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Actual independent leaf averaging and original-law collision cost

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

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    Natural Language Statement

    Lemma

    Finite PMF mixtures preserve the actual leaf masses. Affine collision bounds average over independent leaves without renormalization, skipped pairs remain valid, and squared restriction recovery gives the explicit original-law exponential lower bound.

    Concept map
    110 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 channelsQuantitative collision and squared restrictionexponentsExplicit pruning and scalar agreementexponent marginsPruning 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 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 holesAmbient 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 framesThe binary hole relationPaying the reference-image conditioning anddimension costsIndependent leaf mixtures preserve actualpair-event massesUniform 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 blocksActual independent leaf averaging andoriginal-law collision costActual-mass recovery and residual boundsthrough finite splitsBoolean 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-responseannihilatorsActual paired-witness key spaces satisfy thebaseline hypothesesBinary prescriptions at all endpoints of apaired scalar recipeFull paired witness lists and scalar recipeequationsIndependent parameter averaging withopposite-dependent original cellsSparse 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 quotientspacesRank-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 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 spacesOrthogonality and finite Walsh correlationboundsWitness atoms and their numerical testerrecords
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 5 statements. Each proof establishes one of them relative to its assumptions.

    1 mixture_real_weights proven

    2 original_law_exponential_collision proven

    3 real_pair_mixture proven

    4 skipped_collision_bound proven

    5 weighted_collision_lower proven

    Lean source view on GitHub

    1import Lax342547.IndependentMixtures
    2import Lax342547.ParameterCellAveraging
    3import Lax342547.CollisionCosts
    4import Lax342547.CellAveraging
    5
    6/-!
    7---
    8title: Actual independent leaf averaging and original-law collision cost
    9type: lemma
    10---
    11Finite PMF mixtures preserve the actual leaf masses. Affine collision bounds average over independent leaves without renormalization, skipped pairs remain valid, and squared restriction recovery gives the explicit original-law exponential lower bound.
    12-/
    13
    14namespace Lax342547.LeafCollision
    15
    16open Lax342547.RealCellLaws Lax342547.PairRecovery Lax342547.ParameterCellAveraging
    17open scoped ENNReal
    18
    19axiom mixture_real_weights {Ω J : Type} [Fintype J]
    20 (p : J → PMF Ω) (p₀ : PMF Ω) (w : J → ℝ≥0∞)
    21 (hw : ∀ j, w j ≠ ⊤) (hp : ∀ a, p₀ a = ∑ j, w j * p j a) (a : Ω) :
    22 weights p₀ a = ∑ j, (w j).toReal * weights (p j) a
    23
    24axiom real_pair_mixture {ΩA ΩB J K : Type}
    25 [Fintype ΩA] [Fintype ΩB] [Fintype J] [Fintype K]
    26 (p : J → PMF ΩA) (q : K → PMF ΩB) (p₀ : PMF ΩA) (q₀ : PMF ΩB)
    27 (w : J → ℝ≥0∞) (v : K → ℝ≥0∞)
    28 (hw : ∀ j, w j ≠ ⊤) (hv : ∀ k, v k ≠ ⊤)
    29 (hp : ∀ a, p₀ a = ∑ j, w j * p j a) (hq : ∀ b, q₀ b = ∑ k, v k * q k b)
    30 (H : ΩA → ΩB → Prop) :
    31 pairMass (weights p₀) (weights q₀) H =
    32 average (fun j => (w j).toReal) (fun k => (v k).toReal)
    33 (fun j k => pairMass (weights (p j)) (weights (q k)) H)
    34
    35axiom weighted_collision_lower {ΩA ΩB J K : Type}
    36 [Fintype ΩA] [Fintype ΩB] [Fintype J] [Fintype K]
    37 (p : J → PMF ΩA) (q : K → PMF ΩB) (p₀ : PMF ΩA) (q₀ : PMF ΩB)
    38 (w : J → ℝ≥0∞) (v : K → ℝ≥0∞)
    39 (hw : ∀ j, w j ≠ ⊤) (hv : ∀ k, v k ≠ ⊤)
    40 (hwsum : ∑ j, (w j).toReal = 1) (hvsum : ∑ k, (v k).toReal = 1)
    41 (hp : ∀ a, p₀ a = ∑ j, w j * p j a) (hq : ∀ b, q₀ b = ∑ k, v k * q k b)
    42 (H : ΩA → ΩB → Prop) (overlap : J → K → ℝ) (keys error agreement : ℝ)
    43 (hbound : ∀ j k, agreement*(overlap j k-error)/keys ≤
    44 pairMass (weights (p j)) (weights (q k)) H) :
    45 agreement*(average (fun j => (w j).toReal) (fun k => (v k).toReal) overlap-error)/keys ≤
    46 pairMass (weights p₀) (weights q₀) H
    47
    48axiom skipped_collision_bound {ΩA ΩB : Type} [Fintype ΩA] [Fintype ΩB]
    49 (p : PMF ΩA) (q : PMF ΩB) (H : ΩA → ΩB → Prop)
    50 (use : Prop) (overlap keys error agreement : ℝ)
    51 (hkeys : 0 < keys) (herror : 0 ≤ error) (hagreement : 0 ≤ agreement)
    52 (hbound : use → agreement*(overlap-error)/keys ≤ pairMass (weights p) (weights q) H) : by
    53 classical
    54 exact agreement*((if use then overlap else 0)-error)/keys ≤ pairMass (weights p) (weights q) H
    55
    56axiom original_law_exponential_collision {Ω J : Type} [Fintype Ω] [Fintype J]
    57 (σ : PMF Ω) (S : Set Ω) (hS : ∃ o ∈ S, o ∈ σ.support)
    58 (p : J → PMF Ω) (w : J → ℝ≥0∞) (hw : ∀ j, w j ≠ ⊤)
    59 (hwsum : ∑ j, (w j).toReal = 1)
    60 (hp : ∀ a, (σ.filter S hS) a = ∑ j, w j * p j a)
    61 (H : Ω → Ω → Prop) (overlap : J → J → ℝ) (k η N keys error agreement : ℝ)
    62 (hkeys : 0 < keys) (hkeybound : keys ≤ (2:ℝ)^(k*N))
    63 (hoverlap : (2:ℝ)^(-η*N) ≤ average (fun j => (w j).toReal) (fun j => (w j).toReal) overlap)
    64 (herror : error ≤ (2:ℝ)^(-N/200)) (hscale : 1 ≤ (1/200-η)*N)
    65 (hagreement : (2:ℝ)^(-N/50) ≤ agreement)
    66 (hmass : (2:ℝ)^(-η*N) ≤ (σ.toOuterMeasure S).toReal)
    67 (hbound : ∀ j l, agreement*(overlap j l-error)/keys ≤ pairMass (weights (p j)) (weights (p l)) H) :
    68 (2:ℝ)^(-((k+1/50+3*η)*N)-1) ≤
    69 ((independentPair σ σ).toOuterMeasure {ab | H ab.1 ab.2}).toReal
    70
    71end Lax342547.LeafCollision
    72
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