correcting function are absent from the runtime input of this map. · OAI.MaxCutGames.Reduction.UniformReduction.output
output def: The actual simple, unweighted output. Its common integer scale is printed; loops and both orientations have already been handled by exact aggregation. · OAI.OptimalMaxCut.LongCode.Game.output
output def: A computable generator preserving the list and its order. · OAI.RationalHitting.ExecutableGenerator.output
output def: The actual output constructor depends only on finite tables, not on the correcting map or its equivariance proof. · OAI.UniqueGamesTheorem.Decoder.MatrixGap.Parameters.output
output def: The executable output list has one row for each radix edge-tuple address. · OAI.UniqueGamesTheorem.Explicit.ProductOutput.output
output def: A total executable map with finite vector data and no nonlinear-map input. · OAI.UniqueGamesTheorem.Integration.TableReduction.output
output def: The semantic output, with the control state updated after each input bit. · OAI.UniqueGamesTheorem.Reduction.MachineTransducer.output
output def: All proof-only numerical parameters and the correcting function are absent from the runtime input of this map. · OAI.UniqueGamesTheorem.Reduction.UniformReduction.output
outputBits def: Serialized output in the shared generic graph codec. · OAI.MaxCutGames.Foundations.PCP.PoweringTables.outputBits
outputBits def: Serialized output in the shared generic graph codec. · OAI.UniqueGamesTheorem.Foundations.PCP.PoweringTables.outputBits
outputBlock def: A single discarded-number diagonal block of U(ρA ⊗ ρB)U†. The sums are finite because passive mixing preserves number in each paired mode. The two unrestricted density matrices allow all internal entanglement. · OAI.EntropyPhotonNumber.outputBlock
OutputBound def: This records graph size, every numeric bit length, and total output length. The polynomial is quantified inside the choice of the fixed factor. · OAI.UniformSparsestCut.OutputBound
outputDualChange def: The output-dual matrix uses the inverse basis change, as required for tensors. · OAI.MatrixMultiplication.AuxiliarySeparation.DeterminantFiltration.outputDualChange
outputGraph def: The fixed-width output construction, with private-wire length absorbed. · OAI.WLIdentification.ProductCircuit.outputGraph
outputInstance def: The correcting function is not evaluated when producing an edge list. · OAI.MaxCutGames.Decoder.TableKeysAddressGame.outputInstance
outputInstance def: Only canonical words, bounded radix arithmetic, and the explicit outcome list occur in the runtime path. The correcting function is not inspected. · OAI.MaxCutGames.Reduction.AddressGame.outputInstance
outputInstance def: The correcting function is not evaluated when producing an edge list. · OAI.UniqueGamesTheorem.Decoder.TableKeysAddressGame.outputInstance
outputInstance def: Only canonical words, bounded radix arithmetic, and the explicit outcome list occur in the runtime path. The correcting function is not inspected. · OAI.UniqueGamesTheorem.Reduction.AddressGame.outputInstance
outputInstanceWithEnumeration def: Executable finite output, still allowing loops and parallel occurrences. · OAI.MaxCutGames.Decoder.TableKeysGame.outputInstanceWithEnumeration
outputInstanceWithEnumeration def: This construction computes its vertex numbering, all outcomes, and every forward/inverse permutation entry from the explicitly supplied finite data. · OAI.MaxCutGames.Reduction.ActualGame.outputInstanceWithEnumeration
outputInstanceWithEnumeration def: Executable finite output, still allowing loops and parallel occurrences. · OAI.UniqueGamesTheorem.Decoder.TableKeysGame.outputInstanceWithEnumeration
outputInstanceWithEnumeration def: This construction computes its vertex numbering, all outcomes, and every forward/inverse permutation entry from the explicitly supplied finite data. · OAI.UniqueGamesTheorem.Reduction.ActualGame.outputInstanceWithEnumeration
outputInTime def: The complete run produces the literal full-table target code, clears all private stacks, resets its finite state, and halts. · OAI.MaxCutGames.Explicit.MachineProductRuntime.outputInTime
outputInTime def: The complete actual run, including the physical cleanup and output reversal. · OAI.MaxCutGames.Explicit.MachineUniformRun.outputInTime
outputInTime def: Copy, cleanup, and reversal all run on the actual finite transition program. · OAI.UniqueGamesTheorem.Explicit.MachineFixedCopies.outputInTime
