theorem
MathlibNt.SieveTheory.SwitchingPrinciple.SuzukiLemma144KappaOne.sigmaEleven_add_sigmaTwelve_suzukiVProduct_le_finiteSourceLayer_add_qD
{S : BoundingSieve}
{H : Section13HatLayers}
{β C K d Δ s τ σ : ℝ}
{N D znat : ℕ}
(hH : Section13HatContract H β)
(hsdom : s ∈ SuzukiFiniteContinuousLayers.KappaOneModel.parityDomain β N)
(hτdom : τ - 1 ∈ SuzukiFiniteContinuousLayers.KappaOneModel.parityDomain β (N - 1))
(hsτ : s ≤ τ)
(hτσ : τ ≤ σ)
(hD : 1 < ↑D)
(hz2 : 2 ≤ ↑znat)
(hv2 : 2 ≤ ↑D ^ (1 / τ))
(hw2 : 2 ≤ ↑D ^ (1 / σ))
(hwv : ↑D ^ (1 / σ) ≤ ↑D ^ (1 / τ))
(hvz : ↑D ^ (1 / τ) ≤ ↑znat)
(hz : ↑znat = ↑D ^ (1 / s))
(hτ : H.betaHat + (ErrorSign.ofDepth N).epsilon < τ)
(hK : 2 ≤ K)
(hlocal : HasDimensionOneLocalProductBound S K)
(hii : Claim14_6_MonotoneQPremise H (↑D) d Δ σ)
(hC : 0 ≤ C)
(hΔ : 0 ≤ Δ)
(hCeil : ∀ p ∈ S.prodPrimes.primeFactors, ↑D ^ (1 / σ) ≤ ↑p → ↑p < ↑D ^ (1 / τ) → 2 ≤ p ∧ 2 * p ≤ D)
(hT :
∀ p ∈ S.prodPrimes.primeFactors,
↑D ^ (1 / σ) ≤ ↑p →
↑p < ↑D ^ (1 / τ) → 0 ≤ H.T (ErrorSign.ofDepth (N - 1)) (SuzukiLemma144Equation1410.inheritedCoordinate D p))
(h1413 :
∀ p ∈ S.prodPrimes.primeFactors,
↑D ^ (1 / σ) ≤ ↑p →
↑p < ↑D ^ (1 / τ) → Claim14_13PointwisePremise H N (↑D) d Δ (Real.log ↑D / Real.log ↑p) (↑D / ↑p))
:
SuzukiLemma144Equation1410.sigmaEleven (suzukiSupportedBelow S znat) (⇑S.nu) (fun (p : ℕ) => suzukiVProduct S ↑p)
(suzukiVProduct S ↑znat) β N D σ τ + SuzukiLemma144Equation1410.sigmaTwelve S.prodPrimes.primeFactors (⇑S.nu) (fun (p : ℕ) => suzukiVProduct S ↑p)
(fun (n D' : ℕ) (x : ℝ) => errorEnvelope H n (↑D') d x) (suzukiVProduct S ↑znat) C K Δ N D σ τ ≤ suzukiVProduct S ↑znat * (SuzukiFiniteContinuousLayers.finiteSourceLayer 1 β N s + 6 * K ^ 2 * SuzukiFiniteContinuousLayers.finiteSourceLayer 1 β (N - 1) (τ - 1) / Real.log (↑D ^ (1 / σ)) * (τ / s)) + C * Real.exp √K * suzukiVProduct S ↑znat * Real.log ↑D ^ (-Δ) * ((1 / s * ∫ (t : ℝ) in τ..σ, qD H (ErrorSign.ofDepth N).opposite (↑D) d Δ t) + 6 * K ^ 2 * qD H (ErrorSign.ofDepth N).opposite (↑D) d Δ τ / Real.log (↑D ^ (1 / σ)) * (τ / s))
Concrete middle-range provider for the literal Σ₁₁ + Σ₁₂ terms from
(14.10). Both terms use Suzuki's Euler product, and the error term is the
Section-13 envelope; there is no abstract middle-range or reverse-error
premise.