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Removing selected atoms leaves only unselected component labels

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

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    Natural Language Statement

    Lemma

    Freshness identifies the finite sum of selected atoms with exactly the selected label blocks. Subtracting that sum deletes those blocks. The remaining supports are subsets of the original bounded supports and retain the original Boolean moment and sum-of-block-ranks budgets.

    Concept map
    63 concepts
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    Exact-image bounds for independent affinecolumnsOrdered atom products in the actualgradient formPoint atoms and finite flavor distributionsBarred response spaces and the actualobstruction rank budgetThe exact space of binary base momentsRank 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 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 basecoefficientsRetractions 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 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 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 2 statements. Each proof establishes one of them relative to its assumptions.

    1 residual_reconstruction proven

    2 unselected_obstruction proven

    Lean source view on GitHub

    1import Lax342547.SelectedRemoval
    2
    3/-!
    4---
    5title: Removing selected atoms leaves only unselected component labels
    6type: lemma
    7---
    8Freshness identifies the finite sum of selected atoms with exactly the
    9selected label blocks. Subtracting that sum deletes those blocks. The
    10remaining supports are subsets of the original bounded supports and
    11retain the original Boolean moment and sum-of-block-ranks budgets.
    12-/
    13
    14namespace Lax342547.ResidualLabels
    15
    16open Lax342547.MomentSpace Lax342547.TagGeometry Lax342547.ConcreteGeometry
    17open Lax342547.ConcreteCut Lax342547.CutProfiles Lax342547.PairedWitnesses
    18open Lax342547.ExactPins Lax342547.PinLabelExclusions Lax342547.BaseMoments
    19open Lax342547.LabelRelations Lax342547.TensorBlocks Lax342547.SelectedRemoval
    20
    21variable {k n b degree r : ℕ} {hr : 2 * r ≤ n} {H N : Type}
    22
    23noncomputable def chosenLabels (W : Lists k n b degree r hr) (i : Fin 2) : Finset (Fin b → Binary) :=
    24 Finset.univ.image (fun u : Σ z, Fin (W.length i z) => (W.left i u.1 u.2).label)
    25
    26axiom residual_reconstruction (W : Lists k n b degree r hr)
    27 (x : Fin 2 → Profile k n b degree)
    28 (L : Fin 2 → Component (Tag k) → Finset (Fin b → Binary))
    29 (Z : Fin 2 → Component (Tag k) → (Fin b → Binary) →
    30 Matrix (Option (Base k n)) (Option (Base k n)) Binary)
    31 (c : ∀ i : Fin 2, (Σ z, Fin (W.length i z)) → Binary)
    32 (hM : ∀ i e, (x i).val e = ∑ s ∈ L i e, block selectorEval s (Z i e s))
    33 (hc : ∀ i u e, supported (L i e) (Z i e) (W.left i u.1 u.2).label =
    34 if (W.left i u.1 u.2).tag ∈ e.val then c i u • baseMoment (W.left i u.1 u.2).base else 0) :
    35 ∀ i e, ((x - selectedPart W c) i).val e =
    36 ∑ s ∈ L i e \ chosenLabels W i, block selectorEval s (Z i e s)
    37
    38axiom unselected_obstruction {K : ℕ} [Fintype H] (hk : 0 < k)
    39 (W : Lists k n b degree r hr)
    40 (P : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    41 (hK : P.rank ≤ K) (A : Fin 2 → Finset (Fin b → Binary))
    42 (hA : Covers P A (2 * K + 28)) (hfresh : ∀ i z t, (W.left i z t).label ∉ A i)
    43 (U V : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary))
    44 (x : Fin 2 → Profile k n b degree) (h : Lax342547.PureObstructions.Annihilates W P U V x)
    45 (hdegree : 6 * (2 * K + 28) + 4 ≤ degree) :
    46 ∃ c : ∀ i : Fin 2, (Σ z, Fin (W.length i z)) → Binary,
    47 ∃ L : Fin 2 → Component (Tag k) → Finset (Fin b → Binary),
    48 ∃ Z : Fin 2 → Component (Tag k) → (Fin b → Binary) →
    49 Matrix (Option (Base k n)) (Option (Base k n)) Binary,
    50 Lax342547.PureObstructions.Annihilates W P U V (x - selectedPart W c) ∧
    51 ∀ i e, (L i e).card ≤ 2 * K + 28 ∧
    52 (∀ s ∈ L i e, s ∉ chosenLabels W i ∧ IsBaseMoment (Z i e s)) ∧
    53 (∑ s ∈ L i e, (Z i e s).rank) ≤ 2 * K + 28 ∧
    54 ((x - selectedPart W c) i).val e = ∑ s ∈ L i e, block selectorEval s (Z i e s)
    55
    56end Lax342547.ResidualLabels
    57
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