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Phase density budgets for selected channel groups

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

proven

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    Natural Language Statement

    Lemma

    Original target counting and selected endpoint-channel density costs are paid separately in the explicit phase margin.

    Concept map
    62 concepts; 1 descendant hidden
    100%
    Actual projected span deficitSampling after exposed coordinatesAdaptive sampling on disjoint coordinatesetsUnion over adaptive component coversExact uniform bilinear character meanWalsh operator bounds with explicit bilinearrankRank of a lifted tensor sumIndependent binary channel charactersActual 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 familyDyadic span deficit estimatesA 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 even moment expansionFinite independent sampling and vertexexception tailsGreedy 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 statisticsDimension deficits after an arbitrary modemapExposed mode dimension budgetExplicit no-cover moment parameter marginsBoolean point moments with restricted basecoordinatesFinite moment probability and rank splitA low rank sum supplies an actual adaptivecoverNo-cover phase momentsRank growth without an adaptive modecoverSquared restriction cost for independent uniteventsFinite exposure partitionsProjected 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 restrictionsOriginal 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 modesMonotonicity of span deficitsMode space span deficitsNonzero tensor count from span deficitsRank of an actual linear map sumCharacters of all independent tensor channelsAdmissible pair lawsOrthogonality and finite Walsh correlationbounds
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

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    1import Lax342547.MomentBudgets
    2
    3/-!
    4---
    5title: Phase density budgets for selected channel groups
    6type: lemma
    7---
    8Original target counting and selected endpoint-channel density costs are paid separately in the explicit phase margin.
    9-/
    10
    11namespace Lax342547.GroupPhaseScale
    12
    13
    14
    15axiom group_phase_density_budget (r h m N Γ E G : ℕ) (L : ℝ) (hL : 0 ≤ L)
    16 (hcap : L ≤ (2 : ℝ)^(m+N)) (hΓ : 2*r+10 ≤ Γ)
    17 (hN : m+r*E+G*(h*h+2) ≤ N) :
    18 L*((r+1 : ℝ)^E*(2 : ℝ)^(2*N*r)*(2 : ℝ)^(G*(h*h+2))/(2 : ℝ)^(Γ*N)) ≤
    19 1/(2 : ℝ)^(8*N)
    20
    21end Lax342547.GroupPhaseScale
    22
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