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No-cover phase moments

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

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    Natural Language Statement

    Lemma

    Actual componentwise binary channel moments and phase-tail bounds from finite rank growth, under the explicit small-cover exclusion.

    Concept map
    60 concepts
    100%
    Actual projected span deficitSampling after exposed coordinatesAdaptive sampling on disjoint coordinatesetsUnion over adaptive component coversExact uniform bilinear character meanWalsh operator bounds with explicit bilinearrankRank of a lifted tensor sumIndependent binary channel charactersActual 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 even moment expansionFinite independent sampling and vertexexception tailsGreedy mode space exposureSmall actual greedy exposure tailsNo-cover rank growth on arbitrary finiteindicesActual 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 restrictionsOriginal 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 ADescendants are omitted for concepts with more than 10 descendants.
    Evidence

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

    1 no_cover_even_moment proven

    2 no_cover_phase_tail proven

    3 sum_matrix_rank proven

    Lean source view on GitHub

    1import Lax342547.ComponentMoments
    2import Lax342547.IndexedRankGrowth
    3
    4/-!
    5---
    6title: No-cover phase moments
    7type: lemma
    8---
    9Actual componentwise binary channel moments and phase-tail bounds from finite rank growth, under the explicit small-cover exclusion.
    10-/
    11
    12namespace Lax342547.NoCoverMoments
    13
    14open Lax342547.MomentSpace Lax342547.RelativeEntropy Lax342547.FiniteSampling Lax342547.RetainedImages
    15open Lax342547.ChannelCharacters Lax342547.TensorCharacters Lax342547.ComponentSpaces
    16open Lax342547.ComponentDuals Lax342547.CoverProjection
    17open scoped BigOperators
    18
    19axiom sum_matrix_rank {I J ι : Type} [Fintype I] [Fintype J] [Fintype ι]
    20 (A : ι → Matrix I J Binary) :
    21 Module.finrank Binary (LinearMap.range (∑ i, (A i).mulVecLin)) = (∑ i, A i).rank
    22
    23axiom no_cover_even_moment {e I J Ω : Type} [Fintype e] [Fintype I] [Fintype J] [Fintype Ω]
    24 [DecidableEq e] [DecidableEq I] [DecidableEq J]
    25 (μ : Ω → ℝ) (A : Ω → e → Matrix I J Binary) (ψ : Ω → ℝ) (r h t k : ℕ) (p : ℝ)
    26 (hμ : Probability μ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1) (hr : 1 ≤ r)
    27 (hA : ∀ x, ∑ i, (A x i).rank ≤ r) (hψ : ∀ x, |ψ x| ≤ 1)
    28 (hsize : 2*t = 2^k) (hk : 100*r+12 ≤ k)
    29 (hcover : ∀ S : Submodule Binary (e → I → Binary),
    30 ∀ T : Submodule Binary (Module.Dual Binary (e → J → Binary)),
    31 Componentwise S → DualComponentwise T →
    32 (Module.finrank Binary S : ℝ) ≤ (r : ℝ)*(2*t) →
    33 (Module.finrank Binary T : ℝ) ≤ (r : ℝ)*(2*t) →
    34 cellMass μ (fun x => projection (LinearMap.piMap (fun i => (A x i).mulVecLin)) S T = 0) ≤ p) :
    35 (∑ c, productLaw (fun _ : e × Fin h => channelLaw) c*
    36 |∑ x, μ x*ψ x*tensorCharacter h (A x) c|^(2*t)) ≤
    37 (4 : ℝ)^(2*t)*p^(2^(k-(100*r+10)))+1/(2 : ℝ)^(h*2^(k-(100*r+12)))
    38
    39axiom no_cover_phase_tail {e I J Ω : Type} [Fintype e] [Fintype I] [Fintype J] [Fintype Ω]
    40 [DecidableEq e] [DecidableEq I] [DecidableEq J]
    41 (μ : Ω → ℝ) (A : Ω → e → Matrix I J Binary) (ψ : Ω → ℝ) (r h t k : ℕ) (p θ : ℝ)
    42 (hμ : Probability μ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1) (hr : 1 ≤ r)
    43 (hA : ∀ x, ∑ i, (A x i).rank ≤ r) (hψ : ∀ x, |ψ x| ≤ 1)
    44 (hsize : 2*t = 2^k) (hk : 100*r+12 ≤ k)
    45 (hcover : ∀ S : Submodule Binary (e → I → Binary),
    46 ∀ T : Submodule Binary (Module.Dual Binary (e → J → Binary)),
    47 Componentwise S → DualComponentwise T →
    48 (Module.finrank Binary S : ℝ) ≤ (r : ℝ)*(2*t) →
    49 (Module.finrank Binary T : ℝ) ≤ (r : ℝ)*(2*t) →
    50 cellMass μ (fun x => projection (LinearMap.piMap (fun i => (A x i).mulVecLin)) S T = 0) ≤ p)
    51 (hθ : 0 < θ) :
    52 cellMass (productLaw (fun _ : e × Fin h => channelLaw))
    53 (fun c => θ ≤ |∑ x, μ x*ψ x*tensorCharacter h (A x) c|) ≤
    54 ((4 : ℝ)^(2*t)*p^(2^(k-(100*r+10)))+1/(2 : ℝ)^(h*2^(k-(100*r+12))))/θ^(2*t)
    55
    56end Lax342547.NoCoverMoments
    57
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