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Actual table injection and ambient pin failures

Lax342547.TableInjection · concepts/Lax342547/TableInjection.lean · lax-342547

proven

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    Natural Language Statement

    Lemma

    Restrict the actual cross Gram forms to the numerical table spaces. Each of the sixteen component tests is precisely an ambient primal-image injection modulo the opposite unit's same-sign protected channels.

    Concept map
    103 concepts; 3 descendants hidden
    100%
    Actual projected span deficitChannel bounds with adaptive protectedcoefficient spacesSampling after exposed coordinatesAdaptive sampling on disjoint coordinatesetsUnion over adaptive component coversExact-image bounds for independent affinecolumnsFinite amplified-test averagingExact uniform bilinear character meanWalsh operator bounds with explicit bilinearrankRank of a lifted tensor sumIndependent binary channel charactersActual raw frame channel phase tailsJoint channel law and itsindependent-column densityActual channel moments and phase tailsCombined column and row mode exposureRows of diagonal tensor mapsMoment estimates for actual componenttensor phasesComponentwise mode spaces and diagonaltensorsConcrete coordinates, quadratic testers, andthe self-Gram formThe affine minus-column law at a fixed plusframeExact conditioning costs and recovery offinite probability massesEntropy progress for residual pair lawsIndependence of distinct sample positionsJoin-stable classes of mode coversCount tensors killed by exposureActual exposure cover projectionCut profiles and the constant kernelRank of a diagonal familyDyadic span deficit estimatesA small dyadic tail scaleActual dyadic deficit recurrenceEntropy along feasible mixture lines,including new supportFull feasible support and finite informationprojectionExact image pins in nominal coefficientspacesProjection removes the exposed termsCount tested index occurrencesFinite pin-injection exceptionFinite linear images and their uniform-lawdensity boundsFinite even moment expansionFinite independent sampling and vertexexception tailsAmbient symmetries and frame marginalsThe symmetric binary gradient formGram-conditioned columns and theirrank-failure probabilityTwo-sided Gram normalization forindividually injective framesGreedy independence of subspacesGreedy mode space exposureSmall actual greedy exposure tailsThe binary hole relationPaying the reference-image conditioning anddimension costsGreedy independent failure samplesAmplified channel failures on independentpinned spacesNo-cover rank growth on arbitrary finiteindicesUniform injective frames and channeltranspose failureActual deficits indexed by a distinct listFinite list tail statisticsFactorization and counting of low-rankbinary matricesDimension deficits after an arbitrary modemapExposed mode dimension budgetBoolean point moments with restricted basecoordinatesFinite moment probability and rank splitA low rank sum supplies an actual adaptivecoverNo-cover phase momentsRank growth without an adaptive modecoverSquared restriction cost for independent uniteventsFinite exposure partitionsActual phase averages from small exceptionaltailsExact images mixing independent injectiveframesJoint minus images after exposing severalplus framesProjected nonzero terms in the actualremaining sumPrimal projections and preservation ofeffective spacesDimensions of projected mode spacesExact probabilities for independent protectedchannel imagesWalsh bounds for independent image lawsand separated phasesRank of a tensor killed in two quotientspacesRank loss under two restrictionsActual frame observations realize thenominal channel contractionsRaw matrix frames and their tensorrealizationThe finite uniform raw-vertex lawOriginal retained-cell laws from finite PMFsJoint reference image caps across both signsand all drawsThe full reference cap for exact pin eventsFinite relative entropy and support costsRank loss under restriction of a bilinear formImage caps inside original retained cellsFinite sequential testsOne exposure controls both modesCounting all bounded-dimensional protectedspacesNumerical cross tables, injection flags, andunary admissibilityMonotonicity of span deficitsMode space span deficitsNonzero tensor count from span deficitsRank of an actual linear map sumTable contractions on effective profiles andtheir full extensionsActual table injection and ambient pinfailuresTable coordinates and private channelcomplementsMajority intersections in the cyclic taggeometryCharacters of all independent tensor channelsWell-defined channel contractions onprojected tensor spacesCounting component tensors with boundedtotal rankUniform raw channel phase tails overbounded-rank targetsAdmissible pair lawsUniversal protected pin witnessesOrthogonality and finite Walsh correlationbounds
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

