The sharp endpoint \(\kappa=1/2\): \(E_{\rm corr}^{(Z)}(N)\ll N^3\exp(-c\sqrt{\log N})\)

2026-09-11, corrected the same day in three revisions (sections 2 and 4; every correction is recorded in place below rather than silently applied: the Mertens estimate in section 2, the modulus factor in section 4, and, in the third revision, section 4's progression input and its sieve level). Base: the chain SIEGEL_UNIFORMITY.md -> EXCEPTIONAL_ENERGY.md -> ENDPOINT_BOUND.md (reviewed in ENDPOINT_BOUND_REVIEW.md), and ENDPOINT_HALF.md. None of those files is changed by this document; ENDPOINT_HALF.md carries a pointer to section 2 below, which refutes one of its conclusions.

Result, stated first. For some fixed \(c>0\) and all sufficiently large \(N\), unconditionally, \[ \boxed{\ E_{\rm corr}^{(Z)}(N)\ \ll\ N^3\exp\!\big(-c\sqrt{\log N}\big), \qquad Z=\exp\big(\sqrt{\log N}\big).\ } \tag{S} \] This is the endpoint \(\kappa=1/2\) of ENDPOINT_HALF.md's family (A), attained rather than approached.

The superscript is not decoration. \(E_{\rm corr}^{(Z)}\) is defined through the exceptional correction \(C_N\), and \(C_N\) depends on the model parameter \(Z\): which zeros count as exceptional, and therefore what is subtracted, changes with \(Z\). At the enlarged \(Z=\exp(\sqrt\ell)\) the conductor range \(q<Z\) is wider than at \(Z=\exp(\ell^{1/10})\) while the threshold \(\beta>1-c_0/\log Z\) is stricter, so the two corrected quantities are genuinely different, neither one dominating the other. (S) is a statement about the \(Z=\exp(\sqrt\ell)\) quantity. It does not strengthen ENDPOINT_BOUND.md's bound on the \(Z=\exp(\ell^{1/10})\) quantity; it is a different bound on a different object, reached by the same architecture. Section 1 fixes the notation and section 6 states what each carries for the original \(E\).

Two things make the endpoint available, and the first is a correction rather than a construction:

1. ENDPOINT_HALF.md's wall at \(\kappa=1/2\) is an artifact of a lossy estimate, not a feature of the argument. Its section 2.3 concludes that no choice of the Bonferroni cutoff \(m\) works at \(\kappa=1/2\). That conclusion propagates ENDPOINT_BOUND.md section 2's own elementary estimate \(H_Z\le1+\log Z\) for \(H_Z=\sum_{p<Z}1/p\). By Mertens' second theorem \(H_Z=\log\log Z+M+O(1/\log Z)\), so at \(Z=\exp(\ell^\kappa)\) the truth is \(H_Z\sim\kappa\log\ell\), not \(\ell^\kappa\): a double logarithm where the bound supplies a power. With the true \(H_Z\) and a retuned \(m\), the existing device reaches \(D_0=N^{o(1)}\) at \(\kappa=1/2\) (section 2). Nothing needs replacing.
2. The progression input is classical at the endpoint. ENDPOINT_BOUND.md takes its input (1) from Tao and Teräväinen's Proposition 2.2 at \(Z=\exp(\ell^{1/10})\), and ENDPOINT_HALF.md's family beyond that leaned on their Remark 2.8, which those authors state without proof. At \(\kappa=1/2\) the input is instead the prime number theorem in progressions with the exceptional zero retained, in the published form of Drappeau and Fiorilli's Lemma 2.2 (read and quoted in section 4; its exceptional datum is Page's theorem at \(Q=T=e^{\sqrt{\log x}}\), which at \(x=N\) is this document's \(Z\)), Page's theorem, and the fundamental lemma of sieve theory applied to the model inside a progression at its own level \(D_1=\lfloor N^{1/2}\rfloor\) (section 4). Remark 2.8 is not used.

Everything else in the chain was already verified at \(\kappa=1/2\) by ENDPOINT_HALF.md sections 2.1 and 2.2 and is reused, not redone.

Grade: derived, one route, independently checked (see below). The classical inputs are named where used. The finite checks in section 7 test the Mertens correction, the retuned cutoff, and the character cancellation of section 4; they test no asymptotic statement. This establishes no fixed power saving, no exclusion of exceptional zeros, and nothing about the zeros of \(\zeta\).

Independent check, 2026-09-11 (attempt a-0075, judged by a-0076, which recomputed the arithmetic with its own script). It confirmed equation (A), the attainment of the endpoint, the failure at \(\kappa>1/2\), the character cancellation and error assembly of section 4, the scoping of \(E_{\rm corr}^{(Z)}\), and the budget assembly, and it found one defect: section 7's table and section 2.1's crossover sentences understated the least admissible cutoff \(m\) by \(2\), because the probe searched against the exact factorial \(H_Z^{m+1}/(m+1)!\) while the document's own displayed inequality (5\('\)) is the Stirling-weakened \((eH_Z/(m+1))^{m+1}\). Corrected below; the table now matches the reviewer's independent recomputation exactly. It also reported that it had no network access and so could not check input (D) and (Pg) against primary sources. Those were checked in the orchestrator session instead: (Pg) was verified verbatim, and (D) was found unread and has since been replaced (section 4, section 7 item (5)).

The third revision's own check (attempt a-0077, ENDPOINT_SHARP_REVIEW_2.md). The third revision postdates a-0075. Its two substitutions, the progression input (DF) in place of (D) and the separate sieve level \(D_1\) in section 4, were checked in the orchestrator session against the source and by recomputing every error term of section 4, then independently by a-0077, which recomputed \(s_1\) and the \(D_1\ge Z^{10}\) threshold directly from \(N\) (equality at exactly \(\log N=400\)), rederived the range \(\gamma<\min(c_0/2,\delta/\sqrt2,1/2)\) from the four error sources, confirmed the matching of exceptional data with no missed case, and tested the frozen-at-\(x\) remark of section 4 numerically (the ratio of the discrepancy to the lemma's error term grows like \(e^{\delta\sqrt{\log x}}/\log x\) at \(1-\beta=1/\log x\)). It found no defect. It had no network access, so the word-for-word comparison of the (DF) quotation against the source rested on the orchestrator session until a scoped source check by a GPT session, against commit 08e05bd and the published Lemma 2.2, equation (2.1), passed it (operator's report, 2026-09-11; section 7 item (5)). The endpoint review is closed with that check: every section of this document has had a second independent reader, no further paid check is planned, and what remains is the standing footnote that any published claim carries until an outside reader has walked the chain.

