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Uniform quantitative thresholds for affine-slice variance

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

proven

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    Natural Language Statement

    Lemma

    The variance budget is bounded by a fixed constant times 2^(-floor(n/2)) whenever the ambient dimension is at least n. One threshold supplies every Gram comparison margin and any requested positive error, before choosing coefficient laws, prescribed Grams, or common tests.

    Concept map
    44 concepts
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    Exact-image bounds for independent affinecolumnsIndependent nominal affine slices from thefree blockUniform quantitative thresholds foraffine-slice varianceConditioned affine-slice variance under theactual raw lawWalsh operator bounds with explicit bilinearrankConcrete coordinates, quadratic testers, andthe self-Gram formActual conditioned affine coefficientdependence boundsExact finite conditioning of separated FouriertestsExact conditioning costs and recovery offinite probability massesEntropy progress for residual pair lawsQuantitative comparison after cross-batchGram conditioningIndependent characters of actual mutualGram entriesCut profiles and the constant kernelFinite empirical variance from atwo-coefficient comparisonEntropy along feasible mixture lines,including new supportFull feasible support and finite informationprojectionFinite linear images and their uniform-lawdensity boundsFinite independent sampling and vertexexception tailsFinite scalar agreement from binarycharacter boundsAmbient symmetries and frame marginalsUniform tuple images on prescribed GramorbitsThe symmetric binary gradient formGram-conditioned columns and theirrank-failure probabilityTwo-sided Gram normalization forindividually injective framesActual reference-batch laws for mutual GramtestsTwo-batch comparison across independentcomponent groupsThe binary hole relationExact joining of two prescribed Gram orbitsJoint-injectivity loss for two actual referencebatchesBoolean point moments with restricted basecoordinatesSquared restriction cost for independent uniteventsWalsh bounds for independent image lawsand separated phasesBounded tests under the actual two-batchGram lawActual raw frame observations with uniformtwo-batch decayRaw matrix frames and their tensorrealizationActual grouped raw observations witharbitrary common testsThe finite uniform raw-vertex lawOriginal retained-cell laws from finite PMFsFinite relative entropy and support costsImage caps inside original retained cellsBit rank and Walsh bounds for globalvector-slot pairingsMajority intersections in the cyclic taggeometryAdmissible pair lawsOrthogonality 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 budget_decay proven

    2 uniform_threshold proven

    Lean source view on GitHub

    1import Lax342547.AffineSliceVariance
    2
    3/-!
    4---
    5title: Uniform quantitative thresholds for affine-slice variance
    6type: lemma
    7---
    8The variance budget is bounded by a fixed constant times 2^(-floor(n/2))
    9whenever the ambient dimension is at least n. One threshold supplies every
    10Gram comparison margin and any requested positive error, before choosing
    11coefficient laws, prescribed Grams, or common tests.
    12-/
    13
    14namespace Lax342547.AffineSliceScales
    15noncomputable section
    16open Lax342547.RawFrameComparison
    17
    18def budget (S D : Type) [Fintype S] [Fintype D] (N n : ℕ) (β : ℝ) : ℝ :=
    19 (Fintype.card S : ℝ)*errorBound N (Option D) (Option D) (Option D) (Option D)+
    20 4*(((2 : ℝ)^(2*Fintype.card D+1)/(2 : ℝ)^n)/β^2)
    21
    22def scale (S D : Type) [Fintype S] [Fintype D] (β : ℝ) : ℝ :=
    23 (Fintype.card S : ℝ)*Lax342547.RawFrameComparison.scale (Option D) (Option D) (Option D) (Option D)+
    24 4*((2 : ℝ)^(2*Fintype.card D+1)/β^2)
    25
    26axiom budget_decay (S D : Type) [Fintype S] [Fintype D] (N n : ℕ) (β : ℝ) (hn : n ≤ N) :
    27 budget S D N n β ≤ scale S D β/(2 : ℝ)^(n/2)
    28
    29axiom uniform_threshold (S D : Type) [Fintype S] [Fintype D] (β ε : ℝ) (hε : 0 < ε) :
    30 ∃ n₀ : ℕ,∀ n ≥ n₀,∀ N ≥ n,
    31 Fintype.card (Option D)+Fintype.card (Option D)+1 ≤ N ∧
    32 ((2 : ℝ)^(Fintype.card (Option D)*Fintype.card (Option D)+2))*
    33 ((2 : ℝ)^(Fintype.card (Option D)*Fintype.card (Option D)+2))/Real.sqrt ((2 : ℝ)^N) ≤
    34 1/(2 : ℝ)^(Fintype.card (Lax342547.CrossGramBasis.Slots (Option D) (Option D) (Option D) (Option D))+1) ∧
    35 budget S D N n β ≤ ε
    36
    37end
    38end Lax342547.AffineSliceScales
    39
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