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Entrywise product rank and incompatible tuples

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

proven

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    Natural Language Statement

    Lemma

    Rank-factor expansions prove the entrywise product rank bound and the actual binary incompatibility obstruction.

    Concept map
    79 concepts; 2 descendants hidden
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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 budgetThe 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 testsConcrete 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 plusframeCut 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 columnsBaseline contractions on the selected atomsThe symmetric binary gradient formGram-conditioned columns and theirrank-failure probabilityTwo-sided Gram normalization forindividually injective framesEntrywise product rank and incompatibletuplesThe 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 cutprofilesExtracting fresh label coefficients throughpin quotientsRank 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 labelsMatrix 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 pinvectorsConsistent symmetric binary prescriptions ontwo witness listsTable contractions on effective profiles andtheir full extensionsTable coordinates and private channelcomplementsMajority intersections in the cyclic taggeometryPure 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 A
    Evidence

    This concept declares 5 statements. Each proof establishes one of them relative to its assumptions.

    1 all_one_rank proven

    2 compatible_pair_in_tuple proven

    3 hadamard_rank proven

    4 incompatible_tuple_bound proven

    5 one_add_rank proven

    Lean source view on GitHub

    1import Lax342547.RankFactors
    2import Lax342547.RankedProjection
    3import Mathlib.LinearAlgebra.Matrix.Hadamard
    4
    5/-!
    6---
    7title: Entrywise product rank and incompatible tuples
    8type: lemma
    9---
    10Rank-factor expansions prove the entrywise product rank bound and the actual binary incompatibility obstruction.
    11-/
    12
    13namespace Lax342547.HadamardRank
    14
    15open Lax342547.MomentSpace
    16open scoped BigOperators Matrix
    17
    18axiom hadamard_rank {I J : Type} [Fintype I] [Fintype J]
    19 (A B : Matrix I J Binary) : (A ⊙ B).rank ≤ A.rank*B.rank
    20
    21axiom all_one_rank {I J : Type} [Fintype I] [Fintype J] :
    22 (Matrix.of (fun _ : I => fun _ : J => (1 : Binary))).rank ≤ 1
    23
    24axiom one_add_rank {I J : Type} [Fintype I] [Fintype J] (A : Matrix I J Binary) :
    25 ((Matrix.of (fun _ : I => fun _ : J => (1 : Binary)))+A).rank ≤ 1+A.rank
    26
    27axiom incompatible_tuple_bound {Ω ι : Type} [Fintype Ω] [Fintype ι] [DecidableEq ι]
    28 (F : Matrix Ω Ω Binary) (x : ι → Ω) (hdiag : ∀ a, F a a = 0)
    29 (hinc : ∀ i j, i ≠ j → ¬ (F (x i) (x j) = 0 ∧ F (x j) (x i) = 0)) :
    30 Fintype.card ι ≤ (1+F.rank)^2
    31
    32axiom compatible_pair_in_tuple {Ω ι : Type} [Fintype Ω] [Fintype ι] [DecidableEq ι]
    33 (F : Matrix Ω Ω Binary) (x : ι → Ω) (hdiag : ∀ a, F a a = 0)
    34 (hsize : (1+F.rank)^2 < Fintype.card ι) :
    35 ∃ i j, i ≠ j ∧ F (x i) (x j) = 0 ∧ F (x j) (x i) = 0
    36
    37end Lax342547.HadamardRank
    38
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