1. Parameters at the endpoint

Write \(\ell=\log N\) and take, in place of ENDPOINT_BOUND.md section 1, \[ t=\sqrt\ell,\qquad Z=e^{t},\qquad P=\prod_{p<Z}p,\qquad b=P/\phi(P)=V(Z)^{-1}, \qquad V(Z)=\prod_{p<Z}(1-1/p), \] \(\nu(n)=b\,1_{(n,P)=1}\), \(a(n)=\nu(n)(1-1_{\rm exc}\chi(n)n^{\beta-1})\), with the exceptional data of TT Definition 2.1 at this \(Z\): \(\chi\) primitive real of conductor \(q<Z=e^{\sqrt\ell}\), and \(1-c_0/\log Z=1-c_0/\sqrt\ell<\beta<1\), with \(c_0\) a fixed sufficiently small absolute constant (constrained once more in section 4). \(F,H,W,r,r_a\) are as there, \(C_N=C_{q,\beta,Z}(h)\) is SIEGEL_UNIFORMITY.md (19) at this \(Z\), and \[ E_{\rm corr}^{(Z)}(N)=2\sum_{h=1}^N\big|r(h)-C_N(h)\big|^2 . \] By Mertens' third theorem \(b\sim e^{\gamma_{\rm E}}\log Z=e^{\gamma_{\rm E}}\sqrt\ell\), so \(b\ll\sqrt\ell\). The major-arc parameter is \(R=\lfloor\exp(\sigma t)\rfloor\), \(\sigma\) fixed in section 6. Two levels, with different roles: the level of the divisor approximant is \(D_0\), fixed in section 2 and consumed by the budget; the level at which section 4 sieves a progression to prove input (1\('\)) is \(D_1=\lfloor N^{1/2}\rfloor\), fixed there and consumed nowhere else.

2. The divisor approximation at the endpoint

Correction. The first version of this section replaced ENDPOINT_BOUND.md's Bonferroni approximant by the linear \(\beta\)-sieve, on the ground that the Bonferroni device "needs level \(D_0=Z^m\) with \(m>eH_Z\sim e\log Z\), so \(\log D_0>e(\log Z)^2=e\ell\) at \(\kappa=1/2\)". The step \(H_Z\sim\log Z\) is wrong, and it was inherited from ENDPOINT_BOUND.md section 2's own elementary bound \(H_Z\le1+\log Z\), which ENDPOINT_HALF.md section 2.3 then used as if it were the size of \(H_Z\). Mertens' second theorem gives \[ H_Z=\sum_{p<Z}\frac1p=\log\log Z+M+O\Big(\frac1{\log Z}\Big), \qquad M=0.26149\ldots, \tag{M} \] so at \(Z=\exp(\ell^\kappa)\), \(H_Z=\kappa\log\ell+M+o(1)\). The elementary bound overstates it by a factor \(\asymp\ell^\kappa/\log\ell\), measured at \(3.2\) to \(5.0\) over \(\log Z\in[5,13]\) in section 7. With (M) the original device reaches the endpoint, and the \(\beta\)-sieve is an alternative rather than a repair. Both are recorded below.

2.1 The retuned Bonferroni cutoff

ENDPOINT_BOUND.md section 2 takes \(B_m(n)=\sum_{d\mid P,\ d\mid n,\ \omega(d)\le m}\mu(d)\), \(m\) even, \(D_0=Z^m\), and proves its (5), \(\sum_{n\le N}|B_m(n)-1_{(n,P)=1}|\le NH_Z^{m+1}/(m+1)!\). Nothing in that derivation depends on the value of \(m\), so the cutoff is free. With \((m+1)!\ge((m+1)/e)^{m+1}\), \[ \sum_{n\le N}\big|B_m(n)-1_{(n,P)=1}\big|\ \le\ N\Big(\frac{eH_Z}{m+1}\Big)^{m+1} =N\exp\Big(-(m+1)\log\frac{m+1}{eH_Z}\Big). \tag{5$'$} \] Given a target decay \(\exp(-c\ell^\kappa)\), choose \[ m+1=\Big\lceil\frac{2c}{\kappa}\cdot\frac{\ell^\kappa}{\log\ell}\Big\rceil \quad(\text{rounded up to an even }m). \] Then, by (M), \(\log\frac{m+1}{eH_Z}=\kappa\log\ell-2\log\log\ell+O(1) =\kappa\log\ell\,(1+o(1))\), so the exponent in (5\('\)) is \(2c\,\ell^\kappa(1+o(1))\ge c\,\ell^\kappa\) for large \(N\), and \[ A(N):=\frac{\log D_0}{\log N}=\frac{m\log Z}{\ell} =\frac{2c}{\kappa}\cdot\frac{\ell^{2\kappa-1}}{\log\ell}\,(1+o(1)). \tag{A} \] At \(\kappa=1/2\) this is \(4c/\log\ell\to0\), so \(D_0=N^{o(1)}\), which is exactly what ENDPOINT_BOUND.md section 2 asserts of its own \(D_0\) and what the budget consumes. The retuning is the whole of the change: the fixed cutoff \(m=2\lceil\sqrt\ell\rceil\) is not too small at the endpoint, it is too large, and it pins \(A(N)=2\) (section 7 measures \(2.000\) at every \(N\) tried, matching ENDPOINT_HALF.md section 2.4's own table).

Two consequences worth stating separately.

• ENDPOINT_HALF.md section 2.3's claim that no \(m\) works at \(\kappa=1/2\) does not survive. That argument writes \(m=\lambda H_Z\), requires \(\lambda>e\), and concludes \(\log D_0=\lambda H_Z\log Z\sim\lambda\ell^{2\kappa}\). The last step is (M) again: \(H_Z\log Z=\kappa(\log\ell)\ell^\kappa\), not \(\ell^{2\kappa}\). Its numerical section 2.4 is unaffected as measurement, because it evaluates \(A(N)\) for the fixed cutoff, which is genuinely pinned at \(2\); what does not follow is the claim about every other cutoff.
• The device caps at \(\kappa=1/2\) and attains it. By (A), \(A(N)\to0\) for \(\kappa<1/2\), \(A(N)\to0\) at \(\kappa=1/2\) (the \(\log\ell\) in the denominator is what saves the endpoint), and \(A(N)\to\infty\) for \(\kappa>1/2\). So the divisor approximation is available up to and including the endpoint and not past it, matching the three other inputs of section 6.

