theorem MathlibNt.SieveTheory.exists_caseII_endpoint_common_threshold_uniform {H : SwitchingPrinciple.SuzukiLemma144KappaOne.Section13HatLayers} (hH : SwitchingPrinciple.SuzukiLemma144KappaOne.Section13HatSourceContract H) (K C d Δ : ℝ) (hK : 0 ≤ K) (_hC : 0 ≤ C) (hΔ0 : 0 < Δ) (hΔ1 : Δ < 1) (hd : 7 / (1 - Δ) < d) : ∃ (D0 : ℝ), 1 < D0 ∧ ∀ (N : ℕ) (D : ℝ), Odd N → 3 ≤ N → D0 ≤ D → caseIISharpPositiveEndpointRelativeCoeff N D d Δ (SwitchingPrinciple.SuzukiLemma144KappaOne.sourceSigma D d) C K ≤ 4 * ((1 - Δ) / (32 * SwitchingPrinciple.SuzukiLemma144KappaOne.sourceSigma D d))

The four positive Case-II endpoint terms admit one large-D threshold uniformly for every odd depth N ≥ 3.

theorem MathlibNt.SieveTheory.exists_caseIIConcreteRoundedRelativeBracket_gap_threshold_uniform {H : SwitchingPrinciple.SuzukiLemma144KappaOne.Section13HatLayers} (hH : SwitchingPrinciple.SuzukiLemma144KappaOne.Section13HatSourceContract H) (K C d Δ : ℝ) (hK : 0 ≤ K) (hC : 0 ≤ C) (hΔ0 : 0 < Δ) (hΔ1 : Δ < 1) (hd : 7 / (1 - Δ) < d) : ∃ (D0 : ℝ), 1 < D0 ∧ ∀ (N : ℕ) (D : ℝ), Odd N → 3 ≤ N → D0 ≤ D → 27 * (1 - Δ) / (32 * SwitchingPrinciple.SuzukiLemma144KappaOne.sourceSigma D d) ≤ 1 - caseIIConcreteRoundedRelativeBracket N D d Δ (SwitchingPrinciple.SuzukiLemma144KappaOne.sourceSigma D d) C K

One cutoff, independent of the odd depth, preserves the quantitative 27/32 Case-II rounded-bracket gap.