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Actual primal-channel scalar tests and retained-cell four-hole probability

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

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    Natural Language Statement

    Lemma

    The two nominal orientations of every raw component realize the actual cross forms. The paper primal-channel entry records are linear Gram tests, and their joint Fourier agreement implies all four actual holes under the constructed target gradients.

    Concept map
    115 concepts; 4 descendants hidden
    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 responseSimultaneous assembly of opposite endpointsand reciprocal unit rolesAllowed channel changes and the actual tableinjection testsRank-controlled factorization throughallowed orthogonal channelsCoefficient phases as dot products ofrank-factor imagesJoint retained-cell characters across allcomponentsSimultaneous scalar Gram agreement acrosscomponentsA 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 matricesCut profiles and the constant kernelThe full linearized response on pairs ofactual cut profilesFull response obstructions are effectiveprofiles plus selected atomsExact image pins in nominal coefficientspacesFinite linear images and their uniform-lawdensity boundsActual gradient agreement and matched keysproduce all four cross holesFinite 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 columnsExact agreement of zero quotient charactersWhole-space gradient realization preservingthe actual frozen entriesUnary records determine every frozen crossentryBaseline 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 framesScalar agreement after quotient characterestimatesThe binary hole relationPaying the reference-image conditioning anddimension costsUniform injective frames and channeltranspose failureFresh 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 productsSquared restriction cost for independent uniteventsEndpoint projections of paired pure-responseannihilatorsPaired-frame orbit under the primal anddual actionsActual 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 pinsActual primal-channel scalar tests andretained-cell four-hole probabilityFinite 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 framesSplit quotient projections preserve coefficientcharactersPrimal 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 factors independent modulo the actualfrozen spacesRank 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 eventsRemoving 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 formActual cross Gram characters on frozenunary cellsImage caps inside original retained cellsSimultaneous agreement of linear cross Gramtests on 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 spacesOrthogonality and finite Walsh correlationboundsWitness atoms and their numerical testerrecords
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 7 statements. Each proof establishes one of them relative to its assumptions.

