[{"type":"thinking","thinking":"§2 is about 43 lines (107-149) — roughly 1.2 pages. The user suggested shortening it to a paragraph unless §3 depends on it. §3 doesn't use the Weil/certificate machinery (the Li theorem uses Lagarias's contour). The sanity-gate feature (Davenport–Heilbronn) IS thematically connected to §4 (obstruction analysis) but not load-bearing for §3.\n\nCompress §2 to a compact background subsection (~15 lines): keep (i) one-paragraph description of the certificate method + state-of-the-art, (ii) the sanity gate feature, (iii) a pointer that the details/companion note carry it. Remove the R(ψ) display and the detailed rank-trace account (they belong to the companion paper).\n\nNow let me plan all the paperA.tex edits concretely:\n\n**Edit 1 — title date**: `\\date{Started: 2026-08-17; revised 2026-08-20}` → `\\date{Started: 2026-08-17; revised 2026-08-26}`.\n\n**Edit 2 — abstract**:\nCurrent (after my earlier edit):\n```\n[proved, conditional on computation] The Keiper--Li coefficients of the\nRiemann $\\xi$-function satisfy $\\lambda_n > 0$ for all\n$1 \\le n \\le (T_0-1)^2 \\approx 9\\times10^{24}$, conditional on the\ninterval-arithmetic verification of RH to height $T_0 = 3\\times10^{12}$\n(Platt--Trudgian) and on finite Li-coefficient computations, with every\nconstant traced. This is the first assembled explicit chain of its\nkind; by Voros's sharpening it is the maximal content of the Li route\nshort of RH. Moreover the full reduction is machine-verified in Lean~4\n(package \\texttt{NguyenRH1}): the theorem is proved from exactly eight\nnamed hypotheses --- the two computational certificates above and six\ndeclared analytic inputs (Trudgian's explicit zero-counting bounds in\ntheir unit-window, two-unit, and global forms, a Platt-class\nzero-location certificate, and one short-segment interval constant),\nwith the traced contour bound $|S_f(n)| \\le B(n)$ on $10^5 \\le n \\le\n(T_0-1)^2$ itself machine-proved from those six --- with axiom profile\nexactly the three standard axioms (\\texttt{propext},\n\\texttt{Classical.choice}, \\texttt{Quot.sound}) and no \\texttt{sorry}\nanywhere in the development's Li chain.\n```\n\nNew:\n```\n[proved, conditional on computation] The Keiper--Li coefficients of the\nRiemann $\\xi$-function satisfy $\\lambda_n > 0$ for all\n$1 \\le n \\le (T_0-1)^2 \\approx 9\\times10^{24}$, conditional on the\ninterval-arithmetic verification of RH to height $T_0 = 3\\times10^{12}$\n(Platt--Trudgian) and on finite Li-coefficient computations (Johansson's\nrigorous table to $10^5$ and this project's to $3000$), with every\nconstant traced. We have found no previous explicit assembly of this\nchain in the literature; by Voros's sharpening, positivity of all\nsufficiently large $\\lambda_n$ would itself imply RH, so no result\nshort of RH can assert an infinite positive tail. Moreover the full\nreduction is machine-verified in Lean~4 (package \\texttt{NguyenRH1}):\nthe theorem is proved from exactly seven named hypotheses --- the two\ncomputational certificates above and five declared analytic inputs\n(Trudgian's explicit zero-counting bounds in their unit-window,\ntwo-unit, and global forms, a Platt-class zero-location certificate,\nand one short-segment interval constant), with the traced contour bound\n$|S_f(n)| \\le B(n)$ on $10^5 \\le n \\le (T_0-1)^2$ itself machine-proved\nfrom those five --- with axiom profile exactly the three standard\naxioms (\\texttt{propext}, \\texttt{Classical.choice},\n\\texttt{Quot.sound}) and no \\texttt{sorry} anywhere in the\ndevelopment's Li chain.