Review of ENDPOINT_SHARP.md

Independent check, 2026-09-11. Read in order: ENDPOINT_SHARP.md, ENDPOINT_BOUND.md (section 2 and equations (5), (10), (13), (14), (17), (18)), ENDPOINT_HALF.md (including its correction notice), and SIEGEL_UNIFORMITY.md sections 3-5. Two scripts were written from scratch for this review and are not derived from endpoint_sharp_mertens_probe.py or endpoint_sharp_probe.py:

• review_a0075_mertens_check.py (item 1)
• review_a0075_character_check.py (item 2)

Both were run with /opt/zeta-venv/bin/python; raw output is reproduced inline below rather than saved to a separate JSON file.

Item 1: the Mertens correction, section 2.1

Verdict: defective, in the finite illustration only. The asymptotic claim, equation (A), is confirmed.

The algebra of equation (A) was rechecked by hand. Writing \(l=\log N\), \(m+1=\lceil(2c/\kappa)\,l^\kappa/\log l\rceil\), and using Mertens' theorem \(H_Z=\kappa\log l+M+o(1)\) at \(Z=\exp(l^\kappa)\):

\[ \log\frac{m+1}{eH_Z}=\kappa\log l-2\log\log l+O(1)=\kappa\log l\,(1+o(1)), \]

so the exponent in (5') is \((m+1)\cdot\kappa\log l(1+o(1))=2c\,l^\kappa(1+o(1))\ge c\,l^\kappa\) for large \(N\), and

\[ A(N)=\frac{m\log Z}{l}=\frac{m\,l^\kappa}{l} =\frac{(2c/\kappa)\,l^\kappa/\log l\cdot l^\kappa}{l} =\frac{2c}{\kappa}\cdot\frac{l^{2\kappa-1}}{\log l}(1+o(1)), \]

which is exactly the document's (A). At \(\kappa=1/2\) this is \(4c/\log l\to0\). This part is correct: the endpoint is attained, and the argument that no fixed \(m\) works past \(\kappa=1/2\) (ENDPOINT_HALF.md section 2.3) is correctly identified as an artifact of the elementary bound \(H_Z\le1+\log Z\), not a real obstruction. My own script reproduces ENDPOINT_SHARP.md's own section 7 item (1) table almost exactly, using an independent exact-prime sieve up to \(Z=e^{13}\) rather than reusing the project's probe:

logZ= 5  exact=1.8798  mertens=1.8709  elementary(1+logZ)=6
logZ= 7  exact=2.2124  mertens=2.2074  elementary(1+logZ)=8
logZ= 9  exact=2.4598  mertens=2.4587  elementary(1+logZ)=10
logZ=11  exact=2.6599  mertens=2.6594  elementary(1+logZ)=12
logZ=13  exact=2.8266  mertens=2.8264  elementary(1+logZ)=14

(document: 1.880, 2.212, 2.460, 2.660, 2.827 against Mertens; matches to the stated precision.)

The defect. Section 7 item (2)'s table, and the two crossover sentences that quote it in section 2.1 ("\(A(N)<1\) from about \(\log N=400\)" and "\(A(N)<1/2\) only from about \(\log N=4.7\times10^4\)"), report a "retuned \(m\)" that is too small by exactly 2 at every row with \(\log N\in\{100,400,10^4,10^6\}\), and so does not actually meet the stated target decay \(\exp(-\sqrt l)\) at those rows. I verified this at \(\log N=100\) with an exact prime sieve up to \(Z=e^{10}\approx22026\) (2466 primes, 40-digit arithmetic, no Mertens approximation involved):

H_Z = 2.564650626927288538780604720642885262695
m=12: exponent = 8.10   (needs >= sqrt(100) = 10)   -- FAILS
m=13: exponent = 9.76   (odd, not admissible anyway) -- FAILS
m=14: exponent = 11.49  (needs >= 10)                -- first m that works

So the document's own \(m=12\) for this row gives decay only \(\exp(-8.10)\), not the claimed \(\exp(-\sqrt{100})=\exp(-10)\); the least even \(m\) that actually reaches the target is \(14\), not \(12\). The same pattern (document's \(m\) two below the true least even \(m\)) holds at \(\log N=400\) (document 20, true 22), \(10^4\) (document 62, true 64), and \(10^6\) (document 346, true 348; here I used Mertens' formula for \(H_Z\) since exact enumeration to \(Z=e^{1000}\) is not possible, matching what the document itself must have done at that scale). At \(\log N=10^{10}\) and \(10^{20}\) the document's numbers (16078 and \(6.2\times10^8\)) do match the true least even \(m\) exactly in my independent search, so the error is confined to the four smaller rows; the finite-precision search that generated section 7's table appears to have an off-by-one-step bug that only shows up before the asymptotic regime.

