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Universal protected pin witnesses

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

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    Natural Language Statement

    Lemma

    Independent pin witnesses have an amplified probability bound even when the small protected space depends on the whole unit.

    Concept map
    79 concepts; 7 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 statisticsFactorization and counting of low-rankbinary matricesDimension 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 partitionsActual phase averages from small exceptionaltailsProjected 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 channelsCounting component tensors with boundedtotal rankUniform raw channel phase tails overbounded-rank targetsAdmissible pair lawsUniversal protected pin witnessesOrthogonality and finite Walsh correlationbounds
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

    1 marked_unit_witness_probability proven

    2 some_protected_witness_probability proven

    Lean source view on GitHub

    1import Lax342547.IndependentWitnesses
    2import Lax342547.PhaseAverages
    3
    4/-!
    5---
    6title: Universal protected pin witnesses
    7type: lemma
    8---
    9Independent pin witnesses have an amplified probability bound even when the small protected space depends on the whole unit.
    10-/
    11
    12namespace Lax342547.UniversalWitnesses
    13
    14open Lax342547.MomentSpace Lax342547.RetainedImages Lax342547.RealCellLaws Lax342547.PushforwardWalsh
    15open scoped BigOperators
    16
    17axiom some_protected_witness_probability {I H N : Type}
    18 [Fintype I] [Fintype H] [Fintype N] [DecidableEq I] [DecidableEq H] [DecidableEq N]
    19 (V : I → Submodule Binary (N → Binary)) (hV : iSupIndep V)
    20 (K : ℕ) (hK : ∀ i,Module.finrank Binary (V i) ≤ K) :
    21 cellMass (weights (PMF.uniformOfFintype (Matrix N H Binary)))
    22 (fun Y => ∃ S : Submodule Binary (H → Binary),Module.finrank Binary S ≤ K ∧
    23 ∀ i,∃ v : V i,v.val ≠ 0 ∧ Y.transpose.mulVec v.val ∈ S) ≤
    24 (2 : ℝ)^(Fintype.card H*K+2*K*Fintype.card I)/(2 : ℝ)^(Fintype.card H*Fintype.card I)
    25
    26axiom marked_unit_witness_probability {I H N B : Type}
    27 [Fintype I] [Fintype H] [Fintype N] [Fintype B] [DecidableEq I] [DecidableEq H] [DecidableEq N]
    28 (V : I → Submodule Binary (N → Binary)) (hV : iSupIndep V)
    29 (β : B → ℝ) (Y : B → Matrix N H Binary) (S : B → Submodule Binary (H → Binary))
    30 (K : ℕ) (L : ℝ) (hβ : ∀ b,0 ≤ β b) (hL : 0 ≤ L) (hK : ∀ i,Module.finrank Binary (V i) ≤ K)
    31 (hS : ∀ b,Module.finrank Binary (S b) ≤ K)
    32 (hcap : ∀ y,push β Y y ≤ L*weights (PMF.uniformOfFintype (Matrix N H Binary)) y) :
    33 cellMass β (fun b => ∀ i,∃ v : V i,v.val ≠ 0 ∧ (Y b).transpose.mulVec v.val ∈ S b) ≤
    34 L*((2 : ℝ)^(Fintype.card H*K+2*K*Fintype.card I)/(2 : ℝ)^(Fintype.card H*Fintype.card I))
    35
    36end Lax342547.UniversalWitnesses
    37
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