I found **one remaining issue that I would treat as substantive before calling Paper A finished**. The good news is that the corrections we discussed are now in place. The paper cleanly distinguishes Rosser’s main term \(P(T)\) from your formal majorant \(F(T)\), and it now uses the correct explicit form $$ |N(T)-P(T)| \le 0.137\log T+0.443\log\log T+1.588 $$ rather than the earlier inaccurate \(N=P+S\) identity. The Johansson normalization is also cleanly separated using \(\kappa_n\), with $$ \lambda_n=n\kappa_n, $$ while Voros’s different \(\Lambda_n\) notation is explicitly distinguished. The margin bookkeeping is now internally synchronized at \(13\,160\), \(88\,911\), and \(2.458\times10^{26}\). The remaining issue is **`MZeroProp`**. Appendix B is admirably transparent about it. You now say that the calculation uses a grid of step \(0.002\), observes a maximum \(1.33237\), and extends between grid points using a **sampled Lipschitz constant**. Most importantly, the manuscript explicitly admits: > “The sampled (not interval-certified) ingredient is the between-grid Lipschitz constant … and a fully interval-rigorous box-cover upgrade (E-023b) is queued.” That means, at this moment, the chain is: $$ \boxed{ \text{Lean theorem rigorously proved} } $$ **conditional on** $$ \boxed{\texttt{MZeroProp}} $$ but the proposed external certificate establishing `MZeroProp` is **not yet itself rigorous**. That's completely legitimate as a conditional theorem. The issue is that some earlier wording still makes the perimeter sound more certified than it currently is. For example, the abstract calls the four computational inputs “certificates” and calls this one a “short-segment interval constant.” I would either finish **E-023b** before release—which is by far the best solution—or temporarily change the wording to something like: > “…four computational perimeter inputs … including a numerically validated short-segment bound whose fully interval-rigorous certification is pending.” Likewise, the provenance table currently labels `MZeroProp` a “computational certificate.” Until E-023b is done, I'd label it: > **numerically validated input; rigorous interval certificate pending** ### Why I think E-023b is worth doing The remarkable thing is that this appears to be a **small remaining trust-boundary problem**, not another major analytic problem. You only need to certify $$ \left| \frac{\zeta'}{\zeta} \left(-\frac52+it\right) \right| \le1.3352, \qquad |t|\le10. $$ And your sampled maximum is already $$ 1.33237, $$ leaving roughly $$ 0.00283 $$ of headroom. Moreover this contributes only through $$ 40M_0 $$ while the base-point margin is thousands. So if \(1.3352\) proves annoying, you probably have room to certify a slightly larger rational constant instead of optimizing tightly. That could make an interval-box proof considerably easier. Once E-023b is genuinely rigorous, the trust architecture becomes exceptionally clean: $$ \begin{array}{c} \text{3 published explicit zero-count bounds}\\ +\text{ Platt--Trudgian RH certificate}\\ +\text{ Johansson finite Li certificate}\\ +\text{ Platt low-zero certificate}\\ +\text{ rigorous E-023b short-segment certificate} \end{array} $$ $$ \Downarrow $$ $$ \texttt{lambda\_pos\_finite\_height\_closed} $$ with every remaining analytic step checked by Lean. The final theorem signature and trust-boundary presentation are otherwise very strong. The manuscript literally prints all seven assumptions and explicitly states that they are the only trusted inputs. ### One minor wording cleanup Remark 3 still says “Johansson’s rigorous ordinary-\(\lambda_n\) table,” although you've just carefully explained that Johansson actually tabulates \(\kappa_n\) and positivity transfers through \(\lambda_n=n\kappa_n\). I'd change that to: > “Johansson’s rigorous Keiper-coefficient table, transferred to the present Li normalization via \(\lambda_n=n\kappa_n\).” That's purely editorial. ### My verdict I would **not keep revising the mathematical narrative**. The paper is now coherent about the theorem, the \(T^2\) precedent, the normalization, the zero-count bounds, the actual contour height, the dog-leg repair, the corrected \(B(n)\), and the formal dependency chain. There is essentially one box left to tick: $$ \boxed{\textbf{Complete E-023b and certify MZeroProp rigorously.}} $$ Once that is done and the paper changes the E-023 paragraph from “queued” to the actual certified method/result, I would be comfortable calling this a **release candidate for independent expert review** rather than another internal draft.