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Four actual holes inside an original retained cell pair

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

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    Natural Language Statement

    Lemma

    The original PMF leaf laws supply both block image caps. Separate unary records freeze the target directions, and the checked gradient realization constructs a target whose primal agreement implies all four actual holes.

    Concept map
    134 concepts; 1 descendant 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 responsePositive representatives of original retainedcellsSimultaneous 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 columnsOriginal record cells determine the commonfrozen gradient targetExact 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 failureFrozen pin and key records with nominalcolumn budgetsAppend fresh directions to the independentkey tupleExact leaf entropy bounds joint key andadditional imagesFresh point-ray spans are disjoint from thenominal table spacesLow-rank Boolean moments have boundedlabel supportTriangle relations separate into individuallabel blocksJoint matrix-image caps for retained leaflawsJoint fresh-image entropy on originalretained leaf cellsExtracting a leaf with all remainingexact-image capsAll-rank tuple image caps from exact-pin leafentropyBoolean 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 basecoefficientsExact-pin leaf entropy is invariant undernominal axis reindexingSparse 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 phasesActual query keys independent modulo oldpinsActual query frozen spaces and linear recordexponentsReference keys and matched keys for actualpaired listsThe actual paired-list key slot budgetCompression 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 crossorientationsBoth actual cross orientations haveretained-cell image entropyActual frame observations realize thenominal channel contractionsRaw matrix frames and their tensorrealizationThe finite uniform raw-vertex lawActual query-key image caps on raw leavesFour actual holes inside an original retainedcell pairOriginal 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 drawsConstrained key reference spaces havepositive densityThe 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 cellsRight endpoint query-key caps via paired-listreindexingSimultaneous 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 2 statements. Each proof establishes one of them relative to its assumptions.

