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Finite pin-injection exception

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

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    Natural Language Statement

    Lemma

    Greedy independent failure probes, adaptive protected witnesses, and actual finite Fubini give an exception bound for whole-unit channel injection.

    Concept map
    84 concepts; 6 descendants hidden
    100%
    Actual projected span deficitChannel bounds with adaptive protectedcoefficient spacesSampling after exposed coordinatesAdaptive sampling on disjoint coordinatesetsUnion over adaptive component coversExact-image bounds for independent affinecolumnsFinite amplified-test averagingExact 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 pin-injection exceptionFinite 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 independence of subspacesGreedy mode space exposureSmall actual greedy exposure tailsGreedy independent failure samplesAmplified channel failures on independentpinned spacesNo-cover rank growth on arbitrary finiteindicesUniform injective frames and channeltranspose failureActual 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 spacesExact probabilities for independent protectedchannel imagesWalsh 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 cellsFinite sequential testsOne exposure controls both modesCounting all bounded-dimensional protectedspacesMonotonicity 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 lawsUniversal protected pin witnessesOrthogonality and finite Walsh correlationbounds
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    Each proof establishes this claim relative to its assumptions.

    Lean source view on GitHub

    1import Lax342547.SequentialTests
    2import Lax342547.IndependentSampling
    3import Lax342547.AmplifiedTests
    4import Lax342547.UniversalWitnesses
    5import Lax342547.IndependentWitnesses
    6import Lax342547.FiniteSampling
    7import Mathlib.LinearAlgebra.DFinsupp
    8
    9/-!
    10---
    11title: Finite pin-injection exception
    12type: lemma
    13---
    14Greedy independent failure probes, adaptive protected witnesses, and actual finite Fubini give an exception bound for whole-unit channel injection.
    15-/
    16
    17namespace Lax342547.FiniteInjection
    18
    19open Lax342547.MomentSpace Lax342547.RelativeEntropy Lax342547.RetainedImages Lax342547.FiniteSampling
    20open Lax342547.RealCellLaws Lax342547.PushforwardWalsh
    21open scoped BigOperators
    22
    23def pinFailure {N H : Type} [Fintype N] (V : Submodule Binary (N → Binary))
    24 (Y : Matrix N H Binary) (S : Submodule Binary (H → Binary)) : Prop :=
    25 ∃ v : V,v.val ≠ 0 ∧ Y.transpose.mulVec v.val ∈ S
    26
    27axiom finite_injection_exception {A B N H : Type}
    28 [Fintype A] [Fintype B] [Fintype N] [Fintype H] [DecidableEq N] [DecidableEq H]
    29 (α : A → ℝ) (β : B → ℝ) (V : A → Submodule Binary (N → Binary))
    30 (Y : B → Matrix N H Binary) (S : B → Submodule Binary (H → Binary))
    31 (K ℓ : ℕ) (L η δ : ℝ) (hα : Probability α) (hβ : Probability β)
    32 (hL : 0 ≤ L) (hgap : δ < η) (hV : ∀ a,Module.finrank Binary (V a) ≤ K)
    33 (hS : ∀ b,Module.finrank Binary (S b) ≤ K)
    34 (hcap : ∀ y,push β Y y ≤ L*weights (PMF.uniformOfFintype (Matrix N H Binary)) y)
    35 (havoid : ∀ j,j < ℓ → ∀ t : Fin j → A,
    36 cellMass α (fun a => ¬ Disjoint (V a) (⨆ i : Fin j,V (t i))) ≤ δ) :
    37 cellMass β (fun b => η ≤ cellMass α (fun a => pinFailure (V a) (Y b) (S b))) ≤
    38 (L*((2 : ℝ)^(Fintype.card H*K+2*K*ℓ)/(2 : ℝ)^(Fintype.card H*ℓ)))/(η-δ)^ℓ
    39
    40end Lax342547.FiniteInjection
    41
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