outputInTime def: The complete run produces the literal full-table target code, clears all private stacks, resets its finite state, and halts. · OAI.UniqueGamesTheorem.Explicit.MachineProductRuntime.outputInTime
outputInTime def: The complete actual run, including the physical cleanup and output reversal. · OAI.UniqueGamesTheorem.Explicit.MachineUniformRun.outputInTime
outputLaw def: This is the shared-seed mixture used by Game.localEmbeddingLaw. · OAI.MaxCutGames.Foundations.Repetition.CompletedSampling.outputLaw
outputLaw def: This is the shared-seed mixture used by Game.localEmbeddingLaw. · OAI.UniqueGamesTheorem.Foundations.Repetition.CompletedSampling.outputLaw
outputList def: n entries of n bits each, in the n² wires immediately after the input. · OAI.ExactQuantumFactoring.outputList
outputMass def: The exact mass assigned by the finite uniform proposal. · OAI.ContingencyTables.FairBits.outputMass
outputMass def: The exact first-success-or-fallback mass from the manuscript. · OAI.ContingencyTables.FirstSuccess.outputMass
outputNet def: Runtime output selection retains the entire unused ordinary computation even on the rare branch. Neither success statistics nor future auxiliary data enter this network. · OAI.ExactQuantumFactoring.Completion.outputNet
outputOffset def: The entire bounded contribution to the emitted row: output port plus q times output cloud, with q = d*d. · OAI.MaxCutGames.Foundations.PCP.ExpanderRowControl.outputOffset
outputOffset def: The entire bounded contribution to the emitted row: output port plus q times output cloud, with q = d*d. · OAI.UniqueGamesTheorem.Foundations.PCP.ExpanderRowControl.outputOffset
outputOn def: Completed output decoding keeps the FULL rare guessed transition word. No truncation, success flag, or later true-order certificate enters this decoder. · OAI.ExactQuantumFactoring.OrderSlots.outputOn
outputPolynomial def: One fixed polynomial includes every declared address and every edge table. · OAI.MaxCutGames.Reduction.AddressOutputSize.outputPolynomial
outputPolynomial def: An explicit polynomial for the actual serialized game. · OAI.MaxCutGames.Reduction.OutputSize.outputPolynomial
outputPolynomial def: One fixed polynomial includes every declared address and every edge table. · OAI.UniqueGamesTheorem.Reduction.AddressOutputSize.outputPolynomial
outputPolynomial def: An explicit polynomial for the actual serialized game. · OAI.UniqueGamesTheorem.Reduction.OutputSize.outputPolynomial
outputProbability def: Born probability for an arbitrary fixed input basis state. · OAI.QAC.outputProbability
outputRow def: A row indexed by its actual place in the emitted edge list. · OAI.UniqueGamesTheorem.Explicit.ProductOutput.outputRow
outputsInTime def: Exact execution includes literal initialization, the physical Horner subroutine, input consumption, and cleanup of every coefficient tape. · OAI.MaxCutGames.Foundations.Complexity.CookLevin.PolynomialMachine.outputsInTime
outputsInTime def: The extra cost is bounded from the actual raw execution, independently of the contents of its dirty private tapes and finite state. · OAI.MaxCutGames.Foundations.Complexity.FinalCNFCleanup.outputsInTime
outputsInTime def: No assumption about cleanup execution or clean raw work tapes is required. · OAI.MaxCutGames.Foundations.Complexity.MachineCanonicalOutput.outputsInTime
outputsInTime def: Exact execution includes literal initialization, the physical Horner subroutine, input consumption, and cleanup of every coefficient tape. · OAI.UniqueGamesTheorem.Foundations.Complexity.CookLevin.PolynomialMachine.outputsInTime
outputsInTime def: The extra cost is bounded from the actual raw execution, independently of the contents of its dirty private tapes and finite state. · OAI.UniqueGamesTheorem.Foundations.Complexity.FinalCNFCleanup.outputsInTime
outputsInTime def: No assumption about cleanup execution or clean raw work tapes is required. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineCanonicalOutput.outputsInTime
OutputsTable def: A halted tape contains the row-major binary matrix encoding. · OAI.ContingencyTables.Algorithms.OutputsTable
outputZeroCount def: The measured zero-outcome projector is a product projection on one qubit. · OAI.QAC.outputZeroCount
Outside def: The remaining part of a draw, satisfying the outside observation. · OAI.MaxCutGames.Soundness.ZeroInformation.Outside
Outside def: The remaining part of a draw, satisfying the outside observation. · OAI.UniqueGamesTheorem.Soundness.ZeroInformation.Outside
