← Session 36 — ii′-6-DL dog-leg + side bounds; Paper A review
My view: **the revised Paper A is dramatically stronger than the earlier version**. It now reads like a theorem paper rather than a research-status report. The most important improvement is that the formerly assumed contour bound is now presented as machine-proved, including the dog-leg repair and final fold to `sfBoundProp`. Appendix B makes that explicit and names the actual Lean capstones.
I would **not submit this exact PDF yet**, though. There are a few issues I would fix before calling it final.
* **The biggest issue is Step 5.** It says ordinary Keiper–Li positivity through \(5\times10^5\) comes from Misguich–Voros [10]. But reference [10] is explicitly *Discretized Keiper/Li approach to the Riemann hypothesis*. I would not allow that citation to carry the finite splice unless you have checked, theorem-by-theorem, that the \(5\times10^5\) computation is for the **ordinary \(\lambda_n\)** appearing in your theorem rather than a discretized/modified sequence. This is potentially the most important remaining paper-level vulnerability. If your actual Lean theorem instead takes `LiVerifiedUpTo` as an explicit certificate, state that directly and cite the real certificate.
* **The theorem statement and the declared trust boundary do not yet agree cleanly.** Theorem 2 says only “Assume RH verified to height \(T_0\)” before concluding positivity. But the abstract says the result is conditional on RH verification **and finite Li computations**, and Appendix B says the closed theorem has multiple named perimeter hypotheses. The mathematical theorem should expose its actual assumptions, or you should give a separate corollary explaining how every perimeter hypothesis is instantiated from cited results/certificates. Right now a referee could reasonably say that Theorem 2 claims less conditionality than the formal theorem actually has.
* **The “exactly eight named hypotheses” count needs checking.** The text says two computational inputs plus six analytic inputs, but the analytic list that follows visibly names three zero-count props, `NoLowZeros`, and `MZeroProp`—five items. This is exactly the sort of small discrepancy reviewers use to question the larger dependency audit. I would copy the actual final Lean signature verbatim into Appendix B and count from that, rather than describe it from memory.
* **Tone down two novelty statements.** The abstract currently says “This is the first assembled explicit chain of its kind” and “by Voros’s sharpening it is the maximal content of the Li route short of RH.” The introduction repeats the “maximal content” formulation. I would change “first” to something like “We have found no previous explicit assembly…” until the novelty audit is finished. And I would replace “maximal content” with the narrower, defensible statement that **eventual positivity of all sufficiently large Li coefficients would itself imply RH**, so a result short of RH cannot assert an infinite positive tail. That says exactly what Voros gives you without philosophically claiming that no stronger finite theorem could exist.
* **The F-D correction is mathematically important enough to deserve its own short subsection, not a dense footnote.** The present correction note packs F-A, F-B, F-D and F-E into one large footnote. I would promote the dog-leg portion into something like **“3.1 Repair of the right-horizontal estimate”**. State the failed implication
$$
e/2>1,
$$
then the exact deformation
$$
H_{\rm low}=H_{\rm high}-V_{\rm right}+V_{\rm left},
$$
followed by the safe high-tail contraction. This is one of the paper's most interesting contributions; burying it makes the proof harder to trust and harder to cite.
* **Be very precise about Lagarias.** Instead of “the right-tail charge … inherited from Lagarias’s Thm 6.1 is broken,” I would write something like: “The geometric-decrease conclusion following (6.12) is not implied by the displayed bounds when \(T\asymp\sqrt n\), since the resulting majorant ratio is \(e/2>1\). In the zeta specialization we replace this step by the following zero-free contour deformation.” That distinguishes an apparent proof gap from a claim that Lagarias’s theorem itself is false.
One thing I particularly like is the new Step 2. The displayed \(B(n)\) now visibly includes the honest \(4F(T)\), the two-unit count \(Z_2(T)\), and the dog-leg term `DL(n)`. That makes the paper substantially more trustworthy because the corrections are no longer hidden in the formal code.
The Appendix B section is also much better than before. Naming `partialFraction_identity`, the tight log-derivative bound, `sf_eq_resW1`, the dog-leg theorem, `LiSideBounds`, and finally `sfBoundProp` gives a skeptical reader an actual proof map. I would make that dependency chain even more visual in the final version:
$$
\text{partial fractions}
\to
S_f=\operatorname{Res}_1
\to
\text{rectangle identity}
\to
\text{four side bounds}
\to
\boxed{\texttt{sfBoundProp}}
\to
\boxed{\texttt{lambda\_pos\_finite\_height\_closed}}.
$$
I would also seriously consider shortening Section 2. The several pages on the Weil/Gabor certificate program are interesting, but they are only indirectly related to the Li theorem. For a referee evaluating this theorem, the strongest paper is probably one that gets to the Li coefficient machinery quickly. The certificate research could be summarized in a paragraph and moved to a companion paper unless a specific theorem in Section 3 actually depends on it.
Finally, update the front-page revision date. It still says “revised 2026-08-20,” even though the text clearly incorporates the much later dog-leg closure and `sfBoundProp` result.
Please revise Paper A based on my comments above.