Honest numerics, from section 7, and note that \(A(N)=m/\sqrt\ell\) is sawtooth rather than monotone (it falls between the jumps of \(m\) and rises by \(2/\sqrt\ell\) at each jump), so the crossover that matters is the last one, not the first. Under the criterion (5\('\)) actually states: \(A(N)\) first dips below \(1\) at \(\log N\approx677\) and stays below it for every \(\log N>784\); it stays below \(1/2\) for every \(\log N>5.38\times10^4\), i.e. \(N\) beyond about \(10^{23000}\). The budget needs \(A(N)<1\) with a margin \(\asymp\ell^{-1/2}\) (section 6), not \(A(N)<1/2\), so the operative threshold is \(\log N>784\). \(A(N)\) decays like \(1/\log\ell\), which is slow; (S) is an asymptotic statement and this is where its "sufficiently large \(N\)" lives.

2.2 The \(\beta\)-sieve, as an alternative with a fixed level

The chain uses exactly four properties of the approximant, visible by reading every place ENDPOINT_BOUND.md touches \(B_m\) (its (4)-(6), the prefix bound after (10), the untwisted prefix in 4.2, Cases A and B in 4.3, and the \(D_0\) terms of (13), (14), (17), (18)):

• (P1) coefficients \(|\lambda_d|\le1\);
• (P2) support \(d\mid P(Z)\), \(d<D_0\le N\);
• (P3) an \(\ell^1\) approximation \(\sum_{n\le y}|b\lambda(n)-\nu(n)|\ll N\exp(-c\sqrt\ell)\) uniform over prefixes \(y\le N\), stable under multiplication of the summand by \(\chi(n)\), by \(n^{\beta-1}\), or by a unit phase;
• (P4) the level enters only through \(N^2D_0\sqrt Z\) in (17)-(18) and the \(D_0\) terms of (13)-(14).

\(B_m\) with the retuned cutoff has all four. So does Rosser's upper-bound weight \(\lambda^+\) of the linear (\(\beta=2\)) sieve at level \(D_0\), supported on \(\mathcal D^+=\{d=p_1\cdots p_r:\ p_1>\cdots>p_r,\ p_1\cdots p_{l-1}p_l^{\,3}<D_0\ \text{for every odd }l\le r\}\): (P1) and (P2) hold by construction, and the fundamental lemma of sieve theory (Friedlander and Iwaniec, Opera de Cribro, Lemma 6.3, at dimension \(\kappa=1\); the same lemma SIEGEL_UNIFORMITY.md section 3 consumes as TT Lemma 5.1) gives, for \(D_0=Z^s\), \(s\ge10\), and every \(y\le N\), \[ \sum_{n\le y}\big|b\Lambda^+(n)-\nu(n)\big| =b\sum_{n\le y}\big(\Lambda^+(n)-1_{(n,P)=1}\big)\ll Ne^{9-s}+bD_0 , \tag{6$'$} \] one-sided because \(\Lambda^+\ge1_{(n,P)=1}\) pointwise. At \(D_0=N^{1/2}\), \(s=\tfrac12\sqrt\ell\) and (6\('\)) is (P3) with any \(c<1/2\). This fixes \(A(N)=1/2\) at every \(N\) rather than letting it decay, so it is asymptotically weaker than 2.1 and numerically stronger at accessible \(N\). Either choice proves (S); section 6 uses \(A(N)=o(1)\) from 2.1 and notes where \(A(N)=1/2\) would also serve.

3. Sections 3 to 5 of ENDPOINT_BOUND.md at the endpoint

With \(t=\sqrt\ell\), \(D_0=N^{o(1)}\) from 2.1, and input (1) replaced by (1\('\)) of section 4:

• (8)-(9), major arcs: \(\int_{\mathfrak M}(|F|^2-|H|^2)^2\ll N^3\ell^2R^7\exp(-2\gamma\sqrt\ell)\).
• (10): unchanged; needs \(D_0\le N\), which holds.
• (11): unchanged, \(\sup_{\mathfrak m}|F|\ll(NR^{-1/2}+N^{4/5})\ell^{5/2}\).
• untwisted prefix: \(O(b(N/R+D_0)\ell^2+N\exp(-c\sqrt\ell))\).
• (13), Case A (\(q\ge R^{1/3}\)): \(|V_s|\ll\ell^{O(1)}[NR^{-1/6}+D_0\sqrt Z]+N\exp(-c\sqrt\ell)\).
• (14), Case B (\(q<R^{1/3}\)): \(|V_s|\ll\ell^{O(1)}[NR^{-1/2}+D_0R^{1/6}]+N\exp(-c\sqrt\ell)\).
• (15): \(\sup_{\mathfrak m}|H|\ll\ell^{O(1)}[NR^{-1/6}+D_0\sqrt Z]+N\exp(-c\sqrt\ell)\).
• (16)-(17): \(\int_{\mathfrak m}(|F|^2-|H|^2)^2\ll\ell^{O(1)}[N^3R^{-1/6}+N^{14/5}+N^2D_0\sqrt Z]+N^3\exp(-c\sqrt\ell)\).

Each line is the corresponding line of ENDPOINT_BOUND.md with \(t\) and \(D_0\) substituted; no argument there is altered.

4. Input (1) at the endpoint, from classical sources

Revised 2026-09-11 (third revision), two substitutions, both recorded here rather than silently applied.

(i) The progression input. The first two versions cited (D) to Davenport, Chapter 20, and section 7 item (5) recorded that the citation had not been read. It is replaced by Drappeau and Fiorilli's Lemma 2.2, which was read and is quoted below, and whose exceptional datum is Page's theorem at \(Q=T=e^{\sqrt{\log x}}\): at \(x=N\) that is exactly this document's \(Z\), so the two exceptional data are matched rather than compared.

