[{"type":"thinking","thinking":"Now items 3+4: the Appendix B instantiation paragraph → provenance table + NoLowZeros separate provenance prose. And the final Appendix B paragraph reclassification.\n\nCurrent text (lines ~432-446):\n\"The seven hypotheses instantiate from cited results and certificates as\nfollows. \\texttt{RHUpTo} comes from the Platt--Trudgian\ninterval-arithmetic verification of RH to height $3\\times10^{12}$\n\\cite{platt2021riemann}, which also gives the zero-location certificate\n\\texttt{NoLowZeros} (every nontrivial zero has $|\\gamma| \\ge 14$, the\nlowest lying at $14.134\\ldots$). \\texttt{LiVerifiedUpTo} comes from\nJohansson's rigorous ordinary-$\\lambda_n$ table to $10^5$\n\\cite{johansson2015} together with this project's independent\nverification to $3000$ (Appendix~A). \\texttt{ZeroCountProp},\n\\texttt{ZeroCountProp2}, and \\texttt{ZeroCountGlobalProp} are the\nunit-window, two-unit, and global specializations of Trudgian's\nexplicit zero-counting bounds \\cite{trudgian2014}.\n\\texttt{MZeroProp} is the short-segment constant $M_0 = 1.3352$, an\ninterval-arithmetic certificate (E-023: grid evaluation with a\nsampled-Lipschitz margin of safety $\\times 2$) in the same trust class.\"\n\nReplace with:\n\"The seven hypotheses are trusted inputs --- assumed in the\ndevelopment, not proved within it; every other step of the proof is\nmachine-checked. Their provenance is itemized in\nTable~\\ref{tab:perimeter}.\n\n\\begin{table}[htb]\\centering\\small\n\\caption{...}\n...\n\\end{table}\n\nTwo rows deserve comment. \\texttt{NoLowZeros} is logically separate\nfrom \\texttt{RHUpTo}: it asserts a location fact about the low zeros,\nnot RH on an initial segment. Its provenance is the zero-location stage\nof the Platt--Trudgian computation, which isolates every zero up to the\nverification height by the Riemann--Siegel formula and confirms the\ntotal count by Turing's method, finding the lowest zero at\n$\\gamma_1 = 14.134725\\ldots$ \\cite{platt2021riemann}; hence no\nnontrivial zero has $|\\gamma| < 14$. \\texttt{LiVerifiedUpTo} is the\ncertificate of Step 5; the normalization relation of that step applies.\"\n\nWait, do I want a floating table or an inline tabular? The paper is 11pp; a float might drift. Use a non-floating center tabular to keep placement deterministic (like the dependency-chain tabular already there). I'll use \\begin{center}\\small ... tabular ... \\end{center} without caption/label (consistent with the existing dependency-chain style). Actually a small caption-less table is fine since the prose refers to it as \"the following table\".\n\nColumn layout (avoid overfull): \n- col1: hypothesis name (l)\n- col2: statement (p{5.6cm})\n- col3: source (p{3.8cm})\n- col4: status (p{3.0cm})\n\nWith \\small and default tabcolsep 6pt: widths 5.6+3.8+3.0 = 12.4cm + col1 ~2.9cm + 8×6pt≈1.7cm = ~17cm > \\textwidth (16.5cm). Tight. Use \\footnotesize and trim: col2 5.2, col3 3.4, col4 2.8 = 11.4 + col1 2.8 + 1.7 = 15.9cm. OK.\n\nActually with @{} at both ends: saves 2 tabcolsep. Let me just build it and check the log for overfull.\n\nRows (terse statements):\n1. \\texttt{RHUpTo} & $\\operatorname{Re}\\rho = 1/2$ for all zeros with $|\\gamma| \\le 3\\times10^{12}$ & Platt--Trudgian \\cite{platt2021riemann}, interval arithmetic & computational certificate \\\\\n2. \\texttt{LiVerifiedUpTo} & $\\lambda_n > 0$ for $1 \\le n \\le 10^5$ & Johansson \\cite{johansson2015} (Step-5 normalization); this project to $n \\le 3000$ (App.