noncomputable def MathlibNt.SieveTheory.caseIIRoundedTransportErr (S : BoundingSieve) (H : SwitchingPrinciple.SuzukiLemma144KappaOne.Section13HatLayers) (N : ℕ) (D yr zr d Δ σ C K B0 s : ℝ) : ℝ

The complete endpoint remainder after transporting the cubic endpoint from yr = D^(1/3) to zr = D^(1/s). In contrast with the legacy natural-cutoff wrapper, both analytic cutoff coordinates are the exact real roots.

Equations

• MathlibNt.SieveTheory.caseIIRoundedTransportErr S H N D yr zr d Δ σ C K B0 s = B0 + MathlibNt.SieveTheory.SwitchingPrinciple.suzukiVProduct S zr * (3 / s * (K / Real.log yr) * MathlibNt.SieveTheory.SuzukiFiniteContinuousLayers.finiteSourceLayer 1 2 N 3 + 3 / s * (1 + K / Real.log yr) * MathlibNt.SieveTheory.caseIIEndpointSigma11 K N D σ + 3 / s * (1 + K / Real.log yr) * MathlibNt.SieveTheory.caseIIEndpointQD H N D d Δ σ C K)

theorem MathlibNt.SieveTheory.caseII_total_le_concrete_finiteSourceLayer_add_rawBase_natCeil (S : BoundingSieve) {K s endpointErr : ℝ} {N D y z : ℕ} (hN : Odd N) (hD : Real.exp 1 ≤ ↑D) (hs : 0 < s) (hs3 : s ≤ 3) (hK : 0 ≤ K) (hyz : y ≤ z) (hyLower : (y - 1) ^ 3 < D) (hyUpper : D ≤ y ^ 3) (hzceil : z = ⌈↑D ^ (1 / s)⌉₊) (hzr2 : 2 ≤ ↑D ^ (1 / s)) (hlocal : SwitchingPrinciple.HasDimensionOneLocalProductBound S K) (hendpoint : ∑ n ∈ sourceParityIndices N, suzukiSourceV S n D y ≤ SwitchingPrinciple.suzukiVProduct S (↑D ^ (1 / s)) * (3 / s * SuzukiFiniteContinuousLayers.finiteSourceLayer 1 2 N 3) + endpointErr) : ∑ n ∈ sourceParityIndices N, suzukiSourceV S n D z ≤ SwitchingPrinciple.suzukiVProduct S ↑z * SuzukiFiniteContinuousLayers.finiteSourceLayer 1 2 N s + endpointErr + SwitchingPrinciple.suzukiVProduct S ↑z * (9 * K / (s * Real.log ↑D))

Sharp source-native Case-II assembly when the target cutoff is the natural ceiling of the exact real power coordinate.

