Lax Lean Archive

While this submission is a draft, it cannot be used by other submissions.

Terminal exceptions from real capacity cuts

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

proven

Loading review…

Community review

Flags

Each flag is tied to a public ORCID identity and explains why this concept may be incorrect.

No flags have been submitted.

    Community review

    Flag this concept

    State precisely what appears incorrect. This explanation will be public under your ORCID name.

    No source line selected.

    Natural Language Statement

    Lemma

    A residual pair set admitting no probability law with the marginal and pair caps is covered by a small raw vertex exception and a small product-law pair exception. This proves the finite content of Lemma 3.3.

    Concept map
    113 concepts; 9 descendants 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 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 correctionsBipartite capacity networksBounded allowed derivatives preserving theactual linearized responseActual-mass averaging over retainedrecord-cell pairsSimultaneous assembly of opposite endpointsand reciprocal unit rolesAllowed channel changes and the actual tableinjection testsRank-controlled factorization throughallowed orthogonal channelsPruning small unary record cellsA 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 plusframeExact conditioning costs and recovery offinite probability massesCompression 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 atomsEntropy along feasible mixture lines,including new supportFull feasible support and finite informationprojectionExact image pins in nominal coefficientspacesFinite linear images and their uniform-lawdensity boundsReal max-flow min-cut from residualreachabilityActual gradient agreement and matched keysproduce all four cross holesAmbient symmetries and frame marginalsFresh key directions are independent modulotable spacesThe simultaneous gradient corrections retainevery frozen pin and key entryBaseline bilinear extensions retaining allfrozen rows and columnsWhole-space gradient realization preservingthe actual frozen entriesBaseline contractions on the selected atomsAll four whole-space gradient equations fromassembled channel changesThe symmetric binary gradient formDeterministic whole-space gradientrealization from the genuine recipehypothesesGram-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 failureAccepted key subprobabilities and uniformoverlapFresh 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 partPure corrections realized by actual nonlinearchannel productsSquared restriction cost for independent uniteventsEndpoint 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 pinsFinite primal-channel records with the paperbit count and four-hole implicationRetractions 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 cutprofilesWalsh bounds for independent image lawsand separated phasesCompression 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 lawOriginal retained-cell laws from finite PMFsCompact real-capacity flows and amaximizing flowNonlinear channel realization of the actualwhole-space recipe residualGradient 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 eventsFinite relative entropy and support costsResidual walk augmentation for real-capacityflowsRemoving 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 formImage caps inside original retained cellsSelected 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 taggeometryExplicit low-rank matrices for thewhole-space gradient targetThe fifteen-rank witness tester targetPure bilinear responses detect quotienttensorsQuotient extractors isolate individual tensorlabel blocksWell-defined channel contractions onprojected tensor spacesTerminal exceptions from real capacity cutsAdmissible pair lawsOrthogonality and finite Walsh correlationboundsWitness atoms and their numerical testerrecords
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 3 statements. Each proof establishes one of them relative to its assumptions.

    1 bipartite_cut_capacity proven

    2 law_iff_masked proven

    3 terminal_cut proven

    Lean source view on GitHub

    1import Lax342547.BipartiteFlows
    2import Lax342547.UnitLaws
    3import Lax342547.CellAveraging
    4
    5/-!
    6---
    7title: Terminal exceptions from real capacity cuts
    8type: lemma
    9---
    10A residual pair set admitting no probability law with the marginal and pair caps is covered by a small raw vertex exception and a small product-law pair exception. This proves the finite content of Lemma 3.3.
    11-/
    12
    13namespace Lax342547.TerminalCuts
    14
    15open Lax342547.BipartiteFlows Lax342547.FlowCuts Lax342547.RetainedImages Lax342547.CellAveraging
    16open scoped BigOperators
    17
    18noncomputable def pairCapacity {Ω : Type} (μ : Ω → ℝ) (R : Ω → Ω → Prop) (κ : ℝ) : Ω → Ω → ℝ := by
    19 classical
    20 exact fun x y => if R x y then κ*(μ x*μ y) else 0
    21
    22axiom law_iff_masked {Ω : Type} [Fintype Ω]
    23 (μ : Ω → ℝ) (R : Ω → Ω → Prop) (M κ : ℝ) (ρ : Ω → Ω → ℝ)
    24 (hμ : ∀ x, 0 ≤ μ x) (hκ : 0 ≤ κ) :
    25 Law (fun x => M*μ x) (fun y => M*μ y) (pairCapacity μ R κ) ρ ↔
    26 Lax342547.UnitLaws.Feasible μ R M κ (fun xy => ρ xy.1 xy.2)
    27
    28axiom bipartite_cut_capacity {A B : Type} [Fintype A] [Fintype B]
    29 (a : A → ℝ) (b : B → ℝ) (p : A → B → ℝ) (Z : Finset (Vertex A B))
    30 (hs : source ∈ Z) (ht : sink ∉ Z) : by
    31 classical
    32 exact cutCapacity (capacity a b p) Z =
    33 (∑ x, if left x ∉ Z then a x else 0)+(∑ y, if right y ∈ Z then b y else 0)+
    34 (∑ x, ∑ y, if left x ∈ Z ∧ right y ∉ Z then p x y else 0)
    35
    36axiom terminal_cut {Ω : Type} [Fintype Ω]
    37 (μ : Ω → ℝ) (R : Ω → Ω → Prop) (M κ : ℝ)
    38 (hμ : ∀ x, 0 ≤ μ x) (hM : 0 < M) (hκ : 0 < κ)
    39 (hn : ¬ ∃ ρ, Lax342547.UnitLaws.Feasible μ R M κ ρ) :
    40 ∃ S : Ω → Prop, ∃ E : Ω → Ω → Prop,
    41 cellMass μ S < 1/M ∧ pairEventMass μ μ E < 1/κ ∧
    42 (∀ x y, R x y → S x ∨ S y ∨ E x y)
    43
    44end Lax342547.TerminalCuts
    45
    Show ProofShow ProofShow Proof
    Builds on
    Used by
    From Mathlib

    none

    Discussion

    Ask a question or add context. Endorsements and structured flags are kept in the review panel above.

    Loading discussion…