outside def: The raw unprocessed drawing is a fixed closed obstacle at each stage. · OAI.PlanarL1.HeightTree.outside
OutsideBall def: The ball reached by at most r edges after the first outside neighbour of x, with all subsequent vertices outside X. · OAI.CycleClique.Construction.OutsideBall
outsideClassDenseSet def: The standard incompatible padding gives a dense set in the whole Cohen forcing. This is the precise requirement to be met by the intermediate ordinary extension. · OAI.PartitionConsistency.ProductForcing.outsideClassDenseSet
outsideClassSet def: A class outside the entire local-name evaluation list, decided in a literal stronger diagram that also contains all the condition coordinates. The right factor of the root is recorded and is necessarily nontrivial. · OAI.PartitionConsistency.ProductForcing.outsideClassSet
outsideEquiv def: Use precisely the source's upper support bound for all remaining degrees. · OAI.LittleFinitistic.OrdinarySimulation.FullTensorTerm.outsideEquiv
OutsideLocal def: Persistent exclusion from the evaluation list of all local names at p. This statement is expressed in the actual symmetric forcing relation. · OAI.PartitionConsistency.ProductForcing.OutsideLocal
outsideNearTops def: The extra tops come from nearby next-level cells outside the immediate parent, and all belong to the same block ancestor. · OAI.LipschitzHilbert.AngularGrid.outsideNearTops
outsideNeighborPairFiberEquiv def: The full rank-one neighbor-sampling fiber is the functional kernel. · OAI.MaxCutGames.Inverse.KMSBasisComparison.outsideNeighborPairFiberEquiv
outsideNeighborPairFiberEquiv def: The full rank-one neighbor-sampling fiber is the functional kernel. · OAI.UniqueGamesTheorem.Inverse.KMSBasisComparison.outsideNeighborPairFiberEquiv
outsideNeighborSample def: The full-neighbor sampler on its exact good-event factor space. · OAI.MaxCutGames.Inverse.KMSBasisComparison.outsideNeighborSample
outsideNeighborSample def: The full-neighbor sampler on its exact good-event factor space. · OAI.UniqueGamesTheorem.Inverse.KMSBasisComparison.outsideNeighborSample
OutsidePath def: A simple outside path with d internal vertices. Parameter zero is the joining edge; positive parameters agree with the boundary-graph version. · OAI.CycleClique.Construction.OutsidePath
outsideSelectedLabel def: Every remaining coordinate of the first word is an exposed labelled sample. · OAI.Problem337.RandomExposure.outsideSelectedLabel
outsideTag def: none marks every genuine target cell, while the two exterior orientations remain separately identifiable. · OAI.HyperbolicObstruction.PictureSwitch.PolygonTarget.Boundary.outsideTag
outSpin def: Spin zero follows the positive source wire direction: west out of an up center, and middle/south into it. Spin two is the reverse direction. · OAI.CriticalHoneycomb.Transfer.ChordOrientations.outSpin
outward def: Only the side facing away from the input is outward. · OAI.OneWayLiveness.CellNetwork.outward
overBase def: Underlying R-modules of any module presheaf on Spec R. · OAI.AffineExterior.overBase
overBaseChangeHomotopy def: A homotopy over the base pulls back to an actual simplicial homotopy. It does not require a fibration or preservation of pullbacks by realization. · OAI.Thomason.Construction.overBaseChangeHomotopy
overBaseChangeMap def: The actual pullback map of a morphism over a fixed base. · OAI.Thomason.Construction.overBaseChangeMap
overflowNodes def: Overflow count includes an overflowing full-prefix node that is dropped. · OAI.LowSpaceSubsetSum.overflowNodes
overFunctor def: The cone construction as a functor to the actual slice over [1]. · OAI.Thomason.Construction.StrictOmegaCategory.TerminalCone.overFunctor
Overlap def: The nonempty intersections of the intermediate partitions. · OAI.FiniteAlgebraicEnvelopes.TwistedBrinThompson.Table.Overlap
overlapEquiv def: The overlap on the partner block is exactly the image of the first overlap. · OAI.TorsionFreeZeroDivisors.BlockAlignment.overlapEquiv
OverlapPath structure: A bounded nondecreasing representative of an admissible overlap quantile. · OAI.InvariantIsing.OverlapPath
OverlapPrescription structure: A finite overlap prescription, before transitive closure. none specifies no overlap; some (a,b) identifies (i,a*x) with (j,b*x) for each x. · OAI.PartitionConsistency.DiagramForcing.OverlapPrescription
overlapRows def: The common entries of two master assignments, counted by array type. · OAI.HyperbolicObstruction.TableModel.overlapRows
