{
  "file_path": "D:\\Python\\Quantum\\research\\paper\\paperA.tex",
  "content": "% ============================================================\n% NGUYEN-RH-1 \u2014 PAPER A (split 2026-08-20, session 013, from draft.tex)\n% An explicit finite-height Li-positivity theorem, machine-verified.\n% Self-contained artifact per the session-012-end-notes review:\n% \u00a73.1 + Appendices A.1/B of the combined draft, retargeted.\n% The spectral-line material lives in paperB.tex (in preparation).\n% ============================================================\n\\documentclass[11pt]{article}\n\\usepackage{amsmath,amssymb,amsthm,mathtools}\n\\usepackage{microtype}\n\\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue]{hyperref}\n\n% The prose is dense with unhyphenatable compounds (Li-positivity,\n% [proved, ...] tags). Give TeX room to spread the slack instead of\n% running past the right margin.\n\\emergencystretch=2em\n\n\\newtheorem{theorem}{Theorem}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\theoremstyle{remark}\n\\newtheorem{remark}[theorem]{Remark}\n\n\\title{An explicit finite-height Li-positivity theorem\\\\\nwith a machine-verified assembly}\n\\author{Kimi K3\\thanks{model from Moonshot AI}\n\\and Claude Code\\thanks{harness from Anthropic}\n\\and Dan T Nguyen\\thanks{developer from Research Ai}}\n\\date{Started: 2026-08-17; revised 2026-08-20}\n\n\\begin{document}\n\\maketitle\n\n\\begin{abstract}\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 three\nnamed hypotheses --- the two computational certificates above and a\nsingle named analytic hypothesis, the traced contour bound\n$|S_f(n)| \\le B(n)$ on $10^5 \\le n \\le (T_0-1)^2$ --- with axiom\nprofile exactly 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\\end{abstract}\n\n\\section{Introduction}\n\nThe Riemann hypothesis (RH) asserts that every nontrivial zero of\n$\\zeta(s)$ has real part $1/2$. The strongest unconditional density\nresult to date is the (machine-verified) theorem that at least $2/3$\nof the nontrivial zeros are simple and on the critical line, and at\nleast $5/6$ of the zeros in a given range are distinct\n\\cite{anthropic2026riemann}; the underlying certificate method has a\ndocumented bandwidth-one ceiling near $0.682$ of the zeros. Two\nclassical reformulations frame most modern attacks: \\emph{Weil\npositivity} --- positivity of the Weil quadratic form on all of\n$C_c^2(\\mathbb R)$ is equivalent to RH \\cite{weil1952fonctions,\nbombieri2000weil}, and the negative index of its truncations counts\noff-line zero pairs \\cite{bombieri2000weil} --- and \\emph{Li's\ncriterion} --- $\\lambda_n > 0$ for all $n$ is equivalent to RH\n\\cite{li1997, bombieri1999lagarias}, with Voros's sharpening that\npositivity of any tail $\\{\\lambda_n\\}_{n \\ge N_0}$ already implies RH\n\\cite{voros2006sharpenings}.\n\nThis paper reports one theorem-level outcome of a systematic research\nprogram on RH: an explicit finite-height Li-positivity theorem,\nassembled with every constant traced, and with its reduction\nmachine-verified in Lean~4 modulo one named analytic hypothesis and a\ndeclared computational perimeter. A companion note (in preparation)\ncovers the spectral-line results of the same program (an exact\nfactorization on Suzuki's operator line and a valley theorem for the\nassociated model spaces). Every claim carries an explicit correctness\ntag, and every numerical input is produced by a deterministic,\narchived script.\n\n\\begin{theorem}[R-004; conditional on computation]\n\\label{thm:finite-height-li-intro}\nAssume the Riemann hypothesis verified to height\n$T_0 = 3\\times 10^{12}$ (Platt--Trudgian \\cite{platt2021riemann},\ninterval arithmetic). Then the Keiper--Li coefficients satisfy\n$\\lambda_n > 0$ for all $1 \\le n \\le (T_0-1)^2 \\approx 9\\times10^{24}$.