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Tiny-cover four-bit parity bounds

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

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    Natural Language Statement

    Lemma

    Four even-support characters exactly recover the compatible-pair probability; twelve remaining characters contribute an explicit error, preserving inverse-polynomial mass.

    Concept map
    118 concepts
    100%
    Exact-image bounds for independent affinecolumnsOrdered atom products in the actualgradient formPoint atoms and finite flavor distributionsSparse unselected primal vectors in barredspaces lie in the pinsBarred response spaces and the actualobstruction rank budgetCompression preserving allowed pointmoments and the actual cut domainThe exact space of binary base momentsRank control for the frozen baseline onprimal inputsFormal ordered product bits realizesymmetric correctionsBounded allowed derivatives preserving theactual linearized responseActual-mass averaging over retainedrecord-cell pairsSimultaneous assembly of opposite endpointsand reciprocal unit rolesAllowed channel changes and the actual tableinjection testsRank-controlled factorization throughallowed orthogonal channelsPruning small unary record cellsA bounded-rank pure correction for theactual scalar recipeConcrete cut-space testers and the orderedmixer formConcrete coordinates, quadratic testers, andthe self-Gram formNumerical recipes prescribe the concretegradients on witness atomsThe affine minus-column law at a fixed plusframeExact conditioning costs and recovery offinite probability massesCompression that fixes pin/key vectors andpreserves target matricesEntropy progress for residual pair lawsIndependence of distinct sample positionsCut profiles and the constant kernelThe full linearized response on pairs ofactual cut profilesDisjoint pair exception tailsFull response obstructions are effectiveprofiles plus selected atomsEntropy along feasible mixture lines,including new supportFull feasible support and finite informationprojectionExact image pins in nominal coefficientspacesFinite linear images and their uniform-lawdensity boundsFinite independent sampling and vertexexception tailsActual gradient agreement and matched keysproduce all four cross holesTiny-cover four-bit parity boundsFinite scalar agreement from binarycharacter boundsAmbient symmetries and frame marginalsFresh key directions are independent modulotable spacesThe simultaneous gradient corrections retainevery frozen pin and key entryBaseline bilinear extensions retaining allfrozen rows and columnsWhole-space gradient realization preservingthe actual frozen entriesBaseline contractions on the selected atomsAll four whole-space gradient equations fromassembled channel changesThe symmetric binary gradient formDeterministic whole-space gradientrealization from the genuine recipehypothesesGram-conditioned columns and theirrank-failure probabilityTwo-sided Gram normalization forindividually injective framesEntrywise product rank and incompatibletuplesThe binary hole relationPaying the reference-image conditioning anddimension costsUniform injective frames and channeltranspose failureAccepted key subprobabilities and uniformoverlapFresh point-ray spans are disjoint from thenominal table spacesLow-rank Boolean moments have boundedlabel supportTriangle relations separate into individuallabel blocksBoolean point moments with restricted basecoordinatesThe 3K+28 baseline bound in the actualnominal coordinatesPrimal blocks of the nominal spaces andtheir bounded table partPure corrections realized by actual nonlinearchannel productsWeighted compatibility from inevitablesample pairsSquared restriction cost for independent uniteventsEndpoint projections of paired pure-responseannihilatorsActual paired-witness key spaces satisfy thebaseline hypothesesBinary prescriptions at all endpoints of apaired scalar recipeFull paired witness lists and scalar recipeequationsSparse pin exclusions with arbitrary basecoefficientsSparse residual contractions belong to theactual primal pinsFinite primal-channel records with the paperbit count and four-hole implicationRetractions with bounded rank on theprimal inputsExact images mixing independent injectiveframesJoint minus images after exposing severalplus framesPrimal projections and preservation ofeffective spacesPure obstructions on the actual pair of cutprofilesWalsh bounds for independent image lawsand separated phasesCompression on the actual barred nominalquotientsExtracting fresh label coefficients throughpin quotientsRank of a tensor killed in two quotientspacesFull rank coefficient factorizationsRank-controlled pure forms on the actualbarred quotientsBounded baselines for both actual crossorientationsActual frame observations realize thenominal channel contractionsRaw matrix frames and their tensorrealizationThe finite uniform raw-vertex lawOriginal retained-cell laws from finite PMFsNonlinear channel realization of the actualwhole-space recipe residualGradient residuals vanish on effective profilesand selected atomsCoupled scalar recipes give consistent atomgradientsJoint reference image caps across both signsand all drawsThe full reference cap for exact pin eventsFinite relative entropy and support costsRemoving selected atoms leaves onlyunselected component labelsThe bounded pure remainder of an actualscalar recipeMatrix representations and the boundedresidual rank ingredientsUnrestricted linearized solutions for actualscalar recipesRank loss under restriction of a bilinear formImage caps inside original retained cellsSelected tensor blocks of actual pureobstructionsA selected affine ray determines its momentblockSubtracting selected atoms preserves pureannihilationSelected label coefficients agree across thecut profileInterpolation of finitely many binary selectorlabelsNumerical cross tables, injection flags, andunary admissibilityUnselected sparse vectors cannot concealfresh key coefficientsA uniform label budget for all sparse pinvectorsLow-rank tester routing along tag starsConsistent symmetric binary prescriptions ontwo witness listsTable contractions on effective profiles andtheir full extensionsTable coordinates and private channelcomplementsMajority intersections in the cyclic taggeometryExplicit low-rank matrices for thewhole-space gradient targetThe fifteen-rank witness tester targetPure bilinear responses detect quotienttensorsQuotient extractors isolate individual tensorlabel blocksWell-defined channel contractions onprojected tensor spacesInverse-polynomial tiny-cover compatibilityAdmissible pair lawsOrthogonality and finite Walsh correlationboundsWitness atoms and their numerical testerrecords
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 8 statements. Each proof establishes one of them relative to its assumptions.

