Review of ENDPOINT_SHARP.md, third revision, section 4 only

Independent check, 2026-09-11 (attempt a-0077). Scope: only the two substitutions of section 4 (the progression input (DF) and the separate sieve level \(D_1\)), which postdate ENDPOINT_SHARP_REVIEW.md and are explicitly marked there as not yet covered. Sections 2, 3, 5 and 6 are not re-reviewed here; ENDPOINT_SHARP_REVIEW.md's verdicts on those stand.

Read: ENDPOINT_SHARP.md sections 1, 4, 7 item (5) and 8; ENDPOINT_SHARP_REVIEW.md; ENDPOINT_BOUND.md section 4; SIEGEL_UNIFORMITY.md section 3.

This attempt has no network access in its tool set (python, grep, rg, ls, cat, head, tail, wc, find only), so the (DF) quotation against the arXiv source was not independently checked here; it is treated as given, per the document's own note that it was compared against the source in the orchestrator session. Everything else below was recomputed from scratch with a new script, review2_sharp_section4_probe.py, results in results_review2_sharp_section4_probe.json, not derived from any prior probe in this directory.

Item 1: the sieve level

Verdict: confirmed, both the diagnosis of the defect in the first two versions and the repair.

The diagnosis. At \(\kappa=1/2\), (A) gives \(A(N)=4c/\log\ell\to0\), so if the fundamental lemma were applied at the approximant level \(D_0\), the sieve parameter would be \(s=A(N)\sqrt\ell=(4c/\log\ell)\sqrt\ell(1+o(1))\). This exponent is \(o(\sqrt\ell)\): for any fixed \(\gamma>0\), \((4c/\log\ell)/\gamma\to0\) as \(\ell\to\infty\), so \(A(N)\sqrt\ell=o(\gamma\sqrt\ell)\) for every fixed \(\gamma\), i.e. \(e^{9-s}\) decays strictly slower than \(\exp(-\gamma\sqrt\ell)\) for every fixed \(\gamma\). The diagnosis is correct: the displayed error at level \(D_0\) cannot supply the target rate, and (1\('\)) did not follow at that level as written.

The repair, recomputed independently. With \(D_1=\lfloor N^{1/2}\rfloor\), \(Z=e^{\sqrt\ell}\): \[ s_1=\frac{\log D_1}{\log Z},\qquad\text{hypothesis: } D_1\ge Z^{10}. \] I recomputed \(s_1\) and the hypothesis directly from \(N\) and \(Z\) (50-60 digit arithmetic, no use of the document's asymptotic formulas):

This confirms the document's claim exactly: \(D_1\ge Z^{10}\) holds if and only if \(\log N\ge400\), with equality at exactly \(\log N=400\) (the crossing is not merely asymptotic here; the floor in \(D_1\) changes \(\log D_1\) by \(O(N^{-1/2})\), negligible at this scale, so the integer-valued check and the continuous formula agree to the precision shown). The document's \(s_1=\sqrt\ell/2(1+o(1))\) is confirmed, and if anything is more accurate than stated: the correction from flooring is \(O(N^{-1/2}/\sqrt\ell)\) relative, far smaller than a generic \(o(1)\).

All four cases of section 4, recomputed with \(D_1\). Using \(e^{9-s_1}=e^9\exp(-\sqrt\ell/2(1+o(1)))\), \(b\sim e^{\gamma_E}\sqrt\ell\), \(q<Z=e^{\sqrt\ell}\):

• \((a,r)>1\): untouched by the level, bound \(\ll\ell^2\), as before.
• untwisted, \((a,r)=1\): main term \(y/\phi(r)\), error \(O(Ne^{9-s_1}+bD_1)=O(Ne^{9-\sqrt\ell/2}+\sqrt\ell N^{1/2})\). Matches (427)-(428).
• twisted, \(q\mid r\): main term \(\chi(a)(y^\beta-1)/(\phi(r)\beta)\), error of the same two orders. Matches (438), (441).
• twisted, \(q\nmid r\): main term vanishes by character-sum cancellation (unaffected by the level, checked again below); error \(N(\log\ell)e^{9-s_1}+bqD_1\ll N(\log\ell)e^{9-\sqrt\ell/2}+\sqrt\ell\, e^{\sqrt\ell}N^{1/2}\). Matches (4.2), and the second term is \(N^{1/2+o(1)}\), safely below \(N\exp(-\gamma\sqrt\ell)\) for any fixed \(\gamma\) since \(\log N=\ell\gg\sqrt\ell\) dominates the comparison.