    1 realized_retained_four_holes proven

    2 retained_four_holes proven

    Lean source view on GitHub

    1import Lax342547.RawCellEntropy
    2import Lax342547.CellRepresentatives
    3import Lax342547.GradientRealization
    4
    5/-!
    6---
    7title: Four actual holes inside an original retained cell pair
    8type: lemma
    9---
    10The original PMF leaf laws supply both block image caps. Separate unary records freeze the target directions, and the checked gradient realization constructs a target whose primal agreement implies all four actual holes.
    11-/
    12
    13namespace Lax342547.RawRetainedCells
    14
    15open Lax342547.MomentSpace Lax342547.ConcreteGeometry Lax342547.TagGeometry Lax342547.CutProfiles
    16open Lax342547.ConcreteCut Lax342547.PairedWitnesses Lax342547.PairedRecipes
    17open Lax342547.ExactPins Lax342547.ReferencePins Lax342547.QueryReference Lax342547.QueryIndependence
    18open Lax342547.RawBaselines Lax342547.JoinedRecords Lax342547.RealCellLaws Lax342547.FrozenCellDirections
    19open Lax342547.TableContractions Lax342547.KeySpans Lax342547.PrimalGramTests Lax342547.HoleRelation
    20open scoped ENNReal
    21
    22axiom retained_four_holes {k n b degree r J : ℕ} {hr : 2 * r ≤ n} {H N : Type}
    23 [Fintype H] [Fintype N] {M : Moment k n b degree}
    24 (W : Lists k n b degree r hr) (pA pB : PMF (Unit (H := H) (N := N) (E := M)))
    25 (P Q : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    26 (A B : Fin 2 → Finset (Fin b → Binary)) (hA : Excludes P A) (hB : Excludes Q B)
    27 (hfresh : FreshAgainst W A B)
    28 (C D : Unit (H := H) (N := N) (E := M) → Prop)
    29 (refA refB : Unit (H := H) (N := N) (E := M)) (hAr : C refA) (hBr : D refB)
    30 (hC : ∀ o, C o → o ∈ P.event observation ∧ leftTuple W o = leftTuple W refA ∧
    31 leftRecord W Q refB o = leftRecord W Q refB refA)
    32 (hD : ∀ o, D o → o ∈ Q.event observation ∧ rightTuple W o = rightTuple W refB ∧
    33 rightRecord W P refA o = rightRecord W P refA refB)
    34 (hkey : leftTuple W refA = rightTuple W refB)
    35 (G : CrossForms (Component (Tag k)) (Coordinate k n b degree) H)
    36 (hf : Frozen G W P Q refA refB)
    37 (testers : Testers (k := k) (b := b) (degree := degree) hr)
    38 (L R : Fin J → Component (Tag k) → Component (Tag k) → Moment k n b degree)
    39 (holes : HoleData (Component (Tag k) → Lax342547.RawFrames.Frame (Coordinate k n b degree) H N M)
    40 (Profile k n b degree) (Component (Tag k) → Matrix N N Binary))
    41 (hrole : holes.a = testers.role) (hgradient : holes.T = gradient testers L R)
    42 (hU : ∀ o, holes.U o = (Lax342547.RawFrames.profileMap o).comp (Profile k n b degree).subtype)
    43 (hu : ∀ o, holes.u o = Lax342547.RawFrames.profileContraction o)
    44 (hroles : ∀ i z, testers.role (leftWitness W i z) + testers.role (rightWitness W i z) = 1)
    45 (hleft : ∀ i z x, fullContraction G (Profile k n b degree) i z x =
    46 (testers.role + gradient testers L R (leftWitness W i z)) x)
    47 (hright : ∀ i z x, fullContraction G.flip (Profile k n b degree) z i x =
    48 (testers.role + gradient testers L R (rightWitness W i z)) x)
    49 (ζ : ℝ) (hζ : 0 ≤ ζ) (hζsmall : ζ ≤ 1/1000)
    50 (hk : 2*ζ*Fintype.card (Lax342547.QuerySlots.Slot W) ≤ (1/500 : ℝ))
    51 (hmassA : (2 : ℝ)^(-((Fintype.card (Lax342547.QuerySlots.Slot W) : ℝ)+1/100)*Fintype.card N) ≤
    52 (pA.toOuterMeasure {o | C o}).toReal)
    53 (hmassB : (2 : ℝ)^(-((Fintype.card (Lax342547.QuerySlots.Slot W) : ℝ)+1/100)*Fintype.card N) ≤
    54 (pB.toOuterMeasure {o | D o}).toReal)
    55 (hcapA : ∀ T : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N,
    56 1 ≤ P.relativeRank T → pA.toOuterMeasure (T.event observation) ≤
    57 ((2 : ℝ≥0∞)^(-(1-2*ζ)*Fintype.card N))^(P.relativeRank T))
    58 (hcapB : ∀ T : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N,
    59 1 ≤ Q.relativeRank T → pB.toOuterMeasure (T.event observation) ≤
    60 ((2 : ℝ≥0∞)^(-(1-2*ζ)*Fintype.card N))^(Q.relativeRank T))
    61 (hsmall : (2 : ℝ)^(-(45/100 : ℝ)*Fintype.card N) ≤
    62 1/(2 : ℝ)^(16 * Fintype.card (Component (Tag k)) * Fintype.card H * Fintype.card (Coordinate k n b degree)+1)) : by
