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Simultaneous assembly of opposite endpoints and reciprocal unit roles

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

proven

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    Natural Language Statement

    Lemma

    The two cross forms assemble all four families of barred channel changes. Each updated primal-to-channel map receives precisely its own correction; opposite endpoint channels and reciprocal-role changes contribute zero. The exact formulas cover both cross orientations and their flips.

    Concept map
    89 concepts
    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 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 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 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

    This concept declares 5 statements. Each proof establishes one of them relative to its assumptions.

    1 assembled_minus proven

    2 assembled_plus proven

    3 channel_change_coordinate proven

    4 reciprocal_minus proven

    5 reciprocal_plus proven

    Lean source view on GitHub

    1import Lax342547.RecipeChannels
    2
    3/-!
    4---
    5title: Simultaneous assembly of opposite endpoints and reciprocal unit roles
    6type: lemma
    7---
    8The two cross forms assemble all four families of barred channel changes. Each updated primal-to-channel map receives precisely its own correction; opposite endpoint channels and reciprocal-role changes contribute zero. The exact formulas cover both cross orientations and their flips.
    9-/
    10
    11namespace Lax342547.ChannelAssembly
    12
    13open Lax342547.MomentSpace Lax342547.PairedAnnihilators Lax342547.ChannelChanges
    14open Lax342547.ResponseMatrices Lax342547.TableSpaces Lax342547.TableContractions
    15open Lax342547.DerivativeResponses Lax342547.TagGeometry Lax342547.ConcreteGeometry
    16open Lax342547.ConcreteCut Lax342547.CutProfiles Lax342547.PairedWitnesses
    17open Lax342547.ExactPins Lax342547.BarredSpaces Lax342547.PairedKeys
    18
    19noncomputable def assemble {k n b degree r : ℕ} {hr : 2 * r ≤ n} {H N : Type} [Fintype H]
    20 (W : Lists k n b degree r hr)
    21 (P Q : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    22 (U V U' V' : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary))
    23 (F : CrossForms (Component (Tag k)) (Coordinate k n b degree) H)
    24 (δ : ∀ z, PlusParameters W P Q U z) (ε : ∀ z, MinusParameters W P Q V z)
    25 (δ' : ∀ i, PlusParameters (Lax342547.PairedRecipes.flip W) Q P U' i) (ε' : ∀ i, MinusParameters (Lax342547.PairedRecipes.flip W) Q P V' i) :
    26 CrossForms (Component (Tag k)) (Coordinate k n b degree) H :=
    27 { forward := fun e => F.forward e +
    28 (∑ z, channelChange (barred W P (U e) (e, true))
    29 (protectedChannel Q z (e, false)) (δ z e) z) +
    30 ∑ i, (channelChange (barred (Lax342547.PairedRecipes.flip W) Q (V' e) (e, false))
    31 (protectedChannel P i (e, true)) (ε' i e) i).flip
    32 reverse := fun e => F.reverse e +
    33 (∑ i, channelChange (barred (Lax342547.PairedRecipes.flip W) Q (U' e) (e, true))
    34 (protectedChannel P i (e, false)) (δ' i e) i) +
    35 ∑ z, (channelChange (barred W P (V e) (e, false))
    36 (protectedChannel Q z (e, true)) (ε z e) z).flip }
    37
    38axiom channel_change_coordinate {B H : Type} [Fintype H]
    39 (D : Submodule Binary (Nominal B H)) (S : Submodule Binary (H → Binary))
    40 (δ : (Nominal B H ⧸ D) →ₗ[Binary] perpendicular S) (z j : Fin 2)
