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Uniform raw channel phase tails over bounded-rank targets

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

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    Natural Language Statement

    Lemma

    The actual raw frame phase-tail estimate holds simultaneously over every tensor target of bounded total rank, retaining the sharp rank-allocation count and the channel density cost.

    Concept map
    72 concepts
    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 finiteindicesActual 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 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 channelsCounting component tensors with boundedtotal rankUniform raw channel phase tails overbounded-rank targetsAdmissible 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 2 statements. Each proof establishes one of them relative to its assumptions.

    1 no_cover_raw_family_tail proven

    2 no_cover_uniform_target_tail proven

    Lean source view on GitHub

    1import Lax342547.ChannelColumns
    2import Lax342547.TotalRankCount
    3
    4/-!
    5---
    6title: Uniform raw channel phase tails over bounded-rank targets
    7type: lemma
    8---
    9The actual raw frame phase-tail estimate holds simultaneously over every tensor target of bounded total rank, retaining the sharp rank-allocation count and the channel density cost.
    10-/
    11
    12namespace Lax342547.UniformPhaseTails
    13
    14open Lax342547.MomentSpace Lax342547.RawFrames Lax342547.ChannelDensity Lax342547.ChannelCharacters
    15open Lax342547.RealCellLaws Lax342547.FiniteSampling Lax342547.TensorCharacters
    16open Lax342547.PushforwardWalsh Lax342547.RetainedImages Lax342547.ChannelColumns
    17open Lax342547.RelativeEntropy Lax342547.ComponentSpaces Lax342547.ComponentDuals Lax342547.CoverProjection
    18open scoped BigOperators
    19
    20axiom no_cover_raw_family_tail {e B N Ω C : Type} [Fintype e] [Fintype B] [Fintype N] [Fintype Ω] [Fintype C]
    21 [DecidableEq e] [DecidableEq N] (E : Matrix B B Binary)
    22 (μ : Ω → ℝ) (A : Ω → e → Matrix N N Binary) (ψ : C → Ω → ℝ) (r h t k : ℕ) (p θ : ℝ)
    23 [Nonempty (Frame B (Fin h) N E)] (hN : 2*h+1 ≤ Fintype.card N)
    24 (hμ : Probability μ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1) (hr : 1 ≤ r)
    25 (hA : ∀ x, ∑ i, (A x i).rank ≤ r) (hψ : ∀ c x, |ψ c x| ≤ 1)
    26 (hsize : 2*t = 2^k) (hk : 100*r+12 ≤ k) (hθ : 0 < θ)
    27 (hcover : ∀ S : Submodule Binary (e → N → Binary),
    28 ∀ T : Submodule Binary (Module.Dual Binary (e → N → Binary)),
    29 Componentwise S → DualComponentwise T →
    30 (Module.finrank Binary S : ℝ) ≤ (r : ℝ)*(2*t) →
    31 (Module.finrank Binary T : ℝ) ≤ (r : ℝ)*(2*t) →
    32 cellMass μ (fun x => projection (LinearMap.piMap (fun i => (A x i).mulVecLin)) S T = 0) ≤ p) :
    33 cellMass (weights (Lax342547.RawLaw.uniformLaw E))
    34 (fun o => ∃ c : C, θ ≤ |∑ x, μ x*ψ c x*tensorCharacter h (A x) (columnsEquiv h (fun a => channels (o a)))|) ≤
    35 (Fintype.card C : ℝ)*(2 : ℝ)^(Fintype.card e*(h*h+2))*
    36 (((4 : ℝ)^(2*t)*p^(2^(k-(100*r+10)))+1/(2 : ℝ)^(h*2^(k-(100*r+12))))/θ^(2*t))
    37
    38axiom no_cover_uniform_target_tail {e B N Ω : Type} [Fintype e] [Fintype B] [Fintype N] [Fintype Ω]
    39 [DecidableEq e] [DecidableEq N] (E : Matrix B B Binary)
    40 (μ : Ω → ℝ) (A : Ω → e → Matrix N N Binary) (ψ : (e → Matrix N N Binary) → Ω → ℝ) (r h t k : ℕ) (p θ : ℝ)
    41 [Nonempty (Frame B (Fin h) N E)] (hN : 2*h+1 ≤ Fintype.card N)
    42 (hμ : Probability μ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1) (hr : 1 ≤ r)
    43 (hA : ∀ x, ∑ i, (A x i).rank ≤ r) (hψ : ∀ S, (∑ i, (S i).rank) ≤ r → ∀ x, |ψ S x| ≤ 1)
    44 (hsize : 2*t = 2^k) (hk : 100*r+12 ≤ k) (hθ : 0 < θ)
    45 (hcover : ∀ S : Submodule Binary (e → N → Binary),
    46 ∀ T : Submodule Binary (Module.Dual Binary (e → N → 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 cellMass (weights (Lax342547.RawLaw.uniformLaw E))
    52 (fun o => ∃ S : e → Matrix N N Binary, (∑ i, (S i).rank) ≤ r ∧
    53 θ ≤ |∑ x, μ x*ψ S x*tensorCharacter h (A x) (columnsEquiv h (fun a => channels (o a)))|) ≤
    54 (r+1 : ℝ)^Fintype.card e*(2 : ℝ)^(2*Fintype.card N*r)*
    55 (2 : ℝ)^(Fintype.card e*(h*h+2))*
    56 (((4 : ℝ)^(2*t)*p^(2^(k-(100*r+10)))+1/(2 : ℝ)^(h*2^(k-(100*r+12))))/θ^(2*t))
    57
    58end Lax342547.UniformPhaseTails
    59
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