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Amplified channel failures on independent pinned spaces

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

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    Natural Language Statement

    Lemma

    Nonzero choices from actual independent pinned spaces form independent witnesses; counting all choices and adaptive protected spaces gives the amplified failure exponent.

    Concept map
    74 concepts; 8 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 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 tailsAmplified channel failures on independentpinned spacesNo-cover rank growth on arbitrary finiteindicesUniform 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 spacesExact probabilities for independent protectedchannel imagesWalsh 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 modesCounting all bounded-dimensional protectedspacesMonotonicity 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 adaptive_independent_witness_probability proven

    2 independent_witness_matrix proven

    3 nonzero_witness_card proven

    Lean source view on GitHub

    1import Lax342547.AdaptiveProtection
    2import Mathlib.LinearAlgebra.DFinsupp
    3
    4/-!
    5---
    6title: Amplified channel failures on independent pinned spaces
    7type: lemma
    8---
    9Nonzero choices from actual independent pinned spaces form independent witnesses; counting all choices and adaptive protected spaces gives the amplified failure exponent.
    10-/
    11
    12namespace Lax342547.IndependentWitnesses
    13
    14open Lax342547.MomentSpace Lax342547.RetainedImages Lax342547.RealCellLaws
    15open scoped BigOperators
    16
    17axiom independent_witness_matrix {I N : Type} [Fintype I]
    18 (V : I → Submodule Binary (N → Binary)) (hV : iSupIndep V)
    19 (v : ∀ i,V i) (hv : ∀ i,(v i).val ≠ 0) :
    20 Function.Injective (Matrix.of (fun n i => (v i).val n)).mulVec
    21
    22axiom nonzero_witness_card {I N : Type} [Fintype I] [Fintype N]
    23 (V : I → Submodule Binary (N → Binary)) (K : ℕ) (hK : ∀ i,Module.finrank Binary (V i) ≤ K) :
    24 Nat.card (∀ i,{v : V i // v.val ≠ 0}) ≤ 2^(K*Fintype.card I)
    25
    26axiom adaptive_independent_witness_probability {I H N : Type}
    27 [Fintype I] [Fintype H] [Fintype N] [DecidableEq I] [DecidableEq H] [DecidableEq N]
    28 (V : I → Submodule Binary (N → Binary)) (hV : iSupIndep V)
    29 (S : Matrix N H Binary → Submodule Binary (H → Binary)) (K : ℕ)
    30 (hK : ∀ i,Module.finrank Binary (V i) ≤ K) (hS : ∀ Y,Module.finrank Binary (S Y) ≤ K) :
    31 cellMass (weights (PMF.uniformOfFintype (Matrix N H Binary)))
    32 (fun Y => ∀ i,∃ v : V i,v.val ≠ 0 ∧ Y.transpose.mulVec v.val ∈ S Y) ≤
    33 (2 : ℝ)^(Fintype.card H*K+2*K*Fintype.card I)/(2 : ℝ)^(Fintype.card H*Fintype.card I)
    34
    35end Lax342547.IndependentWitnesses
    36
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