2026-09-12, attempt a-0082. Two scripts written fresh for this review, not reusing sharp_exponent_probe.py: review_a0082_sharp_exponent_check.py (numerical checks of section 5's two items, by a different method: SciPy's QUADPACK oscillatory-weight quadrature rather than the probe's own routine) and review_a0082_budget_check.py (symbolic recomputation of the seven exponents of (7)). Results quoted inline below.
Summary of verdicts: (1) confirmed, (2) confirmed, (3) confirmed, (4) confirmed, with one point (section 2.1) that I agree corrects a real miss in an earlier check, (5) confirmed, with one defective intermediate equation that does not change the theorem. No defect found changes the boxed conclusions (S''') or the exponent constant $2c_0$.
(1) Section 1.1: the dyadic reduction
Verdict: confirmed.
The Farey condition: $|\alpha-a/q|\le 1/(q\sqrt N)\le 1/q^2$ holds exactly for $q\le\sqrt N$, an equality at $q=\sqrt N$, so Vaughan's bound $|F(\alpha)|\ll (Nq^{-1/2}+N^{4/5}+\sqrt{Nq})\ell^{O(1)}$ applies on every $I'{q,a}$ used here. For $q\sim Q$ with $Q$ ranging up to $R^2\ell^9$ (subpolynomial in $N$, since $R=N^{o(1)}$), the term $Nq^{-1/2}$ dominates $N^{4/5}$ and $\sqrt{Nq}$ for all large $N$, so $\sup{q\sim Q,\,I'_{q,a}}|F|^2\ll N^2\ell^{O(1)}/Q$ is the right leading order; the document's stated power $\ell^8$ is one internally consistent choice of exponent for Vaughan's bound (some other documents in this hunt use $\ell^{5/2}$ for $|F|$, giving $\ell^5$ after squaring; this is a bookkeeping choice of which version of Vaughan's theorem is quoted, not an error, and it does not matter for the argument since only $\ell^{O(1)}$ is ever used downstream).
Disjointness gives $\sum_{q\sim Q}\sum_a\int_{I'{q,a}}|F|^2\le\int{\mathbb T}|F|^2=d_N\ll N\ell$ (Chebyshev), so $Z'Q\ll(N^2\ell^8/Q)(N\ell)= N^3\ell^9/Q$, exactly as stated. The comparison $N^3\ell^9/Q\le N^3/R^2$ for $Q\ge R^2\ell^9$ is immediate algebra. The remark that arcs with $q\le R$ are inside $\mathfrak M$ up to their outer parts is a correct observation about geometry (the Farey arc $I'{q,a}$, of radius $1/(q\sqrt N)$, is far wider than the major arc $I_{r,a}$ of radius $R/(qN)$ when $q\le R\ll\sqrt N$, by a factor $\sqrt N/R\to\infty$), and it plays no load-bearing role in the estimate itself, since the Vaughan bound used for $Z'_Q$ does not distinguish major from minor.
I recomputed the reduction to blocks $R<Q\le R^2\ell^9$ independently and it is correct: nothing in section 1.1 needs a defect notice.
(2) Section 1.2: the exact obstruction
Verdict: confirmed, including the negative claim that no known input supplies the missing power of $Q$.
The identity and Gallagher step. $\sum_{x<n\le x+h}c_n(q,a)=\sum_b e(ab/q)\Delta_b(x,h)$ follows from $\sum_b^*e(ab/q)=\mu(q)$ applied to the $\mu(q)/\phi(q)$ part of $c_n(q,a)$ and the definition of $\Delta_b$ termwise; this is the same manipulation as RANK3_Z_COMPONENT.md section 4's (2)-(3) (there proved for the arc radius $Q/(qN)$ rather than $1/(q\sqrt N)$, but the algebra is identical). Gallagher's lemma with $\Theta=1/(q\sqrt N)$, $h_q=q\sqrt N/2=1/(2\Theta)$ is a direct instance of (G). Orthogonality over $a$ to reach (1), and the substitution into (2)-(3), is standard Cauchy-Schwarz-free bookkeeping and I recomputed it: with $\sup|R_{q,a}|^2\ll N^2\ell^8/Q$ from item (1) above, $Z'^{\rm res}Q\ll(N\ell^8/Q^2)\sum{q\sim Q}V_q(h_q)$ matches (2), and dividing the target $N^3Q^{-2+\varepsilon}$ through gives (3) exactly.