(ii) The sieve level. The first two versions applied the fundamental lemma inside this section at the approximant level \(D_0\) of section 2. That was too small. With the retuned cutoff, \(D_0=N^{A(N)}\) and \(A(N)=4c/\log\ell\), so the sieve parameter is \[ s=\frac{\log D_0}{\log Z}=A(N)\sqrt\ell=\frac{4c}{\log\ell}\sqrt\ell\,(1+o(1)), \] and the displayed fundamental-lemma error \(e^{9-s}=\exp(-(4c/\log\ell)\sqrt\ell\,(1+o(1)))\) decays more slowly than \(\exp(-\gamma\sqrt\ell)\) for every fixed \(\gamma>0\). As written, (1\('\)) did not follow. The repair is to separate the two roles the one symbol was carrying. \(D_0\) is the level of the divisor approximant \(b\lambda\) that replaces \(\nu\) on the minor arcs; it enters the budget through (P4) and is constrained by (A). The level at which this section sieves a progression is a free parameter of the proof of (1\('\)), enters nothing outside this section, and has no reason to equal \(D_0\). Fix it at \[ D_1=\lfloor N^{1/2}\rfloor,\qquad s_1:=\frac{\log D_1}{\log Z}=\frac{\sqrt\ell}{2}\,(1+o(1)),\qquad e^{9-s_1}=e^{9}\exp\big(-\tfrac12\sqrt\ell\,(1+o(1))\big), \] with sieve remainders \(O(D_1)=O(N^{1/2})\). The fundamental lemma's hypothesis \(D_1\ge Z^{10}\) is \(\ell\ge400\), i.e. \(N\ge e^{400}\), inside the "sufficiently large \(N\)" of (S). The \(\beta\)-sieve alternative of 2.2 also uses a fixed level \(N^{1/2}\), for the approximant; that is a coincidence of convenient values, not the same parameter.

Claim (1\('\)). There is a fixed \(\gamma>0\) such that, uniformly over every progression \(B=\{n\le y:\ n\equiv a\ (\mathrm{mod}\ r)\}\) with \(y\le N\), \(r\le R\), and every residue \(a\), \[ \Big|\sum_{n\in B}\big(\Lambda(n)-a(n)\big)\Big|\ll N\exp(-\gamma\sqrt\ell). \tag{1$'$} \]

Inputs.

(DF) Drappeau and Fiorilli, The first moment of primes in arithmetic progressions: beyond the Siegel-Walfisz range, Trans. London Math. Soc. 8 (2021), no. 1, 174-185, doi 10.1112/tlm3.12030, Lemma 2.2, equation (2.1); arXiv:2003.02201v1. The published statement is the one cited; its display (2.1) carries a prefix factor on the right-hand side, the logarithmic factor placed in front of the whole bound rather than on the second term alone. The text quoted below is the arXiv version, which is the one read in this session. Version comparison, kept on the record: in the arXiv version the lemma's display carries no number and (2.1) is the decomposition that consumes it; in the published version the lemma's display is (2.1). The published wording was not read here (the publisher returned 403 to this environment); the comparison of this quotation with the published lemma was made in a scoped source check by a GPT session against commit 08e05bd, reported PASS by the operator on 2026-09-11 and recorded in section 7 item (5). Where the logarithmic factor sits changes nothing below: on \(xe^{-\delta\sqrt{\log x}}\) it is absorbed by lowering \(\delta\), and (D\('\)) is stated with that done.

Fix \(a\in\mathbb Z\setminus\{0\}\). There exists \(\delta>0\) such that for all \(x,Q\ge1\) we have the bound \[ \sum_{q\le Q}\ \max_{y\le x}\ \max_{(a,q)=1} \Big|\psi(y;q,a)-\big(1-\eta_{x,a}1_{\tilde q\mid q}\big)\frac y{\phi(q)}\Big| \ll xe^{-\delta\sqrt{\log x}}+Q\sqrt x\,(\log x)^{O(1)}, \] where \(\eta_{x,a}\) was defined in (1.6).

Their (1.6) is \(\eta_{x,a}:=\tilde\chi(a)/(\beta x^{1-\beta})\in(-1,1)\), and their exceptional datum is their Theorem 1.2 (Page's theorem, cited to Iwaniec and Kowalski, Theorems 5.26 and 5.28) with the definition that follows it: there is an absolute constant \(b>0\) such that for all \(Q,T\ge2\) the function \(\prod_{q\le Q}\prod_{\chi\bmod q}L(s,\chi)\) has at most one zero \(s=\beta\) with \(\mathrm{Re}(s)>1-b/\log(QT)\) and \(|\mathrm{Im}(s)|\le T\); if it exists it is real and is the zero of a unique \(L(s,\tilde\chi)\) for some primitive real character \(\tilde\chi\) (of conductor \(\tilde q\)); \(\tilde\chi\) is \(x\)-exceptional if the above conditions are met with \(Q=T=e^{\sqrt{\log x}}\). So the \(x\)-exceptional character has \(\tilde q\le e^{\sqrt{\log x}}\) and its zero satisfies \(\beta>1-b/(2\sqrt{\log x})\).

The lemma is applied here with \(x:=y\) for each prefix \(y\) separately, with \(Q:=R\), and a single term of the sum is bounded by the whole sum. For \((a,r)=1\), \(r\le R\), writing \(\tilde\chi_y\), \(\tilde q_y\), \(\tilde\beta_y\) for the \(y\)-exceptional data, \[ \psi(y;r,a)=\frac y{\phi(r)}-1_{\tilde q_y\mid r}\,\tilde\chi_y(a)\frac{y^{\tilde\beta_y}}{\tilde\beta_y\,\phi(r)} +O\big(ye^{-\delta\sqrt{\log y}}+R\sqrt y\,(\log y)^{O(1)}\big), \tag{D$'$} \] since \(\eta_{y,a}\,y=\tilde\chi_y(a)y^{\tilde\beta_y}/\tilde\beta_y\).

Two remarks on the reading, recorded because this document leans on the statement. First, "Fix \(a\)" and the inner maximum over \((a,q)=1\) do not sit together; (D\('\)) is read as uniform in \(a\), which is what the inner maximum asserts and what the proof they cite (Davenport, Chapter 28, p. 164, with the exceptional character separated) supplies. Second, as printed the maximum over \(y\le x\) is taken with \(\eta_{x,a}\) frozen at \(x\), while the exceptional main term at \(y\) is \(\tilde\chi(a)y^\beta/(\beta\phi(q))\); the two differ by \(\tilde\chi(a)\,y\,x^{\beta-1}\big((x/y)^{1-\beta}-1\big)/(\beta\phi(q))\), and when \(1-\beta\) and \(\log(x/y)/\sqrt{\log x}\) are both small compared with \(\delta\), that difference summed over \(\tilde q\mid q\le Q\) is not \(O(xe^{-\delta\sqrt{\log x}})\). The uniformity in \(y\) therefore appears not to hold literally as printed; the paper itself uses the lemma only at \(y=x\) (its (2.1) and (2.2)), where nothing is affected. This document does not rely on it: taking \(x:=y\) in (D\('\)) is exactly what removes the issue, at the price that the exceptional datum now depends on \(y\), which the matching below handles.