\n```\n\n**Edit 3 — intro para (line ~75 and ~91-96)**:\n- \"machine-verified in Lean~4 modulo a declared perimeter (computational certificates and explicit zero-counting bounds)\" ✓ (already fixed — keep, maybe adjust \"computational certificates\" fine).\n- \"By Voros's sharpening no infinite-range partial statement exists, so this is the maximal content of the Li route short of RH; the explicit chain --- Lagarias's contour argument ... --- appears not to have been assembled before.\" → \"By Voros's sharpening, eventual positivity of all sufficiently large $\\lambda_n$ would itself imply RH, so a result short of RH cannot assert an infinite positive tail; an explicit finite-height statement is the natural form of partial progress on this route. The statement itself has an unpublished precursor (Oesterlé, 2000, cited in \\cite{voros2017li}: RH to height $T_0$ implies $\\lambda_n \\ge 0$ for $n \\le T_0^2$) and a published weaker form (Lagarias's window with a $\\log^2$ denominator, Remark~(i)); the explicit, fully constant-traced, machine-verified assembly --- Lagarias's contour argument \\cite[Thm~6.1]{lagarias2007li} with Trudgian's explicit zero-counting bounds \\cite{trudgian2014}, an elementary archimedean lower bound, a proved margin convexity, and the finite verifications --- we have found nowhere else in the literature.\"\n\n**Edit 4 — §2 compression** (lines 107-149): replace with ~15 lines.\n\n**Edit 5 — Theorem statement (thm:finite-height-li, lines 158-167)**:\n```\n\\begin{theorem}[R-004; conditional on computation]\n\\label{thm:finite-height-li}\nAssume (i) the Riemann hypothesis verified to height\n$T_0 = 3\\times 10^{12}$ (Platt--Trudgian \\cite{platt2021riemann},\ninterval arithmetic), and (ii) $\\lambda_n > 0$ for all $n \\le 10^5$\n(the finite computations of Step~5). Then\n\\[\n\\lambda_n > 0 \\qquad \\text{for all } 1 \\le n \\le N_1 := (T_0-1)^2\n= 8\\,999\\,999\\,999\\,994\\,000\\,000\\,000\\,001 \\approx 9.0\\times 10^{24}.\n\\]\n\\end{theorem}\n```\nAlso the intro's Theorem (thm:finite-height-li-intro at line 83-89) — same assumption expansion.\n\n**Edit 6 — §3.1 new subsection (the F-D repair)**: after the Step 2 footnote's correction note (which I'll shorten to F-A/F-B/F-E), add:\n\n```\n\\subsection{Repair of the right-horizontal estimate}\n\\label{subsec:dogleg}\n\nThe geometric-decrease conclusion following \\cite[(6.12)]{lagarias2007li}\nis not implied by the displayed bounds when $T \\asymp \\sqrt n$: the\nper-step majorant ratio there is $e/2 > 1$ (the zeta factor halves per\nunit step while the kernel ratio is $\\le e$), so the right-tail envelope\ngrows like $e^{0.066\\sqrt n}$ rather than decreasing ($\\sim 6\\times\n10^{10}$ at $n = 10^5$ measured; E-031). In the zeta specialization we\nreplace this step by the following zero-free contour deformation\n(D-073/D-074).\n\nKeep the cut height $T = \\sqrt n$ (so the bridge and the endpoint\n$N_1 = (T_0-1)^2$ are untouched) and deform only the right horizontal\nsegment $[3, 2\\sqrt n]$ at height $T$ into a \\emph{dog-leg}: lift\nvertically from $3 + iT$ to $3 + iU$ with $U = \\tfrac32\\sqrt n$\n(the left vertical), traverse horizontally at height $U$ (the high\nsegment), and drop back at $2\\sqrt n$ (the right vertical). The\nrectangle this traverses lies entirely in $\\Re w \\ge 3$, where $\\zeta$\nis zero-free and pole-free \\emph{unconditionally}, so Cauchy's theorem\ngives the exact identity\n\\[\nH_{\\mathrm{low}} \\;=\\; H_{\\mathrm{high}} - V_{\\mathrm{right}} +\n  V_{\\mathrm{left}}\n\\]\n(\\texttt{upper\\_right\\_horizontal\\_eq\\_dogleg} in the formal\ndevelopment). Each dog-leg piece has an honest bound: the left vertical\ncontributes $\\le \\tfrac{\\sqrt n}{2}(e^{5/2}+1)\\cdot \\tfrac12$; the high\nsegment contracts geometrically (the repaired stepping ratio\n$q = e^{4/9}/2 < 1$, certified) and contributes $\\le 8.38$; the right\nvertical is exponentially small, $\\le \\tfrac14 I_{\\mathrm{Maj}}(n)$.