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Reference keys and matched keys for actual paired lists

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

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    Natural Language Statement

    Lemma

    Global key tuples contain both nominal signs of every actual incident witness direction. Their equality gives matched keys; the constrained reference space has the exact slot cardinality bound and a dimension-independent positive density.

    Concept map
    87 concepts; 10 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 columnsBaseline 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 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 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 cutprofilesReference 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 lawGradient 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 5 statements. Each proof establishes one of them relative to its assumptions.

    1 component_position_budget proven

    2 grouped_slot_count proven

    3 key_space_density proven

    4 key_space_size proven

    5 tuple_equality_matched_keys proven

    Lean source view on GitHub

    1import Lax342547.QuerySlots
    2import Lax342547.ReferenceKeys
    3import Lax342547.PairedWitnesses
    4
    5/-!
    6---
    7title: Reference keys and matched keys for actual paired lists
    8type: lemma
    9---
    10Global key tuples contain both nominal signs of every actual incident witness direction. Their equality gives matched keys; the constrained reference space has the exact slot cardinality bound and a dimension-independent positive density.
    11-/
    12
    13namespace Lax342547.QueryReference
    14
    15open Lax342547.MomentSpace Lax342547.TagGeometry Lax342547.CutProfiles
    16open Lax342547.PairedWitnesses Lax342547.QuerySlots Lax342547.ReferenceKeys
    17open Lax342547.ConcreteGeometry
    18open scoped ENNReal
    19
    20abbrev ComponentPosition {k n b degree r : ℕ} {hr : 2 * r ≤ n}
    21 (W : Lists k n b degree r hr) (e : Component (Tag k)) :=
    22 {p : Lax342547.QuerySlots.Position W // (W.left p.1 p.2.1 p.2.2).tag ∈ e.val}
    23
    24noncomputable def groupedSlots {k n b degree r : ℕ} {hr : 2 * r ≤ n}
    25 (W : Lists k n b degree r hr) :
    26 Slot W ≃ (Σ e : Component (Tag k), ComponentPosition W e × Bool) where
    27 toFun x := ⟨x.2.1.val, ⟨⟨x.1, x.2.1.property⟩, x.2.2⟩⟩
    28 invFun x := ⟨x.2.1.val, ⟨⟨x.1, x.2.1.property⟩, x.2.2⟩⟩
    29 left_inv _ := rfl
    30 right_inv _ := rfl
    31
    32noncomputable def gram {k n b degree r : ℕ} {hr : 2 * r ≤ n}
    33 (W : Lists k n b degree r hr) (e : Component (Tag k)) :
    34 Matrix (ComponentPosition W e) (ComponentPosition W e) Binary := by
    35 classical
    36 exact fun p q => if p.val = q.val then diagonalBit hr (W.left p.val.1 p.val.2.1 p.val.2.2) else 0
    37
    38noncomputable def leftTuple {k n b degree r : ℕ} {hr : 2 * r ≤ n} {H N : Type}
    39 [Fintype H] [Fintype N] {E : Moment k n b degree}
    40 (W : Lists k n b degree r hr) (o : Unit (H := H) (N := N) (E := E)) :
    41 Tuples (Component (Tag k)) (ComponentPosition W) N := fun e =>
    42 ⟨fun n p => (o p.val.1 e).P.mulVec (W.left p.val.1 p.val.2.1 p.val.2.2).vector n,
    43 fun n p => (o p.val.1 e).Q.mulVec (W.left p.val.1 p.val.2.1 p.val.2.2).vector n⟩
    44
    45noncomputable def rightTuple {k n b degree r : ℕ} {hr : 2 * r ≤ n} {H N : Type}
    46 [Fintype H] [Fintype N] {E : Moment k n b degree}
    47 (W : Lists k n b degree r hr) (o : Unit (H := H) (N := N) (E := E)) :
    48 Tuples (Component (Tag k)) (ComponentPosition W) N := fun e =>
    49 ⟨fun n p => (o p.val.2.1 e).P.mulVec (W.right p.val.1 p.val.2.1 p.val.2.2).vector n,
    50 fun n p => (o p.val.2.1 e).Q.mulVec (W.right p.val.1 p.val.2.1 p.val.2.2).vector n⟩
    51
    52axiom tuple_equality_matched_keys {k n b degree r : ℕ} {hr : 2 * r ≤ n} {H N : Type}
    53 [Fintype H] [Fintype N] {E : Moment k n b degree}
    54 (W : Lists k n b degree r hr) (oA oB : Unit (H := H) (N := N) (E := E))
    55 (h : leftTuple W oA = rightTuple W oB) : MatchedKeys W oA oB
    56
    57axiom component_position_budget {k n b degree r : ℕ} {hr : 2 * r ≤ n}
    58 (W : Lists k n b degree r hr) (e : Component (Tag k)) : by
    59 classical
    60 exact Fintype.card (ComponentPosition W e) ≤ 28
    61
    62axiom grouped_slot_count {k n b degree r : ℕ} {hr : 2 * r ≤ n}
    63 (W : Lists k n b degree r hr) : by
    64 classical
    65 exact Fintype.card (Slot W) = 2 * ∑ e, Fintype.card (ComponentPosition W e)
    66
    67axiom key_space_size {k n b degree r : ℕ} {hr : 2 * r ≤ n} {N : Type} [Fintype N]
    68 (W : Lists k n b degree r hr) : by
    69 classical
    70 exact Fintype.card (space (ComponentPosition W) N (gram W)) ≤
    71 2 ^ (Fintype.card (Slot W) * Fintype.card N)
    72
    73axiom key_space_density {k n b degree r : ℕ} {hr : 2 * r ≤ n} {N : Type} [Fintype N]
    74 (W : Lists k n b degree r hr) (hN : 57 ≤ Fintype.card N) : by
    75 classical
    76 exact (∏ e, 1 / (2 : ℝ≥0∞)^
    77 (Fintype.card (ComponentPosition W e) * Fintype.card (ComponentPosition W e) + 2)) ≤
    78 (PMF.uniformOfFintype (Tuples (Component (Tag k)) (ComponentPosition W) N)).toOuterMeasure
    79 (space (ComponentPosition W) N (gram W))
    80
    81end Lax342547.QueryReference
    82
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