The three inputs. Trivial: $\sum_b|\Delta_b|^2\le(\sum_b|\Delta_b|)^2\ll (h\ell)^2$ is Cauchy-Schwarz on $\phi(q)+O(1)$ terms folded into the $\ell$; summing over $x\le N$ and $q\sim Q$ gives $N\cdot Q\cdot h^2\ell^2 \asymp N^2Q^3\ell^2$, off by $Q^3$ from (3) as stated.
Koukoulopoulos: I checked the hypothesis $Q^2\le h/N^{1/3+\varepsilon}$ at $h\asymp Q\sqrt N$ directly: this is $Q\le N^{1/6-\varepsilon/2}$, which holds for all large $N$ since $Q\le R^2\ell^9=N^{o(1)}$; so the theorem's range does contain these moduli, exactly as claimed, and this is the correct reading of RANK3_Z_COMPONENT.md section 4's own applicability range (there stated for $q\le L^C$, a genuinely smaller range than what is used here, but that document's Corollary is stated for general $q,h$ in the displayed range and does not require $q$ itself to be polylogarithmic, only the inequality $Q_K^2\le h/x^{1-2/c+\varepsilon}$, which is satisfied here). The resulting $\sum_{q\sim Q}V_q\ll h^2N\ell^{-A}\asymp N^2Q^2\ell^{-A}$, off by $Q^2$: I confirm the stated reason, that the first-moment tool $\sum_b|\Delta_b|^2\le E(x,h;q)\sum_b|\Delta_b|\ll hE(x,h;q)$ cannot recover the factor $h$ that a genuine second-moment tool would.
The multiplicative large sieve step: the classical inequality $\sum_{q\le 2Q}(q/\phi(q))\sum^*_\chi|\sum_na_n\chi(n)|^2\le(N+4Q^2)\sum|a_n|^2$ (Montgomery-Vaughan form of the large sieve for characters) applied with $a_n=\Lambda(n)\mathbf 1_{x<n\le x+h}$, for which $\sum|a_n|^2\ll h\ell$ (only primes contribute, each $O(\ell)$, at density $1/\ell$), gives $(h+Q^2)h\ell$ as claimed. I recomputed the passage to $\sum_{q\sim Q} \sum_b|\Delta_b|^2\ll\ell^2(h+Q^2)h/Q$ (one $\ell$ from $\sum a_n^2$, one from converting $q/\phi(q)$-weighted sums to $1/\phi(q)$-weighted sums by dividing by $q\asymp Q$, and reduction from all characters to primitive ones): the arithmetic is internally consistent, and integrating over $x\le N$ with $h_q\asymp Q\sqrt N$ gives $\sum_{q\sim Q}V_q(h_q)\ll N\ell^2h_q^2/Q \asymp N^2Q\ell^2$, matching (4) exactly. Substituting into (2): $Z'^{\rm res}_Q\ll(N\ell^8/Q^2)(N^2Q\ell^2)=N^3\ell^{10}/Q$, which I confirm is Vaughan's order up to the log power (the document's own $\ell^9$ bound in section 1.1 uses a different accounting of the log powers; the extra one power of $\ell$ here is harmless since only $\ell^{O(1)}$ is ever claimed). This is off (3) by $Q^{1-\varepsilon}$ exactly as stated, and the stated reason (the large sieve is only sharp when the character count $Q^2$ matches the sequence length $h$, and here $h=Q\sqrt N\gg Q^2$ since $Q= N^{o(1)}$) is the correct diagnosis: the large sieve inequality has genuine slack $h/Q^2$ in this regime, this is not a case where a sharper application of the same tool would close the gap.
The obstruction statement itself. I recomputed $\log(q(N/h)) = \log(q\cdot\sqrt N/Q)$ at $q\sim Q$: this is $\log(\sqrt N) +O(1)=\ell/2+O(1)$, independent of $Q$ (the $Q$ cancels between the modulus and the height), so (ZF) gives $\beta\le1-c/(\ell/2)(1+o(1))$ and $N^{2(\beta-1)}\ge e^{-4c}(1+o(1))$, a constant with no dependence on $Q$, confirming that this specific tool structurally cannot ever deliver a power of $Q$ at this interval length, for any choice of $Q$.
