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Uniform primal-span avoidance

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

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    Natural Language Statement

    Lemma

    Exact random matrix vector laws and finite subspace counts bound intersections of the complete primal image with a fixed space.

    Concept map
    75 concepts; 8 descendants hidden
    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 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 exceptionaltailsUniform primal-span avoidanceProjected 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 A
    Evidence

    This concept declares 6 statements. Each proof establishes one of them relative to its assumptions.

    1 capped_matrix_span_hit proven

    2 evalMatrix_surjective proven

    3 fixed_vector_subspace_mass proven

    4 matrix_span_hit_mass proven

    5 matrix_vector_uniform proven

    6 uniform_subspace_mass proven

    Lean source view on GitHub

    1import Lax342547.FiniteLinearLaw
    2import Lax342547.ChannelColumns
    3import Lax342547.InjectiveFrames
    4import Lax342547.PhaseAverages
    5
    6/-!
    7---
    8title: Uniform primal-span avoidance
    9type: lemma
    10---
    11Exact random matrix vector laws and finite subspace counts bound intersections of the complete primal image with a fixed space.
    12-/
    13
    14namespace Lax342547.PrimalAvoidance
    15
    16open Lax342547.MomentSpace Lax342547.RetainedImages Lax342547.RealCellLaws Lax342547.PushforwardWalsh
    17open scoped BigOperators ENNReal
    18
    19noncomputable def evalMatrix {B N : Type} [Fintype B] (c : B → Binary) :
    20 Matrix N B Binary →ₗ[Binary] (N → Binary) where
    21 toFun A := A.mulVec c
    22 map_add' A D := Matrix.add_mulVec A D c
    23 map_smul' a A := by simp [Matrix.smul_mulVec]
    24
    25axiom evalMatrix_surjective {B N : Type} [Fintype B] [DecidableEq B]
    26 (c : B → Binary) (hc : c ≠ 0) : Function.Surjective (evalMatrix (N := N) c)
    27
    28axiom matrix_vector_uniform {B N : Type} [Fintype B] [Fintype N]
    29 [DecidableEq B] [DecidableEq N] (c : B → Binary) (hc : c ≠ 0) :
    30 (PMF.uniformOfFintype (Matrix N B Binary)).map (fun A : Matrix N B Binary => A.mulVec c) =
    31 PMF.uniformOfFintype (N → Binary)
    32
    33axiom uniform_subspace_mass {N : Type} [Fintype N] [DecidableEq N] (W : Submodule Binary (N → Binary)) :
    34 cellMass (weights (PMF.uniformOfFintype (N → Binary))) (fun v => v ∈ W) =
    35 (2 : ℝ)^Module.finrank Binary W/(2 : ℝ)^Fintype.card N
    36
    37axiom fixed_vector_subspace_mass {B N : Type} [Fintype B] [Fintype N]
    38 [DecidableEq B] [DecidableEq N] (c : B → Binary) (hc : c ≠ 0) (W : Submodule Binary (N → Binary)) :
    39 cellMass (weights (PMF.uniformOfFintype (Matrix N B Binary))) (fun A => A.mulVec c ∈ W) =
    40 (2 : ℝ)^Module.finrank Binary W/(2 : ℝ)^Fintype.card N
    41
    42axiom matrix_span_hit_mass {B N : Type} [Fintype B] [Fintype N]
    43 [DecidableEq B] [DecidableEq N] (W : Submodule Binary (N → Binary)) :
    44 cellMass (weights (PMF.uniformOfFintype (Matrix N B Binary)))
    45 (fun A => ∃ c : B → Binary,c ≠ 0 ∧ A.mulVec c ∈ W) ≤
    46 (2 : ℝ)^(Fintype.card B+Module.finrank Binary W)/(2 : ℝ)^Fintype.card N
    47
    48axiom capped_matrix_span_hit {B N : Type} [Fintype B] [Fintype N]
    49 [DecidableEq B] [DecidableEq N] (μ : Matrix N B Binary → ℝ) (L : ℝ) (hL : 0 ≤ L)
    50 (hcap : ∀ A,μ A ≤ L*weights (PMF.uniformOfFintype (Matrix N B Binary)) A)
    51 (W : Submodule Binary (N → Binary)) :
    52 cellMass μ (fun A => ∃ c : B → Binary,c ≠ 0 ∧ A.mulVec c ∈ W) ≤
    53 L*((2 : ℝ)^(Fintype.card B+Module.finrank Binary W)/(2 : ℝ)^Fintype.card N)
    54
    55end Lax342547.PrimalAvoidance
    56
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