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Actual phase averages from small exceptional tails

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

proven

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    Natural Language Statement

    Lemma

    Absolute values of the actual tensor characters and finite density transfer recover averaged phase cancellation from checked tails without normalizing away exceptional mass.

    Concept map
    73 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 partitionsActual phase averages from small exceptionaltailsProjected 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 4 statements. Each proof establishes one of them relative to its assumptions.

    1 average_from_phase_tail proven

    2 conditional_phase_abs_le_one proven

    3 finite_event_density proven

    4 tensor_character_abs proven

    Lean source view on GitHub

    1import Lax342547.UniformPhaseTails
    2
    3/-!
    4---
    5title: Actual phase averages from small exceptional tails
    6type: lemma
    7---
    8Absolute values of the actual tensor characters and finite density transfer recover averaged phase cancellation from checked tails without normalizing away exceptional mass.
    9-/
    10
    11namespace Lax342547.PhaseAverages
    12
    13open Lax342547.MomentSpace Lax342547.Walsh Lax342547.ChannelCharacters Lax342547.TensorCharacters
    14open Lax342547.RelativeEntropy Lax342547.RetainedImages
    15open scoped BigOperators
    16
    17axiom tensor_character_abs {e I J : Type} [Fintype e] [Fintype I] [Fintype J]
    18 (h : ℕ) (A : e → Matrix I J Binary) (c : e × Fin h → (I → Binary) × (J → Binary)) :
    19 |tensorCharacter h A c| = 1
    20
    21axiom conditional_phase_abs_le_one {e I J Ω : Type} [Fintype e] [Fintype I] [Fintype J] [Fintype Ω]
    22 (μ : Ω → ℝ) (A : Ω → e → Matrix I J Binary) (ψ : Ω → ℝ)
    23 (h : ℕ) (c : e × Fin h → (I → Binary) × (J → Binary))
    24 (hμ : Probability μ) (hψ : ∀ x, |ψ x| ≤ 1) :
    25 |∑ x, μ x*ψ x*tensorCharacter h (A x) c| ≤ 1
    26
    27axiom finite_event_density {Ω : Type} [Fintype Ω] (α β : Ω → ℝ) (L : ℝ)
    28 (hcap : ∀ x, β x ≤ L*α x) (E : Ω → Prop) : cellMass β E ≤ L*cellMass α E
    29
    30axiom average_from_phase_tail {Ω : Type} [Fintype Ω] (β : Ω → ℝ) (f : Ω → ℝ)
    31 (θ : ℝ) (hβ : Probability β) (hf : ∀ x, |f x| ≤ 1) (hθ : 0 ≤ θ) :
    32 |∑ x, β x*f x| ≤ θ+cellMass β (fun x => θ ≤ |f x|)
    33
    34end Lax342547.PhaseAverages
    35
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