\n\\end{theorem}\n\nBy Voros's sharpening no infinite-range partial statement exists, so\nthis is the maximal content of the Li route short of RH; the explicit\nchain --- Lagarias's contour argument \\cite{lagarias2007li} with\nTrudgian's explicit zero-counting bounds \\cite{trudgian2014}, an\nelementary archimedean lower bound, a proved margin convexity, and the\nfinite verifications --- appears not to have been assembled before.\n\nSection~2 recalls the certificate-method background and the sanity\ngate. Section~3 states and proves the theorem. Section~4 collects the\nobstruction analysis delimiting the method. Appendix~A describes the\nnumerical machinery; Appendix~B the formal-verification status.\n\n\\section{Background: the Weil-form certificate method}\n\nWe recall the machinery behind the state-of-the-art density theorem\n\\cite{anthropic2026riemann}, which is the launchpad and the sanity\ngate for everything below; details and notation follow that paper.\n\n\\emph{The Weil explicit formula} \\cite{weil1952fonctions} expresses,\nfor a test function $\\psi$ supported in $[-\\tfrac12,\\tfrac12]$ (the\n\\emph{bandwidth-one} class), a quadratic form on the zeros as a\nfunctional of the primes. Compressing it via a Gabor (Gaussian\nwindowed exponential) frame at the critical density $h = 2\\pi/L$\nyields a finite matrix identity $\\tilde G = P + Q$: a zero-side\nmatrix $\\tilde G$ built from the zeros in a window of size $L$\ndecomposes into a prime-side matrix $P$ (computed unconditionally\nfrom Montgomery's second moment / pair-correlation input\n\\cite{montgomery1973pair}) and a controlled remainder $Q$.\n\n\\emph{The rank--trace inequality.} Writing $\\tilde G$ in inertia form,\na rank--trace comparison between the positive eigenspace of $\\tilde G$\nand the positive part of $P$ converts the second-moment input\n$\\|\\tilde G\\|_{HS}^2 = (R(\\psi) + o(1))\\,N$ into a lower bound on the\nnumber of simple on-line zeros in the window. The strength of the\ncertificate is governed by the \\emph{window ratio}\n\\[\nR(\\psi) \\;=\\; \\frac{\\int \\psi^2 \\;+\\;\n\\iint |u-v|\\,\\psi(u)\\psi(v)\\,du\\,dv}{\\bigl(\\int \\psi\\bigr)^2}\n\\qquad (\\psi \\in C([-\\tfrac12,\\tfrac12]) \\text{ even}, \\ \\psi > 0),\n\\]\nwith $R(\\psi_0) = 4/3$ for the flat window\n$\\psi_0 = \\mathbf 1_{[-1/2,1/2]}$ and\n$R(\\psi_{MT}) = \\tfrac12 + \\tfrac{1}{\\sqrt2}\\cot\\tfrac{1}{\\sqrt2}\n= 1.32749\\ldots$ for the Montgomery--Taylor window (both verified\nsymbolically and against $60$-digit reference values in this project,\nE-001), giving the certified fractions $2 - R(\\psi)$: $0.6725$ of\nzeros simple and on the line, $0.83625$ distinct. A sharpness analysis\nlocates the bandwidth-one ceiling of the method at $\\approx 0.682$,\nand shows the obstructions to going further are structural\n(Section~4).\n\nOne feature of this machinery matters here: the certificate fails, as\nit must, for the Davenport--Heilbronn and Epstein zeta functions\n(which violate the RH-analogue); the main theorem of this paper has\nbeen required to pass this gate structurally (project file D-053).\n\n\\section{The finite-height Li theorem (R-004)}\n\nLet $\\lambda_n = \\sum_{\\rho}{}'\\,\\bigl[1-\\bigl(1-\\tfrac1\\rho\\bigr)^n\\bigr]$\nbe the Keiper--Li coefficients of the Riemann $\\xi$-function. Li's\ncriterion states $\\lambda_n > 0$ for all $n \\geq 1$ if and only if the\nRiemann hypothesis holds \\cite{li1997,bombieri1999lagarias}. Voros\n\\cite{voros2006sharpenings} showed that ``$\\lambda_n > 0$ for all\n$n \\geq N_0$'' is already equivalent to RH for any $N_0$, so no\npartial infinite-range statement exists; the maximal content of the Li\nroute short of RH is an \\emph{explicit finite-height} statement.