Correcting the table changes the reported \(A\) values (all understated by the same margin: \(1.4\) not \(1.2\) at \(\log N=100\); \(1.1\) not \(1.0\) at \(400\); \(0.64\) not \(0.62\) at \(10^4\); \(0.348\) not \(0.346\) at \(10^6\)) and moves both quoted crossovers later. Using the correct least-even-\(m\) search:

l=400     m=22   A=1.1000
l=676     m=26   A=1.0000   <- true crossover for A<1
l=47000   m=110  A=0.5074
l=53824   m=116  A=0.5000   <- true crossover for A<1/2
l=60000   m=120  A=0.4899

So: the claimed crossovers are not right. \(A(N)<1\) first holds around \(\log N\approx676\), not \(400\) (the document's own \(400\) row in fact has \(A=1.1>1\) once computed with an \(m\) that really meets the target); \(A(N)<1/2\) first holds around \(\log N\approx5.4\times10^4\), not \(4.7\times10^4\) (at \(4.7\times10^4\) the correct value is \(A\approx0.507>1/2\), just short). Both errors run in the same direction: the document is about 1.15-1.7x too optimistic about how soon the decay sets in.

This does not touch the two things that actually matter for the proof: equation (A) itself is algebraically correct (checked above), \(A(N)\to0\) as \(l\to\infty\) at \(\kappa=1/2\) is correct, and \(\kappa>1/2\) genuinely fails by the same equation (the numerator exponent \(2\kappa-1>0\) then, so \(A(N)\to\infty\)). The defect is confined to the specific integers in section 7's second table and the two sentences in section 2.1 that repeat them; the corrected statement is: \(A(N)<1\) from \(\log N\gtrsim676\), and \(A(N)<1/2\) from \(\log N\gtrsim5.4\times10^4\).

Item 2: the modulus factor, section 4, case \(q\nmid r\)

Verdict: confirmed.

Cancellation claim. I rebuilt real primitive characters from scratch (not using a library Dirichlet-character routine): the Jacobi symbol for odd squarefree conductors, and explicit \(\chi_4\), \(\chi_8^{\pm}\) for the two 2-power conductors, combined multiplicatively by CRT for conductors \(q=m\), \(4m\), \(8m\). For every such \(\chi\) of conductor \(q\le200\) (119 characters), every proper divisor \(g\mid q\), and every residue \(a\bmod g\) with \((a,g)=1\), I summed \(\chi(c)\) over the coset \(\{c\bmod q:(c,q)=1,\ c\equiv a\ (g)\}\): 3540 cosets tested, zero exceptions. This is 37x more cases than the document's own 96-coset check in section 7 item (3), and it agrees with the claim exactly: the sum always vanishes, because \(\chi\) primitive of conductor \(q\) cannot be trivial on \(K=\ker((\mathbb Z/q)^\to(\mathbb Z/g)^)\) when \(g<q\) (triviality there would make \(\chi\) induced from a character mod \(g\), contradicting primitivity of conductor \(q\)).

I also checked, separately, that the "\((a,g)>1\)" branch the document disposes of in one line is in fact unreachable rather than merely rare: since \(g=\gcd(q,r)\mid r\), \((a,r)=1\) forces \((a,g)=1\) always (a common factor of \(a\) and \(g\) would be a common factor of \(a\) and \(r\)). Brute force over \(q,r<60\) and \(a\) up to \(65\) found 0 violations, as expected. This is a harmless redundancy in the text, not an error.

Error bound (4.1)-(4.2). I redid the algebra by hand. From \(X=u/M\), \(W=V(Z)M/\phi(M)\), \(M=rq/g\):

\[ \frac{\phi(q)}{\phi(g)}XW\le\frac{q}{g}\cdot\frac{ug}{rq}\cdot V(Z)\frac M{\phi(M)} =\frac ur\,V(Z)\frac M{\phi(M)}\ll\frac ur\,V(Z)\log l, \]

using \(M\le rq<Z^2\) so \(M/\phi(M)\ll\log\log M\ll\log l\); this matches the document's bound exactly. Continuing with \(u/r\le N\), \(bV(Z)=1\), \(b\ll\sqrt l\), \(q<Z=e^{\sqrt l}\), and the total-variation bound on \(u\mapsto u^{\beta-1}\):

\[ b\Big|\int_1^yu^{\beta-1}dT(u)\Big|\ll N(\log l)e^{9-s}+bqD_0\ll N(\log l)e^{9-s}+l^{1/2}e^{\sqrt l}D_0, \]

again matching (4.2) exactly. Absorption at both level choices checks out asymptotically: with \(D_0=N^{o(1)}\), the second term is \(N^{o(1)}e^{(1+o(1))\sqrt l}\), and since \(N=e^l\) with \(l\gg\sqrt l\) this is \(\ll N\exp(-\gamma\sqrt l)\) for any fixed \(\gamma\) once \(N\) is large; with the \(\beta\)-sieve's \(D_0=N^{1/2}\), the second term is \(N^{1/2+o(1)}e^{\sqrt l}=e^{l/2+O(\sqrt l)}\), again \(\ll e^{l-\gamma\sqrt l}\) for large \(l\) since \(l/2<l(1-o(1))\). Both hold with a full power of \(N\) to spare, as claimed.

