Case I: source-large-log uniform cutoff #
The qualitative (14.23) scalar estimate was previously instantiated at fixed
K, so its eventual threshold could still depend on K. The source has
already chosen C1 before K and enters Case I only after
C1 * K^Theta < log D. We spend the power 2 / Theta to absorb the exact
K^2 coefficient and apply the source-scalar estimate with the strengthened
exponent Delta + 2 / Theta. Thus the cutoff below is chosen before
K, every depth and coordinate, and the later Claim-14.5 constant.
theorem
MathlibNt.SieveTheory.exists_caseI1423_sourceScalar_sourceLargeLog_uniform
{d Δ Θ : ℝ}
(hΔ0 : 0 < Δ)
(hΘ : 0 < Θ)
(hmargin : Δ + 2 / Θ < 1)
(hd : 7 / (1 - (Δ + 2 / Θ)) < d)
:
∃ (C1min : ℝ),
1 ≤ C1min ∧ ∀ (C1 K D : ℝ),
C1min ≤ C1 →
2 ≤ K →
2 ≤ D →
C1 * K ^ Θ < Real.log D →
K ^ 2 * SwitchingPrinciple.SuzukiLemma144KappaOne.sourceSigma D d ^ 3 * Real.log (Real.exp 1 * SwitchingPrinciple.SuzukiLemma144KappaOne.sourceSigma D d) * Real.log (Real.log D) / Real.log D ^ (1 - Δ) ≤ 1
Quantitative replacement for the fixed-K eventuality hidden in the old
endpoint API. The hypotheses are precisely the strict exponent margin needed
when the K^2 in (14.23) is paid by log D > C1*K^Theta.