\n\n\\begin{theorem}[R-004; conditional on computation]\n\\label{thm:finite-height-li}\nAssume the Riemann hypothesis verified to height\n$T_0 = 3\\times 10^{12}$ (Platt--Trudgian \\cite{platt2021riemann},\ninterval arithmetic). 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\n\\begin{proof}[Proof sketch (full assembly: D-037)]\nBombieri--Lagarias \\cite{bombieri1999lagarias} give\n$\\lambda_n = S_\\infty(n) - S_f(n) + 1$ with $S_\\infty$ the archimedean\nand $S_f$ the finite-place contribution.\n\n\\emph{Step 1 (archimedean lower bound).} From Lagarias's formula\n\\cite[(4.11)]{lagarias2007li},\n$S_\\infty(n) = \\sum_{j=1}^n (-1)^j \\binom{n}{j} a_j$ with\n$a_1 = (\\log 4\\pi + \\gamma)/2$ and $a_j = (1-2^{-j})\\zeta(j)\n= \\sum_{m \\text{ odd}} m^{-j}$ ($j \\ge 2$), the $j$-sum is a finite\ndifference and one obtains the \\emph{exact} identity\n$S_\\infty(n) = (1-a_1)n - 1 + \\sum_{m \\ge 3,\\, m \\text{ odd}}\n\\bigl[(1-\\tfrac1m)^n - 1 + \\tfrac{n}{m}\\bigr]$.\nBernoulli's inequality and the explicit harmonic bounds\n$\\log M + \\gamma + \\tfrac{1}{2M} - \\tfrac{1}{12M^2} \\le H_M \\le\n\\log M + \\gamma + \\tfrac{1}{2M}$ give, for $n \\ge 20$,\n\\[\nS_\\infty(n) \\;\\ge\\; \\tfrac12 n\\log n + (C_1 - \\tfrac{\\gamma}{2})\\,n - 4,\n\\qquad C_1 = \\tfrac{\\gamma - 1 - \\log 2\\pi}{2}.\n\\]\n(Machine-validated against exact values for $20 \\le n \\le 5000$.)\n\n\\emph{Step 2 (finite-part upper bound).} Tracing Lagarias's contour\nargument \\cite[Thm~6.1]{lagarias2007li} with explicit constants\n(Trudgian's $N(T)$ and $S(T)$ bounds \\cite{trudgian2014}, partial\nfractions for $\\zeta'/\\zeta$ with the reference-point estimate\n$|R(s)| \\le 10.5\\log(|t|{+}2) + 48$ for $-5/2 \\le \\sigma \\le 3$,\n$|t| \\ge 10$ (D-056), kernel bounds $e^4+1$, a measured\nshort-segment constant $M_0 = 1.3352$) gives\n$|S_f(n)| \\le B(n)$ with\n\\[\nB(n) = 2N(\\sqrt n{+}1) + 4T\\bigl[\\tfrac23 C_0(T) + 10.5\\log(T{+}2) + 48\\bigr]\n+ 40M_0 + 2(e^4{+}1)\\bigl[7H(T) + 3.3\\bigr] + I_I + 1,\n\\]\n$T = \\sqrt n + 1$, $C_0(t)$ the explicit unit-interval zero count\n(the honest Trudgian form, including the $\\log\\log$ term),\n$H(T) = 3C_0(T)^2 + 10.5\\log(T{+}2) + 48$, $I_I$ an explicitly\nbounded boundary-integral term (D-037/D-056) and the final $1$ the\ntrivial-zero contribution.\n\n\\emph{Step 3 (bridge).} For $n \\le (T_0-1)^2$ every zero with\n$|\\gamma| \\le \\sqrt n + 1$ is on the line, so\n$|1-(1-1/\\rho)^n| \\le 2$ per zero and\n$|\\lambda_n(T)| \\le 2N(T)$; all other contour terms are\n\\emph{unconditional}.\n\n\\emph{Step 4 (margin).}\n$\\mathrm{margin}(n) := \\tfrac12 n\\log n + (C_1 - \\gamma/2)n - 4 - B(n) + 1$\nis convex for $n \\ge 20$ (proved: chain-rule reduction to eleven\none-variable sign conditions, each discharged elementarily \u2014 D-057),\nand $\\mathrm{margin}(8\\times 10^4) = 15\\,764 > 0$ with\n$\\mathrm{margin}(10^5) > \\mathrm{margin}(8\\times 10^4)$; by convexity\n(secant slopes increase) hence\n$\\mathrm{margin}(n) > 0$ for all $n \\in [10^5, N_1]$\\footnote{The\nmargin value at $8\\times10^4$ has a documented trail: $16\\,020$ with\nthe Trudgian $F + S_b$ zero-count majorant (E-028); $15\\,764$ with the\nmildly enlarged majorant consumed by the machine verification --- a\nnegative-weight term of $F + S_b$ dropped so that every convexity\nblock has provably negative $\\psi$ (the true majorant's $\\psi$ crosses\nzero near $T \\approx 16$), at a traced cost of $\\approx 0.9\\sqrt n$,\ni.e.\\ $256$ of the $16\\,020$; and $\\ge 8\\,001$ as the certified\nrational interval lower bound that the machine discharge actually\nconsumes (\\texttt{NguyenRH1},\n\\texttt{marginTilde\\_FAlt\\_80000\\_pos}).}\n(also computed to $n = 4\\times10^5$ on a unit grid and to $N_1$ on a\nlogarithmic grid; $\\mathrm{margin}(N_1) = 2.458\\times 10^{26}$).\n\n\\emph{Step 5 (finite verification).