Concrete slow-weight removal from the actual extracted dispersion sum

The degree-five variation loss is derived from actual floor-retained tuples. Empty blocks are treated separately. All arithmetic coefficients, phases, support conditions and the coupled cutoff remain inside genuine prefixes.

noncomputable def MathlibNt.AnalyticNumberTheory.LargeSieve.LiLiuPrereqFouvry.wAnalyticVariationConstant (Z : ℝ) : ℝ

Equations

• MathlibNt.AnalyticNumberTheory.LargeSieve.LiLiuPrereqFouvry.wAnalyticVariationConstant Z = 2048 * (6 + 2 * Real.pi) ^ 5 * (1 + 112 * Z) ^ 5

theorem MathlibNt.AnalyticNumberTheory.LargeSieve.LiLiuPrereqFouvry.wAnalyticVariationConstant_nonneg {Z : ℝ} (hZ : 0 ≤ Z) : 0 ≤ wAnalyticVariationConstant Z

theorem MathlibNt.AnalyticNumberTheory.LargeSieve.LiLiuPrereqFouvry.wAnalyticGridWeight_variation_bound {M T x Z y : ℝ} (hM : 0 < M) (hT : 0 < T) (hZ : 0 ≤ Z) (hx : x = 4 * M * T) (hy : y ∈ Set.Icc (1 / 2) 3) {N Q : Finset ℕ} (hN : ∀ n ∈ N, T ≤ ↑n) (hQ : ∀ q ∈ Q, 0 < q) {a : ℤ} (ha : |↑a| ≤ x) {P : WOriginalTuple → Prop} {R S ξ : ℝ} {b : ℕ} {K : WExtractedKey} {j : Fin 5 → ℕ} {positive : Bool} {t : WExtractedTuple × ℤ} (ht : t ∈ wAnalyticDyadicBlock (wExtractedKeyFiber (wFloorCutoff M Z) N Q a P R S ξ b K) j positive) : LiLiuPrereqFouvry.Rectangle.variation (wAnalyticBoxLo j) (wAnalyticBoxHi j) (wAnalyticGridWeight K j positive a (M * y)) ≤ wAnalyticVariationConstant Z

theorem MathlibNt.AnalyticNumberTheory.LargeSieve.LiLiuPrereqFouvry.wAnalyticDyadicWeightedBlock_norm_le_prefix {M T x Z y : ℝ} (hM : 0 < M) (hT : 0 < T) (hZ : 0 ≤ Z) (hx : x = 4 * M * T) (hy : y ∈ Set.Icc (1 / 2) 3) {N Q : Finset ℕ} (hN : ∀ n ∈ N, T ≤ ↑n) (hQ : ∀ q ∈ Q, 0 < q) {a : ℤ} (ha : |↑a| ≤ x) (β c₁ γ ζ : ℕ → ℝ) (P : WOriginalTuple → Prop) (R S ξ : ℝ) (b : ℕ) (K : WExtractedKey) (j : Fin 5 → ℕ) (positive : Bool) : have U := wExtractedKeyFiber (wFloorCutoff M Z) N Q a P R S ξ b K; ‖wAnalyticDyadicWeightedBlock U K j positive β c₁ γ ζ a (M * y)‖ ≤ wBlockAmplitude K j * wAnalyticVariationConstant Z * wAnalyticBlockPrefixMax (wAnalyticDyadicBlock U j positive) j β c₁ γ ζ a

noncomputable def MathlibNt.AnalyticNumberTheory.LargeSieve.LiLiuPrereqFouvry.wAnalyticKeyPrefixMajorant (U : Finset (WExtractedTuple × ℤ)) (K : WExtractedKey) (β c₁ γ ζ : ℕ → ℝ) (a : ℤ) : ℝ

An explicitly constructed arithmetic quantity: dyadic reciprocal prefactors times attained, genuinely rectangular arithmetic prefix maxima.

Equations

• MathlibNt.AnalyticNumberTheory.LargeSieve.LiLiuPrereqFouvry.wAnalyticKeyPrefixMajorant U K β c₁ γ ζ a = ∑ j ∈ Finset.image MathlibNt.AnalyticNumberTheory.LargeSieve.LiLiuPrereqFouvry.wAnalyticDyadicKey U, MathlibNt.AnalyticNumberTheory.LargeSieve.LiLiuPrereqFouvry.wBlockAmplitude K j * (MathlibNt.AnalyticNumberTheory.LargeSieve.LiLiuPrereqFouvry.wAnalyticBlockPrefixMax (MathlibNt.AnalyticNumberTheory.LargeSieve.LiLiuPrereqFouvry.wAnalyticDyadicBlock U j true) j β c₁ γ ζ a + MathlibNt.AnalyticNumberTheory.LargeSieve.LiLiuPrereqFouvry.wAnalyticBlockPrefixMax (MathlibNt.AnalyticNumberTheory.LargeSieve.LiLiuPrereqFouvry.wAnalyticDyadicBlock U j false) j β c₁ γ ζ a)