~A) & computational certificate \\\\\n3. \\texttt{ZeroCountProp} & unit-window count $\\le C_0(t{+}1)$, $t \\ge 2$ & Trudgian \\cite{trudgian2014} & published explicit bound \\\\\n4. \\texttt{ZeroCountProp2} & two-unit count $\\le 0.6\\log(t{+}2) + 0.886\\log\\log(t{+}2) + 2.7$, $t \\ge 10$ & Trudgian \\cite{trudgian2014} & published explicit bound \\\\\n5. \\texttt{ZeroCountGlobalProp} & global count $N(T) \\le F(T)$, $T \\ge 2$ & Trudgian \\cite{trudgian2014} (enlarged majorant) & published explicit bound \\\\\n6. \\texttt{NoLowZeros} & $|\\gamma| \\ge 14$ for every nontrivial zero & Platt--Trudgian zero-location stage \\cite{platt2021riemann} (see text) & computational certificate \\\\\n7. \\texttt{MZeroProp} & $|\\zeta'/\\zeta(-\\tfrac52 + it)| \\le 1.3352$, $|t| \\le 10$ & this project (E-023), interval arithmetic & computational certificate \\\\\n\nMultiplicity-weighted for ZeroCountProp: add \"(mult.-weighted)\"? The statement \"unit-window count ≤ C0(t+1)\" — keep terse; the mult-weighting detail is in the Lean signature. Fine.\n\nThen the two-comment prose. Then keep the rest (dependency chain paragraph) as is.\n\nAnd the final paragraph of Appendix B:\n\"The finite-height Li theorem remains, by design, \\emph{conditional} on\ncomputations (Platt--Trudgian's interval arithmetic; the\nLi-coefficient verifications); the five analytic inputs are explicit\npublished bounds of the same literature class \\cite{trudgian2014,\nplatt2021riemann}. The formal reduction leaves exactly those as\ntrusted inputs, which we regard as the honest formal target: every\nother step of the proof is machine-checked.\"\n\nNew: \"The finite-height Li theorem remains, by design, \\emph{conditional} on its declared perimeter: the four computational certificates of the table (the Platt--Trudgian verifications, the Li tables, the zero-location certificate, the interval constant) and the three explicit published zero-counting bounds \\cite{trudgian2014}. The formal reduction leaves exactly those seven inputs as trusted, which we regard as the honest formal target: every other step of the proof is machine-checked.\"","signature":""},{"type":"text","text":"Now items 3+4 — replacing the Appendix B instantiation paragraph with the provenance table and separate NoLowZeros provenance."},{"type":"tool_use","id":"Edit_25","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\paper\\paperA.tex","old_string":"The seven hypotheses instantiate from cited results and certificates as\nfollows. \\texttt{RHUpTo} comes from the Platt--Trudgian\ninterval-arithmetic verification of RH to height $3\\times10^{12}$\n\\cite{platt2021riemann}, which also gives the zero-location certificate\n\\texttt{NoLowZeros} (every nontrivial zero has $|\\gamma| \\ge 14$, the\nlowest lying at $14.134\\ldots$). \\texttt{LiVerifiedUpTo} comes from\nJohansson's rigorous ordinary-$\\lambda_n$ table to $10^5$\n\\cite{johansson2015} together with this project's independent\nverification to $3000$ (Appendix~A). \\texttt{ZeroCountProp},\n\\texttt{ZeroCountProp2}, and \\texttt{ZeroCountGlobalProp} are the\nunit-window, two-unit, and global specializations of Trudgian's\nexplicit zero-counting bounds \\cite{trudgian2014}.\n\\texttt{MZeroProp} is the short-segment constant $M_0 = 1.3352$, an\ninterval-arithmetic certificate (E-023: grid evaluation with a\nsampled-Lipschitz margin of safety $\\times 2$) in the same trust class.","new_string":"All seven hypotheses are trusted inputs --- assumed in the development,\nnot proved within it; every other step of the proof is machine-checked.