\nThe charge replacing the broken ``$3.3$'' is therefore\n$\\mathrm{DL}(n) = \\tfrac{\\sqrt n+1}{4}(e^{5/2}+1) + 8.38 +\n\\tfrac14 I_{\\mathrm{Maj}}(n)$ per side, with margin essentially\nunchanged (Section~3, Step 4: margins $13\\,160$ / $88\\,911$ /\n$2.458\\times10^{26}$ at $8\\times10^4$ / $10^5$ / $N_1$). We note for\ncompleteness that the per-side bound at $T = \\sqrt n$ appears to hold\nempirically by a strong hidden cancellation (E-032: true integral vs\nenvelope ratio down to $5.6\\times10^{-10}$); whether that cancellation\nadmits a proof is an open question we do not rely on.\n```\n\nAnd the Step 2 footnote shrinks to F-A/F-B/F-E (i/ii/iv) with a pointer to §3.1 for (iii).\n\n**Edit 7 — Step 5 rewrite** (lines ~236-241):\n```\n\\emph{Step 5 (finite verification; the declared certificate\n\\texttt{LiVerifiedUpTo($10^5$)}).} The base range $n < 10^5$ is covered\nby rigorous high-precision computation of the ordinary Keiper--Li\ncoefficients: for all $n \\le 10^5$ by Johansson \\cite{johansson2015}\n(interval-arithmetic error bounds throughout, the full table\n$\\lambda_0,\\dots,\\lambda_{100000}$ all positive and matching the\nRH-true asymptotic, with $\\lambda_{100000} = 4.6258\\ldots$ certified to\n$\\sim 2900$ digits), and independently for all $n \\le 3000$ by this\nproject (two methods --- the arithmetic side via Bombieri--Lagarias\nThm~2 and the zero side --- agreeing to $4.8\\times10^{-25}$ relative,\nand $\\lambda_1$ agreeing with its closed form to better than\n$10^{-28}$ relative). The crossover $N_0 = 8\\times 10^4$ is covered by\na factor $1.25$.\\footnote{An earlier draft cited Voros's discretized\nKeiper--Li computations \\cite{voros2017li} (with Misguich) for this\nrange; those concern a \\emph{deformed} sequence $\\Lambda_n$, not the\nordinary $\\lambda_n$, and the citation has been replaced by Johansson's\nrigorous ordinary-$\\lambda_n$ table.}\n```\n\n**Edit 8 — remark (iii) (line ~251-253)**: \"The dependence on computation is explicit: Platt--Trudgian's interval-arithmetic verification, and the finite Li-coefficient verifications.\" → expand: \"(iii) The dependence on computation is explicit and checkable: Platt--Trudgian's interval-arithmetic verification \\cite{platt2021riemann} and Johansson's rigorous ordinary-$\\lambda_n$ table \\cite{johansson2015}, supplemented by this project's independent verification to $n \\le 3000$ (E-011/E-015).\"\n\n**Edit 9 — Appendix B rewrite** (the closure section):\n- Fix \"eight\" → \"seven\", \"six\" → \"five\".\n- Add the verbatim Lean signature (verbatim block).\n- Add the instantiation list (how each hypothesis is discharged from cited results).\n- Add the dependency-chain display.\n- Update the LiVerifiedUpTo description.\n\nDraft:\n```\nThe 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 is machine-proved from exactly seven named hypotheses.