    41 (v : Nominal B H) (h : H) : by
    42 classical
    43 exact channelChange D S δ z v (channelEmbedding j (Pi.single h 1)) =
    44 if z = j then (δ (D.mkQ v)).val h else 0
    45
    46axiom assembled_plus {k n b degree r : ℕ} {hr : 2 * r ≤ n} {H N : Type} [Fintype H]
    47 (W : Lists k n b degree r hr)
    48 (P Q : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    49 (U V U' V' : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary))
    50 (F : CrossForms (Component (Tag k)) (Coordinate k n b degree) H)
    51 (δ : ∀ z, PlusParameters W P Q U z) (ε : ∀ z, MinusParameters W P Q V z)
    52 (δ' : ∀ i, PlusParameters (Lax342547.PairedRecipes.flip W) Q P U' i) (ε' : ∀ i, MinusParameters (Lax342547.PairedRecipes.flip W) Q P V' i)
    53 (e : Component (Tag k)) (i z : Fin 2) :
    54 fullPlus (assemble W P Q U V U' V' F δ ε δ' ε') e i z = fullPlus F e i z +
    55 restriction (barred W P (U e) (e, true)) (protectedChannel Q z (e, false)) (δ z e) i
    56
    57axiom assembled_minus {k n b degree r : ℕ} {hr : 2 * r ≤ n} {H N : Type} [Fintype H]
    58 (W : Lists k n b degree r hr)
    59 (P Q : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    60 (U V U' V' : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary))
    61 (F : CrossForms (Component (Tag k)) (Coordinate k n b degree) H)
    62 (δ : ∀ z, PlusParameters W P Q U z) (ε : ∀ z, MinusParameters W P Q V z)
    63 (δ' : ∀ i, PlusParameters (Lax342547.PairedRecipes.flip W) Q P U' i) (ε' : ∀ i, MinusParameters (Lax342547.PairedRecipes.flip W) Q P V' i)
    64 (e : Component (Tag k)) (i z : Fin 2) :
    65 fullMinus (assemble W P Q U V U' V' F δ ε δ' ε') e i z = fullMinus F e i z +
    66 restriction (barred W P (V e) (e, false)) (protectedChannel Q z (e, true)) (ε z e) i
    67
    68axiom reciprocal_plus {k n b degree r : ℕ} {hr : 2 * r ≤ n} {H N : Type} [Fintype H]
    69 (W : Lists k n b degree r hr)
    70 (P Q : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    71 (U V U' V' : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary))
    72 (F : CrossForms (Component (Tag k)) (Coordinate k n b degree) H)
    73 (δ : ∀ z, PlusParameters W P Q U z) (ε : ∀ z, MinusParameters W P Q V z)
    74 (δ' : ∀ i, PlusParameters (Lax342547.PairedRecipes.flip W) Q P U' i) (ε' : ∀ i, MinusParameters (Lax342547.PairedRecipes.flip W) Q P V' i)
    75 (e : Component (Tag k)) (i z : Fin 2) :
    76 fullPlus (assemble W P Q U V U' V' F δ ε δ' ε').flip e z i = fullPlus F.flip e z i +
    77 restriction (barred (Lax342547.PairedRecipes.flip W) Q (U' e) (e, true)) (protectedChannel P i (e, false)) (δ' i e) z
    78
    79axiom reciprocal_minus {k n b degree r : ℕ} {hr : 2 * r ≤ n} {H N : Type} [Fintype H]
    80 (W : Lists k n b degree r hr)
    81 (P Q : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    82 (U V U' V' : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary))
    83 (F : CrossForms (Component (Tag k)) (Coordinate k n b degree) H)
    84 (δ : ∀ z, PlusParameters W P Q U z) (ε : ∀ z, MinusParameters W P Q V z)
    85 (δ' : ∀ i, PlusParameters (Lax342547.PairedRecipes.flip W) Q P U' i) (ε' : ∀ i, MinusParameters (Lax342547.PairedRecipes.flip W) Q P V' i)
    86 (e : Component (Tag k)) (i z : Fin 2) :
    87 fullMinus (assemble W P Q U V U' V' F δ ε δ' ε').flip e z i = fullMinus F.flip e z i +
    88 restriction (barred (Lax342547.PairedRecipes.flip W) Q (V' e) (e, false)) (protectedChannel P i (e, true)) (ε' i e) z
    89
    90end Lax342547.ChannelAssembly
    91
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