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Coefficient phases as dot products of rank-factor images

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

proven

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    Natural Language Statement

    Lemma

    A binary coefficient tensor contracted with a cross Gram matrix is the dot product of its two rank-factor image tuples, flattened over rank positions and ambient coordinates.

    Concept map
    92 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 channelsCoefficient phases as dot products ofrank-factor imagesA 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 boundsFinite scalar agreement from binarycharacter boundsAmbient symmetries and frame marginalsFresh key directions are independent modulotable spacesBaseline bilinear extensions retaining allfrozen rows and columnsBaseline contractions on the selected atomsThe symmetric binary gradient formGram-conditioned columns and theirrank-failure probabilityTwo-sided Gram normalization forindividually injective framesScalar agreement after quotient characterestimatesThe 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 framesPrimal projections and preservation ofeffective spacesPure obstructions on the actual pair of cutprofilesWalsh bounds for independent image lawsand separated phasesCompression on the actual barred nominalquotientsExtracting fresh label coefficients throughpin quotientsRank of a tensor killed in two quotientspacesRank-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 formImage caps inside original retained cellsSelected 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 spacesOrthogonality and finite Walsh correlationboundsWitness 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 2 statements. Each proof establishes one of them relative to its assumptions.

    1 coefficient_trace proven

    2 independent_factor_phase proven

    Lean source view on GitHub

    1import Lax342547.TargetMatrices
    2import Lax342547.GramTests
    3
    4/-!
    5---
    6title: Coefficient phases as dot products of rank-factor images
    7type: lemma
    8---
    9A binary coefficient tensor contracted with a cross Gram matrix is the dot product of its two rank-factor image tuples, flattened over rank positions and ambient coordinates.
    10-/
    11
    12namespace Lax342547.CoefficientPhase
    13
    14open Lax342547.MomentSpace Lax342547.Walsh
    15
    16noncomputable def coefficientPair {I J N : Type} [Fintype I] [Fintype J] [Fintype N]
    17 (C : Matrix I J Binary) (X : Matrix N I Binary) (Y : Matrix N J Binary) : Binary :=
    18 ∑ i, ∑ j, C i j * (X.transpose * Y) i j
    19
    20noncomputable def factorImage {I R N : Type} [Fintype I]
    21 (A : Matrix I R Binary) (X : Matrix N I Binary) : R × N → Binary :=
    22 fun rn => (X * A) rn.2 rn.1
    23
    24axiom coefficient_trace {I J N : Type} [Fintype I] [Fintype J] [Fintype N]
    25 (C : Matrix I J Binary) (X : Matrix N I Binary) (Y : Matrix N J Binary) :
    26 coefficientPair C X Y = Matrix.trace (C.transpose * (X.transpose * Y))
    27
    28axiom independent_factor_phase {I J R N : Type}
    29 [Fintype I] [Fintype J] [Fintype R] [Fintype N]
    30 (A : Matrix I R Binary) (B : Matrix J R Binary) (X : Matrix N I Binary) (Y : Matrix N J Binary) :
    31 coefficientPair (A * B.transpose) X Y = dotProduct (factorImage A X) (factorImage B Y)
    32
    33end Lax342547.CoefficientPhase
    34
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