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Rank factors independent modulo the actual frozen spaces

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

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    Natural Language Statement

    Lemma

    Projecting the coefficient tensor to complements of the frozen spaces gives rank factors whose quotient images are injective. This is the exact independence needed for conditional image entropy.

    Concept map
    90 concepts
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    Exact-image bounds for independent affinecolumnsOrdered atom products in the actualgradient formPoint atoms and finite flavor distributionsSparse unselected primal vectors in barredspaces lie in the pinsBarred response spaces and the actualobstruction rank budgetCompression preserving allowed pointmoments and the actual cut domainThe exact space of binary base momentsRank control for the frozen baseline onprimal inputsFormal ordered product bits realizesymmetric correctionsBounded allowed derivatives preserving theactual linearized responseAllowed channel changes and the actual tableinjection testsRank-controlled factorization throughallowed orthogonal channelsA bounded-rank pure correction for theactual scalar recipeConcrete cut-space testers and the orderedmixer formConcrete coordinates, quadratic testers, andthe self-Gram formNumerical recipes prescribe the concretegradients on witness atomsThe affine minus-column law at a fixed plusframeCompression that fixes pin/key vectors andpreserves target matricesCut profiles and the constant kernelThe full linearized response on pairs ofactual cut profilesFull response obstructions are effectiveprofiles plus selected atomsExact image pins in nominal coefficientspacesFinite linear images and their uniform-lawdensity boundsAmbient symmetries and frame marginalsFresh key directions are independent modulotable spacesBaseline bilinear extensions retaining allfrozen rows and columnsExact agreement of zero quotient charactersBaseline contractions on the selected atomsThe symmetric binary gradient formGram-conditioned columns and theirrank-failure probabilityTwo-sided Gram normalization forindividually injective framesThe binary hole relationPaying the reference-image conditioning anddimension costsUniform injective frames and channeltranspose failureFresh point-ray spans are disjoint from thenominal table spacesLow-rank Boolean moments have boundedlabel supportTriangle relations separate into individuallabel blocksBoolean point moments with restricted basecoordinatesThe 3K+28 baseline bound in the actualnominal coordinatesPrimal blocks of the nominal spaces andtheir bounded table partEndpoint projections of paired pure-responseannihilatorsActual paired-witness key spaces satisfy thebaseline hypothesesBinary prescriptions at all endpoints of apaired scalar recipeFull paired witness lists and scalar recipeequationsSparse pin exclusions with arbitrary basecoefficientsSparse residual contractions belong to theactual primal pinsRetractions with bounded rank on theprimal inputsExact images mixing independent injectiveframesJoint minus images after exposing severalplus framesSplit quotient projections preserve coefficientcharactersPrimal projections and preservation ofeffective spacesPure obstructions on the actual pair of cutprofilesCompression on the actual barred nominalquotientsExtracting fresh label coefficients throughpin quotientsRank factors independent modulo the actualfrozen spacesRank of a tensor killed in two quotientspacesFull rank coefficient factorizationsRank-controlled pure forms on the actualbarred quotientsBounded baselines for both actual crossorientationsActual frame observations realize thenominal channel contractionsRaw matrix frames and their tensorrealizationThe finite uniform raw-vertex lawGradient residuals vanish on effective profilesand selected atomsCoupled scalar recipes give consistent atomgradientsJoint reference image caps across both signsand all drawsThe full reference cap for exact pin eventsRemoving selected atoms leaves onlyunselected component labelsThe bounded pure remainder of an actualscalar recipeMatrix representations and the boundedresidual rank ingredientsUnrestricted linearized solutions for actualscalar recipesRank loss under restriction of a bilinear formSelected tensor blocks of actual pureobstructionsA selected affine ray determines its momentblockSubtracting selected atoms preserves pureannihilationSelected label coefficients agree across thecut profileInterpolation of finitely many binary selectorlabelsNumerical cross tables, injection flags, andunary admissibilityUnselected sparse vectors cannot concealfresh key coefficientsA uniform label budget for all sparse pinvectorsLow-rank tester routing along tag starsConsistent symmetric binary prescriptions ontwo witness listsTable contractions on effective profiles andtheir full extensionsTable coordinates and private channelcomplementsMajority intersections in the cyclic taggeometryExplicit low-rank matrices for thewhole-space gradient targetThe fifteen-rank witness tester targetPure bilinear responses detect quotienttensorsQuotient extractors isolate individual tensorlabel blocksWell-defined channel contractions onprojected tensor spacesWitness atoms and their numerical testerrecords
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.
    Evidence

    This concept declares 3 statements. Each proof establishes one of them relative to its assumptions.

    1 factor_range proven

    2 projection_range_quotient_injective proven

    3 quotient_independent_factors proven

    Lean source view on GitHub

    1import Lax342547.ProjectedCoefficients
    2import Lax342547.RankFactors
    3
    4/-!
    5---
    6title: Rank factors independent modulo the actual frozen spaces
    7type: lemma
    8---
    9Projecting the coefficient tensor to complements of the frozen spaces gives rank factors whose quotient images are injective. This is the exact independence needed for conditional image entropy.
    10-/
    11
    12namespace Lax342547.QuotientFactors
    13
    14open Lax342547.MomentSpace
    15
    16axiom factor_range {I J R : Type} [Fintype I] [Fintype J] [Fintype R]
    17 (C : Matrix I J Binary) (A : Matrix I R Binary) (B : Matrix J R Binary)
    18 (hC : A * B.transpose = C) (hA : Function.Injective A.mulVec)
    19 (hr : Fintype.card R = C.rank) : LinearMap.range A.mulVecLin = LinearMap.range C.mulVecLin
    20
    21axiom projection_range_quotient_injective {V U : Type}
    22 [AddCommGroup V] [Module Binary V] [AddCommGroup U] [Module Binary U]
    23 (D : Submodule Binary V) (p : V →ₗ[Binary] V)
    24 (hp : LinearMap.ker p = D) (hpid : p.comp p = p)
    25 (A : U →ₗ[Binary] V) (hA : Function.Injective A)
    26 (hrange : LinearMap.range A ≤ LinearMap.range p) : Function.Injective (D.mkQ.comp A)
    27
    28axiom quotient_independent_factors {I : Type} [Fintype I]
    29 (C : Matrix I I Binary) (D E : Submodule Binary (I → Binary))
    30 (p q : (I → Binary) →ₗ[Binary] (I → Binary))
    31 (hp : LinearMap.ker p = D) (hq : LinearMap.ker q = E)
    32 (hpid : p.comp p = p) (hqid : q.comp q = q) : by
    33 classical
    34 let Z := p.toMatrix' * C * q.toMatrix'.transpose
    35 exact ∃ A : Matrix I (Fin Z.rank) Binary, ∃ B : Matrix I (Fin Z.rank) Binary,
    36 A * B.transpose = Z ∧ Function.Injective (D.mkQ.comp A.mulVecLin) ∧
    37 Function.Injective (E.mkQ.comp B.mulVecLin)
    38
    39end Lax342547.QuotientFactors
    40
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