[{"type":"thinking","thinking":"Now the paper appendix. Current text (draft.tex, Formal verification section): the paragraph about \"The remaining tiers, in order of increasing formalization cost: (ii) the traced contour bound...\" — I should update to reflect tier (ii) step 1. Per the audit rule (D-060 §5): any edit to draft.tex re-opens the audit for the edited lines — so keep the edit precise and honest.\n\nEdit plan: after the design-note paragraph (the measured-to-traced one), insert a new paragraph on tier (ii) step 1; and adjust the \"remaining tiers\" paragraph's (ii) sentence to reflect the new state.\n\nNew paragraph draft:\n\n```\nThe second tier has been split along the design of\nsession-012-notes (assembly first): the finite-height Li theorem\n\\texttt{lambda\\_pos\\_finite\\_height} is now machine-proved \\emph{modulo\none named analytic hypothesis}, stated explicitly in the development as\n\\texttt{SfBoundProp}: $|S_f(n)| \\le B(n)$ for all\n$10^5 \\le n \\le (T_0-1)^2$, with $B(n)$ the traced bound of \\S3.1. The\nremaining hypotheses are exactly the declared computational perimeter\n(Platt--Trudgian; the Li verifications) and two elementary evaluations\nof the explicit margin function ($\\widetilde{\\mathrm{margin}}(8\\cdot10^4)\n> 0$ and $\\widetilde{\\mathrm{margin}}(10^5) >\n\\widetilde{\\mathrm{margin}}(8\\cdot10^4)$, whose interval-arithmetic\ndischarge is a finite computation). Inside \\texttt{SfBoundProp}, the\nexplicit $N(T), S(T)$ bounds of Trudgian are treated as trusted\nperimeter (published literature, declared); the contour identity,\nkernel bounds, and the reference-point $R$-bound are plain complex\nanalysis, formalizable in \\textsc{Mathlib}. Axiom profile unchanged:\nthe three standard axioms only, no \\texttt{sorryAx}.\n```\n\nHmm — need to be careful with the paper's voice and precision. Let me check what draft §3.1 says about the Lean names... The Formal verification section references \\texttt{NguyenRH1}. Also the phrase \"session-012-notes\" shouldn't appear in the paper (internal name). Rewrite:\n\n\"The second tier is complete in its assembly form: ...\" Let me write it properly:\n\n```\nThe second tier is 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 above, two\nelementary evaluations of the explicit margin function\n($\\widetilde{\\mathrm{margin}}(8{\\cdot}10^4) > 0$ and\n$\\widetilde{\\mathrm{margin}}(10^5) > \\widetilde{\\mathrm{margin}}(8{\\cdot}10^4)$,\na finite interval-arithmetic computation), and a single named analytic\nhypothesis \\texttt{SfBoundProp} --- the traced contour bound\n$|S_f(n)| \\le B(n)$ of \\S3.1, taken on the range\n$10^5 \\le n \\le (T_0-1)^2$. Within \\texttt{SfBoundProp}, the explicit\nzero-counting bounds of Trudgian (2014) are treated as trusted\nperimeter on the same footing as the Platt--Trudgian certificate; the\nremaining content (the rectangle contour identity, the kernel bounds,\nand the reference-point $R$-bound of \\S3.1) is plain complex analysis.\nThe axiom profile is unchanged: the three standard axioms, no\n\\texttt{sorryAx}.\n```\n\nWait — \"the reference-point R-bound of §3.1\" — is the R-bound in draft §3.1? D-056 content — the paper's §3.1 sketches the B(n) assembly presumably. Fine to say \"of §3.1\" loosely — check §3.1 content briefly? The subsection at line 179 is \"An explicit finite-height Li-positivity theorem (R-004)\". The R-bound may or may not be detailed there. Safer to not cite specific sub-content: \"...is plain complex analysis (the rectangle contour identity, kernel estimates, and partial-fraction bounds for ζ′/ζ).\" Also \"Trudgian (2014)\" needs a citation — check references.bib for Trudgian/Platt keys.","signature":""},{"type":"text","text":"Now the paper appendix update. Checking the bibliography keys and §3.1's framing first."},{"type":"tool_use","id":"Bash_100","name":"Bash","input":{"command":"grep -n \"@\" D:/Python/Quantum/research/paper/references.bib | head -25; echo ---; sed -n '179,200p' D:/Python/Quantum/research/paper/draft.tex","description":"Check bib keys and R-004 section framing"},"extra_content":null}]