Correction notice, 2026-09-11. Section 2.3's conclusion that no choice of the Bonferroni cutoff \(m\) works at \(\kappa=1/2\) does not hold. That argument estimates \(H_Z=\sum_{p<Z}1/p\) through ENDPOINT_BOUND.md section 2's elementary bound \(H_Z\le1+\log Z\), giving \(H_Z\sim\ell^\kappa\); Mertens' second theorem gives \(H_Z=\log\log Z+M+O(1/\log Z)\sim\kappa\log\ell\), a double logarithm where the bound supplies a power. With the true \(H_Z\) and a cutoff retuned to \(m\asymp\ell^\kappa/\log\ell\), the same device reaches \(D_0=N^{o(1)}\) at \(\kappa=1/2\). Section 2.4's measurements are unaffected: they evaluate the fixed cutoff \(m=2\lceil\sqrt\ell\rceil\), which is genuinely pinned at \(\log D_0/\log N=2\). See ENDPOINT_SHARP.md section 2, which attains the endpoint and carries the retuning, the correction, and its finite checks. The rest of this file, including sections 2.1, 2.2 and 1's reading of Remark 2.8, stands.
2026-09-10. Base: the chain SIEGEL_UNIFORMITY.md -> EXCEPTIONAL_ENERGY.md -> LOCALIZED_MIXED_ENERGY.md -> ENDPOINT_BOUND.md (reviewed in ENDPOINT_BOUND_REVIEW.md). All of those files are unchanged by this pass.
Verdict. No. Substituted verbatim as printed in the source, the larger model parameter breaks the chain at one identifiable place: ENDPOINT_BOUND.md's Section 2 (the bounded-order/Bonferroni divisor approximation), specifically its fixed cutoff \(m=2\lceil\sqrt{\log N}\rceil\) feeding into the \(N^2D_0\sqrt Z\) term of equation (18). With that cutoff unchanged, \(D_0=Z^m\) stops being \(N^{o(1)}\) and becomes \(\asymp N^2\), so the single term \(N^2D_0\sqrt Z\) alone reaches order \(N^{4+o(1)}\), which is larger than the trivial bound on \(E_{\rm corr}\) itself. This is the first step in the chain that breaks.
Retuning \(m\) (rather than the whole architecture) recovers a chain that tolerates every \(Q=\exp((\log N)^\kappa)\) with \(\kappa<1/2\) strictly, at the cost of reverting from ENDPOINT_BOUND.md's sharp single exponent to an \(\epsilon\)-indexed family, the same shape the pre-endpoint SIEGEL_UNIFORMITY.md result already had at \(\kappa<1/10\): \[ E_{\rm corr}(N)\ll_\kappa N^3\exp(-c_\kappa(\log N)^\kappa), \qquad\text{every fixed }0<\kappa<\tfrac12. \tag{A} \] The value \(\kappa=1/2\) itself is not attained by this route; the supremum \(1/2\) is open. This matches, rather than beats, what Remark 2.8's own wording already concedes about Proposition 2.2 under the substitution: an exponent "\(1/2-\epsilon\)", not a sharp \(1/2\). Either way, (A) is still an exponential-in-a-fractional-power-of-\(\log N\) saving over \(N^3\), not the near-quadratic \(N^{2+\epsilon}\) that CORRECTED_RH_BRIDGE.md needs for RH-sufficiency. That target remains completely out of reach on this route, independent of the wall identified here.
This is a handwritten deduction, checked against the source text quoted below and against one finite numerical illustration; it is not a formal verification.
1. What Remark 2.8 actually says
Fetched from the paper itself (arXiv:2107.02158v4, HTML rendering, Section 2, printed immediately after Proposition 2.2):
Proposition 2.2 states, for every arithmetic progression \(P\subset[N]\), \[ \sum_{n\in P}(\mu(n)-\mu_{\rm Siegel}(n))\ll N\exp(-c\log^{1/10}N), \] and the analogous bound (2.5) for \(\Lambda-\Lambda_{\rm Siegel}\), both using the default Siegel-model parameter \(Q=\exp((\log N)^{1/10})\) fixed earlier in Section 2 (project name \(Z\)).