The open-problem question. I am not aware of an unconditional variance bound for $\psi$ (or $\theta$) in intervals of length $h\asymp\sqrt N$ with a power saving over the trivial $h^2N$. The unconditional results I know that give a power saving over trivial for short-interval variance (via zero-density estimates, e.g. the Selberg/Montgomery/Huxley line that RANK3_Z_COMPONENT.md section 4 itself surveys and withdraws an incorrect claim about) apply only for $h\ge N^{\theta}$ with $\theta$ bounded away from $1/2$ from above (the density-estimate threshold there is $\theta\ge 1-2/c\ge 1/6$ to $1/3$ for the exponent's range of validity, but that governs how small $h$ can be relative to $N$ for the density method to give any power saving over a fixed polylog baseline, not a bound at $h\asymp\sqrt N=N^{1/2}$ itself beating $h^2N$ by a power of $N$ or of $Q$). Selberg's classical result gives such a saving only conditionally, under RH. This matches the document's claim, and I do not know a counterexample or a missed reference; what would settle this negative claim more strongly is either an unconditional short-interval variance theorem at length $\asymp \sqrt N$ with any fixed power saving over $h^2N$ (which I do not believe exists in the literature) or a proof that none can exist unconditionally without new input on zeros (which the document does not attempt and does not claim).
No defect found in section 1.2.
(3) Section 1.3: the alternate dissection at $Q_2=R^2\ell$
Verdict: confirmed.
The height bound: for $R<q\le Q_2$ on the arcs of the $Q_2$-dissection, $N|\theta|\le Q_2/q<Q_2/R=R\ell$, matching "heights at most $R\log N$" exactly (with $q>R$ used to replace $Q_2/q$ by its bound at $q=R$). MAJOR_ARC_EXPLICIT.md section 2's construction only uses the modulus $r$ and the arc radius to fix $T$ and bound the far-zero sum; it does not require $r\le$ any specific cutoff beyond $r<Z$ (used to discard prime powers dividing $r$) and $T\le N$, both satisfied here with $Q_2=R^2\ell =N^{o(1)}<Z=e^{\sqrt\ell}$, so the substitution $R\to Q_2$ in section 2's inputs is legitimate.
The shape of (5): summing $|P_{q,a}|^4$ over the enlarged major arcs gives the standard singular-series fourth moment $\ll N^3\ell^{O(1)}/Q_2^2\le N^3\ell^{O(1)}/R^2$ (a genuinely stronger bound than $R^{-1}$, consistent with major-arc quartic moments generally being smaller than minor-arc ones), and the error term from replacing $F$ by its major-arc approximation carries a fourth power of the sup bound from MAJOR_ARC_EXPLICIT.md (3)-(4) at modulus $Q_2$; this is not claimed to be sharp and the document does not use it for anything beyond an order-of-magnitude remark, which is appropriate given the section's stated purpose (showing this route does not move the exponent, not producing a new bound).
The two remarks: first, that the literal target $\int_{\mathfrak m}|F|^4 \ll N^3\ell^{O(1)}R^{-2+\varepsilon}$ is false at any point $a/q_e$ where an exceptional real zero of conductor $R<q_e\le R^2\ell$ sits, is a correct reading of the spike structure of $|I_\beta|^2$ (the same object numerically checked in section 5 item 1 of the source document and independently in item (4) below): a spike of height $\asymp N^{2\beta_e}/q_e$ at a rational point inside $\mathfrak m$ cannot be absorbed into a bound of the stated polynomial shape without subtracting it, which is exactly the corrected integrand's role. Second, the claim that (5) reproduces ARC_SPLIT_BUDGET.md's arc-split architecture at cutoff $Q_2$ rather than a new mechanism is correct by construction: nothing in the derivation of (5) differs in kind from the derivation already carried out for cutoff $R$ in MAJOR_ARC_EXPLICIT.md and ARC_SPLIT_BUDGET.md, only the numerical value of the cutoff, and the balance of exponents in any such architecture is governed by the arc-split trade-off already analyzed, independent of which cutoff is chosen. No new information is produced.
No defect found in section 1.3.
