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The full linearized response on pairs of actual cut profiles

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

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    Natural Language Statement

    Lemma

    The two allowed derivative families factor through the actual barred nominal quotients and take values perpendicular to the opposite pin's protected channels. Their contractions with the fixed baseline maps are added to the independent pure bilinear response.

    Concept map
    68 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 budgetThe exact space of binary base momentsRank control for the frozen baseline onprimal inputsFormal ordered product bits realizesymmetric correctionsAllowed channel changes and the actual tableinjection testsConcrete 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 kernelThe full linearized response on pairs ofactual cut profilesExact 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 columnsThe 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 basecoordinatesPrimal 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 cutprofilesExtracting fresh label coefficients throughpin quotientsRank of a tensor killed in two quotientspacesActual frame observations realize thenominal channel contractionsRaw matrix frames and their tensorrealizationThe finite uniform raw-vertex lawCoupled 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 labelsRank 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 pinvectorsConsistent 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 ADescendants are omitted for concepts with more than 10 descendants.
    Evidence

    This concept declares 11 statements. Each proof establishes one of them relative to its assumptions.

    1 atom_annihilates proven

    2 component_annihilation proven

    3 derivative_annihilation proven

    4 left_test proven

    5 minus_contraction_zero proven

    6 plus_contraction_zero proven

    7 pure_annihilation proven

    8 right_test proven

    9 selected_part_annihilates proven

    10 subtract_annihilates proven

    11 unselected_obstruction proven

    Lean source view on GitHub

    1import Lax342547.ChannelChanges
    2import Lax342547.SelectedRemoval
    3import Lax342547.ResidualLabels
    4
    5/-!
    6---
    7title: The full linearized response on pairs of actual cut profiles
    8type: lemma
    9---
    10The two allowed derivative families factor through the actual barred
    11nominal quotients and take values perpendicular to the opposite pin's
    12protected channels. Their contractions with the fixed baseline maps
    13are added to the independent pure bilinear response.
    14-/
    15
    16namespace Lax342547.DerivativeResponses
    17
    18open Lax342547.MomentSpace Lax342547.TagGeometry Lax342547.ConcreteGeometry
    19open Lax342547.ConcreteCut Lax342547.CutProfiles Lax342547.PairedWitnesses
    20open Lax342547.ExactPins Lax342547.TableSpaces Lax342547.PairedAnnihilators
    21open Lax342547.BarredSpaces Lax342547.ChannelChanges Lax342547.TableContractions
    22open Lax342547.SelectedRemoval
    23
    24variable {k n b degree r : ℕ} {hr : 2 * r ≤ n} {H N : Type} [Fintype H]
    25
    26abbrev PlusParameters (W : Lists k n b degree r hr)
    27 (P Q : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    28 (U : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary)) (z : Fin 2) :=
    29 ∀ e, (Nominal (Coordinate k n b degree) H ⧸ barred W P (U e) (e, true)) →ₗ[Binary]
    30 perpendicular (protectedChannel Q z (e, false))
    31
    32abbrev MinusParameters (W : Lists k n b degree r hr)
    33 (P Q : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    34 (V : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary)) (z : Fin 2) :=
