theorem
MathlibNt.SieveTheory.sigmaEleven_add_sigmaTwelve_suzukiVProduct_le_finiteSourceLayer_add_qD_natCeil_closed
{S : BoundingSieve}
{H : SwitchingPrinciple.SuzukiLemma144KappaOne.Section13HatLayers}
{β C K d Δ s τ σ : ℝ}
{N D z : ℕ}
(hH : SwitchingPrinciple.SuzukiLemma144KappaOne.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 ≤ ↑z)
(hroot2 : 2 ≤ ↑D ^ (1 / s))
(hv2 : 2 ≤ ↑D ^ (1 / τ))
(hw2 : 2 ≤ ↑D ^ (1 / σ))
(hwv : ↑D ^ (1 / σ) ≤ ↑D ^ (1 / τ))
(hvroot : ↑D ^ (1 / τ) ≤ ↑D ^ (1 / s))
(hzceil : z = ⌈↑D ^ (1 / s)⌉₊)
(hτerr : H.betaHat + (SwitchingPrinciple.SuzukiLemma144KappaOne.ErrorSign.ofDepth N).epsilon ≤ τ)
(hK : 2 ≤ K)
(hlocal : SwitchingPrinciple.HasDimensionOneLocalProductBound S K)
(hii : SwitchingPrinciple.SuzukiLemma144KappaOne.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 (SwitchingPrinciple.SuzukiLemma144KappaOne.ErrorSign.ofDepth (N - 1))
(SwitchingPrinciple.SuzukiLemma144Equation1410.inheritedCoordinate D p))
(h1413 :
∀ p ∈ S.prodPrimes.primeFactors,
↑D ^ (1 / σ) ≤ ↑p →
↑p < ↑D ^ (1 / τ) →
SwitchingPrinciple.SuzukiLemma144KappaOne.Claim14_13PointwisePremise H N (↑D) d Δ (Real.log ↑D / Real.log ↑p)
(↑D / ↑p))
:
SwitchingPrinciple.SuzukiLemma144Equation1410.sigmaEleven (SwitchingPrinciple.suzukiSupportedBelow S z) (⇑S.nu)
(fun (p : ℕ) => SwitchingPrinciple.suzukiVProduct S ↑p) (SwitchingPrinciple.suzukiVProduct S ↑z) β N D σ τ + SwitchingPrinciple.SuzukiLemma144Equation1410.sigmaTwelve S.prodPrimes.primeFactors (⇑S.nu)
(fun (p : ℕ) => SwitchingPrinciple.suzukiVProduct S ↑p)
(fun (n D' : ℕ) (x : ℝ) => SwitchingPrinciple.SuzukiLemma144KappaOne.errorEnvelope H n (↑D') d x)
(SwitchingPrinciple.suzukiVProduct S ↑z) C K Δ N D σ τ ≤ SwitchingPrinciple.suzukiVProduct S ↑z * (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 * SwitchingPrinciple.suzukiVProduct S ↑z * Real.log ↑D ^ (-Δ) * ((1 / s * ∫ (t : ℝ) in τ..σ, SwitchingPrinciple.SuzukiLemma144KappaOne.qD H
(SwitchingPrinciple.SuzukiLemma144KappaOne.ErrorSign.ofDepth N).opposite (↑D) d Δ t) + 6 * K ^ 2 * SwitchingPrinciple.SuzukiLemma144KappaOne.qD H
(SwitchingPrinciple.SuzukiLemma144KappaOne.ErrorSign.ofDepth N).opposite (↑D) d Δ τ / Real.log (↑D ^ (1 / σ)) * (τ / s))
Closed-endpoint rounded version of the concrete middle provider. The
natural ceiling transports the strict carrier exactly, while the Σ₁₂ input
allows equality at its lower endpoint.