    1 entry_actual_bits proven

    2 entry_target_bits proven

    3 gram_form_evaluation proven

    4 matrix_pair_outer proven

    5 primal_cell_four_holes proven

    6 primal_entry_agreement proven

    7 raw_oriented_forms proven

    Lean source view on GitHub

    1import Lax342547.ComponentTests
    2import Lax342547.PrimalRecords
    3import Lax342547.PairedFrames
    4import Lax342547.PairRecovery
    5
    6/-!
    7---
    8title: Actual primal-channel scalar tests and retained-cell four-hole probability
    9type: lemma
    10---
    11The two nominal orientations of every raw component realize the actual cross forms. The paper primal-channel entry records are linear Gram tests, and their joint Fourier agreement implies all four actual holes under the constructed target gradients.
    12-/
    13
    14namespace Lax342547.PrimalGramTests
    15
    16open Lax342547.MomentSpace Lax342547.ConcreteGeometry Lax342547.TableSpaces
    17open Lax342547.TableContractions Lax342547.RawContractions Lax342547.ReferencePins
    18open Lax342547.ProductImages Lax342547.RetainedCharacters Lax342547.CoefficientPhase
    19open Lax342547.ComponentTests Lax342547.PrimalRecords Lax342547.RawFrames
    20
    21noncomputable def orientedForms {Comp B H : Type} (F : CrossForms Comp B H)
    22 (a : Comp × Bool) : LinearMap.BilinForm Binary ((Fin 2 × (B ⊕ H)) → Binary) :=
    23 if a.2 then F.forward a.1 else (F.reverse a.1).flip
    24
    25noncomputable def leftMatrices {Comp B H N : Type} [Fintype B] [Fintype H] [Fintype N]
    26 {E : Matrix B B Binary} (o : Fin 2 → Comp → Frame B H N E)
    27 (a : Comp × Bool) : Matrix N (Fin 2 × (B ⊕ H)) Binary := observationMatrix o a
    28
    29noncomputable def rightMatrices {Comp B H N : Type} [Fintype B] [Fintype H] [Fintype N]
    30 {E : Matrix B B Binary} (o : Fin 2 → Comp → Frame B H N E)
    31 (a : Comp × Bool) : Matrix N (Fin 2 × (B ⊕ H)) Binary := observationMatrix o (a.1,!a.2)
    32
    33axiom gram_form_evaluation {I N : Type} [Fintype I] [Fintype N]
    34 (X Y : Matrix N I Binary) (u v : I → Binary) :
    35 actualForm X Y u v = dotProduct (X.mulVec u) (Y.mulVec v)
    36
    37axiom raw_oriented_forms {Comp B H N : Type} [Fintype B] [Fintype H] [Fintype N]
    38 {E : Matrix B B Binary} (oA oB : Fin 2 → Comp → Frame B H N E)
    39 (a : Comp × Bool) :
    40 actualForm (leftMatrices oA a) (rightMatrices oB a) = orientedForms (rawForms oA oB) a
    41
    42axiom matrix_pair_outer {I : Type} [Fintype I]
    43 (T : LinearMap.BilinForm Binary (I → Binary)) (u v : I → Binary) : by
    44 classical
    45 exact matrixPair (LinearMap.BilinForm.toMatrix' T) (Matrix.vecMulVec u v) = T u v
    46
    47noncomputable def entryAxis {Comp B H : Type} (t : EntryIndex Comp B H) : Comp × Bool :=
    48 (t.2.2.1, decide ((t.1 = 0) ↔ (t.2.1 = 0)))
    49
    50noncomputable def entryLeft {Comp B H : Type} (t : EntryIndex Comp B H) : (Fin 2 × (B ⊕ H)) → Binary := by
    51 classical
    52 exact if t.1 = 0 then primalEmbedding t.2.2.2.1 (Pi.single t.2.2.2.2.2.1 1)
    53 else channelEmbedding t.2.2.2.2.1 (Pi.single t.2.2.2.2.2.2 1)
    54
    55noncomputable def entryRight {Comp B H : Type} (t : EntryIndex Comp B H) : (Fin 2 × (B ⊕ H)) → Binary := by
    56 classical
    57 exact if t.1 = 0 then channelEmbedding t.2.2.2.2.1 (Pi.single t.2.2.2.2.2.2 1)
    58 else primalEmbedding t.2.2.2.1 (Pi.single t.2.2.2.2.2.1 1)
    59
    60noncomputable def entryTests {Comp B H : Type} (t : EntryIndex Comp B H)
    61 (a : Comp × Bool) : Matrix (Fin 2 × (B ⊕ H)) (Fin 2 × (B ⊕ H)) Binary := by