} $\\lambda_n > 0$ for $n \\le 3000$\nby exact high-precision computation (this project, two independent\nmethods, agreement to $4.8\\times10^{-25}$ relative), and for\n$n \\le 5\\times10^5$ by Misguich--Voros \\cite{voros2017li}. The ranges\noverlap the $n = 10^5$ split by a factor $5$ (and the crossover\n$N_0 = 8\\times 10^4$ by $6.25$).\n\\end{proof}\n\n\\begin{remark}\n(i) Using Lagarias's printed bridge window $n \\le T^2/(4\\log^2 T)$\ninstead of the traced proof gives the conservative\n$N_1^* = 2.73\\times 10^{21}$; the theorem is robust either way.\n(ii) The traced worst-case bound $|S_f|$ is $\\sim 170\\times$ loose\nagainst the measured curve ($|S_f| \\le 13.52$ for $n \\le 5000$); the\nslack is structural to contour constants, not a gap.\n(iii) The dependence on computation is explicit: Platt--Trudgian's\ninterval-arithmetic verification, and the finite Li-coefficient\nverifications.\n\\end{remark}\n\n\\section{Obstruction analysis}\n% Documented dead ends from research/graveyard/ where they\n% delimit the method (contributes to success criterion 4).\n\nThe following routes are closed with documented obstructions\n(full files in the project archive, graveyard G-001--G-003 and\nderivations D-009--D-013):\n\\begin{itemize}\n\\item \\emph{Certificate extension past $2/3$ in the bandwidth-one\nframework} (Track A): a six-wall map (D-013) shows $>2/3$ is\nimpossible within the current data class; the window-class optimum is\nsaturated in both the $\\xi$ and $\\xi'$ worlds (D-009, D-030), and\nmoment-ladder certificates cap below $2/3$ at every order (G-002).\n\\item \\emph{Multiple-zero bounds as chain input} (G-001): the\n$s_2/p$ substitution symmetry makes depth$\\to$0 off-line pairs\nindistinguishable from double zeros in all two-moment data.\n\\item \\emph{Density theorems $\\to$ de~Bruijn--Newman $\\Lambda$}\n(G-003): one-sided information \u2014 density theorems lower-bound the\non-line fraction while $\\Lambda$ requires an upper bound on the\ncomplex fraction.\n\\item \\emph{$\\xi^{(k)}$-certificate routes} (T4/T4b, D-022, D-031):\nthe strength mechanism (a perfect-square node in the diagonal law)\ndoes not transfer to $\\xi$, and the $s_2/p$ symmetry caps every\nderivative class below $100\\%$.\n\\end{itemize}\n\n\\appendix\n\\section{Numerical experiments}\n% Tables/figures generated from research/numerics/data/ CSVs.\n\nAll experiments are deterministic scripts with CSV+JSON sidecars in the\nproject archive (\\texttt{research/numerics/}); checks as stated.\n\n\\subsection{Li-coefficient machinery (E-011/E-015/E-021/E-023)}\n\\begin{itemize}\n\\item $\\lambda_n > 0$ verified for $n \\le 3000$ from the arithmetic\nside (Bombieri--Lagarias Thm~2, zero-free; \\textsc{flint} Stieltjes +\nArb $\\zeta$, 1000--1450-digit working precision), cross-validated\nagainst the zero-side computation (agreement $4.8\\times10^{-25}$\nrelative) and $\\lambda_1$ agreeing with its closed form to better\nthan $10^{-28}$ relative.\n\\item Exact archimedean part: $S_\\infty(n) = \\tfrac12 n\\log n + C_1 n\n- \\tfrac14 - \\tfrac{1}{24n} + o(1/n)$ measured to $n = 5000$\n($c_0 = -1/4$ identified exactly; residuals match $-1/4 - 1/(24n)$ to\nall printed digits), and $|S_f(n)| \\le 13.52$ for $n \\le 5000$\n(max at $n = 3298$) vs the traced worst-case $\\sim 2280$ there \u2014\nthe contour constants are $\\sim 170\\times$ loose, as documented.\n\\item Margin assembly (E-028, fully traced constants $A = 10.5$,\n$B_0 = 48$): crossover at $n = 75\\,451$; adopted $N_0 = 8\\times10^4$\nwith margin $16\\,020$ and positive derivative ($15\\,764$ with the\nformal development's majorant --- Step-4 footnote); convexity proved\n(D-057) and grid-confirmed to $N_1 \\approx 9\\times10^{24}$;\n$\\mathrm{margin}(N_1) = 2.458\\times10^{26}$. (First-estimate\nconstants $2/2$: crossover $32\\,871$; measured $0.9/1.0$: $30\\,217$.)