theorem MathlibNt.SieveTheory.caseII_total_le_from_caseI_endpoint_explicit_rawBase_natCeil (S : BoundingSieve) (H : SwitchingPrinciple.SuzukiLemma144KappaOne.Section13HatLayers) {N D y z : ℕ} {σ C C1 K ΘK Δ d B0 s : ℝ} (hH : SwitchingPrinciple.SuzukiLemma144KappaOne.Section13HatContract H 2) (hN : Odd N) (hN2 : 2 ≤ N) (hycube : ∀ p ∈ SwitchingPrinciple.suzukiSupportedBelow S y, p ^ 3 < D) (hyceil : y = ⌈↑D ^ (1 / 3)⌉₊) (hyDhalf : ↑y ≤ ↑D / 2) (h3σ : 3 ≤ σ) (hD : Real.exp 1 ≤ ↑D) (hDlarge : 2 * Real.log 2 ≤ Real.log ↑D) (hwy : ↑D ^ (1 / σ) ≤ ↑D ^ (1 / 3)) (hy2 : 2 ≤ ↑y) (hyr2 : 2 ≤ ↑D ^ (1 / 3)) (hw2 : 2 ≤ ↑D ^ (1 / σ)) (hEndpoint : SwitchingPrinciple.SuzukiLemma144KappaOne.Claim14_5Regime 2 (↑D) σ C1 K ΘK σ → ∑ p ∈ SwitchingPrinciple.suzukiSupportedBelow S ⌈↑D ^ (1 / σ)⌉₊, S.nu p * ∑ m ∈ sourceParityIndices (N - 1), suzukiSourceV S m (D ⌈/⌉ p) p ≤ B0) (hnu : ∀ p ∈ SwitchingPrinciple.SuzukiLemma144Equation1410.sigmaOneCarrier (SwitchingPrinciple.suzukiSupportedBelow S y) D σ 3, 0 ≤ S.nu p) (hC : 0 ≤ C) (hlog : ∀ p ∈ SwitchingPrinciple.SuzukiLemma144Equation1410.sigmaOneCarrier (SwitchingPrinciple.suzukiSupportedBelow S y) D σ 3, 0 ≤ Real.log ↑(D ⌈/⌉ p)) (hSourceDomain : ∀ p ∈ SwitchingPrinciple.SuzukiLemma144Equation1410.sigmaOneCarrier (SwitchingPrinciple.suzukiSupportedBelow S y) D σ 3, SwitchingPrinciple.SuzukiLemma144Equation1410.inheritedCoordinate D p ∈ SuzukiFiniteContinuousLayers.KappaOneModel.parityDomain 2 (N - 1) ∧ SwitchingPrinciple.SuzukiLemma144Equation1410.recursiveCoordinate D p ∈ SuzukiFiniteContinuousLayers.KappaOneModel.parityDomain 2 (N - 1)) (hClaim14_6_i : ∀ p ∈ SwitchingPrinciple.SuzukiLemma144Equation1410.sigmaOneCarrier (SwitchingPrinciple.suzukiSupportedBelow S y) D σ 3, SwitchingPrinciple.SuzukiLemma144KappaOne.Claim14_6_MonotoneLambdaPremise H (↑(D ⌈/⌉ p)) d σ) (hErrorDomain : ∀ p ∈ SwitchingPrinciple.SuzukiLemma144Equation1410.sigmaOneCarrier (SwitchingPrinciple.suzukiSupportedBelow S y) D σ 3, SwitchingPrinciple.SuzukiLemma144Equation1410.inheritedCoordinate D p ∈ Set.Icc (H.betaHat + (SwitchingPrinciple.SuzukiLemma144KappaOne.ErrorSign.ofDepth (N - 1)).epsilon) σ ∧ SwitchingPrinciple.SuzukiLemma144Equation1410.recursiveCoordinate D p ∈ Set.Icc (H.betaHat + (SwitchingPrinciple.SuzukiLemma144KappaOne.ErrorSign.ofDepth (N - 1)).epsilon) σ) (hIH : SwitchingPrinciple.SuzukiLemma144Equation1410.NaturalCeilPointwiseInductionContract (SwitchingPrinciple.suzukiSupportedBelow S y) (fun (n D' p : ℕ) => ∑ m ∈ sourceParityIndices n, suzukiSourceV S m D' p) (fun (p : ℕ) => SwitchingPrinciple.suzukiVProduct S ↑p) (fun (n D' : ℕ) (x : ℝ) => SwitchingPrinciple.SuzukiLemma144KappaOne.errorEnvelope H n (↑D') d x) 2 C K Δ N D σ 3) (hErrorThreshold : H.betaHat + (SwitchingPrinciple.SuzukiLemma144KappaOne.ErrorSign.ofDepth N).epsilon < 3) (hK : 2 ≤ K) (hlocal : SwitchingPrinciple.HasDimensionOneLocalProductBound S K) (hClaim14_6_ii : SwitchingPrinciple.SuzukiLemma144KappaOne.Claim14_6_MonotoneQPremise H (↑D) d Δ σ) (hΔ0 : 0 ≤ Δ) (hCeilFull : ∀ p ∈ S.prodPrimes.primeFactors, ↑D ^ (1 / σ) ≤ ↑p → ↑p < ↑D ^ (1 / 3) → 2 ≤ p ∧ 2 * p ≤ D) (hT : ∀ p ∈ S.prodPrimes.primeFactors, ↑D ^ (1 / σ) ≤ ↑p → ↑p < ↑D ^ (1 / 3) → 0 ≤ H.T (SwitchingPrinciple.SuzukiLemma144KappaOne.ErrorSign.ofDepth (N - 1)) (SwitchingPrinciple.SuzukiLemma144Equation1410.inheritedCoordinate D p)) (hClaim14_13 : ∀ p ∈ S.prodPrimes.primeFactors, ↑D ^ (1 / σ) ≤ ↑p → ↑p < ↑D ^ (1 / 3) → SwitchingPrinciple.SuzukiLemma144KappaOne.Claim14_13PointwisePremise H N (↑D) d Δ (Real.log ↑D / Real.log ↑p) (↑D / ↑p)) (hs : 0 < s) (hs3 : s ≤ 3) (hyrzr : ↑D ^ (1 / 3) ≤ ↑D ^ (1 / s)) (hyz : y ≤ z) (hyLower : (y - 1) ^ 3 < D) (hyUpper : D ≤ y ^ 3) (hzceil : z = ⌈↑D ^ (1 / s)⌉₊) (hzr2 : 2 ≤ ↑D ^ (1 / s)) : ∑ n ∈ sourceParityIndices N, suzukiSourceV S n D z ≤ SwitchingPrinciple.suzukiVProduct S ↑z * SuzukiFiniteContinuousLayers.finiteSourceLayer 1 2 N s + caseIIRoundedTransportErr S H N (↑D) (↑D ^ (1 / 3)) (↑D ^ (1 / s)) d Δ σ C K B0 s + SwitchingPrinciple.suzukiVProduct S ↑z * (9 * K / (s * Real.log ↑D))

Double-rounded sharp Case-II endpoint transport.

The natural cutoffs are y = ceil(D^(1/3)) and z = ceil(D^(1/s)). Dimension-one transport and the logarithmic ratio are carried out only at the exact real roots. The two Euler products are then returned exactly to their natural-ceiling cutoffs; no equality between a cast natural cutoff and a real root is assumed.