\nIts final signature, copied verbatim from the development\n(\\texttt{LiClosed.lean}), is:\n\\begin{verbatim}\ntheorem lambda_pos_finite_height_closed\n    (hRH : RHUpTo (3 * 10 ^ 12)) (hfinite : LiVerifiedUpTo (10 ^ 5))\n    (hZC : ZeroCountProp) (hZC2 : ZeroCountProp2)\n    (hZCG : ZeroCountGlobalProp) (hNLZ : NoLowZeros)\n    (hMZ : MZeroProp) :\n    ∀ n : ℕ, 1 ≤ n → (n : ℝ) ≤ (3 * 10 ^ 12 - 1 : ℝ) ^ 2 →\n      0 < (liCoeff n).re\n\\end{verbatim}\nThe seven hypotheses instantiate from cited results and certificates as\nfollows: \\texttt{RHUpTo} from the Platt--Trudgian interval-arithmetic\nverification of RH to height $3\\times10^{12}$ \\cite{platt2021riemann}\n(which also gives the zero-location certificate \\texttt{NoLowZeros}:\nevery nontrivial zero has $|\\gamma| \\ge 14$, as the lowest zero lies at\n$14.134\\ldots$); \\texttt{LiVerifiedUpTo} from Johansson's rigorous\nordinary-$\\lambda_n$ table to $10^5$ \\cite{johansson2015} together with\nthis project's independent verification to $3000$ (Appendix~A);\n\\texttt{ZeroCountProp}, \\texttt{ZeroCountProp2}, and\n\\texttt{ZeroCountGlobalProp} from Trudgian's explicit zero-counting\nbounds \\cite{trudgian2014} (the unit-window, two-unit, and global\nspecializations thereof); and \\texttt{MZeroProp} from the\ninterval-arithmetic certificate E-023 (grid evaluation with a\nsampled-Lipschitz margin of safety $\\times 2$). The proof of\n\\texttt{SfBoundProp} contains the full complex-analytic content of\nStep 2, in the dependency chain\n\\[\n\\text{partial fractions} \\;\\to\\; S_f = \\operatorname{Res}_1 \\;\\to\\;\n\\text{rectangle identity} \\;\\to\\; \\text{four side bounds} \\;\\to\\;\n\\boxed{\\texttt{sfBoundProp}} \\;\\to\\;\n\\boxed{\\texttt{lambda\\_pos\\_finite\\_height\\_closed}},\n\\]\nwith the named capstones \\texttt{rectInt'\\_Jlag},\n\\texttt{partialFraction\\_identity},\n\\texttt{logDeriv\\_riemannZeta\\_sub\\_nearZeros\\_le\\_tight} (the\nreference-point bound $|R(s)| \\le 10.5\\log(|t|{+}2) + 48$),\n\\texttt{sf\\_eq\\_resW1} (the exact identity $S_f(n) = \\mathrm{Res}_1(n)$),\n\\texttt{upper\\_right\\_horizontal\\_eq\\_dogleg} (Section~\\ref{subsec:dogleg}),\n\\texttt{LiSideBounds}, and \\texttt{sfBoundProp}. [rest unchanged: margin evaluations, margin majorant paragraph, conditional-on-computations paragraph with \"six analytic inputs\" → \"five analytic inputs\"]\n```\n\nAlso Appendix B earlier text (lines 345-368 in the previous version — my ii′-7c edit) — need to re-read the current state after my edit and adjust: the \"eight named hypotheses\" → seven; \"six declared analytic inputs\" → five; add the signature/diagram.\n\n**Edit 10 — the AI Contribution Statement**: update the closure declarations (no change needed? — it says \"sfBoundProp, lambda_pos_finite_height_closed\" ✓ fine; maybe add Johansson mention? — no, leave).\n\n**Edit 11 — cover-letter.md + referee-list.md Misguich–Voros mentions** → Johansson.\n\n**Edit 12 — references.bib**: add johansson2015 + oesterle2000.\n\nThen compile (pdflatex + bibtex + pdflatex×2).\n\nLet me also reconsider: the user's point about Theorem vs trust boundary — my fix: the theorem statement gets assumptions (i)+(ii) AND Appendix B gets the instantiation list mapping each perimeter hypothesis to its source. That addresses \"give a separate corollary explaining how every perimeter hypothesis is instantiated from cited results/certificates\" — the instantiation list in App B serves as that corollary. Good.\n\nNow execute. Start with references.bib.","signature":""},{"type":"text","text":"§2 is ~1.2 pp, none of it load-bearing for §3 — it compresses well. Executing all paper edits now, starting with the bibliography."},{"type":"tool_use","id":"Bash_299","name":"Bash","input":{"command":"cd \"D:/Python/Quantum\" && grep -n \"@article{voros2017li\" -A9 research/paper/references.bib | tail -4","description":"Show the end of the voros2017li entry"},"extra_content":null}]