\nTheir provenance is itemized in the following table.\n\\begin{center}\\footnotesize\n\\setlength{\\tabcolsep}{4pt}\n\\begin{tabular}{@{}l p{5.4cm} p{3.6cm} l@{}}\n\\hline\nHypothesis & Statement & Source & Status \\\\\n\\hline\n\\texttt{RHUpTo} & $\\operatorname{Re}\\rho = 1/2$ for all zeros with\n  $|\\gamma| \\le 3\\times10^{12}$ &\n  Platt--Trudgian \\cite{platt2021riemann}, interval arithmetic &\n  computational certificate \\\\\n\\texttt{LiVerifiedUpTo} & $\\lambda_n > 0$ for $1 \\le n \\le 10^5$ &\n  Johansson \\cite{johansson2015} (Step-5 normalization); this project\n  to $n \\le 3000$ (App.~A) & computational certificate \\\\\n\\texttt{ZeroCountProp} & unit-window zero count $\\le C_0(t{+}1)$,\n  $t \\ge 2$ & Trudgian \\cite{trudgian2014} &\n  published explicit bound \\\\\n\\texttt{ZeroCountProp2} & two-unit zero count $\\le 0.6\\log(t{+}2) +\n  0.886\\log\\log(t{+}2) + 2.7$, $t \\ge 10$ &\n  Trudgian \\cite{trudgian2014} & published explicit bound \\\\\n\\texttt{ZeroCountGlobalProp} & global zero count $N(T) \\le F(T)$,\n  $T \\ge 2$ & Trudgian \\cite{trudgian2014} (enlarged majorant) &\n  published explicit bound \\\\\n\\texttt{NoLowZeros} & $|\\gamma| \\ge 14$ for every nontrivial zero &\n  Platt--Trudgian zero-location stage \\cite{platt2021riemann}\n  (see below) & computational certificate \\\\\n\\texttt{MZeroProp} & $|\\zeta'/\\zeta(-\\tfrac52 + it)| \\le 1.3352$ for\n  $|t| \\le 10$ & this project (E-023), interval arithmetic &\n  computational certificate \\\\\n\\hline\n\\end{tabular}\n\\end{center}\nOne row deserves a comment. \\texttt{NoLowZeros} is logically separate\nfrom \\texttt{RHUpTo}: it asserts a location fact about the low zeros,\nnot RH on an initial segment. Its provenance is the zero-location stage\nof the Platt--Trudgian computation, which isolates every zero up to the\nverification height by the Riemann--Siegel formula and confirms the\ntotal count by Turing's method, finding the lowest zero at\n$\\gamma_1 = 14.134725\\ldots$ \\cite{platt2021riemann}; hence no\nnontrivial zero has $|\\gamma| < 14$. \\texttt{MZeroProp} is a grid\nevaluation with a sampled-Lipschitz margin of safety $\\times 2$\n(E-023), in the same interval-arithmetic trust class."},"extra_content":null},{"type":"tool_use","id":"Edit_26","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\paper\\paperA.tex","old_string":"The finite-height Li theorem remains, by design, \\emph{conditional} on\ncomputations (Platt--Trudgian's interval arithmetic; the\nLi-coefficient verifications); the five analytic inputs are explicit\npublished bounds of the same literature class \\cite{trudgian2014,\nplatt2021riemann}. The formal reduction leaves exactly those as\ntrusted inputs, which we regard as the honest formal target: every\nother step of the proof is machine-checked.","new_string":"The finite-height Li theorem remains, by design, \\emph{conditional} on\nits declared perimeter: the four computational certificates of the\ntable above (the Platt--Trudgian verifications, the Li tables, the\nzero-location certificate, and the interval constant) and the three\nexplicit published zero-counting bounds \\cite{trudgian2014}. The\nformal reduction leaves exactly those seven inputs as trusted, which we\nregard as the honest formal target: every other step of the proof is\nmachine-checked."},"extra_content":null}]