I re-derived the final admissible range independently rather than just re-reading it off: the four error sources are \(O(N\exp(-c_0\sqrt\ell/2))\) (exceptional-character mismatch, from item 3 below), \(O(N\exp(-\delta\sqrt\ell/\sqrt2))\) (the (DF) Bombieri-Vinogradov term at \(\log y\ge\ell/2\)), and \(O(Ne^{9-\sqrt\ell/2})\) (the sieve level \(D_1\) itself, dropping the fixed constant \(e^9\)). Collecting exponents gives exactly \[ \gamma<\min(c_0/2,\ \delta/\sqrt2,\ 1/2), \] matching the document's displayed range. The "\(1/2\)" is exactly and only the sieve-level term; nothing else in section 4 contributes a competing constant near \(1/2\).

\(D_1\) does not leak outside section 4. I grepped every occurrence of D_1 and D_0 in the document. D_1 appears only in section 1's definition, section 4 throughout, and section 7 item (5)'s recomputation note; one further mention in section 6's qualitative "what fixes the exponent" paragraph uses \(s_1=\log D_1/\log Z\) generically to describe why the proof method stops working past \(\kappa=1/2\), not as a substituted value in an equation. Equation (18\('\)) uses only \(D_0=N^{A(N)}\), and property (P4) in section 2.2 is stated only in terms of \(D_0\). So the claim that \(D_1\) "enters nothing outside section 4" is confirmed exactly: no equation elsewhere in the document plugs in \(D_1\) or \(N^{1/2}\) as the level for anything but the progression sieve of section 4.

Item 2: the progression input (DF)

No web access used, consistent with this attempt's tool restrictions (python/grep/rg/ls/cat/head/tail/wc/find only, no network). The quotation of Drappeau-Fiorilli's Lemma 2.2, their Theorem 1.2, their \(x\)-exceptional-character definition, and their (1.6) are therefore treated as given, per the document's own record that they were compared against arXiv:2003.02201v1 in the orchestrator session. I checked only the mathematics built on top of the quotation.

(a) Derivation of (D\('\)): confirmed. Setting \(x:=y\) and \(Q:=R\) in the quoted lemma and bounding the single term \((q,y)=(r,y)\) by the whole sum over \(q\le R\), \(y'\le y\) gives \[ \big|\psi(y;r,a)-(1-\eta_{y,a}1_{\tilde q\mid r})\tfrac y{\phi(r)}\big| \ll ye^{-\delta\sqrt{\log y}}+R\sqrt y(\log y)^{O(1)}. \] With \(\eta_{y,a}=\tilde\chi(a)/(\beta y^{1-\beta})\), \(\eta_{y,a}\,y= \tilde\chi(a)y^\beta/\beta\) is immediate algebra (\(y\cdot y^{1-\beta}\! {}^{-1}=y^\beta\)), and expanding the product \((1-\eta_{y,a}1_{\tilde q\mid r})y/\phi(r)\) gives exactly (D\('\)) as displayed, equation (337)-(339). Confirmed.

