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Frozen pin and key records with nominal column budgets

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

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    Natural Language Statement

    Lemma

    Joined pin/key spaces have total dimension bounded by the original pin rank plus the key rank. Unary pairings with their known images cost only nominal columns times that rank, and determine all frozen cross entries. The concrete coordinate count is linear in n.

    Concept map
    94 concepts; 5 descendants hidden
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    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 responseAllowed channel changes and the actual tableinjection testsRank-controlled factorization throughallowed orthogonal channelsA 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 plusframeCompression 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 boundsAmbient symmetries and frame marginalsFresh key directions are independent modulotable spacesBaseline bilinear extensions retaining allfrozen rows and columnsExact agreement of zero quotient charactersUnary records determine every frozen crossentryBaseline contractions on the selected atomsThe symmetric binary gradient formGram-conditioned columns and theirrank-failure probabilityTwo-sided Gram normalization forindividually injective framesThe binary hole relationPaying the reference-image conditioning anddimension costsUniform injective frames and channeltranspose failureFrozen pin and key records with nominalcolumn budgetsFresh point-ray spans are disjoint from thenominal table spacesLow-rank Boolean moments have boundedlabel supportTriangle relations separate into individuallabel blocksExtracting 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 partEndpoint 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 pinsRetractions 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 cutprofilesActual query keys independent modulo oldpinsReference keys and matched keys for actualpaired listsThe actual paired-list key slot budgetCompression 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 lawActual query-key image caps on raw leavesGradient 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 formSelected 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 taggeometryPure bilinear responses detect quotienttensorsQuotient extractors isolate individual tensorlabel blocksWell-defined channel contractions onprojected tensor spacesWitness 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 basis_records_cross_agreement proven

    2 basis_vectors_span proven

    3 coordinate_linear_budget proven

    4 joined_rank_budget proven

    5 joined_record_bound proven

    6 joined_record_count proven

    7 nominal_column_count proven

    Lean source view on GitHub

    1import Lax342547.FrozenRecords
    2import Lax342547.RawQueryImages
    3
    4/-!
    5---
    6title: Frozen pin and key records with nominal column budgets
    7type: lemma
    8---
    9Joined pin/key spaces have total dimension bounded by the original pin rank plus the key rank. Unary pairings with their known images cost only nominal columns times that rank, and determine all frozen cross entries. The concrete coordinate count is linear in n.
    10-/
    11
    12namespace Lax342547.JoinedRecords
    13
    14open Lax342547.MomentSpace Lax342547.ExactPins Lax342547.FrozenRecords
    15open Lax342547.ConcreteGeometry Lax342547.TagGeometry
    16
    17noncomputable def joined {Axis I N : Type} {U : Axis → Type}
    18 [∀ a, AddCommGroup (U a)] [∀ a, Module Binary (U a)]
    19 (P : Pin Axis I N) (keys : ∀ a, U a →ₗ[Binary] (I → Binary)) (a : Axis) :
    20 Submodule Binary (I → Binary) := P.space a ⊔ LinearMap.range (keys a)
    21
    22axiom joined_rank_budget {Axis I N : Type} {U : Axis → Type}
    23 [Fintype Axis] [Fintype I]
    24 [∀ a, AddCommGroup (U a)] [∀ a, Module Binary (U a)] [∀ a, FiniteDimensional Binary (U a)]
    25 (P : Pin Axis I N) (keys : ∀ a, U a →ₗ[Binary] (I → Binary)) :
    26 (∑ a, Module.finrank Binary (joined P keys a)) ≤ P.rank + ∑ a, Module.finrank Binary (U a)
    27
    28abbrev RecordType (Axis I : Type) (dims : Axis → ℕ) (Status : Type) :=
    29 (∀ a, Matrix I (Fin (dims a)) Binary) × Status
    30
    31axiom joined_record_count {Axis I Status : Type}
    32 [Fintype Axis] [Fintype I] [Fintype Status] (dims : Axis → ℕ) : by
    33 classical
    34 exact Fintype.card (RecordType Axis I dims Status) = 2 ^ (Fintype.card I * ∑ a, dims a) * Fintype.card Status
    35
    36axiom joined_record_bound {Axis I N Status : Type} {U : Axis → Type}
    37 [Fintype Axis] [Fintype I] [Fintype Status]
    38 [∀ a, AddCommGroup (U a)] [∀ a, Module Binary (U a)] [∀ a, FiniteDimensional Binary (U a)]
    39 (P : Pin Axis I N) (keys : ∀ a, U a →ₗ[Binary] (I → Binary))
    40 (K k s : ℕ) (hP : P.rank ≤ K) (hkeys : (∑ a, Module.finrank Binary (U a)) ≤ k)
    41 (hstatus : Fintype.card Status ≤ 2^s) : by
    42 classical
    43 exact Fintype.card (RecordType Axis I (fun a => Module.finrank Binary (joined P keys a)) Status) ≤
    44 2 ^ (Fintype.card I * (K+k) + s)
    45
    46axiom nominal_column_count {B H : Type} [Fintype B] [Fintype H] :
    47 Fintype.card (Fin 2 × (B ⊕ H)) = 2 * (Fintype.card B + Fintype.card H)
    48
    49axiom coordinate_linear_budget (k n b degree : ℕ) (hn : 1 ≤ n) :
    50 Fintype.card (Coordinate k n b degree) ≤
    51 Fintype.card (SelectorCoordinates b degree) * (Fintype.card (Tag k)^2 + 4) * n
    52
    53noncomputable def basisVectors {I : Type} [Fintype I]
    54 (S : Submodule Binary (I → Binary)) : Fin (Module.finrank Binary S) → I → Binary :=
    55 fun i => (Module.finBasis Binary S i).val
    56
    57axiom basis_vectors_span {I : Type} [Fintype I]
    58 (S : Submodule Binary (I → Binary)) : Submodule.span Binary (Set.range (basisVectors S)) = S
    59
    60axiom basis_records_cross_agreement {I N : Type} [Fintype I] [Fintype N]
    61 (D E : Submodule Binary (I → Binary))
    62 (A A' B B' : (I → Binary) →ₗ[Binary] (N → Binary))
    63 (leftImages : Fin (Module.finrank Binary D) → N → Binary)
    64 (rightImages : Fin (Module.finrank Binary E) → N → Binary)
    65 (hleft : ∀ x, A (basisVectors D x) = leftImages x ∧ A' (basisVectors D x) = leftImages x)
    66 (hright : ∀ y, B (basisVectors E y) = rightImages y ∧ B' (basisVectors E y) = rightImages y)
    67 (hrecA : record A rightImages = record A' rightImages)
    68 (hrecB : record B leftImages = record B' leftImages) :
    69 (∀ v ∈ D, ∀ w, dotProduct (A v) (B w) = dotProduct (A' v) (B' w)) ∧
    70 (∀ v w, w ∈ E → dotProduct (A v) (B w) = dotProduct (A' v) (B' w))
    71
    72end Lax342547.JoinedRecords
    73
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