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Actual query-key image caps on raw leaves

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

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    Natural Language Statement

    Lemma

    The raw paired-frame observation maps realize the selected query tuples. Their independence modulo old pins transfers exact leaf caps to actual key events.

    Concept map
    91 concepts; 8 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 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 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 actual_left_key_cap proven

    2 key_dimension proven

    3 key_map_coordinate proven

    4 key_observations_left proven

    5 left_tuple_event proven

    6 observation_primal_minus proven

    7 observation_primal_plus proven

    Lean source view on GitHub

    1import Lax342547.QueryIndependence
    2import Lax342547.ReferencePins
    3import Lax342547.LeafImages
    4
    5/-!
    6---
    7title: Actual query-key image caps on raw leaves
    8type: lemma
    9---
    10The raw paired-frame observation maps realize the selected query tuples. Their independence modulo old pins transfers exact leaf caps to actual key events.
    11-/
    12
    13namespace Lax342547.RawQueryImages
    14
    15open Lax342547.MomentSpace Lax342547.RawFrames Lax342547.ReferencePins
    16open Lax342547.TableSpaces Lax342547.ProductImages Lax342547.TagGeometry
    17open Lax342547.ConcreteGeometry Lax342547.PairedWitnesses Lax342547.QueryReference
    18open Lax342547.QueryIndependence Lax342547.PairedKeys Lax342547.KeySpans
    19open scoped ENNReal
    20
    21axiom observation_primal_plus {Comp B H N : Type} [Fintype B] [Fintype H] [Fintype N]
    22 {E : Matrix B B Binary} (o : Fin 2 → Comp → Frame B H N E)
    23 (e : Comp) (i : Fin 2) (v : B → Binary) :
    24 observation o (e,true) (primalEmbedding i v) = (o i e).P.mulVec v
    25
    26axiom observation_primal_minus {Comp B H N : Type} [Fintype B] [Fintype H] [Fintype N]
    27 {E : Matrix B B Binary} (o : Fin 2 → Comp → Frame B H N E)
    28 (e : Comp) (i : Fin 2) (v : B → Binary) :
    29 observation o (e,false) (primalEmbedding i v) = (o i e).Q.mulVec v
    30
    31axiom key_map_coordinate {k n b degree r : ℕ} {hr : 2 * r ≤ n} {H : Type}
    32 (W : Lists k n b degree r hr) (e : Lax342547.CutProfiles.Component (Tag k))
    33 (p : ComponentPosition W e) : by
    34 classical
    35 exact keyMap (H := H) W e (Pi.single p 1) =
    36 primalEmbedding p.val.1 (W.left p.val.1 p.val.2.1 p.val.2.2).vector
    37
    38axiom key_observations_left {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 (e : Lax342547.CutProfiles.Component (Tag k)) (mode : Bool) : by
    42 classical
    43 exact LinearMap.toMatrix' ((observation o (e,mode)).comp (keyMap (H := H) W e)) =
    44 if mode then (leftTuple W o e).1 else (leftTuple W o e).2
    45
    46noncomputable def tupleMaps {k n b degree r : ℕ} {hr : 2 * r ≤ n} {N : Type}
    47 (W : Lists k n b degree r hr) (q : Lax342547.ReferenceKeys.Tuples
    48 (Lax342547.CutProfiles.Component (Tag k)) (ComponentPosition W) N)
    49 (a : Lax342547.CutProfiles.Component (Tag k) × Bool) :
    50 (ComponentPosition W a.1 → Binary) →ₗ[Binary] (N → Binary) := by
    51 classical
    52 exact (if a.2 then (q a.1).1 else (q a.1).2).mulVecLin
    53
    54axiom left_tuple_event {k n b degree r : ℕ} {hr : 2 * r ≤ n} {H N : Type}
    55 [Fintype H] [Fintype N] {E : Moment k n b degree}
    56 (W : Lists k n b degree r hr) (q : Lax342547.ReferenceKeys.Tuples
    57 (Lax342547.CutProfiles.Component (Tag k)) (ComponentPosition W) N)
    58 (o : Unit (H := H) (N := N) (E := E)) :
    59 leftTuple W o = q ↔ ∀ a, (observation o a).comp (keyMap (H := H) W a.1) = tupleMaps W q a
    60
    61axiom key_dimension {k n b degree r : ℕ} {hr : 2 * r ≤ n}
    62 (W : Lists k n b degree r hr) : by
    63 classical
    64 exact (∑ a : Lax342547.CutProfiles.Component (Tag k) × Bool,
    65 Module.finrank Binary (ComponentPosition W a.1 → Binary)) =
    66 Fintype.card (Lax342547.QuerySlots.Slot W)
    67
    68axiom actual_left_key_cap {k n b degree r : ℕ} {hr : 2 * r ≤ n} {H N : Type}
    69 [Fintype H] [Fintype N] {E : Moment k n b degree}
    70 (W : Lists k n b degree r hr)
    71 (p : PMF (Unit (H := H) (N := N) (E := E)))
    72 (P : Lax342547.ExactPins.Pin (Lax342547.CutProfiles.Component (Tag k) × Bool)
    73 (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    74 (A : Fin 2 → Finset (Fin b → Binary)) (hA : Excludes P A)
    75 (hfresh : ∀ i z t, (W.left i z t).label ∉ A i)
    76 (q : Lax342547.ReferenceKeys.Tuples (Lax342547.CutProfiles.Component (Tag k)) (ComponentPosition W) N)
    77 (α : ℝ≥0∞)
    78 (hcap : ∀ Q : Lax342547.ExactPins.Pin (Lax342547.CutProfiles.Component (Tag k) × Bool)
    79 (Fin 2 × (Coordinate k n b degree ⊕ H)) N,
    80 1 ≤ P.relativeRank Q → p.toOuterMeasure (Q.event observation) ≤ α ^ (P.relativeRank Q))
    81 (hr : 1 ≤ Fintype.card (Lax342547.QuerySlots.Slot W)) :
    82 p.toOuterMeasure {o | leftTuple W o = q} ≤ α ^ Fintype.card (Lax342547.QuerySlots.Slot W)
    83
    84end Lax342547.RawQueryImages
    85
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