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Actual query frozen spaces and linear record exponents

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

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    Natural Language Statement

    Lemma

    The actual key-map range is precisely the selected incident key span. Joined pin/key dimensions and the concrete nominal column budget give a record exponent linear in n, with no ambient-image charge.

    Concept map
    95 concepts; 4 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 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 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 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 4 statements. Each proof establishes one of them relative to its assumptions.

    1 actual_joined_rank_budget proven

    2 actual_nominal_column_linear_bound proven

    3 key_map_range proven

    4 record_linear_bound proven

    Lean source view on GitHub

    1import Lax342547.JoinedRecords
    2import Lax342547.RawQueryImages
    3
    4/-!
    5---
    6title: Actual query frozen spaces and linear record exponents
    7type: lemma
    8---
    9The actual key-map range is precisely the selected incident key span. Joined pin/key dimensions and the concrete nominal column budget give a record exponent linear in n, with no ambient-image charge.
    10-/
    11
    12namespace Lax342547.QueryRecordBounds
    13
    14open Lax342547.MomentSpace Lax342547.TagGeometry Lax342547.ConcreteGeometry
    15open Lax342547.PairedWitnesses Lax342547.QueryReference Lax342547.QueryIndependence
    16open Lax342547.JoinedRecords Lax342547.PairedKeys Lax342547.KeySpans
    17
    18axiom key_map_range {k n b degree r : ℕ} {hr : 2 * r ≤ n} {H : Type}
    19 (W : Lists k n b degree r hr) (e : Lax342547.CutProfiles.Component (Tag k)) :
    20 LinearMap.range (keyMap (H := H) W e) = keys (H := H) W e
    21
    22axiom record_linear_bound {Axis I Status : Type}
    23 [Fintype Axis] [Fintype I] [Fintype Status]
    24 (dims : Axis → ℕ) (columnCoef n K k s : ℕ)
    25 (hcolumns : Fintype.card I ≤ columnCoef*n) (hdims : (∑ a, dims a) ≤ K+k)
    26 (hstatus : Fintype.card Status ≤ 2^(s*n)) : by
    27 classical
    28 exact Fintype.card (RecordType Axis I dims Status) ≤ 2^((columnCoef*(K+k)+s)*n)
    29
    30axiom actual_nominal_column_linear_bound {H : Type} [Fintype H]
    31 (k n b degree hcoef : ℕ) (hn : 1 ≤ n) (hH : Fintype.card H ≤ hcoef*n) :
    32 Fintype.card (Fin 2 × (Coordinate k n b degree ⊕ H)) ≤
    33 (2 * (Fintype.card (SelectorCoordinates b degree) * (Fintype.card (Tag k)^2+4) + hcoef)) * n
    34
    35axiom actual_joined_rank_budget {k n b degree r : ℕ} {hr : 2 * r ≤ n} {H N : Type} [Fintype H]
    36 (W : Lists k n b degree r hr)
    37 (P : Lax342547.ExactPins.Pin (Lax342547.CutProfiles.Component (Tag k) × Bool)
    38 (Fin 2 × (Coordinate k n b degree ⊕ H)) N) : by
    39 classical
    40 exact (∑ a : Lax342547.CutProfiles.Component (Tag k) × Bool,
    41 Module.finrank Binary (joined P (fun a => keyMap (H := H) W a.1) a)) ≤
    42 P.rank + Fintype.card (Lax342547.QuerySlots.Slot W)
    43
    44end Lax342547.QueryRecordBounds
    45
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