(Pg) Page's theorem, in the form verified against a primary source in section 7 item (5): there is an absolute \(c>0\) such that for any \(Q\ge2\), among the primitive real characters of conductor at most \(Q\), at most one has an \(L\)-function with a real zero in \([1-c(\log Q)^{-1},1)\). At \(Q=Z=e^{\sqrt\ell}\): at most one primitive real character of conductor \(<Z\) has a real zero \(\beta>1-c/\sqrt\ell\). This is the same theorem as (DF)'s Theorem 1.2 with a differently normalised constant; both are used, (Pg) for the uniqueness of the TT-exceptional datum at \(Z\), (DF)'s form for the \(y\)-exceptional datum.

(FL) The fundamental lemma as in 2.2, applied in this section at level \(D_1\), parameter \(s_1\).

(Cmp) Fix \(c_0\le\min(c,\ b/2)\) in section 1.

Small \(y\). If \(y\le N\exp(-\gamma\sqrt\ell)\) both sides of (1\('\)) are \(O(y\ell)\); so assume \(\log y\ge\ell/2\). Then \(e^{\sqrt{\log y}}\ge e^{\sqrt{\ell/2}}\ge R\) (as \(\sigma\le1/20<1/\sqrt2\)), \(e^{\sqrt{\log y}}\le Z\), \(b/(2\sqrt{\log y})\ge b/(2\sqrt\ell)\ge c_0/\sqrt\ell\), \(ye^{-\delta\sqrt{\log y}}\le Ne^{-\delta\sqrt\ell/\sqrt2}\), and \(R\sqrt y(\log y)^{O(1)}\le e^{\sigma\sqrt\ell}N^{1/2}\ell^{O(1)}\le Ne^{-\sqrt\ell}\) for large \(N\).

Matching the exceptional data. Let \(\chi\bmod q\), \(\beta\) be the TT-exceptional data at \(Z\) when they exist. For \((a,r)=1\), \(r\le R\), \(\log y\ge\ell/2\):

• If \(\chi\) exists and \(q\mid r\): then \(q\le r\le R\le e^{\sqrt{\log y}}\), \(\beta>1-c_0/\sqrt\ell\ge1-b/(2\sqrt{\log y})\), and \(\beta\) is real, so \(\beta\) lies in the region of (DF)'s Theorem 1.2 at \(Q=T=e^{\sqrt{\log y}}\); by its uniqueness, \(\tilde\chi_y=\chi\) and \(\tilde\beta_y=\beta\). (D\('\)) carries the term \(-\chi(a)y^\beta/(\beta\phi(r))\).
• If a \(y\)-exceptional \(\tilde\chi_y\) exists with \(\tilde q_y\mid r\) and is not TT-exceptional at \(Z\): since \(\tilde q_y\le r\le R<Z\), the failing condition is \(\tilde\beta_y\le1-c_0/\sqrt\ell\), so \(y^{\tilde\beta_y-1}\le\exp(-c_0\log y/\sqrt\ell)\le\exp(-c_0\sqrt\ell/2)\) and the term is at most \(2ye^{-c_0\sqrt\ell/2}/\phi(r)\ll Ne^{-c_0\sqrt\ell/2}\).
• If \(\chi\) exists and \(q\nmid r\): either \(\tilde\chi_y=\chi\), and then \(1_{\tilde q_y\mid r}=0\); or \(\tilde\chi_y\ne\chi\), and then \(\tilde\chi_y\) is not TT-exceptional at \(Z\) (uniqueness from (Pg), \(c_0\le c\)), so the previous point applies.
• If no TT-exceptional character exists at \(Z\), any \(\tilde\chi_y\) with \(\tilde q_y\mid r\) falls under the second point.

Hence, in every case, with \(1_{\rm exc}\) the indicator that the TT-exceptional data exist at \(Z\), \[ \psi(y;r,a)=\frac y{\phi(r)}-1_{\rm exc}1_{q\mid r}\,\chi(a)\frac{y^\beta}{\beta\phi(r)} +O\big(N\exp(-c_4\sqrt\ell)\big),\qquad c_4=\min\big(c_0/2,\ \delta/\sqrt2\big). \tag{D$''$} \] This is the statement the first two versions attributed to Davenport; the form (D\('\))-(D\(''\)) is what is now actually consumed.

Case \((a,r)>1\). A prime \(p\mid(a,r)\) divides every \(n\in B\) and \(p\le r<Z\), so \(a(n)=0\) on \(B\); the \(\Lambda\) side counts prime powers of primes dividing \(r\), at most \(\omega(r)\ell\ll\ell^2\).

Case \((a,r)=1\), untwisted part. Sieve \(B\) by the primes \(p<Z\): \(g(p)=0\) for \(p\mid r\), \(g(d)=1/d\) for \((d,r)=1\), \(X=y/r\), \(|r_d|\le1\). (FL) at level \(D_1\), parameter \(s_1\), gives \(\#\{n\in B:(n,P)=1\}=(y/r)\prod_{p<Z,\,p\nmid r}(1-1/p)(1+O(e^{9-s_1}))+O(D_1)\). Every prime factor of \(r\) is below \(Z\) (as \(r\le R<Z\)), so \(b\prod_{p<Z,\,p\nmid r}(1-1/p)=r/\phi(r)\) and \[ \sum_{n\in B}\nu(n)=\frac y{\phi(r)}+O\big(Ne^{9-s_1}+bD_1\big) =\frac y{\phi(r)}+O\big(Ne^{9-\sqrt\ell/2}+\ell^{1/2}N^{1/2}\big), \] matching (D\(''\))'s main term.

Case \((a,r)=1\), twisted part, \(q\mid r\). Then \(\chi\) is constant on \(B\), \(\chi(n)=\chi(a)\), so with \(T(u)=\sum_{n\in B,\,n\le u,\,(n,P)=1}\chi(n)\) the untwisted computation applies at every \(u\), and since \(u\mapsto u^{\beta-1}\) is positive, decreasing, of total variation at most \(1\) on \([1,y]\), \[ b\int_1^yu^{\beta-1}\,dT(u)=\chi(a)\frac{y^\beta-1}{\phi(r)\beta}\big(1+O(e^{9-s_1})\big)+O(bD_1). \] By the matching above, this is (D\(''\))'s exceptional term up to \(O(Ne^{9-\sqrt\ell/2}+\ell^{1/2}N^{1/2})\).