    63 classical
    64 exact 1/(2 : ℝ)^(16 * Fintype.card (Component (Tag k)) * Fintype.card H * Fintype.card (Coordinate k n b degree)+1) ≤
    65 Lax342547.PairRecovery.pairMass (subtypeWeights (weights pA) C) (subtypeWeights (weights pB) D)
    66 (fun x y => ∀ i z, Hole holes (x.val i) (y.val z))
    67
    68axiom realized_retained_four_holes {k n b degree r J K : ℕ} {hr : 2 * r ≤ n} {H N : Type}
    69 [Fintype H] [Fintype N] {M : Moment k n b degree}
    70 (hk0 : 0 < k) (W : Lists k n b degree r hr) (pA pB : PMF (Unit (H := H) (N := N) (E := M)))
    71 (P Q : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    72 (hP : P.rank ≤ K) (hQ : Q.rank ≤ K)
    73 (A B : Fin 2 → Finset (Fin b → Binary))
    74 (hA : Lax342547.PinLabelExclusions.Covers P A (2*K+28))
    75 (hB : Lax342547.PinLabelExclusions.Covers Q B (2*K+28))
    76 (C D : Unit (H := H) (N := N) (E := M) → Prop)
    77 (refA refB : Unit (H := H) (N := N) (E := M)) (hAr : C refA) (hBr : D refB)
    78 (hC : ∀ o, C o → o ∈ P.event observation ∧ leftTuple W o = leftTuple W refA ∧
    79 leftRecord W Q refB o = leftRecord W Q refB refA)
    80 (hD : ∀ o, D o → o ∈ Q.event observation ∧ rightTuple W o = rightTuple W refB ∧
    81 rightRecord W P refA o = rightRecord W P refA refB)
    82 (hkey : leftTuple W refA = rightTuple W refB)
    83 (testers : Testers (k := k) (b := b) (degree := degree) hr)
    84 (L R : Fin J → Component (Tag k) → Component (Tag k) → Moment k n b degree)
    85 (holes : HoleData (Component (Tag k) → Lax342547.RawFrames.Frame (Coordinate k n b degree) H N M)
    86 (Profile k n b degree) (Component (Tag k) → Matrix N N Binary))
    87 (hrole : holes.a = testers.role) (hgradient : holes.T = gradient testers L R)
    88 (hU : ∀ o, holes.U o = (Lax342547.RawFrames.profileMap o).comp (Profile k n b degree).subtype)
    89 (hu : ∀ o, holes.u o = Lax342547.RawFrames.profileContraction o)
    90 (T : Lax342547.SmallTables.Table P Q) (hT : Lax342547.SmallTables.Injecting T)
    91 (hrecipe : ScalarRecipe W testers L R refA refB T A B)
    92 (hgradients : ∀ i, Lax342547.ConcreteRecipes.RecipeGradients testers L R
    93 (endpointAtoms W i) (oppositeP W refB i))
    94 (hgradients' : ∀ z, Lax342547.ConcreteRecipes.RecipeGradients testers L R
    95 (endpointAtoms (flip W) z) (oppositeP (flip W) refA z))
    96 (hadmissible : Lax342547.SmallTables.Admissible T observation observation refA refB)
    97 (hdegree : 6*(2*K+28)+4 ≤ degree)
    98 (hmargin : 28*J*Fintype.card (Component (Tag k)) +
    99 Lax342547.CompressedResidual.quotientBound k b degree K (14*K+140) + 2*K+4*(3*K+28) < 970*r)
    100 (hchannels : 1000*r ≤ Fintype.card H)
    101 (ζ : ℝ) (hζ : 0 ≤ ζ) (hζsmall : ζ ≤ 1/1000)
    102 (hk : 2*ζ*Fintype.card (Lax342547.QuerySlots.Slot W) ≤ (1/500 : ℝ))
    103 (hmassA : (2 : ℝ)^(-((Fintype.card (Lax342547.QuerySlots.Slot W) : ℝ)+1/100)*Fintype.card N) ≤
    104 (pA.toOuterMeasure {o | C o}).toReal)
    105 (hmassB : (2 : ℝ)^(-((Fintype.card (Lax342547.QuerySlots.Slot W) : ℝ)+1/100)*Fintype.card N) ≤
    106 (pB.toOuterMeasure {o | D o}).toReal)
    107 (hcapA : ∀ T : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N,
    108 1 ≤ P.relativeRank T → pA.toOuterMeasure (T.event observation) ≤
    109 ((2 : ℝ≥0∞)^(-(1-2*ζ)*Fintype.card N))^(P.relativeRank T))
    110 (hcapB : ∀ T : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N,
    111 1 ≤ Q.relativeRank T → pB.toOuterMeasure (T.event observation) ≤
    112 ((2 : ℝ≥0∞)^(-(1-2*ζ)*Fintype.card N))^(Q.relativeRank T))
    113 (hsmall : (2 : ℝ)^(-(45/100 : ℝ)*Fintype.card N) ≤
    114 1/(2 : ℝ)^(16 * Fintype.card (Component (Tag k)) * Fintype.card H * Fintype.card (Coordinate k n b degree)+1)) : by
    115 classical
    116 exact 1/(2 : ℝ)^(16 * Fintype.card (Component (Tag k)) * Fintype.card H * Fintype.card (Coordinate k n b degree)+1) ≤
    117 Lax342547.PairRecovery.pairMass (subtypeWeights (weights pA) C) (subtypeWeights (weights pB) D)
    118 (fun x y => ∀ i z, Hole holes (x.val i) (y.val z))
    119
    120end Lax342547.RawRetainedCells
    121
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