Test-Specific Weighted Pair-Correlation Hypothesis: $P_{70}$ Does Not Need the Full Hardy–Littlewood Conjecture

The Montgomery strong prime-pair hypothesis, as recorded by Goldston, requires square-root-level error at every shift; the document points out that this is clearly too strong for a proof that only ever needs one designated extremal test $R_\sigma^\star$ — what's actually needed is just the correct asymptotic for the single linear functional $\int_{1<|\alpha|\le\sigma}\widehat R_\sigma^\star(\alpha)F(\alpha,T)\,d\alpha$. The document defines WSPC (Weighted Strong Pair Correlation, which constrains only one linear functional and has far lower information dimension than the full SPC) and, matched to Claude's actual $O_1$ formula, WPPH (Weighted Prime-Pair Hypothesis, which only requires the one specific weighted double sum Claude actually uses to be correct, allowing errors in different $h$ and different $m$ ranges to cancel against each other). The most striking numerical finding: the optimal test for $P_{70}$ ($\sigma_{70}\approx1.042628$) uses only about 0.114% of its total normalized Fourier mass on the unknown strip $|\alpha|>1$ — the difficulty comes from “crossing the boundary” itself, not from “needing a large amount of new band.” But this dependence rises quickly as the proportion increases: $P_{80}\approx2.89\%$, $P_{90}\approx12.36\%$, $P_{95}\approx24.32\%$, $P_{99}\approx51.07\%$, showing that $P_{70}$ and $P_{99}$ are not “the same thing done a few more times.” The document also gives a practical version: taking $\sigma=1.05$ (rather than sitting exactly at $\sigma_{70}$) buys an error tolerance of $|\mathcal J_{1.05}(T)|\le0.0044$, forming the first support–accuracy frontier.