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Uniform injective-matrix density

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

proven

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    Natural Language Statement

    Lemma

    Exact raw endpoint channel marginals have density at most two over uniform full matrices.

    Concept map
    71 concepts; 8 descendants hidden
    100%
    Actual projected span deficitSampling after exposed coordinatesAdaptive sampling on disjoint coordinatesetsUnion over adaptive component coversExact-image bounds for independent affinecolumnsExact 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 diagonaltensorsExact 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 projectionRank 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 informationprojectionProjection removes the exposed termsCount tested index occurrencesFinite linear images and their uniform-lawdensity boundsFinite even moment expansionFinite independent sampling and vertexexception tailsAmbient symmetries and frame marginalsGram-conditioned columns and theirrank-failure probabilityTwo-sided Gram normalization forindividually injective framesGreedy mode space exposureSmall actual greedy exposure tailsNo-cover rank growth on arbitrary finiteindicesUniform injective-matrix densityUniform injective frames and channeltranspose failureActual deficits indexed by a distinct listFinite list tail statisticsDimension 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 partitionsProjected nonzero terms in the actualremaining sumDimensions of projected mode spacesWalsh bounds for independent image lawsand separated phasesRank of a tensor killed in two quotientspacesRank loss under two restrictionsRaw matrix frames and their tensorrealizationThe finite uniform raw-vertex lawOriginal retained-cell laws from finite PMFsFinite relative entropy and support costsRank loss under restriction of a bilinear formImage caps inside original retained cellsOne exposure controls both modesMonotonicity of span deficitsMode space span deficitsNonzero tensor count from span deficitsRank of an actual linear map sumCharacters of all independent tensor channelsAdmissible pair lawsOrthogonality and finite Walsh correlationbounds
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

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

    1 injective_matrix_point_cap proven

    2 raw_minus_channel_point_cap proven

    3 raw_plus_channel_point_cap proven

    Lean source view on GitHub

    1import Lax342547.InjectiveFrames
    2import Lax342547.ChannelColumns
    3
    4/-!
    5---
    6title: Uniform injective-matrix density
    7type: lemma
    8---
    9Exact raw endpoint channel marginals have density at most two over uniform full matrices.
    10-/
    11
    12namespace Lax342547.InjectiveDensity
    13
    14open Lax342547.MomentSpace Lax342547.FrameSymmetry Lax342547.RawFrames
    15open Lax342547.RealCellLaws Lax342547.PushforwardWalsh
    16open scoped ENNReal
    17
    18axiom injective_matrix_point_cap {H N : Type} [Fintype H] [Fintype N]
    19 [DecidableEq H] [DecidableEq N] [Nonempty (Injection H N)]
    20 (hN : Fintype.card H+1 ≤ Fintype.card N) (Y : Matrix N H Binary) :
    21 push (weights (PMF.uniformOfFintype (Injection H N))) Subtype.val Y ≤
    22 2*weights (PMF.uniformOfFintype (Matrix N H Binary)) Y
    23
    24axiom raw_minus_channel_point_cap {B H N : Type} [Fintype B] [Fintype H] [Fintype N]
    25 [DecidableEq H] [DecidableEq N] (E : Matrix B B Binary)
    26 [Nonempty (Frame B H N E)] [Nonempty (Injection H N)]
    27 (hN : Fintype.card H+1 ≤ Fintype.card N) (Y : Matrix N H Binary) :
    28 push (weights (PMF.uniformOfFintype (Frame B H N E))) (fun F => F.Y) Y ≤
    29 2*weights (PMF.uniformOfFintype (Matrix N H Binary)) Y
    30
    31axiom raw_plus_channel_point_cap {B H N : Type} [Fintype B] [Fintype H] [Fintype N]
    32 [DecidableEq H] [DecidableEq N] (E : Matrix B B Binary)
    33 [Nonempty (Frame B H N E)] [Nonempty (Injection H N)]
    34 (hN : Fintype.card H+1 ≤ Fintype.card N) (Y : Matrix N H Binary) :
    35 push (weights (PMF.uniformOfFintype (Frame B H N E))) (fun F => F.X) Y ≤
    36 2*weights (PMF.uniformOfFintype (Matrix N H Binary)) Y
    37
    38end Lax342547.InjectiveDensity
    39
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