    62 classical
    63 exact if a = entryAxis t then Matrix.vecMulVec (entryLeft t) (entryRight t) else 0
    64
    65axiom entry_target_bits {Comp B H : Type} [Fintype Comp] [Fintype B] [Fintype H]
    66 (F : CrossForms Comp B H) : targetBits entryTests (orientedForms F) = entries F
    67
    68axiom entry_actual_bits {Comp B H N : Type} [Fintype Comp] [Fintype B] [Fintype H] [Fintype N]
    69 {E : Matrix B B Binary} (oA oB : Fin 2 → Comp → Frame B H N E) :
    70 actualBits entryTests (leftMatrices oA) (rightMatrices oB) = entries (rawForms oA oB)
    71
    72axiom primal_entry_agreement {ΩA ΩB Comp B H N : Type}
    73 [Fintype ΩA] [Fintype ΩB] [Fintype Comp] [Fintype B] [Fintype H] [Fintype N]
    74 {M : Matrix B B Binary}
    75 (α : ΩA → ℝ) (β : ΩB → ℝ)
    76 (unitA : ΩA → Fin 2 → Comp → Frame B H N M)
    77 (unitB : ΩB → Fin 2 → Comp → Frame B H N M)
    78 (D E : Comp × Bool → Submodule Binary ((Fin 2 × (B ⊕ H)) → Binary))
    79 (F : CrossForms Comp B H)
    80 (hα : ∀ x, 0 ≤ α x) (hβ : ∀ y, 0 ≤ β y)
    81 (hαsum : ∑ x, α x = 1) (hβsum : ∑ y, β y = 1)
    82 (hD : ∀ x y a v, v ∈ D a → ∀ w,
    83 actualForm (leftMatrices (unitA x) a) (rightMatrices (unitB y) a) v w = orientedForms F a v w)
    84 (hE : ∀ x y a v w, w ∈ E a →
    85 actualForm (leftMatrices (unitA x) a) (rightMatrices (unitB y) a) v w = orientedForms F a v w)
    86 (hcapA : ∀ (r : Comp × Bool → ℕ)
    87 (A : ∀ a, Matrix (Fin 2 × (B ⊕ H)) (Fin (r a)) Binary),
    88 (∀ a, Function.Injective ((D a).mkQ.comp (A a).mulVecLin)) →
    89 ∀ x, Lax342547.PushforwardWalsh.push α
    90 (fun ω => Lax342547.ComponentCharacters.blockImage A (leftMatrices (unitA ω))) x ≤
    91 (2 : ℝ)^(-(95/100 : ℝ)*((∑ a, r a)*Fintype.card N)))
    92 (hcapB : ∀ (r : Comp × Bool → ℕ)
    93 (A : ∀ a, Matrix (Fin 2 × (B ⊕ H)) (Fin (r a)) Binary),
    94 (∀ a, Function.Injective ((E a).mkQ.comp (A a).mulVecLin)) →
    95 ∀ x, Lax342547.PushforwardWalsh.push β
    96 (fun ω => Lax342547.ComponentCharacters.blockImage A (rightMatrices (unitB ω))) x ≤
    97 (2 : ℝ)^(-(95/100 : ℝ)*((∑ a, r a)*Fintype.card N)))
    98 (hsmall : (2 : ℝ)^(-(45/100 : ℝ)*Fintype.card N) ≤
    99 1 / (2 : ℝ)^(16 * Fintype.card Comp * Fintype.card H * Fintype.card B + 1)) :
    100 1 / (2 : ℝ)^(16 * Fintype.card Comp * Fintype.card H * Fintype.card B + 1) ≤
    101 Lax342547.FourierTests.patternMass (fun ab : ΩA × ΩB => α ab.1 * β ab.2)
    102 (fun ab => entries (rawForms (unitA ab.1) (unitB ab.2))) (entries F)
    103
    104axiom primal_cell_four_holes {ΩA ΩB H N : Type} {k n b degree r J : ℕ} {hr : 2 * r ≤ n}
    105 [Fintype ΩA] [Fintype ΩB] [Fintype H] [Fintype N]
    106 {M : Lax342547.ConcreteGeometry.Moment k n b degree}
    107 (α : ΩA → ℝ) (β : ΩB → ℝ)
    108 (unitA : ΩA → Fin 2 → Lax342547.CutProfiles.Component (Lax342547.TagGeometry.Tag k) → Frame (Lax342547.ConcreteGeometry.Coordinate k n b degree) H N M)
    109 (unitB : ΩB → Fin 2 → Lax342547.CutProfiles.Component (Lax342547.TagGeometry.Tag k) → Frame (Lax342547.ConcreteGeometry.Coordinate k n b degree) H N M)
    110 (D E : Lax342547.CutProfiles.Component (Lax342547.TagGeometry.Tag k) × Bool → Submodule Binary ((Fin 2 × (Lax342547.ConcreteGeometry.Coordinate k n b degree ⊕ H)) → Binary))
    111 (F : CrossForms (Lax342547.CutProfiles.Component (Lax342547.TagGeometry.Tag k)) (Lax342547.ConcreteGeometry.Coordinate k n b degree) H)
    112 (hα : ∀ x, 0 ≤ α x) (hβ : ∀ y, 0 ≤ β y)
    113 (hαsum : ∑ x, α x = 1) (hβsum : ∑ y, β y = 1)
    114 (hD : ∀ x y a v, v ∈ D a → ∀ w,