I looked for other places in section 4 where a factor of \(q\), \(\phi(q)\), or a residue-class count could have been dropped: the untwisted part (\(b\prod_{p<Z,p\nmid r}(1-1/p)=r/\phi(r)\)), the \(q\mid r\) twisted part (single residue class, no extra factor needed), and the assembly into (1') at the end of section 4 all carry their multiplicities correctly. No further dropped factor found.

Item 3: are \(E_{\rm corr}^{(Z)}\) at the two values of \(Z\) kept separate

Verdict: confirmed.

Every one of the twelve places ENDPOINT_SHARP.md mentions ENDPOINT_BOUND.md (lines 5, 6, 28, 39, 47, 68, 91, 95, 111, 136, 170, 202, 216, 351, 381, 391 by grep) either (a) reuses the generic architecture with the parameter substituted (sections 2-3, which is legitimate: it is the same lemma applied at a different \(t\) and \(D_0\), not a claim that the two final bounds are the same object), or (b) explicitly disclaims comparability (the boxed-result discussion at the top, and section 6's "For the original \(E\)" paragraph, which states outright that the two versions are "not comparable term by term" because the exceptional class differs). I found no place where a bound proved at one \(Z\) is used to conclude something about the corrected quantity at the other \(Z\), and no place where the two are silently identified.

The description of the two exceptional-class ranges is also right: going from \(Z_1=\exp(l^{1/10})\) to \(Z_2=\exp(\sqrt l)\), the conductor range \(q<Z\) widens (\(Z_2>Z_1\)) while the exceptional threshold \(\beta>1-c_0/\log Z\) tightens (\(c_0/\log Z_2<c_0/\log Z_1\), so fewer zeros qualify as exceptional at the larger \(Z\)). Because \(r(h)\) is fixed (built from the true \(\Lambda\)) but \(C_N\) depends on \(Z\) through both \(\nu\) (support of the sieve, present even with no exceptional zero at all) and the exceptional term when one exists, \(E_{\rm corr}^{(Z_1)}\) and \(E_{\rm corr}^{(Z_2)}\) are different quantities in general, including in the no-exception case; there is no generic inequality forcing one to dominate the other, and the document does not assert one. This is a correct scoping statement, not a proved non-domination theorem, and the document does not claim more than that.

Section 6's statement for the original \(E\) is correctly scoped: it gives the bound at this \(Z\), names the exceptional term's dependence on \(q,\beta\) explicitly (not deleted), and states plainly that it is "not comparable term by term" with ENDPOINT_BOUND.md section 7's version.

Item 4: input (1') from Davenport, Page, and the fundamental lemma

Verdict: unresolved for the primary-source check; the internal derivation logic is confirmed.

I do not have web access in this environment and did not check Davenport's Multiplicative Number Theory Chapter 20 or Page's 1935 theorem against a primary source; I can only say what I checked internally. What would settle this fully is fetching the actual chapter and theorem statement and comparing constants; I did not do that and am saying so plainly rather than guessing.

What I did check, against my own recollection of the standard statements (the general shape of the prime-number-theorem-in-progressions with an explicit exceptional-zero term, and Page's uniqueness theorem for a real exceptional zero among characters of bounded conductor), is that the document's citations (D) and (Pg) are the right shape for those classical results, and that the four-case derivation built on top of them is internally consistent:

• \((a,r)>1\): \(a(n)=0\) on \(B\) is correct (every \(n\in B\) shares a factor \(p\mid(a,r)\) with \(p<Z\), which is inside the sieve modulus \(P\)); the \(\Lambda\)-side bound \(\omega(r)\log y\) is valid, though loose (\(\omega(r)\ll\sqrt l\), not \(\ll l\), for \(r\le R<Z\); the document's \(\ll l^2\) bound still holds, just not tightly, and nothing downstream needs it tight).
• \((a,r)=1\), untwisted: the sieve computation \(b\prod_{p<Z,p\nmid r}(1-1/p)=r/\phi(r)\) is correct because every prime factor of \(r\) is \(<Z\) (since \(r\le R<Z\)).
• \((a,r)=1\), \(q\mid r\): the claim that Davenport's \(\chi_1\) (induced by a primitive \(\chi_1^*\) of conductor \(q_1\mid r\)) must coincide with the document's model character \(\chi\) follows from (Cmp)+(Pg): fixing \(c_0\le c_4\) makes any TT-shaped exceptional zero at this \(Z\) automatically the unique Page-exceptional zero, so if Davenport's middle term is present at all, its character is the same \(\chi\). Given that, \(\chi_1(a)=\chi(a)\) for \((a,r)=1\) is immediate from the definition of an induced character.
• \((a,r)=1\), \(q\nmid r\): checked in item 2 above.

On (Cmp) itself: \(c_0\) here is not a constant inherited from Tao and Teräväinen's own proof of Proposition 2.2 (that proposition is explicitly not used at this endpoint; see the document's own "Remark 2.8 is not used"). It only appears in this document's own definition of "exceptional" (the shape of TT Definition 2.1 is reused, not its proof), so constraining it to \(c_0\le c_4\) is a free choice available to whoever writes this argument, not a violation of an external constraint. I did not trace every other use of \(c_0\) across the rest of the project (that would require reading EXCEPTIONAL_ENERGY.md and LOCALIZED_MIXED_ENERGY.md in full, which the assigned reading list for this review does not include); ENDPOINT_HALF.md section 2.2 already traces the one place those two files use a \(Z\)-dependent range and finds it a bookkeeping substitution, not a break, which this document's section 5 relies on without re-deriving it.

Item 5: the budget assembly, equation (18')

Verdict: confirmed.

Term by term against ENDPOINT_BOUND.md equation (18), with \(t\to\sqrt l\) and \(D_0\to N^{A(N)}\):

• \(N^3R^7e^{-2\gamma t}\to N^3e^{-(2\gamma-7\sigma)\sqrt l}\): matches, using \(R=e^{\sigma\sqrt l}\).
• \(N^3R^{-1/6}\to N^3e^{-\sigma\sqrt l/6}\): matches.
• \(N^{14/5}\): unchanged, matches.
• \(N^2D_0\sqrt Z\to N^{2+A(N)}e^{\sqrt l/2}\): matches, using \(D_0=N^{A(N)}\), \(\sqrt Z=e^{\sqrt l/2}\).
• the two direct exponential terms carry over unchanged.

I rechecked the absorption conditions by hand. Requiring \(N^{2+A(N)}e^{\sqrt l/2}\le N^3e^{-c'\sqrt l}\) reduces to \((2+A(N))l+\sqrt l/2\le3l-c'\sqrt l\), i.e. \(A(N)\le1-(c'+1/2)/\sqrt l\), exactly the condition the document states; since \(A(N)=4c/\log l\to0\) this holds for large \(N\) (both sides converge, left to \(0\), right to \(1\)), and it would hold with more room using the \(\beta\)-sieve's fixed \(A(N)=1/2\) as noted. Requiring \(2\gamma-7\sigma>0\) with \(\sigma\le\gamma/20\) gives \(2\gamma-7\sigma\ge2\gamma-7\gamma/20=33\gamma/20>0\), so the first term does decay; \(\sigma/6>0\) trivially for the second. \(N^{14/5}=N^3e^{-l/5}\) is far below any \(N^3e^{-c'\sqrt l}\) rate since \(l/5\gg\sqrt l\) eventually, so calling it "a fixed power below \(N^3\)" understates how comfortably it is absorbed but is not wrong.

The four-input account of what fixes the exponent at \(1/2\) (divisor approximation via (A); the classical progression error \(\exp(-c\sqrt{\log y})\), which does not improve with \(\kappa\); the fundamental lemma parameter \(s=\log D_0/\log Z\), which stops growing fast enough past \(\kappa=1/2\); and the singular-series truncation, which is the only one of the four that improves with \(\kappa\)) is a correct restatement of what the preceding sections establish; it does not introduce a new equation to check beyond (18').

Summary

The one confirmed defect is narrow: section 7 item (2)'s table and the two sentences in section 2.1 quoting it understate the least even \(m\) needed to reach the stated target decay by 2 at \(\log N\in\{100,400,10^4,10^6\}\), which pushes the two named crossovers (\(A<1\), \(A<1/2\)) later than claimed (about \(\log N\approx676\) and \(\approx5.4\times10^4\), against the document's \(400\) and \(4.7\times10^4\)). Equation (A) itself, the claim that the endpoint \(\kappa=1/2\) is attained, and the claim that \(\kappa>1/2\) is not reachable by this device are all algebraically correct and unaffected by this table error.