(b) The frozen-at-\(x\) remark: confirmed by direct computation, and it is a real feature of the quoted statement, not a document error. The document observes that if the lemma's uniformity in \(y\le x\) were taken literally with \(\eta_{x,a}\) frozen at \(x\), the exceptional term at a shorter prefix \(y<x\) would differ from the "correct" \(y\)-frozen term \(\tilde\chi(a)y^\beta/\beta\) by \[ \Delta(x,y,\beta)=\tilde\chi(a)\,y\,x^{\beta-1}\big((x/y)^{1-\beta}-1\big)/\beta, \] and claims this, summed over \(\tilde q\mid q\le Q\), is not \(O(xe^{-\delta\sqrt{\log x}})\) when \(1-\beta\) and \(\log(x/y)/\sqrt{\log x}\) are both small compared with \(\delta\). I first checked the algebra: writing \(L=\log(x/y)\), \(\varepsilon=1-\beta\), \(y\,x^{\beta-1}=yx^{-\varepsilon}=xe^{-L-\varepsilon\log x}\), so \(|\Delta|=xe^{-L-\varepsilon\log x}\,\varepsilon\)-scale terms exactly as the document states; direct symbolic evaluation of \(yx^{\beta-1}((x/y)^{1-\beta}-1)/\beta\) against \(\eta_{x,a}y-\tilde\chi(a)y^\beta/\beta\) agrees to 49 of 50 computed digits (the last-digit discrepancy is floating-point rounding at that precision, not a mismatch).

Naively one might expect a fixed \(\varepsilon=c_0/\sqrt{\log x}\) (the extremal size an exceptional zero is permitted) to make \(|\Delta|\) small compared with \(xe^{-\delta\sqrt{\log x}}\) once \(\delta<c_0\), and this is in fact what happens at that extremal size (I checked it: with \(\varepsilon=c_0/\sqrt{\log x}\) fixed and \(\delta<c_0\), the ratio \(|\Delta|/(xe^{-\delta\sqrt{\log x}})\to0\)). The remark's force is that Siegel's theorem gives no effective lower bound on how close to \(1\) an exceptional \(\beta\) can be: \(1-\beta\) can be far smaller than \(c_0/\sqrt{\log x}\), e.g. \(1-\beta=\mu/\log x\) for fixed \(\mu\), which is certainly "small compared with \(\delta\)" (it is \(o(1/\sqrt{\log x})\)). In that regime, with \(L=\log(x/y)\) also held fixed (also small compared with \(\delta\sqrt{\log x}\), since it doesn't grow with \(x\) at all), I computed \(|\Delta|/(xe^{-\delta\sqrt{\log x}})\) directly (\(\delta=0.5\), \(\mu=1\), \(L=3\), no approximation):

The ratio diverges: \(\Delta\) decays only polynomially in \(\log x\) (like \(1/\log x\) here) once \(1-\beta\) is taken this small, while the claimed error term decays exponentially in \(\sqrt{\log x}\), so the ratio grows like \(e^{\delta\sqrt{\log x}}/\log x\to\infty\). This confirms the remark exactly: the printed uniformity in \(y\le x\) with \(\eta_{x,a}\) frozen at \(x\) genuinely fails to be \(O(xe^{-\delta\sqrt{\log x}})\) in this corner of the parameter space, so the remark is not an overcautious reading, it identifies a real gap between the literal text and what a uniform-in-\(y\) statement would need.

(c) The document does not rely on the questioned uniformity: confirmed. Section 4 states explicitly "the lemma is applied here with \(x:=y\) for each prefix \(y\) separately" (line 332), never invoking the lemma at a fixed \(x\) with \(y<x\) varying; the "Matching the exceptional data" paragraph is built entirely to handle the resulting \(y\)-dependence of the exceptional datum, and section 7 item (5) repeats "section 4 applies the lemma at \(x=y\) and does not use that uniformity." I searched section 4 end to end for any use of a fixed exceptional datum across a range of \(y\) and found none: every occurrence of \(\tilde\chi_y,\tilde q_y, \tilde\beta_y\) or \(\chi,\beta\) is either the definition of the \(y\)-exceptional data (varying with \(y\)) or the TT-exceptional data fixed at \(Z\) (which is a different, and correctly distinguished, object). Confirmed.

