Provenance
The coordinator supplied the candidate argument directly in the Codex review request on 2026-09-09. At review time it had no endpoint commit or attached file. The reviewer checked that supplied text against the cited primary sources and project dependencies. The exact mathematical version reviewed was then preserved, without a new mathematical step, in:
f659c8d693fd6dea4cc640b420c7ed862bcaf7a6
hunts/prime_pair_error/ENDPOINT_BOUND.mdThis record attaches the review to that first durable proof commit. It does not claim that the proof existed at this commit before the review, and it does not attach the review retroactively to checkpoint #215.
Verdict
PASS. The proof commit above supports the following statement. There are fixed constants \(C,c>0\) such that, for every sufficiently large integer \(N\),
\[ \boxed{E_{\rm corr}(N)\le C N^3 \exp\!\left(-c(\log N)^{1/10}\right).} \]
Here \(E_{\rm corr}\) uses the unchanged exceptional correction \(C_N\) from #212-#215. The assertion is uniform over the exceptional data in Tao and Teräväinen's source range, including the case in which there are no such data.
This is a review of a handwritten mathematical argument. It is not formal verification, external peer review, or a novelty determination.
Checked dependencies
- Tao and Teräväinen, arXiv:2107.02158v4, Proposition 2.2, equation (2.5), gives \[ \sum_{n\in B}(\Lambda(n)-\Lambda_{\rm Siegel}(n)) \ll N\exp(-\gamma(\log N)^{1/10}) \] for every arithmetic progression \(B\subset[N]\). Its proof writes such a progression as a difference of progression prefixes and equation (7.1) states the required endpoint form with the model fixed at the ambient \(N\). The endpoint exponent is obtained from this proposition. It is not obtained by setting \(\kappa=1/10\) in Theorem 2.7.
- Montgomery and Vaughan, Multiplicative Number Theory II, Theorem 17.1 and equations (17.28)-(17.32), give the stated minor-arc estimate for the full von Mangoldt exponential sum and the Type I bound used in the proof. The full \(\Lambda\) retains proper prime powers.
SIEGEL_UNIFORMITY.mdat76ed1155d087d9137306d3632f66dba7555aadc3, equations (18)-(19), gives uniformly in \(1\le h\le N\) \[ r_a(h)=C_N(h)+O\!\left(N\exp(-c_m(\log N)^{1/10})\right). \] That project dependency keeps the sharp endpoint \(N-h\), the original infinite singular series, and every exceptional conductor case.
CORRECTED_RH_BRIDGE.mdatcb0189db3863ac43b2314c7df1008636bd4da477defines the same \(E_{\rm corr}\) and proves the separate implication \[ E_{\rm corr}(N)\ll_\epsilon N^{2+\epsilon} \text{ for every }\epsilon>0\quad\Longrightarrow\quad\mathrm{RH}. \] The endpoint estimate reviewed here does not meet that near-quadratic hypothesis.
Complete budget and parameters
The proof takes
\[ \ell=\log N,\quad t=\ell^{1/10},\quad Z=e^t, \]
\[ m=2\lceil\sqrt\ell\rceil,quad D_0=Z^m,quad 0<\sigma\le\min(\gamma/20,1/20),quad R=\lfloor e^{\sigma t}\rfloor. \]
Before absorption, the complete bound is
\[ \begin{aligned} E_{\rm corr}(N)\ll \ell^{O(1)}\bigl[& N^3R^7e^{-2\gamma t}+N^3R^{-1/6} +N^{14/5}+N^2D_0\sqrt Z\bigr]\\ &+N^3e^{-c_d\sqrt\ell}+N^3e^{-2c_mt}. \end{aligned} \]
Every displayed term is retained. The first two terms become respectively \(N^3e^{-(2\gamma-7\sigma)t}\) and \(N^3e^{-\sigma t/6}\). Also
\[ \log(D_0\sqrt Z)=2\ell^{3/5}+O(\ell^{1/10})=o(\ell), \]
so the remaining polynomial and \(N^{2+o(1)}\) terms are absorbed after decreasing \(c\). For example, after increasing the lower threshold for \(N\), one can take a positive \(c\) no larger than \(\min(\sigma/12,c_m)\).
The review checked the Bonferroni divisor error, the length \(D_0\), all major-arc powers of \(R\), both exceptional-conductor ranges on the minor arcs, rational shifts, non-coprime divisors, sharp prefixes, Abel summation, centering, Parseval normalization, and the final absorption. No \(N^{3-o(1)}\) remainder was removed from the budget.
Original error and remaining gap
This endpoint statement is for \(E_{\rm corr}\). For the original CHHL error, the same proof and the inherited exceptional-energy estimate give
\[ \boxed{\begin{aligned} E(N)\ll{}&N^3\exp\!\left(-c(\log N)^{1/10}\right)\\ &+1_{q\ {\rm odd}}N^{2\beta+1}\frac{q^2}{\phi(q)^4} +N^{4\beta-1}\frac{q^2}{\phi(q)^3}. \end{aligned}} \]
Both exceptional contributions remain when the source-range exceptional data exist. They disappear only in the no-exception case.
The improvement over the previous result is precise: it replaces the family
\[ E_{\rm corr}(N)\ll_\kappa N^3\exp(-c_\kappa(\log N)^\kappa),\qquad\kappa<1/10, \]
by the endpoint exponent \(\kappa=1/10\), with fixed constants. It does not produce any fixed \(\eta>0\) in \(E_{\rm corr}\ll N^{3-\eta}\). The remaining mathematical gap is therefore a fixed power saving, ultimately the near-quadratic family required by the #214 RH-sufficiency bridge. That is a separate future research task.