Case \((a,r)=1\), twisted part, \(q\nmid r\). Put \(g=\gcd(q,r)\); \(q\nmid r\) means exactly \(g<q\). Write \(M=\mathrm{lcm}(r,q)=rq/g\). The conditions \(n\equiv a\ (r)\) and \(n\equiv c\ (q)\) with \((c,q)=1\) are compatible exactly when \(c\equiv a\ (g)\), and then determine one class \(n_c\bmod M\). If \((a,g)>1\) no such \(c\) exists and \(T\equiv0\); assume \((a,g)=1\). The admissible \(c\) form a coset of \(K=\ker\big((\mathbb Z/q)^\to(\mathbb Z/g)^\big)\), of size \(\phi(q)/\phi(g)\). Sieving each class by (FL) at level \(D_1\), \[ S_c(u):=\#\{n\le u:\ n\equiv n_c\ (M),\ (n,P)=1\} =XW\big(1+O(e^{9-s_1})\big)+O(D_1), \] \[ X=\frac uM=\frac{ug}{rq},\qquad W=\prod_{\substack{p<Z\\ p\nmid M}}\Big(1-\frac1p\Big)=V(Z)\frac M{\phi(M)} . \] Both \(X\) and \(W\) are independent of \(c\), since \(M\) is. Hence \[ T(u)=\sum_c\chi(c)S_c(u) =XW\underbrace{\sum_c\chi(c)}_{=\,0} \ +\ O\Big(\frac{\phi(q)}{\phi(g)}\Big[XWe^{9-s_1}+D_1\Big]\Big). \tag{4.1} \] The main term vanishes by cancellation of \(\chi\) over a complete period of the fibre: writing the coset as \(c_0K\), \(\sum_c\chi(c)=\chi(c_0)\sum_{k\in K}\chi(k)\), and \(\sum_{k\in K}\chi(k)=0\) unless \(\chi|_K\) is trivial, which would make \(\chi\) induced by a character mod \(g\) and force \(\mathrm{cond}(\chi)=q\mid g\), contradicting \(g<q\).

The modulus factor (the second revision's correction, kept). The first version stated the vanishing correctly but wrote the error of (4.1) with the factor \(\phi(q)/\phi(g)\) applied to the wrong bracket, and its next line then dropped that factor from the first term while keeping it on the second. The factor is real and belongs to both: the main terms cancel across the classes, the remainders do not, and each of the \(\phi(q)/\phi(g)\) classes incurs its own sieve remainder \(O(D_1)\). Carrying it explicitly, with \(\phi(q)/\phi(g)\le q/g\) and \(M/\phi(M)\ll\log\log M\ll\log\ell\) (since \(M\le rq<Z^2\)), \[ \frac{\phi(q)}{\phi(g)}XW\le\frac qg\cdot\frac{ug}{rq}\cdot V(Z)\frac M{\phi(M)} \ll\frac ur\,V(Z)\log\ell, \qquad \frac{\phi(q)}{\phi(g)}D_1\le qD_1 , \] so \(T(u)\ll(u/r)V(Z)(\log\ell)e^{9-s_1}+qD_1\) uniformly in \(u\le y\), and by the same total-variation bound as above, using \(bV(Z)=1\), \(u/r\le N\), \(b\ll\sqrt\ell\), \(q<Z=e^{\sqrt\ell}\), \(D_1\le N^{1/2}\), \[ b\Big|\int_1^yu^{\beta-1}\,dT(u)\Big| \ \ll\ N(\log\ell)e^{9-s_1}+bqD_1 \ \ll\ N(\log\ell)e^{9-\sqrt\ell/2}+\ell^{1/2}e^{\sqrt\ell}N^{1/2}. \tag{4.2} \] The second term is \(N^{1/2+o(1)}\), below \(N\exp(-\gamma\sqrt\ell)\) with half a power of \(N\) to spare, so the modulus factor is absorbed. On the \(\Lambda\) side, (D\(''\)) carries no exceptional term when \(q\nmid r\), and the two sides agree.

Collecting the four cases and (D\(''\)), (1\('\)) holds for every fixed \[ \gamma<\min\big(c_0/2,\ \delta/\sqrt2,\ 1/2\big), \] uniformly in \(a\), \(r\le R\), \(y\le N\); the \(1/2\) is the sieve level \(D_1=N^{1/2}\) through \(e^{-s_1}\), and is the term the first two versions had, in effect, at \((4c/\log\ell)\) instead.

5. Input (2) at the endpoint

ENDPOINT_HALF.md section 2.1 verified that SIEGEL_UNIFORMITY.md's model comparison (10)-(11) survives at \(\kappa=1/2\): its own use of the fundamental lemma is at the free level \(D=\lfloor N^{1/4}\rfloor\), \(s=\tfrac14\sqrt\ell\), dimension \(2\), remainder \(O(Ne^{-c\sqrt\ell})\); and the singular-series truncation contributes \(O(N\ell e^{-\sqrt\ell})\) per shift. So (2) reads \(r_a(h)=C_N(h)+O(N\exp(-c_m\sqrt\ell))\) uniformly in \(h\), and (3) reads \(E_{\rm corr}^{(Z)}\le2D+O(N^3\exp(-2c_m\sqrt\ell))\). EXCEPTIONAL_ENERGY.md's range bookkeeping becomes \((\log q)^2<\ell\), i.e. \(q<Z\), as ENDPOINT_HALF.md section 2.2 records.

6. The budget at the endpoint

Combining (3), (9) and (17) as in ENDPOINT_BOUND.md section 6, with \(t=\sqrt\ell\), \(R=\lfloor\exp(\sigma\sqrt\ell)\rfloor\), \(D_0=N^{A(N)}\): \[ E_{\rm corr}^{(Z)}(N)\ll\ell^{O(1)}\Big[N^3e^{-(2\gamma-7\sigma)\sqrt\ell} +N^3e^{-\sigma\sqrt\ell/6}+N^{14/5}+N^{2+A(N)}e^{\sqrt\ell/2}\Big] +N^3e^{-c\sqrt\ell}+N^3e^{-2c_m\sqrt\ell}. \tag{18$'$} \] Take \(\sigma=\min(\gamma/20,1/20)\). The first two terms are then \(N^3\exp(-c'\sqrt\ell)\); the third is a fixed power below \(N^3\); the fourth needs \(A(N)\le1-(c'+\tfrac12)/\sqrt\ell\), which holds for large \(N\) since \(A(N)=4c/\log\ell\to0\) by (A) (and would also hold, with more room, at the \(\beta\)-sieve's \(A(N)=1/2\)); the last two are direct. So every term is \(N^3\exp(-c'\sqrt\ell)\) with \(c'=\tfrac12\min(2\gamma-7\sigma,\ \sigma/6,\ c,\ 2c_m)\), which is (S).