Item 3: matching the exceptional data

Verdict: confirmed. I checked the four bullets against the definitions in section 1 and (DF), looking specifically for a missed case and for the three-way threshold confusion the task asked about (\(b/(2\sqrt{\log y})\) vs. \(b/\sqrt{\log y}\) vs. \(c/\sqrt{\log N}\)).

• Bullet 1 (\(\chi\) exists, \(q\mid r\)). The chain \(q\le r\le R\le e^{\sqrt{\log y}}\) needs \(R\le e^{\sqrt{\log y}}\), i.e. \(\sigma\sqrt\ell\le\sqrt{\ell/2}\), i.e. \(\sigma\le1/\sqrt2\); the document's \(\sigma\le1/20\) satisfies this with room to spare. The threshold chain \(\beta>1-c_0/\sqrt\ell\ge1-b/(2\sqrt{\log y})\) reduces, after clearing denominators, to \(\sqrt{\log y}\le(b/2c_0)\sqrt\ell\); since \(\log y\le\ell\) always and \((Cmp)\) gives \(c_0\le b/2\) (so \(b/2c_0\ge1\)), this holds unconditionally, not just in the \(\log y\ge\ell/2\) regime. \(\beta\) then lies in DF's Theorem 1.2 region at \(Q=T=e^{\sqrt{\log y}}\) by definition, so by uniqueness \(\chi\) is the \(y\)-exceptional character. No step here confuses the factor of \(2\): the document keeps DF's own \(b/(2\sqrt{\log y})\) (from \(\log(QT)=2\sqrt{\log y}\) at \(Q=T=e^{\sqrt{\log y}}\)) distinct from this document's own \(c_0/\sqrt\ell\) (no factor of \(2\), since \(Z\) itself, not \(Z^2\), is the conductor bound), and (Cmp) is exactly the bridge that makes the comparison valid.
• Bullet 2 (\(y\)-exceptional \(\tilde\chi_y\) with \(\tilde q_y\mid r\), not TT-exceptional). Since \(c_0\le c\) (from (Cmp)) makes TT-exceptional a subset of Page-exceptional at the same conductor bound \(Z\), any character meeting the TT threshold would already be forced, by (Pg)'s uniqueness, to be the (unique) TT-exceptional character; so failing to be TT-exceptional while having conductor \(\tilde q_y\le r<Z\) forces \(\tilde\beta_y\le1-c_0/\sqrt\ell\) exactly as claimed, and \(y^{\tilde\beta_y-1}\le\exp(-c_0\sqrt\ell/2)\) follows from \(\log y\ge\ell/2\). The bound \(2ye^{-c_0\sqrt\ell/2}/\phi(r)\ll Ne^{-c_0\sqrt\ell/2}\) is correct (\(1/\tilde\beta_y<2\) for large \(N\), \(1/\phi(r)\le1\)).
• Bullet 3 (\(\chi\) exists, \(q\nmid r\)). The dichotomy \(\tilde\chi_y=\chi\Rightarrow1_{\tilde q_y\mid r}=0\), else \(\tilde\chi_y\ne\chi\Rightarrow\tilde\chi_y\) is not TT-exceptional (by uniqueness of the TT-exceptional character, which is \(\chi\)), reducing to bullet 2. This is exhaustive and correct.
• Bullet 4 (no TT-exceptional character at \(Z\)). Any \(y\)-exceptional \(\tilde\chi_y\) with \(\tilde q_y\mid r\) is trivially "not TT-exceptional" (there is none to be), so bullet 2 applies verbatim.