\n\\end{itemize}\n\n\\section{Formal verification}\n\nThe inherited density theorem \\cite{anthropic2026riemann} is formally\nverified in Lean~4 (repository \\texttt{zeta-23-lean}, tag v1.0); this\nproject performed a full \\texttt{lake} rebuild (9010 jobs) and a live\n\\texttt{\\#print axioms} audit of all comparator statements, confirming\nthe published axiom profile verbatim: no axioms beyond \\texttt{propext},\n\\texttt{Classical.choice}, and \\texttt{Quot.sound}, with \\texttt{sorry}\noccurring only in the trusted challenge-statement stubs.\n\nThe new result of this paper is formalized in the companion\npackage \\texttt{NguyenRH1} (same toolchain, path-linked to the vendored\n\\texttt{zeta-23-lean} and its \\textsc{Mathlib} build; no new axioms are\nintroduced anywhere). The first tier is complete: the archimedean lower\nbound for $S_\\infty(n)$ and the margin convexity are machine-proved\n(\\texttt{sInfty\\_lower\\_bound}, \\texttt{marginTilde\\_convex}, and the\nchain-rule identity \\texttt{chain\\_rule\\_sqrt\\_shift}), each with axiom\nprofile exactly the three standard axioms above. A design note on the\nconvexity proof: the pen-and-paper argument uses eleven one-variable\nsign conditions, but the Lean development proves an eight-block\n$\\psi$-table plus a separate analytic bound for the $I_I$ term\n($\\widetilde B_4$); the divergence from the eleven-row table is that\n$\\widetilde B_4$ is not concave, so it is excluded from the $\\psi$\nmachinery and bounded directly. That direct bound is itself a worked\nexample of this project's measured-to-traced conversion: the grid\ncheck of E-029 (maximum $2n\\,|\\widetilde B_4''(n)| = 0.189$ on\n$[20, 10^4]$, with the decay argument covering the tail) is superseded\nby a fully analytic factor-monotonicity proof of the sufficient\ninequality $\\widetilde\\psi''(s) - \\widetilde\\psi'(s)/s \\le 1.316 < 2$\nfor $s \\ge \\sqrt{20}$, i.e.\\ $\\widetilde B_4''(n) < 1/(2n)$ for all\n$n \\ge 20$ (\\texttt{iimaj\\_bracket\\_bound},\n\\texttt{IIMaj\\_deriv2\\_lt}).\n\nThe 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.\n\n\\bibliographystyle{plain}\n\\bibliography{references}\n\\end{document}\n"
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  "filePath": "D:\\Python\\Quantum\\research\\paper\\paperA.tex",
  "content": "% ============================================================\n% NGUYEN-RH-1 \u2014 PAPER A (split 2026-08-20, session 013, from draft.tex)\n% An explicit finite-height Li-positivity theorem, machine-verified.\n% Self-contained artifact per the session-012-end-notes review:\n% \u00a73.1 + Appendices A.1/B of the combined draft, retargeted.\n% The spectral-line material lives in paperB.tex (in preparation).\n% ============================================================\n\\documentclass[11pt]{article}\n\\usepackage{amsmath,amssymb,amsthm,mathtools}\n\\usepackage{microtype}\n\\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue]{hyperref}\n\n% The prose is dense with unhyphenatable compounds (Li-positivity,\n% [proved, ...] tags). Give TeX room to spread the slack instead of\n% running past the right margin.\n\\emergencystretch=2em\n\n\\newtheorem{theorem}{Theorem}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\theoremstyle{remark}\n\\newtheorem{remark}[theorem]{Remark}\n\n\\title{An explicit finite-height Li-positivity theorem\\\\\nwith a machine-verified assembly}\n\\author{Kimi K3\\thanks{model from Moonshot AI}\n\\and Claude Code\\thanks{harness from Anthropic}\n\\and Dan T Nguyen\\thanks{developer from Research Ai}}\n\\date{Started: 2026-08-17; revised 2026-08-20}\n\n\\begin{document}\n\\maketitle\n\n\\begin{abstract}\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 three\nnamed hypotheses --- the two computational certificates above and a\nsingle named analytic hypothesis, the traced contour bound\n$|S_f(n)| \\le B(n)$ on $10^5 \\le n \\le (T_0-1)^2$ --- with axiom\nprofile exactly 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\\end{abstract}\n\n\\section{Introduction}\n\nThe Riemann hypothesis (RH) asserts that every nontrivial zero of\n$\\zeta(s)$ has real part $1/2$. The strongest unconditional density\nresult to date is the (machine-verified) theorem that at least $2/3$\nof the nontrivial zeros are simple and on the critical line, and at\nleast $5/6$ of the zeros in a given range are distinct\n\\cite{anthropic2026riemann}; the underlying certificate method has a\ndocumented bandwidth-one ceiling near $0.682$ of the zeros. Two\nclassical reformulations frame most modern attacks: \\emph{Weil\npositivity} --- positivity of the Weil quadratic form on all of\n$C_c^2(\\mathbb R)$ is equivalent to RH \\cite{weil1952fonctions,\nbombieri2000weil}, and the negative index of its truncations counts\noff-line zero pairs \\cite{bombieri2000weil} --- and \\emph{Li's\ncriterion} --- $\\lambda_n > 0$ for all $n$ is equivalent to RH\n\\cite{li1997, bombieri1999lagarias}, with Voros's sharpening that\npositivity of any tail $\\{\\lambda_n\\}_{n \\ge N_0}$ already implies RH\n\\cite{voros2006sharpenings}.\n\nThis paper reports one theorem-level outcome of a systematic research\nprogram on RH: an explicit finite-height Li-positivity theorem,\nassembled with every constant traced, and with its reduction\nmachine-verified in Lean~4 modulo one named analytic hypothesis and a\ndeclared computational perimeter. A companion note (in preparation)\ncovers the spectral-line results of the same program (an exact\nfactorization on Suzuki's operator line and a valley theorem for the\nassociated model spaces). Every claim carries an explicit correctness\ntag, and every numerical input is produced by a deterministic,\narchived script.\n\n\\begin{theorem}[R-004; conditional on computation]\n\\label{thm:finite-height-li-intro}\nAssume the Riemann hypothesis verified to height\n$T_0 = 3\\times 10^{12}$ (Platt--Trudgian \\cite{platt2021riemann},\ninterval arithmetic). Then the Keiper--Li coefficients satisfy\n$\\lambda_n > 0$ for all $1 \\le n \\le (T_0-1)^2 \\approx 9\\times10^{24}$.\n\\end{theorem}\n\nBy Voros's sharpening no infinite-range partial statement exists, so\nthis is the maximal content of the Li route short of RH; the explicit\nchain --- Lagarias's contour argument \\cite{lagarias2007li} with\nTrudgian's explicit zero-counting bounds \\cite{trudgian2014}, an\nelementary archimedean lower bound, a proved margin convexity, and the\nfinite verifications --- appears not to have been assembled before.\n\nSection~2 recalls the certificate-method background and the sanity\ngate. Section~3 states and proves the theorem. Section~4 collects the\nobstruction analysis delimiting the method. Appendix~A describes the\nnumerical machinery; Appendix~B the formal-verification status.\n\n\\section{Background: the Weil-form certificate method}\n\nWe recall the machinery behind the state-of-the-art density theorem\n\\cite{anthropic2026riemann}, which is the launchpad and the sanity\ngate for everything below; details and notation follow that paper.\n\n\\emph{The Weil explicit formula} \\cite{weil1952fonctions} expresses,\nfor a test function $\\psi$ supported in $[-\\tfrac12,\\tfrac12]$ (the\n\\emph{bandwidth-one} class), a quadratic form on the zeros as a\nfunctional of the primes. Compressing it via a Gabor (Gaussian\nwindowed exponential) frame at the critical density $h = 2\\pi/L$\nyields a finite matrix identity $\\tilde G = P + Q$: a zero-side\nmatrix $\\tilde G$ built from the zeros in a window of size $L$\ndecomposes into a prime-side matrix $P$ (computed unconditionally\nfrom Montgomery's second moment / pair-correlation input\n\\cite{montgomery1973pair}) and a controlled remainder $Q$.