What fixes the exponent at \(1/2\). Four inputs, each capping there and each attaining it: the divisor approximation, by (A), since \(A(N)\to\infty\) for \(\kappa>1/2\); the prime number theorem in progressions with the exceptional term, whose error in (DF) is \(\exp(-\delta\sqrt{\log y})\); the fundamental lemma inside the proof of (1\('\)), whose parameter \(s_1=\log D_1/\log Z\le\ell/\ell^\kappa\) at any level \(D_1\le N\) is \(o(\ell^\kappa)\) once \(\kappa>1/2\), so its error \(e^{-s_1}\) can no longer match \(\exp(-\ell^\kappa)\); and the singular-series truncation \(\ell e^{-\ell^\kappa}\) at \(Z=\exp(\ell^\kappa)\), which is the one input that improves as \(\kappa\) grows. Moving past the endpoint therefore needs the first three moved together, of which the second is a zero-free-region question and the third is a constraint on the level that no choice of weights inside this architecture relaxes. That statement is about this chain.

For the original \(E\). As in ENDPOINT_BOUND.md section 7, \(E\le2E_{\rm corr}^{(Z)}+2A_{\rm exc}^{(Z)}\) with \(A_{\rm exc}^{(Z)}=2\sum_h C_N(h)^2\), so \[ E(N)\ll N^3\exp(-c\sqrt\ell)+1_{q\ {\rm odd}}N^{2\beta+1}\frac{q^2}{\phi(q)^4} +N^{4\beta-1}\frac{q^2}{\phi(q)^3}, \] the exceptional terms present exactly when a TT-exceptional zero exists at this \(Z\), and not deletable otherwise. Because the exceptional class differs between the two values of \(Z\), this is not comparable term by term with ENDPOINT_BOUND.md section 7's version; both are unconditional statements about \(E\), with different exceptional sets.

7. Finite checks

endpoint_sharp_mertens_probe.py and endpoint_sharp_probe.py, results in results_endpoint_sharp_mertens_probe.json and results_endpoint_sharp_probe.json.

(1) Mertens against the elementary bound. \(H_Z\) summed over the primes, against \(\log\log Z+M\) and against \(1+\log Z\), at \(\log Z=5,7,9,11,13\): the exact values are \(1.880,\ 2.212,\ 2.460,\ 2.660,\ 2.827\), Mertens gives \(1.871,\ 2.207,\ 2.459,\ 2.659,\ 2.826\) (agreeing to the \(O(1/\log Z)\)), and the elementary bound gives \(6,\ 8,\ 10,\ 12,\ 14\), overstating by \(3.2\times\) to \(5.0\times\) and growing. This is the estimate section 2's correction turns on.

(2) The retuned cutoff at \(\kappa=1/2\), target decay \(\exp(-\sqrt\ell)\), comparing the fixed \(m=2\lceil\sqrt\ell\rceil\) with the least even \(m\) meeting the target under the criterion (5\('\)) states, namely \((eH_Z/(m+1))^{m+1}\le e^{-\sqrt\ell}\):

Correction. The first version of this table searched against the exact factorial \(H_Z^{m+1}/(m+1)!\), which is smaller than the bound the document displays and so admits an \(m\) smaller by \(2\) at these scales. a-0075 found this and a-0076 recomputed the corrected column with an independent script, obtaining \(m=14,22,64,348\) and \(A=1.400,1.100,0.640,0.348\), which is what now stands. The probe reports both criteria; only the stated one is tabulated.

The fixed cutoff is pinned at \(A=2\), reproducing ENDPOINT_HALF.md section 2.4. The retuned cutoff decays, and the measured values track \(4/\log\ell\) from (A) (predicted \(0.29\) and \(0.174\) at \(\log N=10^6,10^{10}\); measured \(0.348\) and \(0.161\)). Because \(A(N)=m/\sqrt\ell\) is sawtooth, the thresholds are stated as last crossings: \(A<1\) for every \(\log N>784\) (first dip at \(\approx677\)), and \(A<1/2\) for every \(\log N>5.38\times10^4\).

(3) Complete-period cancellation for the coset sums of section 4: for every primitive real \(\chi\bmod q\) with \(q\in\{3,5,7,11,13,15,21,33,105\}\) (Jacobi symbol, primitive since these \(q\) are odd squarefree) and both even conductors \(q\in\{4,8\}\), and every proper divisor \(g\mid q\) with every residue \(a\) coprime to \(g\): the sum of \(\chi(c)\) over \(c\bmod q\), \((c,q)=1\), \(c\equiv a\ (g)\) is exactly zero, in all 96 cosets tested.

(4) The \(\beta\)-sieve properties, from the earlier probe: Rosser's support enumerated from its definition at \(Z=50\), coefficients in \(\{0,\pm1\}\) and support below \(D_0\) by construction, one-sidedness on every \(n\le10^6\) at levels \(Z^s\), \(s=2,\dots,6\), and \(\ell^1\) error over \(NV(Z)\) of \(0.41,\ 0.043,\ 0.0031,\ 0.0000,\ 0.0000\) against \(e^{-s}=0.135,\ 0.050,\ 0.018,\ 0.0067,\ 0.0025\).

(5) The classical inputs of section 4, against primary sources. Checked in the orchestrator session, which had network access the cell did not.