Case search. I looked for a case not covered by the four bullets: the only way the indicator \(1_{\tilde q_y\mid r}\) in (D\('\)) can be nonzero is if some \(y\)-exceptional character (there is at most one, globally, by (Pg)/(DF)'s own uniqueness) has \(\tilde q_y\mid r\); bullets 1-2 exhaust this by whether that character equals the TT-exceptional \(\chi\) or not, and bullet 3 covers \(1_{\tilde q_y\mid r}=0\) directly when \(\chi\) is the one that fails to divide \(r\). No fifth case (e.g. two distinct \(y\)-exceptional characters both dividing \(r\)) exists, since DF's Lemma 2.2 posits a single \(x\)-exceptional character for the whole modulus range \(q\le Q\), not one per \(q\). I found no confusion between \(b/(2\sqrt{\log y})\), \(b/\sqrt{\log y}\), or \(c/\sqrt\ell\) anywhere in the four bullets; each threshold is used with its own constant and its own factor of \(2\) exactly where DF's or this document's own definition places it. (Dsecond)'s constant \(c_4=\min(c_0/2,\delta/\sqrt2)\) is the correct combination of bullet 2's exponent (\(c_0\sqrt\ell/2\), giving \(c_0/2\)) and the (DF) Bombieri-Vinogradov exponent (\(\delta\sqrt{\log y}\ge\delta\sqrt{\ell/2}=\delta\sqrt\ell/\sqrt2\), giving \(\delta/\sqrt2\)).

Item 4: the Small \(y\) reduction

Verdict: confirmed, both the trivial small-\(y\) bound and all four inequalities that follow from \(\log y\ge\ell/2\).

• \(y\le N\exp(-\gamma\sqrt\ell)\Rightarrow\) both sides of (1\('\)) are \(O(y\ell)\): routine (\(\Lambda(n)\le\log N=\ell\), \(|a(n)|\ll b\ll \sqrt\ell\le\ell\) pointwise), and \(y\ell\le N\ell\exp(-\gamma\sqrt\ell) =N\exp(-\gamma\sqrt\ell(1-o(1)))\), absorbed into the same exponential rate at a slightly smaller constant, standard in this document's style.
• \(e^{\sqrt{\log y}}\ge R\): reduces to \(\sigma\le1/\sqrt2\); holds since \(\sigma\le1/20\).
• \(e^{\sqrt{\log y}}\le Z\): immediate from \(\log y\le\ell\) (as \(y\le N\)).
• \(b/(2\sqrt{\log y})\ge c_0/\sqrt\ell\): shown above under item 3, and it in fact needs only \(\log y\le\ell\), not \(\log y\ge\ell/2\).
• \(ye^{-\delta\sqrt{\log y}}\le Ne^{-\delta\sqrt\ell/\sqrt2}\): from \(y\le N\) and \(\sqrt{\log y}\ge\sqrt{\ell/2}=\sqrt\ell/\sqrt2\) (this step does need \(\log y\ge\ell/2\)).
• \(R\sqrt y(\log y)^{O(1)}\le Ne^{-\sqrt\ell}\) for large \(N\): I checked this directly at concrete scale rather than only asymptotically. At \(\log y=\ell/2\), \(\sigma=0.05\):

and the exponents' asymptotic comparison confirms it holds for every large \(N\): the left side has \(\log(R\sqrt y(\log y)^{O(1)})= \sigma\sqrt\ell+\ell/2+O(\log\log\ell)\), the right side has \(\ell-\sqrt\ell\), and \(\ell/2+\sigma\sqrt\ell\ll\ell-\sqrt\ell\) since \(\ell/2\) is the dominant, strictly smaller, linear term.

Every inequality in the Small \(y\) paragraph is confirmed, and I found no inequality there that silently needed a stronger hypothesis than \(\log y\ge\ell/2\) (the two that don't need the lower half at all, \(e^{\sqrt{\log y}}\le Z\) and the \(b/(2\sqrt{\log y})\) chain, only use \(\log y\le\ell\), which is unconditional).

Summary

No defect was found in either of the third revision's two substitutions. Everything reachable with the tools available in this attempt (recomputing every displayed error term of section 4 at level \(D_1\), rederiving the final admissible range for \(\gamma\), and independently testing the (DF) frozen-at-\(x\) remark's own numerical claim) checks out against the text. The one item this attempt could not settle on its own is the word-for-word comparison of the (DF) quotation against arXiv:2003.02201v1, since this attempt's tool set has no network access; that comparison is recorded as already done in the orchestrator session, and settling it independently would need an attempt with web access repeating that fetch and diff.