    1 duplicate_injective proven

    2 even_character_sum proven

    3 even_iff_duplicate proven

    4 four_phase_parity_lower proven

    5 odd_character_card proven

    6 phase_duplicate proven

    7 tiny_four_phase_lower proven

    8 tiny_four_phase_positive proven

    Lean source view on GitHub

    1import Lax342547.FourierTests
    2import Lax342547.TinyCompatibility
    3
    4/-!
    5---
    6title: Tiny-cover four-bit parity bounds
    7type: lemma
    8---
    9Four even-support characters exactly recover the compatible-pair probability; twelve remaining characters contribute an explicit error, preserving inverse-polynomial mass.
    10-/
    11
    12namespace Lax342547.FourPhaseParity
    13
    14open Lax342547.MomentSpace Lax342547.Walsh Lax342547.FourierTests
    15open Lax342547.RelativeEntropy Lax342547.CellAveraging
    16open scoped BigOperators
    17
    18def duplicate (t : Fin 2 → Binary) : Fin 4 → Binary := ![t 0,t 0,t 1,t 1]
    19def parity (b : Fin 4 → Binary) : Fin 2 → Binary := ![b 0+b 1,b 2+b 3]
    20abbrev Even (t : Fin 4 → Binary) : Prop := t 0 = t 1 ∧ t 2 = t 3
    21
    22axiom phase_duplicate (t : Fin 2 → Binary) (b : Fin 4 → Binary) :
    23 phase (duplicate t) b = phase t (parity b)
    24
    25axiom duplicate_injective : Function.Injective duplicate
    26
    27axiom even_iff_duplicate (t : Fin 4 → Binary) :
    28 Even t ↔ ∃ s, duplicate s = t
    29
    30axiom even_character_sum {Ω : Type} [Fintype Ω]
    31 (ρ : Ω → ℝ) (bits : Ω → Fin 4 → Binary) :
    32 (∑ t ∈ Finset.univ.filter Even, characterMean ρ bits 0 t) =
    33 4*patternMass ρ (fun ω => parity (bits ω)) 0
    34
    35axiom odd_character_card : (Finset.univ.filter (fun t : Fin 4 → Binary => ¬ Even t)).card = 12
    36
    37axiom four_phase_parity_lower {Ω : Type} [Fintype Ω]
    38 (ρ : Ω → ℝ) (bits : Ω → Fin 4 → Binary) (ε : ℝ)
    39 (hodd : ∀ t, ¬ Even t → |characterMean ρ bits 0 t| ≤ ε) :
    40 patternMass ρ (fun ω => parity (bits ω)) 0/4-3*ε/4 ≤ patternMass ρ bits 0
    41
    42axiom tiny_four_phase_lower {Ω : Type} [Fintype Ω] [DecidableEq Ω]
    43 (μ : Ω → ℝ) (F : Matrix Ω Ω Binary) (bits : Ω × Ω → Fin 4 → Binary) (d N : ℕ) (ε : ℝ)
    44 (hμ : Probability μ) (hdiag : ∀ a, F a a = 0) (hrank : F.rank ≤ d*N)
    45 (hparity : ∀ a b, parity (bits (a,b)) = ![F a b,F b a])
    46 (hodd : ∀ t, ¬ Even t → |characterMean (fun ab : Ω × Ω => μ ab.1*μ ab.2) bits 0 t| ≤ ε) :
    47 1/(16*(1+(d : ℝ)*N)^4)-3*ε/4 ≤
    48 patternMass (fun ab : Ω × Ω => μ ab.1*μ ab.2) bits 0
    49
    50axiom tiny_four_phase_positive {Ω : Type} [Fintype Ω] [DecidableEq Ω]
    51 (μ : Ω → ℝ) (F : Matrix Ω Ω Binary) (bits : Ω × Ω → Fin 4 → Binary) (d N : ℕ) (ε : ℝ)
    52 (hμ : Probability μ) (hdiag : ∀ a, F a a = 0) (hrank : F.rank ≤ d*N)
    53 (hparity : ∀ a b, parity (bits (a,b)) = ![F a b,F b a])
    54 (hodd : ∀ t, ¬ Even t → |characterMean (fun ab : Ω × Ω => μ ab.1*μ ab.2) bits 0 t| ≤ ε)
    55 (hε : ε ≤ 1/(24*(1+(d : ℝ)*N)^4)) :
    56 1/(32*(1+(d : ℝ)*N)^4) ≤
    57 patternMass (fun ab : Ω × Ω => μ ab.1*μ ab.2) bits 0
    58
    59end Lax342547.FourPhaseParity
    60
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