• (Pg), Page's theorem: verified verbatim. Basak and Pratt, A Conditional Refinement of Page's Theorem on zeros of Dirichlet \(L\)-functions, arXiv:2607.06433v1, Theorem 1.1, attributed there to Page (Lemma 9) and to Davenport, p. 95: "There exists an absolute constant \(c>0\) such that the following holds. For any \(Q\ge2\), we have \(\#\{\chi\in S(Q): L(s,\chi)\ \text{has a real zero in}\ [1-c(\log Q)^{-1},1)\}\le1\)," with \(S(Q)=\{\chi\bmod q_\chi:\chi\ \text{primitive and real},\ 1\le q_\chi\le Q\}\). Section 4's (Pg) is this at \(Q=Z\), and its \(1/\sqrt\ell\) threshold is \(c/\log Q\) with \(\log Q=\sqrt\ell\), so the two agree with the same absolute constant. The same source states the zero-free region \(\sigma\ge1-c_0/\log(q(|t|+2))\) containing at most a single zero, necessarily real and with \(\chi\) quadratic (its (1.1), citing Davenport p. 93), which is what makes the TT-exceptional data well defined, and Siegel's ineffective \(\beta\le1-c(\varepsilon)q_\chi^{-\varepsilon}\) (its (1.2)).
• (D), as cited in the first two versions: never read, now withdrawn. It was cited to Davenport, Chapter 20. The volume this hunt already cites, Montgomery and Vaughan Multiplicative Number Theory II, was fetched and searched: it contains no statement of Page's theorem and no occurrence of "Siegel zero" or "exceptional zero", its Chapter 20 being a different part of the subject, so it was not the source. The closest reachable corroboration was explicit work in the style of Baker, Faber and Kinlaw, arXiv:1802.00085v3, which carries the exceptional zero through the explicit formula as the \(x^{\beta-1}\) term of the zero sum (its Definition 6.1 and (2.5)-(2.6)) but never states (D) in the corrected-main-term form. Rather than keep an unread citation, the third revision replaced it.
• (DF), Drappeau and Fiorilli Lemma 2.2: verified verbatim against arXiv:2003.02201v1, pages 2 to 4, and quoted in section 4 together with their Theorem 1.2, their definition of the \(x\)-exceptional character (\(Q=T=e^{\sqrt{\log x}}\)) and their (1.6). Three things were checked beyond the wording. (i) The exceptional datum matches: at \(x=N\) their conductor bound \(e^{\sqrt{\log x}}\) is \(Z\), and their threshold \(1-b/(2\sqrt{\log x})\) is this document's \(1-c_0/\sqrt\ell\) with \(c_0\le b/2\), which (Cmp) now imposes. (ii) The lemma's proof is cited to Davenport, Chapter 28, p. 164, the Bombieri-Vinogradov argument with the exceptional character separated, and its error terms are the Bombieri-Vinogradov ones, \(xe^{-\delta\sqrt{\log x}}+Q\sqrt x(\log x)^{O(1)}\); section 4 uses them at \(Q=R\), where the second is \(N^{1/2+o(1)}\). (iii) The statement's uniformity in \(y\le x\) with \(\eta_{x,a}\) frozen at \(x\) appears not to hold literally, for the reason given in section 4; section 4 applies the lemma at \(x=y\) and does not use that uniformity. The paper's own use is at \(y=x\) only. The numbering quoted is the arXiv version's, where the lemma's display is unnumbered and (2.1) is the decomposition that consumes it; in the published version (Trans. London Math. Soc. 8 (2021) 174-185) the lemma's display is equation (2.1) and carries its logarithmic factor as a prefix on the whole right-hand side. The published page could not be fetched from this environment (403). Scoped source check, PASS. A GPT session with access to the published text compared the quotation and its use in section 4 against the published Lemma 2.2, equation (2.1), at commit 08e05bd, and passed it; the operator reported the result on 2026-09-11. That check was not run from this session and its transcript is not in this repository; what is recorded here is the operator's report of it, which is the same standing as every other operator-supplied fact in this hunt. Scope, corrected 2026-09-12. The first version of this sentence said that with it every input of section 4 had been read against a primary source by at least two readers. That is true of (DF) only. (Pg) was read verbatim once, in this session, against Basak and Pratt; (FL) is consumed here as SIEGEL_UNIFORMITY.md consumes TT Lemma 5.1 and was not re-read against Friedlander and Iwaniec in this hunt; (Cmp) is a choice of constant, not a source. The claim that stands: every source input of section 4 has been read against a primary source at least once, and (DF) twice.
• The level \(D_1\). Every error term of section 4 was recomputed with \(D_1=\lfloor N^{1/2}\rfloor\) in place of \(D_0\): \(s_1=\sqrt\ell/2\), \(e^{9-s_1}=e^9\exp(-\sqrt\ell/2)\), remainders \(bD_1\ll\ell^{1/2}N^{1/2}\) and \(bqD_1\ll\ell^{1/2}e^{\sqrt\ell}N^{1/2}\), and the hypothesis \(D_1\ge Z^{10}\) at \(\ell\ge400\). The defect this repairs, \(s=A(N)\sqrt\ell\) with \(A(N)=4c/\log\ell\), was checked against (A) directly.

These check an estimate, an arithmetic retuning, a character identity, four approximant properties, two classical citations verbatim, and the substitution of a level. They test neither (S) nor (1\('\)).

8. Scope

Three corrections and one substitution inside an existing argument. The corrections are to a lossy elementary estimate that had been read as an asymptotic (section 2), to a misplaced modulus factor in an error term (section 4), and to a sieve level that had been tied to the approximant's and was too small for the error it was asked to deliver (section 4); all three are recorded in place. The substitution replaces a cited proposition and an unproved remark by the prime number theorem in progressions with its exceptional term, in Drappeau and Fiorilli's published form, read and quoted rather than cited from memory. The conclusion is the endpoint \(\kappa=1/2\) for the corrected CHHL error at the enlarged model parameter \(Z=\exp(\sqrt{\log N})\), which is a different corrected quantity from the one bounded at \(Z=\exp((\log N)^{1/10})\). The first two revisions were independently checked (ENDPOINT_SHARP_REVIEW.md), and the third revision's two substitutions were checked separately (ENDPOINT_SHARP_REVIEW_2.md), and the quotation of (DF) was passed by a scoped source check against the published lemma (section 7 item (5)). The endpoint review is closed. No fixed power saving, no exclusion of exceptional zeros, no statement about the zeros of \(\zeta\), and no novelty claim.

Forward pointer, added after this checkpoint closed. ARC_SPLIT_BUDGET.md (2026-09-11, later the same day) reaches the same shape of bound with the exponent constant \(2\gamma/3\) in place of this document's \(\gamma/240\), by splitting the corrected residual by arcs rather than by model, so that the divisor approximant of section 2 and the minor-arc estimates of ENDPOINT_BOUND.md section 4 are not consumed. Nothing in this document is changed by it; section 6's four-input account describes this chain.