OverlapType abbrev: A fixed, finite overcount of all nonempty matching types. Types with incompatible coordinate constraints simply have zero realizations. · OAI.HyperbolicObstruction.BernoulliCandidates.OverlapType
overlay def: An extension of a small register store with arbitrary untouched registers. · OAI.ExactQuantumFactoring.BitStackProgram.overlay
overlay def: Add the expander's stored ports; every new constraint always accepts. · OAI.MaxCutGames.Foundations.PCP.PreprocessingOverlayTables.overlay
overlay def: Fixed earlier and later structures implement precedence. Label recycling is immaterial to this one-edit operation: an unchanged record has its old label and both other structures are held fixed. · OAI.UniformKServer.KeyEdits.overlay
overlay def: Add the expander's stored ports; every new constraint always accepts. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingOverlayTables.overlay
overlayExpander def: The overlay expander is transported to this exact padded vertex set. · OAI.MaxCutGames.Foundations.PCP.Preprocessing.overlayExpander
overlayExpander def: The overlay expander is transported to this exact padded vertex set. · OAI.UniqueGamesTheorem.Foundations.PCP.Preprocessing.overlayExpander
overlayInTime def: Overlay executes on the two archives physically produced above; only its output stack changes in the retained outside frame. · OAI.MaxCutGames.Foundations.Complexity.MachinePaddedOverlay.overlayInTime
overlayInTime def: Overlay executes on the two archives physically produced above; only its output stack changes in the retained outside frame. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachinePaddedOverlay.overlayInTime
overlayPorts def: Old ports precede expander ports. · OAI.MaxCutGames.Foundations.PCP.PreprocessingOverlayTables.overlayPorts
overlayPorts def: Old ports precede expander ports. · OAI.UniqueGamesTheorem.Foundations.PCP.PreprocessingOverlayTables.overlayPorts
overlayTape def: Overlay reads exactly the two archives written by the previous phases. · OAI.MaxCutGames.Foundations.Complexity.MachinePaddedOverlay.overlayTape
overlayTape def: Overlay reads exactly the two archives written by the previous phases. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachinePaddedOverlay.overlayTape
overLocalEquiv def: Restriction of local linear Hom to a slice is the local linear Hom on that slice. This is the actual restriction identity needed to glue coherence. · OAI.CoherentDual.overLocalEquiv
ownBox def: The actual spatial rectangle and direction of a source packet. · OAI.LipschitzHilbert.PacketLatticeData.ownBox
ownBox def: The actual spatial rectangle and direction of a source packet. · OAI.LipschitzHilbert.SourceLabeledLattice.ownBox
ownDensityTest def: The ordinary density includes each packet's own rectangle. · OAI.LipschitzHilbert.PacketLatticeData.ownDensityTest
ownDensityTest def: The ordinary density includes each packet's own rectangle. · OAI.LipschitzHilbert.SourceLabeledLattice.ownDensityTest
ownedReferenceEquiv def: Forgetting an owner is a bijection, rather than a bounded-multiplicity map. This is why recombination has no top-count loss. · OAI.LipschitzHilbert.SourceLabeledLattice.ownedReferenceEquiv
OwnedTranslations structure: Purely local interfaces for the actual affine vertex translations. The piecewise field is a bijection on vertices, not an asserted game strategy. · OAI.WLIdentification.AffineBlocks.OwnedTranslations
owner def: Every vertical name has a neighboring square, even for singleton factors. · OAI.WLIdentification.RingLayout.owner
ownerInTime def: Actual complete owner execution, including the genuinely empty padding case. The bound is assembled from proven component bounds; its family term is not used as an exact number of transitions. · OAI.MaxCutGames.Foundations.Complexity.MachineRegularOwnerBody.ownerInTime
ownerInTime def: Actual complete owner execution, including the genuinely empty padding case. The bound is assembled from proven component bounds; its family term is not used as an exact number of transitions. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRegularOwnerBody.ownerInTime
ownerTapes def: Original table, affine query, copied scan, owner output, and scratch. · OAI.MaxCutGames.Foundations.Complexity.MachineRegularMetadata.ownerTapes
ownerTapes def: Original table, affine query, copied scan, owner output, and scratch. · OAI.UniqueGamesTheorem.Foundations.Complexity.MachineRegularMetadata.ownerTapes