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.