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Small affine slices force global cross-coefficient rank

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

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    Natural Language Statement

    Lemma

    Every matrix rank has a witness on actual coordinate rows and columns. A sum of s affine products has a cross-coefficient matrix of rank at most 2s. The same bound on every (2s+1)-square coordinate slice forces the global bound. Summed predictor parity then contradicts an identity matrix larger than the sum of these ranks. This is the deterministic conclusion of Lemma 9.5.

    Concept map
    78 concepts; 1 descendant 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 budgetThe 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 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 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 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 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 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 eventsRemoving selected atoms leaves onlyunselected component labelsMatrix 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 selectorlabelsSmall affine slices force globalcross-coefficient rankNumerical 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 A
    Evidence

    This concept declares 8 statements. Each proof establishes one of them relative to its assumptions.

    1 cross_affine_product proven

    2 cross_products_rank proven

    3 cross_rank_of_products proven

    4 independent_column_selection proven

    5 parity_rank_contradiction proven

    6 rank_from_local_products proven

    7 rank_le_of_small_slices proven

    8 rank_witness proven

    Lean source view on GitHub

    1import Lax342547.RankedProjection
    2import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
    3
    4/-!
    5---
    6title: Small affine slices force global cross-coefficient rank
    7type: lemma
    8---
    9Every matrix rank has a witness on actual coordinate rows and columns. A sum
    10of s affine products has a cross-coefficient matrix of rank at most 2s.
    11The same bound on every (2s+1)-square coordinate slice forces the global
    12bound. Summed predictor parity then contradicts an identity matrix larger
    13than the sum of these ranks. This is the deterministic conclusion of Lemma 9.5.
    14-/
    15
    16namespace Lax342547.SmallSliceRank
    17open Lax342547.MomentSpace
    18open scoped BigOperators
    19
    20axiom independent_column_selection {I J : Type} [Fintype I] [Fintype J]
    21 (A : Matrix I J Binary) (t : ℕ) (ht : t ≤ A.rank) :
    22 ∃ c : Fin t → J,Function.Injective c ∧ (A.submatrix id c).rank = t
    23
    24axiom rank_witness {I J : Type} [Fintype I] [Fintype J]
    25 (A : Matrix I J Binary) (t : ℕ) (ht : t ≤ A.rank) :
    26 ∃ r : Fin t → I,∃ c : Fin t → J,
    27 Function.Injective r ∧ Function.Injective c ∧ (A.submatrix r c).rank = t
    28
    29axiom rank_le_of_small_slices {I J : Type} [Fintype I] [Fintype J]
    30 (A : Matrix I J Binary) (r : ℕ)
    31 (h : ∀ i : Fin (r+1) → I,∀ j : Fin (r+1) → J,
    32 Function.Injective i → Function.Injective j → (A.submatrix i j).rank ≤ r) : A.rank ≤ r
    33
    34def crossCoefficients {I J : Type} [DecidableEq I] [DecidableEq J]
    35 (F : (I → Binary) → (J → Binary) → Binary) : Matrix I J Binary :=
    36 fun i j => F (Pi.single i 1) (Pi.single j 1)+F (Pi.single i 1) 0+
    37 F 0 (Pi.single j 1)+F 0 0
    38
    39def affineValue {I J : Type} [Fintype I] [Fintype J]
    40 (c : Binary) (a : I → Binary) (b : J → Binary) (x : I → Binary) (y : J → Binary) : Binary :=
    41 c+dotProduct a x+dotProduct b y
    42
    43axiom cross_affine_product {I J : Type} [Fintype I] [Fintype J] [DecidableEq I] [DecidableEq J]
    44 (c d : Binary) (a u : I → Binary) (b v : J → Binary) :
    45 crossCoefficients (fun x y => affineValue c a b x y*affineValue d u v x y) =
    46 Matrix.vecMulVec a v+Matrix.vecMulVec u b
    47
    48axiom cross_products_rank {P I J : Type} [Fintype P] [Fintype I] [Fintype J]
    49 [DecidableEq I] [DecidableEq J] (c d : P → Binary) (a u : P → I → Binary)
    50 (b v : P → J → Binary) :
    51 (crossCoefficients (fun x y => ∑ p,affineValue (c p) (a p) (b p) x y*
    52 affineValue (d p) (u p) (v p) x y)).rank ≤ 2*Fintype.card P
    53
    54def liftCoordinates {T I : Type} [Fintype T] [DecidableEq I]
    55 (r : T → I) (x : T → Binary) : I → Binary := ∑ t,x t • Pi.single (r t) 1
    56
    57def HasProducts {I J : Type} [Fintype I] [Fintype J]
    58 (s : ℕ) (F : (I → Binary) → (J → Binary) → Binary) : Prop :=
    59 ∃ c d : Fin s → Binary,∃ a u : Fin s → I → Binary,∃ b v : Fin s → J → Binary,
    60 ∀ x y,F x y = ∑ p,affineValue (c p) (a p) (b p) x y*affineValue (d p) (u p) (v p) x y
    61
    62axiom cross_rank_of_products {I J : Type} [Fintype I] [Fintype J]
    63 [DecidableEq I] [DecidableEq J] (s : ℕ) (F : (I → Binary) → (J → Binary) → Binary)
    64 (h : HasProducts s F) : (crossCoefficients F).rank ≤ 2*s
    65
    66axiom rank_from_local_products {I J : Type} [Fintype I] [Fintype J]
    67 [DecidableEq I] [DecidableEq J] (s : ℕ) (F : (I → Binary) → (J → Binary) → Binary)
    68 (h : ∀ r : Fin (2*s+1) → I,∀ c : Fin (2*s+1) → J,
    69 Function.Injective r → Function.Injective c →
    70 HasProducts s (fun x y => F (liftCoordinates r x) (liftCoordinates c y))) :
    71 (crossCoefficients F).rank ≤ 2*s
    72
    73axiom parity_rank_contradiction {T : Type} [Fintype T] (s m : ℕ)
    74 (F : T → (Fin m → Binary) → (Fin m → Binary) → Binary)
    75 (hparity : ∀ x y,(∑ l,F l x y) = dotProduct x y)
    76 (hloc : ∀ l : T,∀ r : Fin (2*s+1) → Fin m,∀ c : Fin (2*s+1) → Fin m,
    77 Function.Injective r → Function.Injective c →
    78 HasProducts s (fun x y => F l (liftCoordinates r x) (liftCoordinates c y)))
    79 (hm : 2*s*Fintype.card T < m) : False
    80
    81end Lax342547.SmallSliceRank
    82
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