(4) Section 2: the mechanism, integrating the Page term over each arc
Verdict: confirmed, including the correction in section 2.1, which I independently re-derive and find necessary; I agree the original bullet in MAJOR_ARC_EXPLICIT.md was a genuine miss, and that the earlier check (a-0080, as recorded in MAJOR_ARC_EXPLICIT_REVIEW.md item 3) endorsed the wrong reading rather than merely failing to look.
Numerical checks, written fresh, not reusing sharp_exponent_probe.py. review_a0082_sharp_exponent_check.py uses SciPy's QUADPACK oscillatory quadrature (weight='cos'/'sin') rather than the probe's own integration routine.
The decay bound (6). At $N\in\{10^4,10^6\}$, $\kappa\in\{0.1,0.3,1.0\}$, 40 values of $N\theta$ from 1 to 200, the worst value of $|I_\beta(\theta)|\cdot N|\theta|/N^\beta$ came out at $0.3615,\ 0.4918,\ 2.0763$ ($N=10^4$) and $0.3543,\ 0.4540,\ 1.4569$ ($N=10^6$), against the stated bound $C=e^{\kappa\sigma}$ of $1.19,\ 1.69,\ 5.73$ and $1.15,\ 1.53,\ 4.16$. These match the document's own $0.36,\ 0.48,\ 2.03$ and $0.35,\ 0.44,\ 1.42$ closely (small residual differences are consistent with slightly different sample grids and quadrature routines), confirming both the bound and the specific numbers independently.
The flatness in $R$. At $N=10^4$, $\int_{|\theta|\le R/(rN)}|K_N(\theta)|^2 |I_\beta(\theta)|^2\,d\theta/N^{2\beta+1}$ for $\kappa\in\{0.3,1.0\}$, $r\in\{1,3,7\}$, $R\in\{5,20,80\}$:
| $\kappa$ | $r=1$ | $r=3$ | $r=7$ |
|---|---|---|---|
| $0.3$ | $0.8145,\ 0.8145,\ 0.8145$ | $0.8139,\ 0.8145,\ 0.8145$ | $0.8101,\ 0.8144,\ 0.8145$ |
| $1.0$ | $1.4559,\ 1.4561,\ 1.4561$ | $1.4540,\ 1.4560,\ 1.4561$ | $1.4444,\ 1.4554,\ 1.4561$ |
(entries are $R=5,20,80$). These match the source document's own table to four decimal places, confirming the "no factor $R$" claim independently.
I also confirmed by hand that the flatness is structurally forced, not a numerical accident: writing $u=N\theta$, the integrand $\min(N,1/(2|\theta|))^2\min(2N^\beta,CN^\beta/(N|\theta|))^2$ is $\asymp N^{2\beta+1}$ on $|u|\lesssim1$ and decays like $u^{-4}$ for $u\gg1$ (from $K_N$'s $u^{-2}$ times $I_\beta$'s $u^{-2}$), so $\int u^{-4}du$ converges and the tail beyond $|u|=O(1)$ contributes a bounded, rapidly vanishing correction; extending the arc radius $R/(rN)$ (i.e. $u$ up to $R/r$) past $O(1)$ therefore adds essentially nothing, which is exactly why the mass sits at $|\theta|\ll1/N$ regardless of $R$.
The four bullets, recomputed.
- First bullet: $|W_P|^2\le(r/\phi(r)^2)|I_{\tilde\beta}|^2$ from $|\tau (\tilde\chi_r)|\le\sqrt r$, and dividing by the extra $\phi(r)^2$ from $|H|\le|K_N|/\phi(r)$ gives $(r/\phi(r)^4)\int|K_N|^2|I_{\tilde\beta}|^2 \ll(r/\phi(r)^4)N^{2\tilde\beta+1}$, matching the numerically confirmed flatness. Summing over $a$ (factor $\phi(r)$) and over $r=\tilde qm\le R$ gives $\sum_m\tilde qm/\phi(\tilde qm)^3$, which converges (since $m/\phi(m)^3\ll(\log\log m)^3/m^2$ is summable) to $\ll\tilde q^{-2} (\tilde q/\phi(\tilde q))^3$, exactly as claimed, and no factor of $R$ appears anywhere in this chain.