    35 ∀ e, (Nominal (Coordinate k n b degree) H ⧸ barred W P (V e) (e, false)) →ₗ[Binary]
    36 perpendicular (protectedChannel Q z (e, true))
    37
    38def restriction {B : Type} (D : Submodule Binary (Nominal B H))
    39 (S : Submodule Binary (H → Binary))
    40 (δ : (Nominal B H ⧸ D) →ₗ[Binary] perpendicular S) (i : Fin 2) :
    41 (B → Binary) →ₗ[Binary] (H → Binary) :=
    42 (((perpendicular S).subtype.comp δ).comp D.mkQ).comp (primalEmbedding i)
    43
    44noncomputable def componentResponse {Comp B : Type} [Fintype B]
    45 (F : CrossForms Comp B H) (e : Comp) (z : Fin 2)
    46 (D E : Submodule Binary (Nominal B H)) (S T : Submodule Binary (H → Binary))
    47 (δF : (Nominal B H ⧸ D) →ₗ[Binary] perpendicular S)
    48 (δG : (Nominal B H ⧸ E) →ₗ[Binary] perpendicular T) :
    49 (Fin 2 → Matrix B B Binary) →ₗ[Binary] Binary :=
    50 ∑ i, (TensorContractions.ambient (fullPlus F e i z) (restriction E T δG i) +
    51 TensorContractions.ambient (restriction D S δF i) (fullMinus F e i z)).comp (LinearMap.proj i)
    52
    53noncomputable def response (W : Lists k n b degree r hr)
    54 (P Q : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    55 (U V : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary))
    56 (F : CrossForms (Component (Tag k)) (Coordinate k n b degree) H) (z : Fin 2)
    57 (δF : PlusParameters W P Q U z) (δG : MinusParameters W P Q V z) :
    58 (Fin 2 → Profile k n b degree) →ₗ[Binary] Binary :=
    59 ∑ e, (componentResponse F e z (barred W P (U e) (e, true))
    60 (barred W P (V e) (e, false)) (protectedChannel Q z (e, false))
    61 (protectedChannel Q z (e, true)) (δF e) (δG e)).comp (PureObstructions.endpointComponent e)
    62
    63noncomputable def fullResponse (W : Lists k n b degree r hr)
    64 (P Q : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    65 (U V : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary))
    66 (F : CrossForms (Component (Tag k)) (Coordinate k n b degree) H) (z : Fin 2)
    67 (β : PureObstructions.Parameters W P U V)
    68 (δF : PlusParameters W P Q U z) (δG : MinusParameters W P Q V z) :
    69 (Fin 2 → Profile k n b degree) →ₗ[Binary] Binary :=
    70 PureObstructions.response W P U V β + response W P Q U V F z δF δG
    71
    72def Annihilates (W : Lists k n b degree r hr)
    73 (P Q : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    74 (U V : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary))
    75 (F : CrossForms (Component (Tag k)) (Coordinate k n b degree) H) (z : Fin 2)
    76 (x : Fin 2 → Profile k n b degree) : Prop :=
    77 ∀ β δF δG, fullResponse W P Q U V F z β δF δG x = 0
    78
    79axiom pure_annihilation (W : Lists k n b degree r hr)
    80 (P Q : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    81 (U V : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary))
    82 (F : CrossForms (Component (Tag k)) (Coordinate k n b degree) H) (z : Fin 2)
    83 (x : Fin 2 → Profile k n b degree) (h : Annihilates W P Q U V F z x) :
    84 Lax342547.PureObstructions.Annihilates W P U V x
    85
    86axiom derivative_annihilation (W : Lists k n b degree r hr)
    87 (P Q : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    88 (U V : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary))
    89 (F : CrossForms (Component (Tag k)) (Coordinate k n b degree) H) (z : Fin 2)
    90 (x : Fin 2 → Profile k n b degree) (h : Annihilates W P Q U V F z x)
    91 (δF : PlusParameters W P Q U z) (δG : MinusParameters W P Q V z) :
    92 response W P Q U V F z δF δG x = 0
    93
    94axiom atom_annihilates (W : Lists k n b degree r hr)
    95 (P Q : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    96 (U V : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary))
    97 (F : CrossForms (Component (Tag k)) (Coordinate k n b degree) H) (z : Fin 2)