Remark 2.8, quoted:
If one redefined the Siegel models \(\mu_{\rm Siegel},\Lambda_{\rm Siegel}\) by assigning the parameter \(Q\) the larger value \(\exp((\log N)^{1/2})\), one could inspect that the exponent of logarithm in (2.12) and (2.13) (and in particular in Proposition 2.2) could be increased to \(1/2-\epsilon\), hence essentially matching the shape of the error term in the classical prime number theorem. For this modification, one would have to tweak the exponents in Section 5 a little; in particular in Proposition 5.2 the exponents \(3/5\) and \(4/5\) would have to be replaced with \(1/2\). As the precise value of the exponent has very little influence on our bounds, we leave the details of this strengthening to the interested reader.
Two things follow directly from this text, before any of the project's own machinery is touched.
(i) The claimed exponent already carries an \(\epsilon\)-loss. The remark's own phrase is "increased to \(1/2-\epsilon\)", the same shape as (2.12)-(2.13) (which are themselves \(\epsilon\)-indexed consequences of the general higher-order theorem, TT Theorem 2.7). Proposition 2.2 as printed and used by ENDPOINT_BOUND_REVIEW.md is not an \(\epsilon\)-family: it is a single bound with a single fixed exponent \(1/10\) and a single constant \(c\); ENDPOINT_BOUND_REVIEW.md's checked dependency list says explicitly "[t]he endpoint exponent is obtained from this proposition. It is not obtained by setting \(\kappa=1/10\) in Theorem 2.7." Nothing in Remark 2.8 asserts that the strengthened statement keeps Proposition 2.2's sharp, non-\(\epsilon\) form; its own wording says the opposite. So even before checking the project's downstream files, citing "Proposition 2.2 at \(Q=\exp((\log N)^{1/2})\)" as a drop-in replacement for [S1]'s (1) in ENDPOINT_BOUND.md Section 1 already means replacing a sharp bound by a family of bounds indexed by \(\epsilon\) -- which is exactly the family shape that ENDPOINT_BOUND.md's whole point was to escape.
(ii) The remark is an unverified aside, not a proved theorem. The authors explicitly did not carry out the retuning of their own Section 5 (Proposition 5.2, exponents \(3/5,4/5\to1/2\)) that the modification requires; they "leave the details... to the interested reader." So even taking (i) at face value, "Proposition 2.2 at \(Q=\exp((\log N)^{1/2})\)" is not a citable statement of the source paper -- it is a claim the paper says is plausible and does not prove. Anything built on it inherits that gap on top of whatever the project's own files require.
Point (ii) is a caveat about the source, separate from the question the task asks: whether the project's own chain (SIEGEL_UNIFORMITY.md through ENDPOINT_BOUND.md), which largely does not reuse Proposition 2.2's proof machinery but instead redoes an independent two-form sieve computation around the same parameter \(Z=Q\), survives the substitution. Section 2 below answers that question directly and finds a harder, load-bearing break inside the project's own arithmetic, independent of (i)-(ii).
2. Tracing \(t=\ell^{1/10}\to\ell^\kappa\) through each file
Write \(\ell=\log N\), and replace every occurrence of \(Z=\exp(\ell^{1/10})\) in the chain by \(Z=\exp(\ell^\kappa)\), \(t=\ell^{1/10}\) by \(t=\ell^\kappa\), tracking each place a specific number (not just the symbol \(1/10\)) was chosen because of the specific value \(\kappa=1/10\).
2.1 SIEGEL_UNIFORMITY.md -- survives up to and including \(\kappa=1/2\)
Section 3's sieve estimate (10)-(11) uses TT Lemma 5.1 (the classical fundamental lemma of the sieve, not paper-specific machinery) at level \(D=\lfloor N^{1/4}\rfloor\), a level chosen freely by the project, not inherited from \(Z\). Its exponent parameter is \(s=\log D/\log Z\asymp\ell^{1-\kappa}/4\); the stated remainder \(O(Ne^{-cL^{9/10}})\) becomes \(O(Ne^{-c\ell^{1-\kappa}})\). For any fixed \(\kappa<1\), \(s\to\infty\) and the fundamental lemma applies; at \(\kappa=1/2\) the remainder is \(O(Ne^{-c\ell^{1/2}})\), matching (not beating, but not breaking) the target order. The other error term, \(D(1+\log D)=O(N^{1/4}\log N)\), is independent of \(Z\) entirely and stays negligible. Section 1's use of TT (2.13) (the general Theorem 2.7 consequence, at \(k=2\)) is exactly the kind of statement Remark 2.8 targets, so it inherits caveats (i)-(ii) above but no new project-level break; it was already an \(\epsilon\)-family before this substitution. Section 2, "A uniform sieve calculation," and Sections 4-5 (exceptional term bookkeeping) use \(q<Z<\sqrt N\); this remains true for any fixed \(\kappa<1\). No numeric parameter here is tied specifically to \(1/10\).
