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Small phases on selected endpoint channel groups

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

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    Natural Language Statement

    Lemma

    The explicit numerical phase margin holds for arbitrary selected endpoint images and whole-unit marks while preserving the original no-cover rank parameter.

    Concept map
    82 concepts
    100%
    Actual projected span deficitSampling after exposed coordinatesAdaptive sampling on disjoint coordinatesetsUnion over adaptive component coversExact-image bounds for independent affinecolumnsExact uniform bilinear character meanWalsh operator bounds with explicit bilinearrankRank of a lifted tensor sumIndependent binary channel charactersActual raw frame channel phase tailsJoint channel law and itsindependent-column densityActual channel moments and phase tailsCombined column and row mode exposureRows of diagonal tensor mapsMoment estimates for actual componenttensor phasesComponentwise mode spaces and diagonaltensorsExact conditioning costs and recovery offinite probability massesEntropy progress for residual pair lawsIndependence of distinct sample positionsJoin-stable classes of mode coversCount tensors killed by exposureActual exposure cover projectionRank of a diagonal familyMoments across endpoint channel groupsOriginal-rank no-cover tails for endpointgroupsDyadic span deficit estimatesFixed dyadic moment orders at the ambientscaleA small dyadic tail scaleActual dyadic deficit recurrenceEntropy along feasible mixture lines,including new supportFull feasible support and finite informationprojectionProjection removes the exposed termsCount tested index occurrencesFinite linear images and their uniform-lawdensity boundsFinite even moment expansionFinite independent sampling and vertexexception tailsFixed-parameter endpoint-group phaseboundsAmbient symmetries and frame marginalsGram-conditioned columns and theirrank-failure probabilityTwo-sided Gram normalization forindividually injective framesGreedy mode space exposureSmall actual greedy exposure tailsPhase density budgets for selected channelgroupsNo-cover rank growth on arbitrary finiteindicesActual deficits indexed by a distinct listFinite list tail statisticsFactorization and counting of low-rankbinary matricesDimension deficits after an arbitrary modemapExposed mode dimension budgetExplicit no-cover moment parameter marginsBoolean point moments with restricted basecoordinatesFinite moment probability and rank splitRaw phase estimates for dependent markedunitsA low rank sum supplies an actual adaptivecoverNo-cover phase momentsRank growth without an adaptive modecoverSquared restriction cost for independent uniteventsFinite exposure partitionsActual phase averages from small exceptionaltailsThe numerical no-cover phase marginProjected nonzero terms in the actualremaining sumDimensions of projected mode spacesWalsh bounds for independent image lawsand separated phasesRank of a tensor killed in two quotientspacesRank loss under two restrictionsRaw matrix frames and their tensorrealizationThe finite uniform raw-vertex lawOriginal retained-cell laws from finite PMFsFinite relative entropy and support costsRank loss under restriction of a bilinear formImage caps inside original retained cellsOne exposure controls both modesSmall phases on selected endpoint channelgroupsMonotonicity of span deficitsMode space span deficitsNonzero tensor count from span deficitsRank of an actual linear map sumCharacters of all independent tensor channelsCounting component tensors with boundedtotal rankUniform raw channel phase tails overbounded-rank targetsAdmissible pair lawsOrthogonality and finite Walsh correlationbounds
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    Each proof establishes this claim relative to its assumptions.

    Lean source view on GitHub

    1import Lax342547.FixedMultiPhase
    2import Lax342547.GroupPhaseScale
    3import Lax342547.PhaseScales
    4import Lax342547.PhaseScales
    5
    6/-!
    7---
    8title: Small phases on selected endpoint channel groups
    9type: lemma
    10---
    11The explicit numerical phase margin holds for arbitrary selected endpoint images and whole-unit marks while preserving the original no-cover rank parameter.
    12-/
    13
    14namespace Lax342547.SmallMultiPhase
    15
    16open Lax342547.MomentSpace Lax342547.RawFrames Lax342547.ChannelDensity Lax342547.ChannelCharacters
    17open Lax342547.RealCellLaws Lax342547.FiniteSampling Lax342547.TensorCharacters
    18open Lax342547.PushforwardWalsh Lax342547.RetainedImages Lax342547.ChannelColumns
    19open Lax342547.RelativeEntropy Lax342547.ComponentSpaces Lax342547.ComponentDuals Lax342547.CoverProjection
    20open scoped BigOperators
    21
    22axiom no_cover_marked_unit_phase_small {e d B N Ω U : Type} [Fintype e] [Fintype d] [Nonempty d] [Fintype U] [Fintype B] [Fintype N] [Fintype Ω]
    23 [DecidableEq e] [DecidableEq d] [DecidableEq N] (E : Matrix B B Binary)
    24 (μ : Ω → ℝ) (A : Ω → e → Matrix N N Binary) (ψ : (e → Matrix N N Binary) → Ω → ℝ) (r h a c P m : ℕ)
    25 [Nonempty (Frame B (Fin h) N E)] (hN : 2*h+1 ≤ Fintype.card N)
    26 (hμ : Probability μ) (hr : 1 ≤ r)
    27 (hA : ∀ x, ∑ i, (A x i).rank ≤ r) (hψ : ∀ S, (∑ i, (S i).rank) ≤ r → ∀ x, |ψ S x| ≤ 1)
    28 (hc : 0 < c) (hscale : c*2^(100*r+12) ≤ Fintype.card N)
    29 (hP : 2^(100*r+10)*(2+a+(2*r+10)*(2*c)+1) ≤ P)
    30 (hh : 4*2^(100*r+10)*(a+(2*r+10)*(2*c)+1) ≤ h)
    31 (β : U → ℝ) (image : U → (e × d) → Frame B (Fin h) N E) (L : ℝ)
    32 (S : U → e → Matrix N N Binary)
    33 (hβ : Probability β) (hL : 0 ≤ L)
    34 (hLcap : L ≤ (2 : ℝ)^(m+Fintype.card N))
    35 (hbig : m+r*Fintype.card e+Fintype.card (e × d)*(h*h+2) ≤ Fintype.card N)
    36 (ha : 14 ≤ a) (hpos : 1 ≤ Fintype.card N)
    37 (hβcap : ∀ o, push β image o ≤ L*weights (Lax342547.RawLaw.uniformLaw E) o)
    38 (hS : ∀ o, (∑ i, (S o i).rank) ≤ r)
    39 (hcover : ∀ S : Submodule Binary (e → N → Binary),
    40 ∀ T : Submodule Binary (Module.Dual Binary (e → N → Binary)),
    41 Componentwise S → DualComponentwise T →
    42 (Module.finrank Binary S : ℝ) ≤ (r : ℝ)*(Fintype.card N/c) →
    43 (Module.finrank Binary T : ℝ) ≤ (r : ℝ)*(Fintype.card N/c) →
    44 cellMass μ (fun x => projection (LinearMap.piMap (fun i => (A x i).mulVecLin)) S T = 0) ≤ 1/(2 : ℝ)^P) :
    45 |∑ o, β o*(∑ x, μ x*ψ (S o) x*tensorCharacter h (fun i : e × d => A x i.1) (columnsEquiv h (fun a => channels (image o a))))| < 1/200
    46
    47end Lax342547.SmallMultiPhase
    48
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