- Second bullet: I recomputed $\sum_{r\le R}r/\phi(r)\ll R$ (average order of $r/\phi(r)$ is a constant), giving $R^3E_{\rm md}^2\cdot O(R)\cdot N^{2\tilde\beta-1}=O(R^4)E_{\rm md}^2N^{2\tilde\beta-1}$, matching the stated $R^4E_{\rm md}^2N^{2\tilde\beta-1}\ell$ (the extra $\ell$ is a safe over-estimate of the average-order constant, harmless).
- Third bullet: $\sum_{r\le R}r^2/\phi(r)^3=\sum_r(1/r)(r/\phi(r))^3\ll \sum_r(\log\log r)^3/r\ll(\log\log R)^3\log R\ll\ell^{O(1)}$, confirming the claimed polylogarithmic bound (a genuine, if generous, bound: the true growth is $\log R\,(\log\log R)^3=O(\sigma\sqrt\ell(\log\ell)^3)$, still inside $\ell^{O(1)}$).
- Fourth bullet ($W_0$): correctly reuses the unmodified sup-times-Parseval bound of
MAJOR_ARC_EXPLICIT.md, since only the Page term needed the refined treatment; consistent with the section's stated scope.
Section 2.1, re-derived from scratch. (ZF) is a per-character statement: for a fixed real $\chi\bmod r$ ($r\le R$), it excludes zeros with $\sigma>1-c/\log(2r)$ except for one. Since $r$ ranges down to small values (e.g. $r=3$), $c/\log(2r)$ can be an $O(1)$ constant, not shrinking with $\ell$: (ZF) alone permits that one character's own exceptional zero to sit anywhere below this comparatively wide threshold. Page's theorem is a cross-character statement: among all characters of conductor $\le R$ and height $\le R^3$, at most one zero anywhere exceeds the narrower threshold $1-b/\log(R^4)$. These two statements do not compose into "every other real character has no exceptional zero at all": Page only forbids a second zero above its own (narrow) threshold; it says nothing about a character having its own zero below Page's threshold but above its own (wider) (ZF) threshold, e.g. somewhere in $(1-c/\log2r,\,1-b/\log R^4]$. Such a zero is (ZF)-exceptional for its own character and not Page-exceptional, exactly as SHARP_EXPONENT.md now states, and it must be included: its contribution after the Gauss sum is $\sqrt r\cdot2N^\beta\le\sqrt r\cdot2Ne^{-(b/(4\sigma)) \sqrt\ell}$, i.e. the added term $e^{-(b/(4\sigma))\sqrt\ell}$ in $\Upsilon(r)$, which I confirm reproduces $Re^{-(b/(2\sigma))\sqrt\ell}$ in the squared budget with exponent $b/(4c_0)-2c_0\ge2c_0$ at $\sigma=2c_0$ exactly under $c_0^2\le b/16$, a condition already inside (H'').
Re-reading MAJOR_ARC_EXPLICIT_REVIEW.md item 3 with this in hand: the earlier check's defense of the original parenthetical, "the cross-character uniqueness needed to rule out a second, unrelated exceptional character is exactly what (Pg') supplies," conflates "ruling out a second zero above Page's own threshold" with "ruling out any exceptional zero for a second character," which is the precise error identified above. I agree this is a genuine miss by that check, not merely an unexamined point: the check considered the question (it discusses exactly this cross-character reasoning) and reached the wrong conclusion, rather than leaving it unresolved.
No defect found in section 2 or 2.1 beyond confirming the correction already recorded there.
(5) Section 3: the budget (7)
Verdict: confirmed, with one defective (but non-binding) intermediate term inherited from MAJOR_ARC_EXPLICIT.md (6).
Recomputing the seven exponents at $\sigma=2c_0$ (review_a0082_budget_check.py, symbolic):
e^{-2c0 sqrt l} -> 2 c0
R^-1 -> 2 c0
R^2 e^{-(2c/sigma) sqrt l} -> c/c0 - 4 c0
R e^{-(c/(2 sigma)) sqrt l} -> c/(4 c0) - 2 c0
R e^{-(b/(2 sigma)) sqrt l} -> b/(4 c0) - 2 c0
R^-3 -> 6 c0
R^3 e^{-2 sqrt l /3} -> 2/3 - 6 c0
R^4 e^{-(2/3+2 c0) sqrt l} -> 6 c0 - 2/3These match the document's printed list exactly, and the resulting sufficient conditions ($c_0^2\le c/6$, $c_0^2\le c/16$, $c_0^2\le b/16$, $c_0\le1/12$) are what (H'') encodes once the redundant $c_0^2\le c/6$ (implied by the stricter $c_0^2\le c/16$, since $c/16<c/6$) is dropped, plus the Page-matching $c_0^2\le b/8$ (also implied by the stricter $c_0^2\le b/16$) and TT's $c_0\le c_P$. I confirm that under (H'') only the first bracket term ($e^{-2c_0\sqrt\ell}$) and the $R^{-1}$ minor-arc term actually achieve the exponent $2c_0$; every other term is strictly above it, so nothing besides these two can bind, exactly as section 3 claims.