2.2 EXCEPTIONAL_ENERGY.md and LOCALIZED_MIXED_ENERGY.md -- survive
EXCEPTIONAL_ENERGY.md Section 5's range bookkeeping, \((\log q)^{10}<\log N<(c_0/\delta)^{10}\) (its equation 27), comes from inverting \(q<Z=\exp(\ell^{1/10})\); at general \(\kappa\) the exponent \(10\) becomes \(1/\kappa\) (e.g. \(2\) at \(\kappa=1/2\)). This is a bookkeeping substitution, not a break: nothing else in that file's period-mass or lower-bound argument (Sections 3-4) depends on \(1/10\) specifically, only on \(q<Z<N\).
LOCALIZED_MIXED_ENERGY.md's own fixed exponents -- the rational-denominator cutoff \(R=N^\rho\) at \(\rho=1/10\) in its equation (13), and the short-kernel cutoff \(L_0=\lfloor N^{3/5}\rfloor\) in (14) -- are chosen freely against \(N\), not against \(Z\); they do not need retuning. Its long-window estimate (19), \(T_N(0)\ll N^3e^{-c\mathcal L(N)}\) with \(\mathcal L(N)=\ell^{3/5}(\log\ell)^{-1/5}\), is the classical prime number theorem rate (TT Theorem 1.3(i)) and does not involve \(Z\) at all; its exponent \(3/5\) already exceeds \(1/2\), so it is not the bottleneck at \(\kappa=1/2\) either. This file's own stated bottleneck (Section 6) is the inherited \(O(N^{3/2}e^{-c\ell^{1/10}})\) model-comparison term from SIEGEL_UNIFORMITY.md/ENDPOINT_BOUND.md, i.e. it is downstream of whatever those two files produce, not an independent obstruction.
2.3 ENDPOINT_BOUND.md Section 2 -- breaks at \(\kappa=1/2\)
This is where the substitution fails, in a part of the chain that is the project's own construction, not TT's. Section 2 approximates \(\nu(n)=b_Z1_{(n,P(Z))=1}\) by the bounded-order Möbius sum \(B_m(n)=\sum_{d\mid P,\,d\mid n,\,\omega(d)\le m}\mu(d)\), with \[ m=2\lceil\sqrt\ell\rceil,\qquad D_0=Z^m, \] fixed once, not as a function of \(\kappa\). Equation (5)-(6) bounds the approximation error by \(N H_Z^{m+1}/(m+1)!\) with \(H_Z\le1+\log Z=1+\ell^\kappa\), and the write-up's own derivation ("the elementary estimate \(H_Z\le1+\log Z\) and \(k!\ge(k/e)^k\)") only checks this decays for \(\kappa=1/10\). The same computation at general \(\kappa\), with \(m=2\lceil\sqrt\ell\rceil\) held fixed, gives ratio \[ \frac{H_Ze}{m+1}\ \sim\ \frac e2\,\ell^{\kappa-1/2}, \] which \(\to0\) (so the Bonferroni error genuinely decays) only for \(\kappa<1/2\); at \(\kappa=1/2\) it tends to the constant \(e/2\approx1.36>1\), so \((H_Ze/(m+1))^{m+1}\) does not shrink as \(N\to\infty\) -- the truncation error stops being an error term at all. Simultaneously, \[ \log D_0 = m\log Z = 2\lceil\sqrt\ell\rceil\cdot\ell^\kappa \ \sim\ 2\ell^{\,\kappa+1/2}. \] At \(\kappa=1/10\) this is \(2\ell^{3/5}=o(\ell)\), exactly what ENDPOINT_BOUND_REVIEW.md's budget section records (\(\log(D_0\sqrt Z)=2\ell^{3/5}+O(\ell^{1/10})=o(\ell)\)), keeping \(D_0=N^{o(1)}\) and the term \(N^2D_0\sqrt Z\) at \(N^{2+o(1)}\), comfortably inside the \(N^3\) budget of equation (18). At \(\kappa=1/2\) this becomes \(\log D_0\sim2\ell\), i.e. \(D_0\asymp N^2\), and the single term \(N^2D_0\sqrt Z\) in (18) reaches order \(N^{4+o(1)}\) -- larger than \(N^3\) outright, let alone smaller than \(E_{\rm corr}\)'s target order. This is the first place the chain breaks: the fixed numeric cutoff \(m=2\lceil\sqrt\ell\rceil\) was tuned for \(\kappa=1/10\) and is simply the wrong cutoff at \(\kappa=1/2\), not a small constant-factor loss but a qualitative one (the divisor-approximation device stops producing any saving at all).
