I found two concrete issues worth fixing before you freeze Paper A. The important one is in the description of the zero-counting input. The paper currently writes, essentially, $$ N(T)=F(T)+S(T) $$ with $$ F(T)=\frac{T}{2\pi}\log\frac{T}{2\pi e}+\frac78, $$ and then attributes the \(0.137,0.443,1.588\) bound to \(S(T)\). Later Appendix B repeats that the three zero-count bounds arise from the “Riemann–von Mangoldt formula \(N(T)=F(T)+S(T)\).” That equality is **not exact with that \(F(T)\)**. The exact Riemann–von Mangoldt formula contains a small additional gamma-factor/remainder term. Trudgian, for example, gives an explicit relation of the form $$ \left| N(T)-\frac{T}{2\pi}\log\frac{T}{2\pi e}-\frac78 \right| \le |S(T)|+\frac{0.2}{T}. $$ ([MaRDI Portal][1]) More importantly, you don't actually need to express the perimeter through \(S(T)\) at all. Rosser's Theorem 19 directly provides the zero-counting estimate $$ \boxed{ \left| N(T)-P(T) \right| \le 0.137\log T +0.443\log\log T +1.588 } $$ for the appropriate range, where $$ P(T)= \frac{T}{2\pi}\log\frac{T}{2\pi e} +\frac78 . $$ That is exactly the kind of input your `ZeroCountProp`, `ZeroCountProp2`, and `ZeroCountGlobalProp` need. Modern formalization references also state Rosser's theorem directly in this \(N(T)-P(T)\) form. ([Alex Kontorovich][2]) So I recommend changing the paper from > “the Riemann–von Mangoldt formula \(N(T)=F(T)+S(T)\) with Rosser's explicit bound…” to something like: > “Rosser's explicit Riemann–von Mangoldt error bound > > $$ > |N(T)-P(T)| > \le 0.137\log T+0.443\log\log T+1.588, > $$ > > where > > $$ > P(T)=\frac{T}{2\pi}\log\frac{T}{2\pi e}+\frac78, > $$ > > together with the certified finite-range bounds described below.” Then derive the unit- and two-unit-window bounds directly from $$ N(T_2)-N(T_1) \le P(T_2)-P(T_1)+R(T_2)+R(T_1). $$ That is cleaner and avoids a technically false exact identity. There is also a **notation collision involving \(\Lambda_n\)**. You now use $$ \Lambda_n $$ for Johansson's Keiper normalization and correctly explain $$ \lambda_n=n\Lambda_n. $$ But footnote 3 then says the earlier Voros/Misguich computations concern “a deformed sequence \(\Lambda_n\), not the ordinary \(\lambda_n\).” That makes the same symbol \(\Lambda_n\) mean **two different sequences** within about a page: $$ \Lambda_n^{\rm Keiper} =\frac{\lambda_n}{n} $$ versus Voros's discretized/deformed coefficients. I would eliminate that immediately. For example, use $$ \kappa_n $$ for Johansson's Keiper-normalized coefficients throughout: $$ \lambda_n=n\kappa_n, $$ and reserve \(\Lambda_n\) for Voros's discretized sequence if that is his notation. Or use explicit superscripts $$ \lambda_n^{K}, \qquad \Lambda_n^{V}. $$ That will make Step 5 much easier to read. There is a smaller notation issue in the zero-counting paragraph too: you currently call the Riemann–von Mangoldt main term \(F(T)\), then a few lines later say “\(F\) is the honest zero-count majorant.” Those should definitely be different symbols. I would use $$ P(T)=\frac{T}{2\pi}\log\frac{T}{2\pi e}+\frac78 $$ for the classical main term and retain \(F(T)\) or \(\widehat N(T)\) for your enlarged formal majorant. Beyond those points, the revision is in very good shape. The corrected margin presentation is now internally consistent: $$ \operatorname{margin}(80\,000)=13\,160, \qquad \operatorname{margin}(100\,000)=88\,911, $$ and the historical trail $$ 16\,020 \to15\,764 \to14\,682 \to13\,160 $$ is documented separately. That is a substantial improvement. The dog-leg section also now looks internally coherent: actual contour height \(\tau_n\), deterministic majorant \(T_n\), elevated height \(U_n=\tfrac32\tau_n\), exact Cauchy deformation, and the three explicit replacement charges. And Appendix B is now unusually transparent for a paper of this type. It prints the final Lean signature and then states explicitly that all seven inputs are assumed rather than proved internally. The provenance table and explanation of `NoLowZeros` are also much better. One thing I would strengthen before publication is `MZeroProp`. The manuscript calls it an interval-arithmetic certificate but describes it as a “grid evaluation with a sampled-Lipschitz margin of safety ×2.” Because this is one of only **seven trusted inputs**, a referee may ask exactly how continuity between samples is certified. Ideally the archive should provide the actual interval subdivision, rigorous derivative/Lipschitz bound, precision, source code, output certificate, and hash. That is not necessarily a flaw—I just think this particular perimeter certificate deserves unusually explicit documentation. So my current verdict is: $$ \boxed{\text{The core Paper A structure now looks very solid.}} $$ I would make three final edits before freezing it: replace the exact \(N=P+S\) statement by Rosser's direct \(|N-P|\) bound, separate the two meanings of \(\Lambda_n\), and separate the classical \(P(T)\) main term from your formal zero-count majorant \(F(T)\). The first of those is a genuine mathematical correction; the latter two are notation/clarity corrections. After those, I would stop iterative internal rewriting and move to an **independent expert audit** of the seven perimeter instantiations and the dog-leg argument. [1]: https://portal.mardi4nfdi.de/wiki/Item%3AQ2436775?utm_source=chatgpt.com "An improved upper bound for the argument of the Riemann zeta-function on the critical line. II - MaRDI portal" [2]: https://alexkontorovich.github.io/PrimeNumberTheoremAnd/blueprint/zeta-chapter.html?utm_source=chatgpt.com "Zeta function estimates" Always verify first before make changes. My suggestion might not be right.