There is, however, **one concrete inconsistency I would fix immediately**. ### The Step 4 margin still contains a stale pre-dog-leg value Step 4 currently says $$ \operatorname{margin}(8\times10^4)=15\,764>0. $$ But the completed dog-leg section says the final corrected margins are $$ 13\,160,\quad88\,911,\quad2.458\times10^{26} $$ at \(8\times10^4\), \(10^5\), and \(N_1\), respectively. Those cannot both be the final `margin` for the same corrected \(B(n)\). The accompanying footnote is also still describing the earlier trail ending at \(15\,764\). Given your final D-074/ii′-6 state, I believe the main Step 4 value should now be the fully corrected one: $$ \boxed{\operatorname{margin}(8\times10^4)=13\,160} $$ or, even better, use the exact certified lower bound that the final Lean theorem actually consumes. Then the footnote can preserve the history: $$ 16\,020 \rightarrow 15\,764 \rightarrow 13\,160 $$ and explain which correction caused each decrease. That is the most obvious remaining internal inconsistency in the PDF. ### I would make one more refinement to the external-input table The table is excellent, but for `ZeroCountProp`, `ZeroCountProp2`, and `ZeroCountGlobalProp`, I would add exact source locations—Theorem/Corollary/equation numbers from Trudgian—rather than only citing “[10].” This matters because these are three of the seven trusted assumptions. A skeptical referee should be able to go directly from $$ \texttt{ZeroCountProp2} $$ to the exact published inequality from which it is instantiated. If one of the Props is not literally printed by Trudgian but is an elementary specialization of his \(N(T)\)/\(S(T)\) result, say so explicitly: > “Derived algebraically from Trudgian [10, Theorem X] by applying the bound at \(T+1\) and \(T-1\).” That would make the perimeter audit exceptionally strong. ### `NoLowZeros` is much clearer now I'm pleased you explicitly separated it logically from `RHUpTo` and explained its provenance rather than saying vaguely that Platt–Trudgian “also gives it.” I would still make sure the sentence > “finding the lowest zero at \(\gamma_1=14.134725\ldots\) [9]” is literally supportable from [9] or from a specific certified dataset referenced by [9]. If not, add the appropriate primary source for the first-zero computation. That's a citation/provenance issue, not a mathematical problem. ### The dog-leg subsection now reads very well This may now be one of the most interesting pieces of the paper. The language is appropriately restrained: > “The geometric-decrease conclusion … is not implied by the displayed bounds…” rather than claiming Lagarias's theorem is false. You then keep the actual residue cut at \(\tau_n\), lift only the right tail to $$ U_n=\frac32\tau_n, $$ and give the exact Cauchy identity $$ H_{\rm low} = H_{\rm high}-V_{\rm right}+V_{\rm left}. $$ This tells the repair in a mathematically clean way. It also makes clear **why you recover \(9\times10^{24}\)** instead of falling back to the \(4\times10^{24}\) global-\(c=1.5\) workaround. I also strongly like that you retain the numerical hidden-cancellation observation but explicitly say: > “whether that cancellation admits a proof is an open question we do not rely on.” That is exactly the right scientific distinction between observation and theorem. ### One wording point in the bibliography Reference [4] currently says: > “rigorous interval-arithmetic computation of the ordinary Keiper–Li coefficients to \(n=10^5\).” Since the body has now carefully distinguished Johansson's **Keiper normalization** \(\Lambda_n\) from your Li normalization \(\lambda_n\), I would make the bibliography wording match: > “Section 4.2: rigorous interval-arithmetic computation of the Keiper coefficients \(\Lambda_n\) to \(n=10^5\).” That avoids reintroducing the normalization ambiguity you just fixed so carefully.