    1 actual_table_injecting_iff proven

    2 admissible_injection_iff proven

    Lean source view on GitHub

    1import Lax342547.RawContractions
    2import Lax342547.FiniteInjection
    3
    4/-!
    5---
    6title: Actual table injection and ambient pin failures
    7type: lemma
    8---
    9Restrict the actual cross Gram forms to the numerical table spaces. Each
    10of the sixteen component tests is precisely an ambient primal-image
    11injection modulo the opposite unit's same-sign protected channels.
    12-/
    13
    14namespace Lax342547.TableInjection
    15
    16open Lax342547.MomentSpace Lax342547.RawFrames Lax342547.ExactPins
    17open Lax342547.TableSpaces Lax342547.SmallTables Lax342547.TableContractions
    18open Lax342547.RawContractions Lax342547.FiniteInjection
    19
    20variable {Comp B H N : Type} [Fintype B] [Fintype H] [Fintype N]
    21 {E : Matrix B B Binary}
    22
    23def rawTable (P Q : Pin (Comp × Bool) (Fin 2 × (B ⊕ H)) N)
    24 (A Bmap : Fin 2 → Comp → Frame B H N E) : Table P Q where
    25 forward e := ((rawForms A Bmap).forward e).compl₁₂
    26 (tableSpace P (e, true)).subtype (tableSpace Q (e, false)).subtype
    27 reverse e := ((rawForms A Bmap).reverse e).compl₁₂
    28 (tableSpace Q (e, true)).subtype (tableSpace P (e, false)).subtype
    29
    30def primalMatrix (A : Fin 2 → Comp → Frame B H N E) (e : Comp) (i : Fin 2)
    31 (s : Bool) : Matrix N B Binary := if s then (A i e).P else (A i e).Q
    32
    33def oppositeChannel (A : Fin 2 → Comp → Frame B H N E) (e : Comp) (i : Fin 2)
    34 (s : Bool) : Matrix N H Binary := if s then (A i e).Y else (A i e).X
    35
    36def primalImage (P : Pin (Comp × Bool) (Fin 2 × (B ⊕ H)) N)
    37 (A : Fin 2 → Comp → Frame B H N E) (e : Comp) (i : Fin 2) (s : Bool) :
    38 Submodule Binary (N → Binary) :=
    39 (pinnedPrimal P i (e, s)).map (primalMatrix A e i s).mulVecLin
    40
    41def Failed (P Q : Pin (Comp × Bool) (Fin 2 × (B ⊕ H)) N)
    42 (A Bmap : Fin 2 → Comp → Frame B H N E) (e : Comp) (i z : Fin 2)
    43 (s role : Bool) : Prop :=
    44 if role then
    45 pinFailure (primalImage Q Bmap e i s) (oppositeChannel A e z s)
    46 (protectedChannel P z (e, s))
    47 else
    48 pinFailure (primalImage P A e i s) (oppositeChannel Bmap e z s)
    49 (protectedChannel Q z (e, s))
    50
    51axiom actual_table_injecting_iff (P Q : Pin (Comp × Bool) (Fin 2 × (B ⊕ H)) N)
    52 (A Bmap : Fin 2 → Comp → Frame B H N E) :
    53 Injecting (rawTable P Q A Bmap) ↔ ∀ e i z s role,¬ Failed P Q A Bmap e i z s role
    54
    55axiom admissible_injection_iff (P Q : Pin (Comp × Bool) (Fin 2 × (B ⊕ H)) N)
    56 (A Bmap : Fin 2 → Comp → Frame B H N E) (T : Table P Q)
    57 (hT : Admissible T Lax342547.ReferencePins.observation Lax342547.ReferencePins.observation A Bmap) :
    58 Injecting T ↔ Injecting (rawTable P Q A Bmap)
    59
    60end Lax342547.TableInjection
    61
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