This does not improve with a different (untuned) choice of \(m\). Write \(m=\lambda H_Z\) for a parameter \(\lambda\) to be chosen (constant or slowly growing). Decay of the Bonferroni error needs \(H_Ze/m=e/\lambda\to0\), i.e. \(\lambda\to\infty\), or at least \(\lambda>e\) for a nontrivial fixed bound. But then \[ \log D_0=m\log Z=\lambda H_Z\log Z\sim\lambda\,\ell^{2\kappa}/\ell^{\,?} \] more precisely, with \(H_Z\sim\ell^\kappa\) and \(\log Z=\ell^\kappa\), \(\log D_0\sim\lambda\ell^{2\kappa}\), so \(A(N):=\log D_0/\log N\sim\lambda\ell^{2\kappa-1}\). For \(\kappa<1/2\), \(\ell^{2\kappa-1}\to0\), so \(\lambda\) can be sent to infinity slowly enough that \(A(N)\to0\) and \(\lambda>e\) eventually -- both requirements are simultaneously satisfiable, and \(D_0=N^{o(1)}\) is recoverable with a retuned \(m\). At \(\kappa=1/2\) exactly, \(\ell^{2\kappa-1}=\ell^0=1\) identically, so \(A(N)\sim\lambda\) is pinned to whatever constant \(\lambda\) is chosen; decay of the Bonferroni error forces \(\lambda>e\), so \(A(N)>e\) unavoidably, giving \(D_0\gtrsim N^e\) and \(N^2D_0\sqrt Z\gtrsim N^{2+e+o(1)}\gg N^3\) no matter how \(m\) is chosen. No choice of \(m\) makes this device produce a saving at \(\kappa=1/2\). For any fixed \(\kappa<1/2\) it works, with the achievable margin \((1/2-\kappa)\) shrinking to zero as \(\kappa\to1/2\).
2.4 Numerical illustration
artifacts/endpoint_half/check.py (one thread, closed form, no distribution-theory input) evaluates \(A(N)=\log D_0/\log N\) for the fixed cutoff \(m=2\lceil\sqrt\ell\rceil\) exactly as printed in ENDPOINT_BOUND.md, at \(\kappa\in\{0.1,0.3,0.45,0.49,0.5,0.6\}\) and \(N=10^{20},10^{100},10^{1000},10^{10000}\), and separately finds, at fixed \(N=10^{60}\), the smallest \(m\) for which the raw ratio \(H_Z^{m+1}/(m+1)!\) first drops below 1, for the same \(\kappa\) values. Both confirm Section 2.3 quantitatively:
- \(\kappa=0.1\): \(A(N)\) falls from \(0.45\) to \(0.036\) as \(N\) runs \(10^{20}\to10^{10000}\) (heading to \(0\), as expected: \(o(1)\)).
- \(\kappa=0.3\): \(A(N)\) falls from \(0.96\) to \(0.27\) (still heading to \(0\), more slowly).
- \(\kappa=0.45\): \(A(N)\) falls from \(1.70\) to \(1.21\) over the same range -- decreasing, consistent with \(o(1)\) asymptotically (\(\ell^{\kappa-1/2}=\ell^{-0.05}\) decays extremely slowly), but nowhere near \(0\) at any \(N\) that could ever be written down. The fixed \(m=2\lceil\sqrt\ell\rceil\) is asymptotically fine but practically the wrong cutoff as \(\kappa\to1/2\); a retuned, \(\kappa\)-dependent \(m\) is genuinely needed, not optional, well before \(\kappa=1/2\) is reached.
- \(\kappa=0.49\) and \(\kappa=0.5\): \(A(N)\) sits at \(1.8\)-\(2.0\) and \(2.0\)-\(2.1\) respectively across the entire tested range, matching the closed form \(A\sim2\ell^{\kappa-1/2}\to2\ell^0=2\) exactly at \(\kappa=1/2\): no decay, a genuine power \(D_0\asymp N^2\).
- \(\kappa=0.6\): \(A(N)\) grows with \(N\) (\(3.0\to5.5\)), matching \(2\ell^{0.1}\to\infty\): the device is not merely stalled but actively diverges past \(\kappa=1/2\).