    115 actualForm (leftMatrices (unitA x) a) (rightMatrices (unitB y) a) v w = orientedForms F a v w)
    116 (hE : ∀ x y a v w, w ∈ E a →
    117 actualForm (leftMatrices (unitA x) a) (rightMatrices (unitB y) a) v w = orientedForms F a v w)
    118 (hcapA : ∀ (r : Lax342547.CutProfiles.Component (Lax342547.TagGeometry.Tag k) × Bool → ℕ)
    119 (A : ∀ a, Matrix (Fin 2 × (Lax342547.ConcreteGeometry.Coordinate k n b degree ⊕ H)) (Fin (r a)) Binary),
    120 (∀ a, Function.Injective ((D a).mkQ.comp (A a).mulVecLin)) →
    121 ∀ x, Lax342547.PushforwardWalsh.push α
    122 (fun ω => Lax342547.ComponentCharacters.blockImage A (leftMatrices (unitA ω))) x ≤
    123 (2 : ℝ)^(-(95/100 : ℝ)*((∑ a, r a)*Fintype.card N)))
    124 (hcapB : ∀ (r : Lax342547.CutProfiles.Component (Lax342547.TagGeometry.Tag k) × Bool → ℕ)
    125 (A : ∀ a, Matrix (Fin 2 × (Lax342547.ConcreteGeometry.Coordinate k n b degree ⊕ H)) (Fin (r a)) Binary),
    126 (∀ a, Function.Injective ((E a).mkQ.comp (A a).mulVecLin)) →
    127 ∀ x, Lax342547.PushforwardWalsh.push β
    128 (fun ω => Lax342547.ComponentCharacters.blockImage A (rightMatrices (unitB ω))) x ≤
    129 (2 : ℝ)^(-(95/100 : ℝ)*((∑ a, r a)*Fintype.card N)))
    130 (W : Lax342547.PairedWitnesses.Lists k n b degree r hr)
    131 (testers : Lax342547.ConcreteCut.Testers (k := k) (b := b) (degree := degree) hr)
    132 (L R : Fin J → Lax342547.CutProfiles.Component (Lax342547.TagGeometry.Tag k) →
    133 Lax342547.CutProfiles.Component (Lax342547.TagGeometry.Tag k) → Lax342547.ConcreteGeometry.Moment k n b degree)
    134 (holes : Lax342547.HoleRelation.HoleData
    135 (Lax342547.CutProfiles.Component (Lax342547.TagGeometry.Tag k) → Frame (Lax342547.ConcreteGeometry.Coordinate k n b degree) H N M)
    136 (Lax342547.ConcreteCut.Profile k n b degree)
    137 (Lax342547.CutProfiles.Component (Lax342547.TagGeometry.Tag k) → Matrix N N Binary))
    138 (hrole : holes.a = testers.role) (hgradient : holes.T = Lax342547.ConcreteCut.gradient testers L R)
    139 (hU : ∀ o, holes.U o = (Lax342547.RawFrames.profileMap o).comp (Lax342547.ConcreteCut.Profile k n b degree).subtype)
    140 (hu : ∀ o, holes.u o = Lax342547.RawFrames.profileContraction o)
    141 (hkeys : ∀ x y, Lax342547.PairedWitnesses.MatchedKeys W (unitA x) (unitB y))
    142 (hroles : ∀ i z, testers.role (Lax342547.PairedWitnesses.leftWitness W i z) +
    143 testers.role (Lax342547.PairedWitnesses.rightWitness W i z) = 1)
    144 (hleft : ∀ i z x, fullContraction F (Lax342547.ConcreteCut.Profile k n b degree) i z x =
    145 (testers.role + Lax342547.ConcreteCut.gradient testers L R (Lax342547.PairedWitnesses.leftWitness W i z)) x)
    146 (hright : ∀ i z x, fullContraction F.flip (Lax342547.ConcreteCut.Profile k n b degree) z i x =
    147 (testers.role + Lax342547.ConcreteCut.gradient testers L R (Lax342547.PairedWitnesses.rightWitness W i z)) x)
    148 (hsmall : (2 : ℝ)^(-(45/100 : ℝ)*Fintype.card N) ≤
    149 1 / (2 : ℝ)^(16 * Fintype.card (Lax342547.CutProfiles.Component (Lax342547.TagGeometry.Tag k)) * Fintype.card H * Fintype.card (Lax342547.ConcreteGeometry.Coordinate k n b degree) + 1)) :
    150 1 / (2 : ℝ)^(16 * Fintype.card (Lax342547.CutProfiles.Component (Lax342547.TagGeometry.Tag k)) * Fintype.card H * Fintype.card (Lax342547.ConcreteGeometry.Coordinate k n b degree) + 1) ≤
    151 Lax342547.PairRecovery.pairMass α β (fun x y => ∀ i z,
    152 Lax342547.HoleRelation.Hole holes (unitA x i) (unitB y z))
    153
    154end Lax342547.PrimalGramTests
    155
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