[{"type":"thinking","thinking":"Batch 2: the table rows, the after-table note (derivation + NoLowZeros), and the closing Appendix B paragraph.","signature":""},{"type":"text","text":"Batch 2: table rows, the derivation + NoLowZeros notes, and the closing paragraph."},{"type":"tool_use","id":"Edit_97","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\paper\\paperA.tex","old_string":"\\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 \\\\","new_string":"\\texttt{ZeroCountProp} & unit-window zero count $\\le C_0(t{+}1)$,\n  $t \\ge 2$ & Rosser \\cite{rosser1941}; derived (see below) &\n  derived 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  Rosser \\cite{rosser1941}; derived (see below) &\n  derived explicit bound \\\\\n\\texttt{ZeroCountGlobalProp} & global zero count $N(T) \\le F(T)$,\n  $T \\ge 2$ & Rosser \\cite{rosser1941} (enlarged majorant); derived\n  (see below) & derived explicit bound \\\\"},"extra_content":null},{"type":"tool_use","id":"Edit_98","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\paper\\paperA.tex","old_string":"One 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.","new_string":"The three zero-counting rows are derived bounds, not printed as such:\nthey are elementary specializations of the Riemann--von Mangoldt\nformula $N(T) = F(T) + S(T)$, $F(T) = \\frac{T}{2\\pi}\\log\\frac{T}{2\\pi\ne} + \\frac78$, combined with Rosser's explicit bound\n$|S(T)| \\le 0.137\\log T + 0.443\\log\\log T + 1.588$, valid for\n$T \\ge 1467$ \\cite{rosser1941} (tabulated with its provenance in\n\\cite[Table~1]{trudgian2014ii}). The two window forms take differences\nof the formula at the window endpoints, and the global form uses\n$N(T) \\le F(T) + |S(T)|$; the additive constants and the enlargement\nof $F$ absorb the ranges below Rosser's threshold via the finite-range\nverifications $|S(T)| \\le 1$ for $T \\le 280$ and $|S(T)| \\le 2$ for\n$T \\le 6.8\\times10^6$ \\cite[Eq.~(1.2)]{trudgian2014ii}.\n\\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 zero-location\ncomputation: Platt \\cite{platt2017isolating} rigorously isolates the\nnontrivial zeros with $0 < \\gamma \\le 3.06\\times10^{10}$ by interval\narithmetic (an independent verification of RH to that height), the\nlowest at $\\gamma_1 = 14.134725\\ldots$ --- a value classical since\nGram \\cite{gram1903} --- so no nontrivial zero has $|\\gamma| < 14$;\nthe Platt--Trudgian verification \\cite{platt2021riemann} extends the\non-line confirmation to height $3\\times10^{12}$ via Turing's method.\n\\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_99","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\paper\\paperA.tex","old_string":"explicit 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.","new_string":"explicit zero-counting bounds derived from the Riemann--von Mangoldt\nformula with Rosser's $S(T)$ bound \\cite{rosser1941}. 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}]