    98 (i : Fin 2) (u : Σ z, Fin (W.length i z)) :
    99 Annihilates W P Q U V F z (Pi.single i (W.left i u.1 u.2).profile)
    100
    101axiom selected_part_annihilates (W : Lists k n b degree r hr)
    102 (P Q : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    103 (U V : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary))
    104 (F : CrossForms (Component (Tag k)) (Coordinate k n b degree) H) (z : Fin 2)
    105 (c : ∀ i : Fin 2, (Σ z, Fin (W.length i z)) → Binary) :
    106 Annihilates W P Q U V F z (selectedPart W c)
    107
    108axiom subtract_annihilates (W : Lists k n b degree r hr)
    109 (P Q : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    110 (U V : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary))
    111 (F : CrossForms (Component (Tag k)) (Coordinate k n b degree) H) (z : Fin 2)
    112 (x : Fin 2 → Profile k n b degree) (h : Annihilates W P Q U V F z x)
    113 (c : ∀ i : Fin 2, (Σ z, Fin (W.length i z)) → Binary) :
    114 Annihilates W P Q U V F z (x - selectedPart W c)
    115
    116axiom component_annihilation (W : Lists k n b degree r hr)
    117 (P Q : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    118 (U V : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary))
    119 (F : CrossForms (Component (Tag k)) (Coordinate k n b degree) H) (z : Fin 2)
    120 (x : Fin 2 → Profile k n b degree) (h : Annihilates W P Q U V F z x)
    121 (e : Component (Tag k))
    122 (δF : (Nominal (Coordinate k n b degree) H ⧸ barred W P (U e) (e, true)) →ₗ[Binary]
    123 perpendicular (protectedChannel Q z (e, false)))
    124 (δG : (Nominal (Coordinate k n b degree) H ⧸ barred W P (V e) (e, false)) →ₗ[Binary]
    125 perpendicular (protectedChannel Q z (e, true))) :
    126 componentResponse F e z (barred W P (U e) (e, true))
    127 (barred W P (V e) (e, false)) (protectedChannel Q z (e, false))
    128 (protectedChannel Q z (e, true)) δF δG (Lax342547.PureObstructions.endpointComponent e x) = 0
    129
    130axiom unselected_obstruction {K : ℕ} (hk : 0 < k)
    131 (W : Lists k n b degree r hr)
    132 (P Q : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    133 (hK : P.rank ≤ K) (A : Fin 2 → Finset (Fin b → Binary))
    134 (hA : Lax342547.PinLabelExclusions.Covers P A (2 * K + 28))
    135 (hfresh : ∀ i z t, (W.left i z t).label ∉ A i)
    136 (U V : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary))
    137 (F : CrossForms (Component (Tag k)) (Coordinate k n b degree) H) (z : Fin 2)
    138 (x : Fin 2 → Profile k n b degree) (h : Annihilates W P Q U V F z x)
    139 (hdegree : 6 * (2 * K + 28) + 4 ≤ degree) :
    140 ∃ c : ∀ i : Fin 2, (Σ z, Fin (W.length i z)) → Binary,
    141 ∃ L : Fin 2 → Component (Tag k) → Finset (Fin b → Binary),
    142 ∃ Z : Fin 2 → Component (Tag k) → (Fin b → Binary) →
    143 Matrix (Option (Base k n)) (Option (Base k n)) Binary,
    144 Annihilates W P Q U V F z (x - selectedPart W c) ∧
    145 ∀ i e, (L i e).card ≤ 2 * K + 28 ∧
    146 (∀ s ∈ L i e, s ∉ Lax342547.ResidualLabels.chosenLabels W i ∧
    147 Lax342547.BaseMoments.IsBaseMoment (Z i e s)) ∧
    148 (∑ s ∈ L i e, (Z i e s).rank) ≤ 2 * K + 28 ∧
    149 ((x - selectedPart W c) i).val e =
    150 ∑ s ∈ L i e, Lax342547.TensorBlocks.block selectorEval s (Z i e s)
    151
    152axiom right_test (W : Lists k n b degree r hr)
    153 (P Q : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    154 (U V : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary))
    155 (F : CrossForms (Component (Tag k)) (Coordinate k n b degree) H) (z : Fin 2)
    156 (x : Fin 2 → Profile k n b degree) (h : Annihilates W P Q U V F z x)
    157 (e : Component (Tag k)) (i : Fin 2)
    158 (ψ : Module.Dual Binary (Vector k n b degree))
