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Whole-space gradient realization preserving the actual frozen entries

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

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    Natural Language Statement

    Lemma

    Under the genuine scalar recipe, injecting table, matched keys, sparse covers and the explicit degree and channel margins, the bounded baseline and both reciprocal recipe solutions assemble into cross forms that preserve every frozen entry and satisfy all four gradients. The original small table may change away from its frozen rows and columns, as permitted by Theorem 6.3.

    Concept map
    92 concepts
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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 responseSimultaneous assembly of opposite endpointsand reciprocal unit rolesAllowed 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 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 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 partPure corrections realized by actual nonlinearchannel productsEndpoint 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 cutprofilesCompression 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 lawNonlinear 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 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 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 spacesWitness atoms and their numerical testerrecords
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.
    Evidence

    Each proof establishes this claim relative to its assumptions.

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    1import Lax342547.FrozenAssembly
    2import Lax342547.PairedRecipes
    3
    4/-!
    5---
    6title: Whole-space gradient realization preserving the actual frozen entries
    7type: lemma
    8---
    9Under the genuine scalar recipe, injecting table, matched keys, sparse covers and the explicit degree and channel margins, the bounded baseline and both reciprocal recipe solutions assemble into cross forms that preserve every frozen entry and satisfy all four gradients. The original small table may change away from its frozen rows and columns, as permitted by Theorem 6.3.
    10-/
    11
    12namespace Lax342547.FrozenGradients
    13
    14open Lax342547.MomentSpace Lax342547.TagGeometry Lax342547.ConcreteGeometry
    15open Lax342547.ConcreteCut Lax342547.CutProfiles Lax342547.PairedWitnesses
    16open Lax342547.ExactPins Lax342547.ProjectedPins Lax342547.PairedAnnihilators
    17open Lax342547.BarredSpaces Lax342547.ChannelChanges Lax342547.TableContractions
    18open Lax342547.DerivativeResponses Lax342547.ResponseMatrices Lax342547.RawBaselines
    19open Lax342547.TableSpaces Lax342547.BaseCompression Lax342547.QuotientCompression
    20open Lax342547.ResidualRealization
    21open Lax342547.CompressedResidual
    22
    23axiom frozen_gradients {k n b degree r : ℕ} {hr : 2 * r ≤ n}
    24 {H N : Type} [Fintype H] [Fintype N] {K : ℕ} {E : Moment k n b degree}
    25 (hk : 0 < k) (W : Lists k n b degree r hr)
    26 (D : Testers (k := k) (b := b) (degree := degree) hr) {copies : ℕ}
    27 (L R : Fin copies → Component (Tag k) → Component (Tag k) → Moment k n b degree)
    28 (oA oB : Unit (H := H) (N := N) (E := E))
    29 (P Q : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    30 (hK : P.rank ≤ K) (hQ : Q.rank ≤ K) (A B : Fin 2 → Finset (Fin b → Binary))
    31 (hA : Lax342547.PinLabelExclusions.Covers P A (2 * K + 28))
    32 (hfresh : ∀ i z t, (W.left i z t).label ∉ A i)
    33 (hB : Lax342547.PinLabelExclusions.Covers Q B (2 * K + 28))
    34 (U V : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary))
    35 (hU : ∀ e j, IsCompl (protectedChannel P j (e, true)) (U e j))
    36 (hV : ∀ e j, IsCompl (protectedChannel P j (e, false)) (V e j))
    37 (U' V' : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary))
    38 (hU' : ∀ e j, IsCompl (protectedChannel Q j (e, true)) (U' e j))
    39 (hV' : ∀ e j, IsCompl (protectedChannel Q j (e, false)) (V' e j))
    40 (T : Lax342547.SmallTables.Table P Q) (hT : Lax342547.SmallTables.Injecting T)
    41 (F : CrossForms (Component (Tag k)) (Coordinate k n b degree) H)
    42 (hF : Extends F T) (hbounded : Bounded F K)
    43 (hrecipe : ScalarRecipe W D L R oA oB T A B)
    44 (hfrozen : Frozen F W P Q oA oB) (hkeys : MatchedKeys W oA oB)
    45 (hgradients : ∀ i, Lax342547.ConcreteRecipes.RecipeGradients D L R
    46 (Lax342547.PairedRecipes.endpointAtoms W i) (Lax342547.PairedRecipes.oppositeP W oB i))
    47 (hgradients' : ∀ z, Lax342547.ConcreteRecipes.RecipeGradients D L R
    48 (Lax342547.PairedRecipes.endpointAtoms (Lax342547.PairedRecipes.flip W) z)
    49 (Lax342547.PairedRecipes.oppositeP (Lax342547.PairedRecipes.flip W) oA z))
    50 (hdegree : 6 * (2 * K + 28) + 4 ≤ degree)
    51 (hmargin : 28 * copies * Fintype.card (Component (Tag k)) +
    52 quotientBound k b degree K (14 * K + 140) + 2 * K + 4 * (3 * K + 28) < 970 * r)
    53 (hchannels : 1000 * r ≤ Fintype.card H) :
    54 ∃ G : CrossForms (Component (Tag k)) (Coordinate k n b degree) H,
    55 Frozen G W P Q oA oB ∧
    56 (∀ i z x, fullContraction G (Profile k n b degree) i z x =
    57 (D.role + gradient D L R (leftWitness W i z)) x) ∧
    58 (∀ i z x, fullContraction G.flip (Profile k n b degree) z i x =
    59 (D.role + gradient D L R (rightWitness W i z)) x)
    60
    61end Lax342547.FrozenGradients
    62
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