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Sparse profiles from raw intersections (Lemma 4.4)

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

proven

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    Natural Language Statement

    Theorem

    The explicit exceptional-set estimate and the deterministic shared-tensor argument combine on the actual concrete moment profiles. The normalized representatives are unique and have the paper's support and chiStar bounds.

    Concept map
    26 concepts
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    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 exists_concrete_scale proven

    2 paper_sparse_parameters proven

    3 raw_intersections proven

    Lean source view on GitHub

    1import Lax342547.RawIntersections
    2import Lax342547.SparseRepresentatives
    3
    4/-!
    5---
    6title: Sparse profiles from raw intersections (Lemma 4.4)
    7type: theorem
    8---
    9The explicit exceptional-set estimate and the deterministic shared-tensor
    10argument combine on the actual concrete moment profiles. The normalized
    11representatives are unique and have the paper's support and chiStar bounds.
    12-/
    13
    14namespace Lax342547.SparseIntersections
    15
    16open Lax342547.MomentSpace Lax342547.TagGeometry Lax342547.ConcreteGeometry
    17open Lax342547.CutProfiles Lax342547.CutSparsity Lax342547.ConcreteCut
    18open Lax342547.RawFrames Lax342547.RawIntersections Lax342547.SparseRepresentatives
    19open scoped ENNReal
    20
    21axiom paper_sparse_parameters :
    22 let g := 10 ^ 9 + 1
    23 let D := 4000 * g
    24 1 ≤ D ∧ 8 * D < g * g ∧ 4 * D / g = 16000
    25
    26axiom exists_concrete_scale (k b degree h : ℕ) :
    27 ∃ M₀ : ℕ, ∀ n : ℕ, 1 ≤ n →
    28 2 * (Fintype.card (Coordinate k n b degree) + h) + 1 ≤ M₀ * n ∧
    29 (4 : ℝ) * (Fintype.card (Coordinate k n b degree) + h : ℕ) ≤ (1 / 4 : ℝ) * (M₀ * n : ℕ) ∧
    30 (8 : ℝ) * Fintype.card (Component (Tag k)) ≤ (1 / 4 : ℝ) * (M₀ * n : ℕ) ∧
    31 ∀ D : ℕ, (Fintype.card (Component (Tag k)) * (2 * (Fintype.card (Coordinate k n b degree) + h)) * (2 * D) : ℕ) ≤
    32 (D : ℝ) * (M₀ * n : ℕ) / 4
    33
    34axiom raw_intersections {k n b degree r h N : ℕ}
    35 (hr : 2 * r ≤ n) (hd : 1 ≤ degree) (M : Fin b → Matrix (Fin n) (Fin n) Binary)
    36 (D : ℕ) (hD : 1 ≤ D) (hsmall : 8 * D < (2 * k + 1) * (2 * k + 1))
    37 (hN : 2 * (Fintype.card (Coordinate k n b degree) + h) + 1 ≤ N)
    38 (hp : (4 : ℝ) * (Fintype.card (Coordinate k n b degree) + h : ℕ) ≤ (1 / 4 : ℝ) * N)
    39 (hc : (8 : ℝ) * Fintype.card (Component (Tag k)) ≤ (1 / 4 : ℝ) * N)
    40 (hcount : (Fintype.card (Component (Tag k)) * (2 * (Fintype.card (Coordinate k n b degree) + h)) * (2 * D) : ℕ) ≤
    41 (D : ℝ) * N / 4)
    42 (σ : PMF (Fin 2 → Component (Tag k) →
    43 Frame (Coordinate k n b degree) (Fin h) (Fin N) (selfGram hr hd M)))
    44 (hσ : ∀ o, σ o ≤ (2 : ℝ≥0∞) ^ (D * N) / Fintype.card (Fin 2 → Component (Tag k) →
    45 Frame (Coordinate k n b degree) (Fin h) (Fin N) (selfGram hr hd M))) :
    46 σ.toOuterMeasure {o | 2 * D < intersectionDimension o} ≤ (2 : ℝ≥0∞) ^ (-((D : ℝ) * N / 4)) ∧
    47 ∀ o : Fin 2 → Component (Tag k) →
    48 Frame (Coordinate k n b degree) (Fin h) (Fin N) (selfGram hr hd M),
    49 intersectionDimension o ≤ 2 * D →
    50 ∀ x y : Profile k n b degree, profileMap (o 0) x.val = profileMap (o 1) y.val →
    51 (∑ e, (x.val e).rank) ≤ 2 * D ∧ (∑ e, (y.val e).rank) ≤ 2 * D ∧
    52 ∃ w v : Representation k n b degree, Normalized x w ∧ Normalized y v ∧
    53 supportSize w.val ≤ 4 * D / (2 * k + 1) ∧ supportSize v.val ≤ 4 * D / (2 * k + 1) ∧
    54 chiStar hr w = chiStar hr v ∧
    55 (∀ u : Representation k n b degree, Normalized x u → u = w) ∧
    56 (∀ u : Representation k n b degree, Normalized y u → u = v)
    57
    58end Lax342547.SparseIntersections
    59
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