\n\n\\emph{The rank--trace inequality.} Writing $\\tilde G$ in inertia form,\na rank--trace comparison between the positive eigenspace of $\\tilde G$\nand the positive part of $P$ converts the second-moment input\n$\\|\\tilde G\\|_{HS}^2 = (R(\\psi) + o(1))\\,N$ into a lower bound on the\nnumber of simple on-line zeros in the window. The strength of the\ncertificate is governed by the \\emph{window ratio}\n\\[\nR(\\psi) \\;=\\; \\frac{\\int \\psi^2 \\;+\\;\n\\iint |u-v|\\,\\psi(u)\\psi(v)\\,du\\,dv}{\\bigl(\\int \\psi\\bigr)^2}\n\\qquad (\\psi \\in C([-\\tfrac12,\\tfrac12]) \\text{ even}, \\ \\psi > 0),\n\\]\nwith $R(\\psi_0) = 4/3$ for the flat window\n$\\psi_0 = \\mathbf 1_{[-1/2,1/2]}$ and\n$R(\\psi_{MT}) = \\tfrac12 + \\tfrac{1}{\\sqrt2}\\cot\\tfrac{1}{\\sqrt2}\n= 1.32749\\ldots$ for the Montgomery--Taylor window (both verified\nsymbolically and against $60$-digit reference values in this project,\nE-001), giving the certified fractions $2 - R(\\psi)$: $0.6725$ of\nzeros simple and on the line, $0.83625$ distinct. A sharpness analysis\nlocates the bandwidth-one ceiling of the method at $\\approx 0.682$,\nand shows the obstructions to going further are structural\n(Section~4).\n\nOne feature of this machinery matters here: the certificate fails, as\nit must, for the Davenport--Heilbronn and Epstein zeta functions\n(which violate the RH-analogue); the main theorem of this paper has\nbeen required to pass this gate structurally (project file D-053).\n\n\\section{The finite-height Li theorem (R-004)}\n\nLet $\\lambda_n = \\sum_{\\rho}{}'\\,\\bigl[1-\\bigl(1-\\tfrac1\\rho\\bigr)^n\\bigr]$\nbe the Keiper--Li coefficients of the Riemann $\\xi$-function. Li's\ncriterion states $\\lambda_n > 0$ for all $n \\geq 1$ if and only if the\nRiemann hypothesis holds \\cite{li1997,bombieri1999lagarias}. Voros\n\\cite{voros2006sharpenings} showed that ``$\\lambda_n > 0$ for all\n$n \\geq N_0$'' is already equivalent to RH for any $N_0$, so no\npartial infinite-range statement exists; the maximal content of the Li\nroute short of RH is an \\emph{explicit finite-height} statement.\n\n\\begin{theorem}[R-004; conditional on computation]\n\\label{thm:finite-height-li}\nAssume the Riemann hypothesis verified to height\n$T_0 = 3\\times 10^{12}$ (Platt--Trudgian \\cite{platt2021riemann},\ninterval arithmetic). 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\n\\begin{proof}[Proof sketch (full assembly: D-037)]\nBombieri--Lagarias \\cite{bombieri1999lagarias} give\n$\\lambda_n = S_\\infty(n) - S_f(n) + 1$ with $S_\\infty$ the archimedean\nand $S_f$ the finite-place contribution.\n\n\\emph{Step 1 (archimedean lower bound).} From Lagarias's formula\n\\cite[(4.11)]{lagarias2007li},\n$S_\\infty(n) = \\sum_{j=1}^n (-1)^j \\binom{n}{j} a_j$ with\n$a_1 = (\\log 4\\pi + \\gamma)/2$ and $a_j = (1-2^{-j})\\zeta(j)\n= \\sum_{m \\text{ odd}} m^{-j}$ ($j \\ge 2$), the $j$-sum is a finite\ndifference and one obtains the \\emph{exact} identity\n$S_\\infty(n) = (1-a_1)n - 1 + \\sum_{m \\ge 3,\\, m \\text{ odd}}\n\\bigl[(1-\\tfrac1m)^n - 1 + \\tfrac{n}{m}\\bigr]$.\nBernoulli's inequality and the explicit harmonic bounds\n$\\log M + \\gamma + \\tfrac{1}{2M} - \\tfrac{1}{12M^2} \\le H_M \\le\n\\log M + \\gamma + \\tfrac{1}{2M}$ give, for $n \\ge 20$,\n\\[\nS_\\infty(n) \\;\\ge\\; \\tfrac12 n\\log n + (C_1 - \\tfrac{\\gamma}{2})\\,n - 4,\n\\qquad C_1 = \\tfrac{\\gamma - 1 - \\log 2\\pi}{2}.\n\\]\n(Machine-validated against exact values for $20 \\le n \\le 5000$.)\n\n\\emph{Step 2 (finite-part upper bound).