[{"type":"text","text":"Now Appendix B — the closure account replacing the \"assembly form\" and \"remaining target\" paragraphs."},{"type":"tool_use","id":"Edit_226","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\paper\\paperA.tex","old_string":"The second tier is 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 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}.\n\nThe remaining formalization target inside this paper is\n(ii$'$) the contour estimate inside \\texttt{SfBoundProp} (requires\nmeromorphic residue machinery on rectangles and partial-fraction bounds\nfor $\\zeta'/\\zeta$ --- a nontrivial but finite library task). The\nfinite-height Li theorem is, by design,\n\\emph{conditional} on computations (Platt--Trudgian's interval\narithmetic; the Li-coefficient verifications); formalizing the\n\\emph{reduction} leaves exactly those computational certificates\nas trusted inputs, which we regard as the honest formal target.","new_string":"The second tier is complete; the named analytic hypothesis is\ndischarged. The traced contour bound $|S_f(n)| \\le B(n)$ of Section~3\n(\\texttt{SfBoundProp}) is machine-proved (tier ii$'$), so the closed\nheadline theorem \\texttt{lambda\\_pos\\_finite\\_height\\_closed} is\nmachine-proved from exactly eight named hypotheses: the two\ncomputational-perimeter inputs of Theorem~\\ref{thm:finite-height-li}\n(Platt--Trudgian; the Li verifications) and six declared analytic\ninputs --- Trudgian's explicit zero-counting bounds \\cite{trudgian2014}\nin their unit-window, two-unit, and global forms\n(\\texttt{ZeroCountProp}, \\texttt{ZeroCountProp2},\n\\texttt{ZeroCountGlobalProp}), a Platt-class zero-location certificate\n(\\texttt{NoLowZeros}), and the short-segment constant $M_0 = 1.3352$\n(\\texttt{MZeroProp}, an interval-arithmetic certificate in the same\ntrust class). The proof of \\texttt{SfBoundProp} contains the full\ncomplex-analytic content of Step 2: meromorphic residue machinery on\nrectangles (\\texttt{rectInt'\\_Jlag}), the partial-fraction identity\nfor $\\zeta'/\\zeta$ (\\texttt{partialFraction\\_identity}), the\nreference-point bound $|R(s)| \\le 10.5\\log(|t|{+}2) + 48$\n(\\texttt{logDeriv\\_riemannZeta\\_sub\\_nearZeros\\_le\\_tight}), the exact\nidentity $S_f(n) = \\mathrm{Res}_1(n)$ (\\texttt{sf\\_eq\\_resW1}), the\nzero-free dog-leg repair of the right tail\n(\\texttt{upper\\_right\\_horizontal\\_eq\\_dogleg} and the\n\\texttt{LiDogLeg} bounds), the four side bounds\n(\\texttt{LiSideBounds}), and the fold (\\texttt{sfBoundProp}). The two\nelementary evaluations of the explicit margin function that the\nassembly consumes ($\\mathrm{margin}(8{\\cdot}10^4) > 0$ 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}). The\nmargin majorant in the development is a mildly enlarged Trudgian-shape\nfunction (a negative-weight term dropped, at a cost of $\\approx\n0.9\\sqrt n$ in the bound) chosen so that every convexity block has\nprovably negative $\\psi$; the axiom profile is unchanged --- the three\nstandard axioms, no \\texttt{sorryAx}, audited declaration-by-declaration\non the full chain.\n\nThe finite-height Li theorem remains, by design, \\emph{conditional} on\ncomputations (Platt--Trudgian's interval arithmetic; the\nLi-coefficient verifications); the six 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."},"extra_content":null}]