theorem MathlibNt.AnalyticNumberTheory.LargeSieve.LiLiuPrereqFouvry.wAnalyticKeyPrefixMajorant_nonneg (U : Finset (WExtractedTuple × ℤ)) (K : WExtractedKey) (β c₁ γ ζ : ℕ → ℝ) (a : ℤ) : 0 ≤ wAnalyticKeyPrefixMajorant U K β c₁ γ ζ a

theorem MathlibNt.AnalyticNumberTheory.LargeSieve.LiLiuPrereqFouvry.wExtractedFloorKeyExponential_norm_le_prefixes {M T x Z y : ℝ} (hM : 0 < M) (hT : 0 < T) (hZ : 0 ≤ Z) (hx : x = 4 * M * T) (hy : y ∈ Set.Icc (1 / 2) 3) {N Q : Finset ℕ} (hN : ∀ n ∈ N, T ≤ ↑n) (hQ : ∀ q ∈ Q, 0 < q) {a : ℤ} (ha : |↑a| ≤ x) (β c₁ γ ζ : ℕ → ℝ) (P : WOriginalTuple → Prop) (R S ξ : ℝ) (b : ℕ) (K : WExtractedKey) : ‖wExtractedKeyExponential (wFloorCutoff M Z) N Q β c₁ γ ζ a P R S ξ b K (M * y)‖ ≤ wAnalyticVariationConstant Z * wAnalyticKeyPrefixMajorant (wExtractedKeyFiber (wFloorCutoff M Z) N Q a P R S ξ b K) K β c₁ γ ζ a

The full actual fixed-key exponential sum has now lost its slow weight at a proved polynomial cost, with no assumed derivative or variation bound.

theorem MathlibNt.AnalyticNumberTheory.LargeSieve.LiLiuPrereqFouvry.wellFactorable_signedError_sq_le_floor_prefix_c2 {ι : Type u_1} {κ k i j : ℕ} (A : ℕ) {T : ι → ℝ} {N : ι → Finset ℕ} {β : ι → ℕ → ℝ} (hSW : BetaCoprimeSWFamily κ T N β) (hT : ∀ (z : ι), 1 ≤ T z) (hN : ∀ (z : ι), ∀ n ∈ N z, T z ≤ ↑n ∧ ↑n ≤ 2 * T z) (hβ : ∀ (z : ι), ∀ n ∈ N z, |β z n| ≤ ↑((fouvryTau k) n)) {ε η : ℝ} (hε : 0 < ε) (hη : 0 < η) : ∀ᶠ (x : ℝ) in Filter.atTop, ∀ (z : ι) (M ν : ℝ), 1 ≤ M → 4 * M * T z = x → ε ≤ ν → ν ≤ 1 / 10 → T z = x ^ ν → ∀ (U : Finset ℕ), (∀ m ∈ U, M ≤ ↑m ∧ ↑m ≤ 2 * M) → ∀ (α c : ℕ → ℝ), (∀ m ∈ U, |α m| ≤ ↑((fouvryTau i) m)) → SignedWellFactorable j (x ^ ((5 - 5 * ν) / 9 - ε)) c → have L := x ^ ((5 - 5 * ν) / 9 - ε); have R₀ := x ^ c2RExponent ν ε; have S₀ := x ^ c2SExponent ν ε; have J := Nat.log 2 ⌈L ^ 2 / M * x ^ η⌉₊; R₀ * S₀ = L ∧ ∃ (γ : ℕ → ℝ) (ζ : ℕ → ℝ), factorSupported R₀ γ ∧ factorSupported S₀ ζ ∧ (∀ (r : ℕ), |γ r| ≤ ↑((fouvryTau j) r)) ∧ (∀ (s : ℕ), |ζ s| ≤ ↑((fouvryTau j) s)) ∧ c = factorConvolution γ ζ ∧ ∀ (a : ℤ), a ≠ 0 → |↑a| ≤ x → ∃ b ≤ J, ∃ K ∈ wExtractedKeyBox (x ^ η), signedError U (N z) (Finset.Ioc 0 ⌊L⌋₊) α (β z) c a ^ 2 ≤ (3072 * M * ∑ m ∈ U, α m ^ 2) * ↑(J + 1) * (x ^ η) ^ 7 * wAnalyticVariationConstant (x ^ η) * wAnalyticKeyPrefixMajorant (wExtractedKeyFiber (wFloorCutoff M (x ^ η)) (N z) (Finset.Ioc 0 ⌊L⌋₊) a (c2FiveSmallMask x η) R₀ S₀ (highOmegaCutoff x) b K) K (betaClean (β z) a) (factorConvolution γ (betaLowOmega ζ (highOmegaCutoff x))) γ ζ a + x ^ 2 / Real.log x ^ A

Original signed distribution error after concrete five-variable partial summation. The Fourier integration point has disappeared; the remaining maxima contain only the exact arithmetic coefficients and phases.