    159 (hψ : Lax342547.ProjectedPins.projected P i (e, false) ⊔ individualKeys W i e ≤ LinearMap.ker ψ)
    160 (c : H → Binary) (hc : c ∈ perpendicular (protectedChannel Q z (e, true))) :
    161 dotProduct (fullPlus F e i z
    162 (((x i).val e).mulVecLin (Lax342547.PrimalContractions.coordinates ψ))) c = 0
    163
    164axiom plus_contraction_zero {R : ℕ} (W : Lists k n b degree r hr)
    165 (P Q : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    166 (A : Fin 2 → Finset (Fin b → Binary))
    167 (hA : Lax342547.PinLabelExclusions.Covers P A R)
    168 (hfresh : ∀ i z t, (W.left i z t).label ∉ A i)
    169 (U V : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary))
    170 (T : Lax342547.SmallTables.Table P Q) (hT : Lax342547.SmallTables.Injecting T)
    171 (F : CrossForms (Component (Tag k)) (Coordinate k n b degree) H) (hF : Extends F T) (z : Fin 2)
    172 (x : Fin 2 → Profile k n b degree) (h : Annihilates W P Q U V F z x)
    173 (e : Component (Tag k)) (i : Fin 2)
    174 (hU : ∀ j, IsCompl (protectedChannel P j (e, true)) (U e j))
    175 (L : Finset (Fin b → Binary)) (hL : L.card ≤ R)
    176 (hdisjoint : Disjoint L (Lax342547.ResidualLabels.chosenLabels W i))
    177 (Z : (Fin b → Binary) → Matrix (Option (Base k n)) (Option (Base k n)) Binary)
    178 (hM : (x i).val e = ∑ s ∈ L, Lax342547.TensorBlocks.block selectorEval s (Z s))
    179 (ψ : Module.Dual Binary (Vector k n b degree))
    180 (hψ : Lax342547.ProjectedPins.projected P i (e, false) ⊔ individualKeys W i e ≤ LinearMap.ker ψ) :
    181 ((x i).val e).mulVecLin (Lax342547.PrimalContractions.coordinates ψ) = 0
    182
    183axiom left_test (W : Lists k n b degree r hr)
    184 (P Q : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    185 (U V : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary))
    186 (F : CrossForms (Component (Tag k)) (Coordinate k n b degree) H) (z : Fin 2)
    187 (x : Fin 2 → Profile k n b degree) (h : Annihilates W P Q U V F z x)
    188 (e : Component (Tag k)) (i : Fin 2)
    189 (ψ : Module.Dual Binary (Vector k n b degree))
    190 (hψ : Lax342547.ProjectedPins.projected P i (e, true) ⊔ individualKeys W i e ≤ LinearMap.ker ψ)
    191 (c : H → Binary) (hc : c ∈ perpendicular (protectedChannel Q z (e, false))) :
    192 dotProduct (fullMinus F e i z
    193 (((x i).val e).transpose.mulVecLin (Lax342547.PrimalContractions.coordinates ψ))) c = 0
    194
    195axiom minus_contraction_zero {R : ℕ} (W : Lists k n b degree r hr)
    196 (P Q : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    197 (A : Fin 2 → Finset (Fin b → Binary))
    198 (hA : Lax342547.PinLabelExclusions.Covers P A R)
    199 (hfresh : ∀ i z t, (W.left i z t).label ∉ A i)
    200 (U V : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary))
    201 (T : Lax342547.SmallTables.Table P Q) (hT : Lax342547.SmallTables.Injecting T)
    202 (F : CrossForms (Component (Tag k)) (Coordinate k n b degree) H) (hF : Extends F T) (z : Fin 2)
    203 (x : Fin 2 → Profile k n b degree) (h : Annihilates W P Q U V F z x)
    204 (e : Component (Tag k)) (i : Fin 2)
    205 (hV : ∀ j, IsCompl (protectedChannel P j (e, false)) (V e j))
    206 (L : Finset (Fin b → Binary)) (hL : L.card ≤ R)
    207 (hdisjoint : Disjoint L (Lax342547.ResidualLabels.chosenLabels W i))
    208 (Z : (Fin b → Binary) → Matrix (Option (Base k n)) (Option (Base k n)) Binary)
    209 (hM : (x i).val e = ∑ s ∈ L, Lax342547.TensorBlocks.block selectorEval s (Z s))
    210 (ψ : Module.Dual Binary (Vector k n b degree))
    211 (hψ : Lax342547.ProjectedPins.projected P i (e, true) ⊔ individualKeys W i e ≤ LinearMap.ker ψ) :
    212 ((x i).val e).transpose.mulVecLin (Lax342547.PrimalContractions.coordinates ψ) = 0
    213
    214end Lax342547.DerivativeResponses
    215
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