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Low-rank tester routing along tag stars

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

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    Natural Language Statement

    Lemma

    Ordered tester matrices have rank at most r. Summing cut moments along a tag star removes its even center multiplicity. Forbidden ordinary centers vanish, and each component receives at most two routed testers.

    Concept map
    84 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 responseAllowed 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 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 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 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 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 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 12 statements. Each proof establishes one of them relative to its assumptions.

    1 incidence_sum proven

    2 interval_center_excluded proven

    3 matrix_add_rank proven

    4 matrix_finset_sum_rank proven

    5 matrix_smul_rank proven

    6 matrix_sum_rank proven

    7 ordered_tester_rank proven

    8 ordinary_tester_disallowed_zero proven

    9 star_cut_sum proven

    10 star_matrix_rank proven

    11 star_matrix_representation proven

    12 tester_matrix_representation proven

    Lean source view on GitHub

    1import Lax342547.ChannelFactorization
    2
    3/-!
    4---
    5title: Low-rank tester routing along tag stars
    6type: lemma
    7---
    8Ordered tester matrices have rank at most r. Summing cut moments along a tag star removes its even center multiplicity. Forbidden ordinary centers vanish, and each component receives at most two routed testers.
    9-/
    10
    11namespace Lax342547.StarRouting
    12
    13open Lax342547.MomentSpace Lax342547.ConcreteGeometry Lax342547.ConcreteCut
    14open Lax342547.TagGeometry Lax342547.CutProfiles Lax342547.Atoms
    15
    16axiom matrix_sum_rank {I B : Type} [Fintype I] [Fintype B]
    17 (M : I → Matrix B B Binary) :
    18 (∑ i, M i).rank ≤ ∑ i, (M i).rank
    19
    20noncomputable def orderedTesterMatrix {k n b degree r : ℕ}
    21 (hr : 2 * r ≤ n) (c : Fin n → Lax342547.TagGeometry.Base k n) : Moment k n b degree := by
    22 classical
    23 exact ∑ ρ, Matrix.vecMulVec
    24 (Pi.single (emptySelector b degree, some (c (pairLeft hr ρ))) 1)
    25 (Pi.single (emptySelector b degree, some (c (pairRight hr ρ))) 1)
    26
    27axiom ordered_tester_rank {k n b degree r : ℕ}
    28 (hr : 2 * r ≤ n) (c : Fin n → Lax342547.TagGeometry.Base k n) :
    29 (orderedTesterMatrix (b := b) (degree := degree) hr c).rank ≤ r
    30
    31axiom tester_matrix_representation {k n b degree r : ℕ}
    32 (hr : 2 * r ≤ n) (c : Fin n → Lax342547.TagGeometry.Base k n) :
    33 matrixPair (orderedTesterMatrix (b := b) (degree := degree) hr c) = blockTester hr c
    34
    35axiom interval_center_excluded {k : ℕ} (d : Tag k) : d ∉ interval k d
    36
    37axiom ordinary_tester_disallowed_zero {k n b degree r : ℕ} (hr : 2 * r ≤ n)
    38 (d t l : Tag k) (hl : l ∉ interval k d) (w : Representation k n b degree) :
    39 blockTester hr (ordinary d t) (w.val l) = 0
    40
    41noncomputable def incidentEquiv {k : ℕ} (l : Tag k) :
    42 {t : Tag k // t ≠ l} ≃ {e : Component (Tag k) // l ∈ e.val} := by
    43 classical
    44 let f : {t : Tag k // t ≠ l} → {e : Component (Tag k) // l ∈ e.val} :=
    45 fun t => ⟨⟨{l, t.val}, Finset.card_pair t.property.symm⟩, by simp⟩
    46 apply Equiv.ofBijective f
    47 constructor
    48 · intro u v h
    49 apply Subtype.ext
    50 have hs : ({l, u.val} : Finset (Tag k)) = {l, v.val} :=
    51 congrArg (fun e : {e : Component (Tag k) // l ∈ e.val} => e.val.val) h
    52 have hu : u.val ∈ ({l, v.val} : Finset (Tag k)) := by rw [← hs]; simp
    53 simpa [u.property] using hu
    54 · intro e
    55 obtain ⟨a, b, hab, he⟩ := Finset.card_eq_two.mp e.val.property
    56 have hl : l = a ∨ l = b := by simpa only [he, Finset.mem_insert, Finset.mem_singleton] using e.property
    57 rcases hl with rfl | rfl
    58 · refine ⟨⟨b, hab.symm⟩, ?_⟩
    59 apply Subtype.ext
    60 exact Subtype.ext he.symm
    61 · refine ⟨⟨a, hab⟩, ?_⟩
    62 apply Subtype.ext
    63 apply Subtype.ext
    64 change ({l, a} : Finset (Tag k)) = e.val.val
    65 rw [he, Finset.pair_comm]
    66
    67axiom star_cut_sum {k : ℕ} {X : Type} [AddCommGroup X] [Module Binary X]
    68 (w : Tag k → X) (l : Tag k) :
    69 (∑ e : {e : Component (Tag k) // l ∈ e.val}, ∑ t ∈ e.val.val, w t) =
    70 ∑ t : {t : Tag k // t ≠ l}, w t.val
    71
    72axiom incidence_sum {k : ℕ} {X : Type} [AddCommMonoid X]
    73 (F : Tag k → Component (Tag k) → X) :
    74 (∑ e, ∑ d ∈ e.val, F d e) = ∑ d, ∑ e : {e : Component (Tag k) // d ∈ e.val}, F d e.val
    75
    76noncomputable def starMatrix {k n b degree r : ℕ} (hr : 2 * r ≤ n)
    77 (c : Tag k → Fin n → Base k n) (coeff : Tag k → Binary) (e : Component (Tag k)) :
    78 Moment k n b degree := ∑ d ∈ e.val, coeff d • orderedTesterMatrix hr (c d)
    79
    80axiom star_matrix_representation {k n b degree r : ℕ} (hr : 2 * r ≤ n)
    81 (c : Tag k → Fin n → Base k n) (coeff : Tag k → Binary)
    82 (w : Representation k n b degree) :
    83 (∑ e, matrixPair (starMatrix (b := b) (degree := degree) hr c coeff e)
    84 ((representationCut k n b degree w) e)) =
    85 ∑ d, coeff d * ∑ l : {l : Tag k // l ≠ d}, blockTester hr (c d) (w.val l.val)
    86
    87axiom matrix_smul_rank {B : Type} [Fintype B] (M : Matrix B B Binary) (a : Binary) :
    88 (a • M).rank ≤ M.rank
    89
    90axiom matrix_add_rank {B : Type} [Fintype B] (M Q : Matrix B B Binary) :
    91 (M + Q).rank ≤ M.rank + Q.rank
    92
    93axiom matrix_finset_sum_rank {I B : Type} [Fintype B] (s : Finset I)
    94 (M : I → Matrix B B Binary) (R : ℕ) (hM : ∀ i ∈ s, (M i).rank ≤ R) :
    95 (∑ i ∈ s, M i).rank ≤ s.card * R
    96
    97axiom star_matrix_rank {k n b degree r : ℕ} (hr : 2 * r ≤ n)
    98 (c : Tag k → Fin n → Base k n) (coeff : Tag k → Binary) (e : Component (Tag k)) :
    99 (starMatrix (b := b) (degree := degree) hr c coeff e).rank ≤ 2 * r
    100
    101end Lax342547.StarRouting
    102
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