Arithmetic Realizability Bridge: From the Unconditional Prime Side at σ=1 to the Arithmetic Requirements of P70-P99

The previous round established how much support $\sigma$ each proportion target needs; this round asks what the arithmetic side actually has to supply for that to be legitimately realized. From the explicit off-diagonal $O_1$ formula in Claude's Proposition 5.6, one derives that the near-diagonal region not adequately suppressed roughly satisfies $T|\log(n/m)|\lesssim1$, which translates to a prime-scale shift scale of $H_\sigma\asymp X^{1-1/\sigma}$ — this step is a scale inference this research derived from the formula, not a formula Claude's original text states directly. Substituting in the $\sigma$ values reconstructed in the previous round: $\sigma_{70}=1.042628\to X^{0.040885}$, $\sigma_{80}=1.257848\to X^{0.204991}$, $\sigma_{90}=1.701455\to X^{0.412268}$, $\sigma_{95}=2.260790\to X^{0.557677}$, $\sigma_{99}=4.187215\to X^{0.761178}$. The document defines a three-tier set of Arithmetic Bridge Hypotheses — ABH-1 (the direct trace hypothesis closest to Claude's proof pipeline), ABH-2 (a partial Strong Pair Correlation stated in zero/pair-statistics language), and ABH-3 (prime-side sufficient conditions, such as strong Hardy–Littlewood or short-interval variance) — and reports one key finding: the $\sigma$ values for $P_{70},P_{80},P_{90}$ all still fall within the $\alpha<2$ range that the classical Montgomery strong Hardy–Littlewood framework, as compiled by Goldston, can supply, but $\sigma_{95}\approx2.26>2$ and $\sigma_{99}\approx4.19>2$ already fall outside this standard region — a more substantive arithmetic regime change than simply going from “$90\%\to95\%$”; the first CPL node that genuinely requires a new arithmetic regime is $P_{95}$, not the intuitively expected $P_{99}$.