Quotient-Weighted Type II Reductions and Product-Modulus Targets
Year: 2026
Type: preprint
Abstract: We isolate the analytic Type II obstructions that arise in quotient-weighted distribution estimates for the companion Chen--Wu--Richert criterion. The refined formulation separates three tasks: a sub-$1/3$ quotient $P_2$ range, which is proved here from Type I estimates and completion; the remaining balanced shifted semiprime-prime problem, i.e. primes of the form $qr+2$ with $q,r$ prime and $y^{1/3} 2/7$ is treated by a direct triple-prime convolution Bombieri--Vinogradov theorem plus an ordinary upper linear sieve, giving constant $4$. We also include a certified special-factor refinement of the fixed $P_4$ residual; the companion coefficient certificate encloses the refined cost in $[0.13631427,0.13710612]$ and certifies a gain $>0.0165$ over the old $P_4$ residual. These two theorem-level refinements together certify remaining tolerance above $1.086$ for the balanced shifted semiprime-prime sector. This version then reformulates that sector in the more intrinsic language of binary shifted convolution: after a high-order Heath--Brown decomposition of $\Lambda(qr+2)$, Type I and non-top branches are reduced to standard convolution-Bombieri--Vinogradov and smooth binary additive-divisor inputs, while the primitive top branch is isolated as a shifted convolution with the special HB-small coefficient generated by $L_z/U_z$, equivalently as $CD=qr+2$ with two HB-small well-factorable weights on the shifted side. For possible full balanced-$P_3$ control, the paper further isolates the simultaneous top all-small branch and the quotient-weighted friable progression estimate that would close it; the quotient coefficient is the only part not matched to Pascadi's stated $5/8$ friable distribution theorem. The paper is a reduction and target-problem document: standard distribution theorems are quoted only where their coefficient classes, local singular factors, and quotient weights match the stated hypotheses.
Author: Ratliff, Marshall
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7161740837
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