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MathlibNt.SieveTheory.LinearSieve.Suzuki.SuzukiLemma144CaseISourceLargeLogUniformCutoff

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 : } (hΔ0 : 0 < Δ) ( : 0 < Θ) (hmargin : Δ + 2 / Θ < 1) (hd : 7 / (1 - (Δ + 2 / Θ)) < d) :
∃ (C1min : ), 1 C1min ∀ (C1 K D : ), C1min C12 K2 DC1 * K ^ Θ < Real.log DK ^ 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.