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Fixed dyadic moment orders at the ambient scale

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

proven

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    Natural Language Statement

    Lemma

    An explicit natural logarithm choice supplies even dyadic moments below N/c and above N/(2*c), with a fixed parameter budget independent of the actual law.

    Concept map
    62 concepts; 4 descendants hidden
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    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 estimatesFixed dyadic moment orders at the ambientscaleA 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 budgetExplicit no-cover moment parameter marginsBoolean 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 A
    Evidence

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

    1 dyadic_budget_with_fixed_parameters proven

    2 exists_dyadic_moment_order proven

    3 fixed_dyadic_budget proven

    Lean source view on GitHub

    1import Lax342547.MomentBudgets
    2import Mathlib.Data.Nat.Log
    3
    4/-!
    5---
    6title: Fixed dyadic moment orders at the ambient scale
    7type: lemma
    8---
    9An explicit natural logarithm choice supplies even dyadic moments below N/c and above N/(2*c), with a fixed parameter budget independent of the actual law.
    10-/
    11
    12namespace Lax342547.DyadicMoments
    13
    14
    15
    16axiom exists_dyadic_moment_order (c n N : ℕ) (hc : 0 < c) (hN : c*2^(n+2) ≤ N) :
    17 ∃ k t : ℕ, n+2 ≤ k ∧ 2*t = 2^k ∧ c*2^k ≤ N ∧ N ≤ (2*c)*2^k
    18
    19axiom fixed_dyadic_budget (c n a Γ N : ℕ) (hc : 0 < c) (hN : c*2^(n+2) ≤ N) :
    20 ∃ k t : ℕ, n+2 ≤ k ∧ 2*t = 2^k ∧ c*2^k ≤ N ∧
    21 (((4 : ℝ)^(2*t)*(1/(2 : ℝ)^(2^n*(2+a+Γ*(2*c)+1)))^(2^(k-n))+
    22 1/(2 : ℝ)^((4*2^n*(a+Γ*(2*c)+1))*2^(k-(n+2))))/(1/(2 : ℝ)^a)^(2*t)) ≤
    23 1/(2 : ℝ)^(Γ*N)
    24
    25axiom dyadic_budget_with_fixed_parameters (c n a Γ P h N : ℕ) (hc : 0 < c)
    26 (hN : c*2^(n+2) ≤ N) (hP : 2^n*(2+a+Γ*(2*c)+1) ≤ P)
    27 (hh : 4*2^n*(a+Γ*(2*c)+1) ≤ h) :
    28 ∃ k t : ℕ, n+2 ≤ k ∧ 2*t = 2^k ∧ c*2^k ≤ N ∧
    29 (((4 : ℝ)^(2*t)*(1/(2 : ℝ)^P)^(2^(k-n))+
    30 1/(2 : ℝ)^(h*2^(k-(n+2))))/(1/(2 : ℝ)^a)^(2*t)) ≤
    31 1/(2 : ℝ)^(Γ*N)
    32
    33end Lax342547.DyadicMoments
    34
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