[{"type":"thinking","thinking":"Now Edit 3: the appendix formal-status paragraph — five hypotheses → three (discharge landed). Current text (from my session-012 edit):\n\n```\nThe second tier is now complete in its assembly form. The theorem\n\\texttt{lambda\\_pos\\_finite\\_height} is machine-proved from five named\nhypotheses: the two computational-perimeter inputs of\nTheorem~\\ref{thm:finite-height-li} (Platt--Trudgian; the Li\nverifications), two elementary evaluations of the explicit margin\nfunction ($\\mathrm{margin}(8{\\cdot}10^4) > 0$ and\n$\\mathrm{margin}(10^5) > \\mathrm{margin}(8{\\cdot}10^4)$, a finite\ninterval-arithmetic computation), and a single named analytic\nhypothesis \\texttt{SfBoundProp}: the traced contour bound\n$|S_f(n)| \\le B(n)$ of Section~3.1 on the range\n$10^5 \\le n \\le (T_0-1)^2$. Inside \\texttt{SfBoundProp}, ...\n```\n\nNew version (three hypotheses, discharge declared):","signature":""},{"type":"text","text":"Now the appendix formal-status paragraph — the headline upgrade to three hypotheses."},{"type":"tool_use","id":"Edit_181","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\paper\\draft.tex","old_string":"The second tier is now complete in its assembly form. The theorem\n\\texttt{lambda\\_pos\\_finite\\_height} is machine-proved from five named\nhypotheses: the two computational-perimeter inputs of\nTheorem~\\ref{thm:finite-height-li} (Platt--Trudgian; the Li\nverifications), two elementary evaluations of the explicit margin\nfunction ($\\mathrm{margin}(8{\\cdot}10^4) > 0$ and\n$\\mathrm{margin}(10^5) > \\mathrm{margin}(8{\\cdot}10^4)$, a finite\ninterval-arithmetic computation), and a single named analytic\nhypothesis \\texttt{SfBoundProp}: the traced contour bound\n$|S_f(n)| \\le B(n)$ of Section~3.1 on the range\n$10^5 \\le n \\le (T_0-1)^2$. Inside \\texttt{SfBoundProp}, the explicit\nzero-counting bounds of \\cite{trudgian2014} are treated as trusted\nperimeter on the same footing as the Platt--Trudgian certificate; the\nremaining content (the rectangle contour identity, the kernel\nestimates, and the reference-point bound for $\\zeta'/\\zeta$) is plain\ncomplex analysis. The margin majorant in the development is a mildly\nenlarged Trudgian-shape function (a negative-weight term dropped, at a\ncost of $\\approx 0.9\\sqrt n$ in the bound) chosen so that every\nconvexity block has provably negative $\\psi$; the axiom profile is\nunchanged --- the three standard axioms, no \\texttt{sorryAx}.","new_string":"The second tier is now complete in its assembly form. The headline\ntheorem \\texttt{lambda\\_pos\\_finite\\_height\\_verified} is\nmachine-proved from exactly three named hypotheses: the two\ncomputational-perimeter inputs of\nTheorem~\\ref{thm:finite-height-li} (Platt--Trudgian; the Li\nverifications) and a single named analytic hypothesis\n\\texttt{SfBoundProp}: the traced contour bound\n$|S_f(n)| \\le B(n)$ of Section~3.1 on the range\n$10^5 \\le n \\le (T_0-1)^2$. The two elementary evaluations of the\nexplicit margin function that the assembly consumes\n($\\mathrm{margin}(8{\\cdot}10^4) > 0$ and\n$\\mathrm{margin}(10^5) > \\mathrm{margin}(8{\\cdot}10^4)$) are discharged\nin the development itself, by certified rational interval bounds on\n$\\log$, $\\exp$, and $\\sqrt{\\cdot}$ (\\texttt{LiBasePoints.lean}).\nInside \\texttt{SfBoundProp}, the explicit\nzero-counting bounds of \\cite{trudgian2014} are treated as trusted\nperimeter on the same footing as the Platt--Trudgian certificate; the\nremaining content (the rectangle contour identity, the kernel\nestimates, and the reference-point bound for $\\zeta'/\\zeta$) is plain\ncomplex analysis. The margin majorant in the development is a mildly\nenlarged Trudgian-shape function (a negative-weight term dropped, at a\ncost of $\\approx 0.9\\sqrt n$ in the bound) chosen so that every\nconvexity block has provably negative $\\psi$; the axiom profile is\nunchanged --- the three standard axioms, no \\texttt{sorryAx}."},"extra_content":null}]