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Gradient residuals vanish on effective profiles and selected atoms

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

proven

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    Natural Language Statement

    Lemma

    The scalar recipe and binary prescriptions make the target difference zero on the effective space and the span of every selected atom at an endpoint. This is the domain on which correctness is already established before the whole-space annihilator argument.

    Concept map
    49 concepts
    100%
    Exact-image bounds for independent affinecolumnsOrdered atom products in the actualgradient formPoint atoms and finite flavor distributionsRank control for the frozen baseline onprimal inputsFormal ordered product bits realizesymmetric correctionsConcrete 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 plusframeCut profiles and the constant kernelExact 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 spacesBoolean point moments with restricted basecoordinatesThe 3K+28 baseline bound in the actualnominal coordinatesPrimal blocks of the nominal spaces andtheir bounded table partActual paired-witness key spaces satisfy thebaseline hypothesesBinary prescriptions at all endpoints of apaired scalar recipeFull paired witness lists and scalar recipeequationsRetractions with bounded rank on theprimal inputsExact images mixing independent injectiveframesJoint minus images after exposing severalplus framesPrimal projections and preservation ofeffective spacesBounded 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 drawsThe full reference cap for exact pin eventsNumerical cross tables, injection flags, andunary admissibilityA uniform label budget for all sparse pinvectorsConsistent symmetric binary prescriptions ontwo witness listsTable contractions on effective profiles andtheir full extensionsTable coordinates and private channelcomplementsMajority intersections in the cyclic taggeometryWell-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

    This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

    1 residual_correct proven

    2 selected_gradients proven

    Lean source view on GitHub

    1import Lax342547.FrozenValues
    2
    3/-!
    4---
    5title: Gradient residuals vanish on effective profiles and selected atoms
    6type: lemma
    7---
    8The scalar recipe and binary prescriptions make the target difference
    9zero on the effective space and the span of every selected atom at an
    10endpoint. This is the domain on which correctness is already established
    11before the whole-space annihilator argument.
    12-/
    13
    14namespace Lax342547.RecipeResiduals
    15
    16open Lax342547.MomentSpace Lax342547.TagGeometry Lax342547.ConcreteGeometry
    17open Lax342547.ConcreteCut Lax342547.CutProfiles Lax342547.PairedWitnesses Lax342547.PairedRecipes
    18open Lax342547.ExactPins Lax342547.ProjectedPins Lax342547.SmallTables
    19open Lax342547.TableContractions Lax342547.RawBaselines Lax342547.ConcreteRecipes
    20
    21variable {k n b degree r : ℕ} {hr : 2 * r ≤ n}
    22
    23noncomputable def residual {H : Type} [Fintype H]
    24 (W : Lists k n b degree r hr) (D : Testers (k := k) (b := b) (degree := degree) hr)
    25 {copies : ℕ} (L R : Fin copies → Component (Tag k) → Component (Tag k) → Moment k n b degree)
    26 (F : CrossForms (Component (Tag k)) (Coordinate k n b degree) H) (i z : Fin 2) :
    27 Profile k n b degree →ₗ[Binary] Binary :=
    28 D.role + gradient D L R (leftWitness W i z) - fullContraction F (Profile k n b degree) i z
    29
    30def correctSpace {H N : Type} (W : Lists k n b degree r hr)
    31 (P : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N) (i : Fin 2) :
    32 Submodule Binary (Profile k n b degree) :=
    33 effective (Profile k n b degree) P i ⊔
    34 Submodule.span Binary (Set.range (fun x : Position W i => (endpointAtoms W i x).profile))
    35
    36variable {H N : Type} [Fintype H] [Fintype N] {E : Moment k n b degree}
    37
    38axiom selected_gradients (W : Lists k n b degree r hr)
    39 (D : Testers (k := k) (b := b) (degree := degree) hr) {copies : ℕ}
    40 (L R : Fin copies → Component (Tag k) → Component (Tag k) → Moment k n b degree)
    41 (oB : Unit (H := H) (N := N) (E := E)) (i : Fin 2)
    42 (h : RecipeGradients D L R (endpointAtoms W i) (oppositeP W oB i))
    43 (z z' : Fin 2) (t : Fin (W.length i z')) :
    44 D.role (W.left i z' t).profile +
    45 gradient D L R (leftWitness W i z) (W.left i z' t).profile =
    46 if z = z' then 0 else ownContraction oB z' (W.right i z' t).profile
    47
    48axiom residual_correct (W : Lists k n b degree r hr)
    49 (D : Testers (k := k) (b := b) (degree := degree) hr) {copies : ℕ}
    50 (L R : Fin copies → Component (Tag k) → Component (Tag k) → Moment k n b degree)
    51 (oA oB : Unit (H := H) (N := N) (E := E))
    52 {P Q : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N}
    53 (T : Table P Q) (A B : Fin 2 → Finset (Fin b → Binary))
    54 (F : CrossForms (Component (Tag k)) (Coordinate k n b degree) H)
    55 (h : ScalarRecipe W D L R oA oB T A B) (htable : Extends F T)
    56 (hfrozen : Frozen F W P Q oA oB) (hkeys : MatchedKeys W oA oB)
    57 (hgradients : ∀ i, RecipeGradients D L R (endpointAtoms W i) (oppositeP W oB i)) :
    58 ∀ i z, correctSpace W P i ≤ LinearMap.ker (residual W D L R F i z)
    59
    60end Lax342547.RecipeResiduals
    61
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