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Subtracting selected atoms preserves pure annihilation

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

proven

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    Natural Language Statement

    Lemma

    Both factors of an incident selected atom are frozen keys. Every pure quotient response therefore kills that atom. Linear combinations of the selected atoms can be subtracted from the pair of cut profiles while preserving every pure-response annihilation test.

    Concept map
    62 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 eventsRank 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 3 statements. Each proof establishes one of them relative to its assumptions.

    1 atom_annihilates proven

    2 selected_part_annihilates proven

    3 subtract_annihilates proven

    Lean source view on GitHub

    1import Lax342547.SelectedScalars
    2
    3/-!
    4---
    5title: Subtracting selected atoms preserves pure annihilation
    6type: lemma
    7---
    8Both factors of an incident selected atom are frozen keys. Every pure
    9quotient response therefore kills that atom. Linear combinations of
    10the selected atoms can be subtracted from the pair of cut profiles
    11while preserving every pure-response annihilation test.
    12-/
    13
    14namespace Lax342547.SelectedRemoval
    15
    16open Lax342547.MomentSpace Lax342547.TagGeometry Lax342547.ConcreteGeometry
    17open Lax342547.ConcreteCut Lax342547.CutProfiles Lax342547.PairedWitnesses
    18open Lax342547.ExactPins Lax342547.PureObstructions
    19
    20variable {k n b degree r : ℕ} {hr : 2 * r ≤ n} {H N : Type}
    21
    22noncomputable def selectedPart (W : Lists k n b degree r hr)
    23 (c : ∀ i : Fin 2, (Σ z, Fin (W.length i z)) → Binary) : Fin 2 → Profile k n b degree :=
    24 fun i => ∑ u, c i u • (W.left i u.1 u.2).profile
    25
    26axiom atom_annihilates (W : Lists k n b degree r hr)
    27 (P : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    28 (U V : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary))
    29 (i : Fin 2) (u : Σ z, Fin (W.length i z)) :
    30 Annihilates W P U V (Pi.single i (W.left i u.1 u.2).profile)
    31
    32axiom selected_part_annihilates (W : Lists k n b degree r hr)
    33 (P : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    34 (U V : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary))
    35 (c : ∀ i : Fin 2, (Σ z, Fin (W.length i z)) → Binary) :
    36 Annihilates W P U V (selectedPart W c)
    37
    38axiom subtract_annihilates (W : Lists k n b degree r hr)
    39 (P : Pin (Component (Tag k) × Bool) (Fin 2 × (Coordinate k n b degree ⊕ H)) N)
    40 (U V : Component (Tag k) → Fin 2 → Submodule Binary (H → Binary))
    41 (x : Fin 2 → Profile k n b degree) (h : Annihilates W P U V x)
    42 (c : ∀ i : Fin 2, (Σ z, Fin (W.length i z)) → Binary) :
    43 Annihilates W P U V (x - selectedPart W c)
    44
    45end Lax342547.SelectedRemoval
    46
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