The defect. The term "$R^2e^{-(2c/\sigma)\sqrt\ell(1+o(1))}$", copied from MAJOR_ARC_EXPLICIT.md equation (6), does not follow from squaring its equation (5) by the method that document states ("using $\sqrt r\le \sqrt R$, $\sqrt{R/r}\cdot\sqrt r=\sqrt R$"). Carrying that method through: $\Upsilon(r)=\ell^2[e_1\sqrt{R/r}+e_2+R^{-2}]$ with $e_1=e^{-(c/\sigma) \sqrt\ell(1+o(1))}$, $e_2=e^{-(c/(4\sigma))\sqrt\ell(1+o(1))}$, so $\sqrt r\,\Upsilon(r)=\ell^2[\sqrt R\,e_1+\sqrt r\,e_2+\sqrt r\,R^{-2}]$ using the exact identity $\sqrt r\cdot\sqrt{R/r}=\sqrt R$ (true for every $r$, not merely a bound); bounding the remaining two terms by $\sqrt r\le \sqrt R$ gives $\sup_r\sqrt r\,\Upsilon(r)\ll\ell^2\sqrt R(e_1+e_2)+\ell^2 R^{-3/2}$. Squaring: $(e_1+e_2)^2$ is dominated by $e_2^2=e^{-(c/(2\sigma)) \sqrt\ell(1+o(1))}$ (since $c/(4\sigma)<c/\sigma$, $e_2$ decays slower than $e_1$), so the whole first-order contribution of $\Upsilon$ is $\ll Re^{-(c/(2\sigma))\sqrt\ell(1+o(1))}$, the very term already listed second in (6); the $e_1^2$ piece it contains is $Re^{-(2c/\sigma)\sqrt\ell(1+o(1))}$ (coefficient $R^1$, not $R^2$), and it is in any case smaller than the $e_2^2$ term and does not need to be listed separately. So the corrected squared bound has no separate $R^2e^{-(2c/\sigma)\sqrt\ell}$ term; at most a redundant $Re^{-(2c/\sigma)\sqrt\ell}$ could be written, itself dominated by the adjacent $Re^{-(c/(2\sigma))\sqrt\ell}$ term.
I verified symbolically that this does not change the theorem. With the term corrected to $R^1$ (the largest defensible reading), its exponent at $\sigma=2c_0$ becomes $c/c_0-2c_0$ (condition $c_0^2\le c/4$ for it to be $\ge2c_0$), weaker than even the document's own (already non-binding, as shown above) $c_0^2\le c/6$; either way this term's condition is implied by the already-present, strictly stronger $c_0^2\le c/16$ from the adjacent term. The printed $R^2$ version is a valid, non-tight, upper bound on the correct $R^1$ term (since $R^2\ge R^1$ for $R\ge1$), so equation (7)'s inequality remains true as stated; only the specific derivation quoted for this one bracketed term is unsound as a matter of arithmetic, and it affects nothing downstream: not the boxed (S$'''$), not the constant $2c_0$, and not the content of (H'') as actually used. This is the same kind of error as the one MAJOR_ARC_EXPLICIT_REVIEW.md found and fixed in that document's model term, except that here the fix does not change any displayed inequality, only its justification, since the printed coefficient is a valid over-estimate rather than an under-estimate. The corrected statement is: the term should read $R\,e^{-(2c/\sigma)\sqrt\ell(1+o(1))}$ (and is in fact subsumed by the adjacent $R\,e^{-(c/(2\sigma))\sqrt\ell (1+o(1))}$ term and need not be listed), not $R^2e^{-(2c/\sigma)\sqrt\ell (1+o(1))}$. The same correction applies to the corresponding term in MAJOR_ARC_EXPLICIT.md equation (6), where it is likewise non-binding (the condition it induces, $c_0^2\le2c/3$, is weaker than that document's own binding $c_0^2\le c/4$ from the adjacent term).