- The minimal-\(m\) table, at fixed \(N=10^{60}\), shows \(m_{\min}/H_Z\) sitting at \(1.9\)-\(2.6\) across every tested \(\kappa\) (pre-asymptotic; the true threshold is \(e\approx2.718\)), and the resulting \(A\) at that minimal \(m\) climbing from \(0.06\) at \(\kappa=0.1\) to \(2.43\)-\(2.64\) at \(\kappa=0.49,0.5\) -- direct confirmation that the minimal viable \(m\) for decay forces \(A\) up toward the constant \(e\), not down toward \(0\), exactly where the algebra of Section 2.3 says it must.
Raw output is saved in artifacts/endpoint_half/result.json.
3. What this means for the budget (18) and the m = 2⌈√(log N)⌉ / D_0√Z term specifically
The task's own question was precisely whether \(m=2\lceil\sqrt{\log N}\rceil\) and the \(D_0\sqrt Z\) term survive. They do not, verbatim, at \(\kappa=1/2\): the single term \(N^2D_0\sqrt Z\) in equation (18) of ENDPOINT_BOUND.md moves from \(N^{2+o(1)}\) (comfortably absorbed) to at least \(N^{4+o(1)}\) (dominating the entire budget and exceeding even the trivial \(O(N^3)\) bound on \(E_{\rm corr}\) coming from \(|r(h)|,|C_N(h)|\le O(N)\) pointwise). Every other term in (18) -- \(N^3R^7e^{-2\gamma t}\), \(N^3R^{-1/6}\), \(N^{14/5}\), and the two remaining exponential terms -- either scales benignly with \(t=\ell^\kappa\) (the major/minor arc terms, Sections 3-4 of ENDPOINT_BOUND.md, which actually improve as \(\kappa\) grows, since \(R=\exp(\sigma t)\) grows faster) or is a fixed power of \(N\) below \(3\) regardless of \(\kappa\) (the Montgomery-Vaughan \(N^{14/5}\) term, from a classical estimate that does not reference \(Z\) at all). The Bonferroni/\(D_0\sqrt Z\) term is the only one that is qualitatively, not just quantitatively, broken by the substitution.
4. Largest \(Q\) the chain tolerates, and the exponent it yields
Retuning \(m\) as a function of \(\kappa\) -- e.g. \(m=\lceil\lambda(N)\cdot\ell^{1/2-\kappa}\cdot\ell^\kappa\rceil\) for any slowly growing \(\lambda(N)\to\infty\) with \(\lambda(N)>e\) eventually, matching the derivation in Section 2.3 -- and correspondingly retuning EXCEPTIONAL_ENERGY.md's range bookkeeping (Section 2.2 above), the chain tolerates \[ Q=\exp((\log N)^\kappa)\qquad\text{for every fixed }0<\kappa<\tfrac12, \] with \(\kappa=1/2\) itself excluded (Section 2.3 shows no choice of \(m\) works there). This yields, chain-wide, \[ E_{\rm corr}(N)\ll_\kappa N^3\exp(-c_\kappa(\log N)^\kappa), \qquad\text{every fixed }0<\kappa<\tfrac12, \] i.e. exactly statement (A) from the top of this file: an improvement over the pre-endpoint family (which only reached \(\kappa<1/10\)), but a reversion from ENDPOINT_BOUND.md's sharp single exponent \(\kappa=1/10\) back to an \(\epsilon\)-indexed family, capped strictly below \(1/2\). Reaching a sharp \(\kappa=1/2\) endpoint -- the literal reading of the task's target \(E_{\rm corr}(N)\ll N^3\exp(-c(\log N)^{1/2-\epsilon})\) with a single substituted budget -- is not available from this chain as constructed, because its own divisor-approximation device (Section 2.3) has no slack left at exactly \(\kappa=1/2\), independent of whether Remark 2.8's unverified claim about Proposition 2.2 itself (Section 1, points (i)-(ii)) is granted or not.
Nothing here excludes a different proof of the minor-arc control TT's Proposition 2.2 supplies -- one that does not route through a bounded-order/Bonferroni divisor approximation of \(\nu\) at all -- from reaching \(\kappa\) closer to or at \(1/2\); Section 2.3 identifies a wall in this specific project construction, not a theorem that no construction can do better.
5. Scope
Written under hunts/prime_pair_error/ only. No file outside this directory was read or touched. No claim of proof, disproof, or external validation beyond the finite check in artifacts/endpoint_half/ is made.