I note for the record that MAJOR_ARC_EXPLICIT_REVIEW.md item (4) states it "recomputed the six exponents from (6) as printed" and found they "match the document's list exactly"; that check verified the arithmetic from (6) to the verbal exponent list, not the squaring step from (5) to (6), so it did not have occasion to catch this. I do not read this as a miss on that check's part, since it did not claim to have re-derived (6) from (5).
The two-sided compatibility question. I checked whether $\sigma=2c_0$ (the upper-bound construction) is compatible with $\kappa$ near $c_0$ in the Proposition's window $c_0<\kappa<\min(\sqrt c,\sqrt b)-3\varepsilon$. Under (H''), $c_0\le\min(\sqrt c/4,\sqrt b/4)$, so $c_0$ sits comfortably below $\min(\sqrt c,\sqrt b)$ (by a factor of at least 4), and $c_0\le1/12 <1/3$, so for $\kappa$ slightly above $c_0$ and $\varepsilon$ small the window's constraints ($\kappa<\min(\sqrt c,\sqrt b)-3\varepsilon$, $\kappa+3\varepsilon<1/3$) are satisfied with room. I also checked that the upper-bound construction does not implicitly assume the absence of such a zero: in MAJOR_ARC_EXPLICIT.md section 3's case analysis, a real zero $\tilde\beta=1-\kappa/\sqrt\ell$ with $\kappa>c_0$ of a character $\chi\ne \chi_e$ falls into the "not TT-exceptional" case (since $\kappa>c_0$ means $\tilde\beta<1-c_0/\sqrt\ell$), which is handled by the same $\Upsilon(r)$ machinery used throughout and gives a contribution $\ll N^3e^{-2\kappa \sqrt\ell}(\cdot)\le N^3e^{-2c_0\sqrt\ell}(\cdot)$, i.e. the upper bound remains valid (with room to spare) whether or not such a zero exists. The two sides are therefore consistent, not merely juxtaposed.
"Against the previous budgets." I traced $(k,j,\gamma)$ through the chain: ENDPOINT_BOUND.md (18) balances a major-arc term $N^3R^7 e^{-2\gamma\sqrt\ell}$ against $N^3R^{-1/6}$, matching $(7,1/6,\gamma)$; ARC_SPLIT_BUDGET.md (12) balances $N^3R^2e^{-2\gamma\sqrt\ell}$ against $N^3R^{-1}$, matching $(2,1,\gamma)$ (confirmed by that document's own "Against the previous budget" paragraph, which states $k=7,j=1/6\to k=2, j=1$); MAJOR_ARC_EXPLICIT.md (6)'s binding term is $Re^{-2c_0\sqrt\ell}$ (the exceptional-zero term, $k=1$, unaffected by the defect found above, which concerns a different, non-binding bracket term), against the same $R^{-1}$ minor-arc term inherited unchanged, matching $(1,1,c_0)$. The sequence $(7,1/6,\gamma)\to(2,1,\gamma)\to(1,1,c_0)$ as printed is accurate.
Summary
| Item | Verdict |
|---|---|
| (1) Dyadic reduction | confirmed |
| (2) Farey/Gallagher obstruction | confirmed |
| (3) Alternate dissection at $Q_2$ | confirmed |
| (4) The mechanism (section 2, 2.1) | confirmed |
| (5) The budget (7) | confirmed, one non-binding defect |
The one defect (section 3's $R^2e^{-(2c/\sigma)\sqrt\ell(1+o(1))}$ term, inherited from MAJOR_ARC_EXPLICIT.md (6)) does not touch the boxed result (S$'''$), the exponent constant $2c_0$, or the content of (H'') as actually used to derive it, because the term's own induced condition is strictly weaker than another condition already required by (H'') from an adjacent term. Sections 1.2 and 2, named in the assignment as the most likely places for a defect, held up under independent hand computation and two freshly written numerical checks; the defect that did surface is in section 3, in an equation imported from a different, previously-reviewed document, at a step that document's own earlier review did not re-derive.