} Tracing Lagarias's contour\nargument \\cite[Thm~6.1]{lagarias2007li} with explicit constants\n(Trudgian's $N(T)$ and $S(T)$ bounds \\cite{trudgian2014}, partial\nfractions for $\\zeta'/\\zeta$ with the reference-point estimate\n$|R(s)| \\le 10.5\\log(|t|{+}2) + 48$ for $-5/2 \\le \\sigma \\le 3$,\n$|t| \\ge 10$ (D-056), kernel bounds $e^4+1$, a measured\nshort-segment constant $M_0 = 1.3352$) gives\n$|S_f(n)| \\le B(n)$ with\n\\[\nB(n) = 2N(\\sqrt n{+}1) + 4T\\bigl[\\tfrac23 C_0(T) + 10.5\\log(T{+}2) + 48\\bigr]\n+ 40M_0 + 2(e^4{+}1)\\bigl[7H(T) + 3.3\\bigr] + I_I + 1,\n\\]\n$T = \\sqrt n + 1$, $C_0(t)$ the explicit unit-interval zero count\n(the honest Trudgian form, including the $\\log\\log$ term),\n$H(T) = 3C_0(T)^2 + 10.5\\log(T{+}2) + 48$, $I_I$ an explicitly\nbounded boundary-integral term (D-037/D-056) and the final $1$ the\ntrivial-zero contribution.\n\n\\emph{Step 3 (bridge).} For $n \\le (T_0-1)^2$ every zero with\n$|\\gamma| \\le \\sqrt n + 1$ is on the line, so\n$|1-(1-1/\\rho)^n| \\le 2$ per zero and\n$|\\lambda_n(T)| \\le 2N(T)$; all other contour terms are\n\\emph{unconditional}.\n\n\\emph{Step 4 (margin).}\n$\\mathrm{margin}(n) := \\tfrac12 n\\log n + (C_1 - \\gamma/2)n - 4 - B(n) + 1$\nis convex for $n \\ge 20$ (proved: chain-rule reduction to eleven\none-variable sign conditions, each discharged elementarily \u2014 D-057),\nand $\\mathrm{margin}(8\\times 10^4) = 15\\,764 > 0$ with\n$\\mathrm{margin}(10^5) > \\mathrm{margin}(8\\times 10^4)$; by convexity\n(secant slopes increase) hence\n$\\mathrm{margin}(n) > 0$ for all $n \\in [10^5, N_1]$\\footnote{The\nmargin value at $8\\times10^4$ has a documented trail: $16\\,020$ with\nthe Trudgian $F + S_b$ zero-count majorant (E-028); $15\\,764$ with the\nmildly enlarged majorant consumed by the machine verification --- a\nnegative-weight term of $F + S_b$ dropped so that every convexity\nblock has provably negative $\\psi$ (the true majorant's $\\psi$ crosses\nzero near $T \\approx 16$), at a traced cost of $\\approx 0.9\\sqrt n$,\ni.e.\\ $256$ of the $16\\,020$; and $\\ge 8\\,001$ as the certified\nrational interval lower bound that the machine discharge actually\nconsumes (\\texttt{NguyenRH1},\n\\texttt{marginTilde\\_FAlt\\_80000\\_pos}).}\n(also computed to $n = 4\\times10^5$ on a unit grid and to $N_1$ on a\nlogarithmic grid; $\\mathrm{margin}(N_1) = 2.458\\times 10^{26}$).\n\n\\emph{Step 5 (finite verification).} $\\lambda_n > 0$ for $n \\le 3000$\nby exact high-precision computation (this project, two independent\nmethods, agreement to $4.8\\times10^{-25}$ relative), and for\n$n \\le 5\\times10^5$ by Misguich--Voros \\cite{voros2017li}. The ranges\noverlap the $n = 10^5$ split by a factor $5$ (and the crossover\n$N_0 = 8\\times 10^4$ by $6.25$).\n\\end{proof}\n\n\\begin{remark}\n(i) Using Lagarias's printed bridge window $n \\le T^2/(4\\log^2 T)$\ninstead of the traced proof gives the conservative\n$N_1^* = 2.73\\times 10^{21}$; the theorem is robust either way.\n(ii) The traced worst-case bound $|S_f|$ is $\\sim 170\\times$ loose\nagainst the measured curve ($|S_f| \\le 13.52$ for $n \\le 5000$); the\nslack is structural to contour constants, not a gap.\n(iii) The dependence on computation is explicit: Platt--Trudgian's\ninterval-arithmetic verification, and the finite Li-coefficient\nverifications.\n\\end{remark}\n\n\\section{Obstruction analysis}\n% Documented dead ends from research/graveyard/ where they\n% delimit the method (contributes to success criterion 4).\n\nThe following routes are closed with documented obstructions\n(full files in the project archive, graveyard G-001--G-003 and\nderivations D-009--D-013):\n\\begin{itemize}\n\\item \\emph{Certificate extension past $2/3$ in the bandwidth-one\nframework} (Track A): a six-wall map (D-013) shows $>2/3$ is\nimpossible within the current data class; the window-class optimum is\nsaturated in both the $\\xi$ and $\\xi'$ worlds (D-009, D-030), and\nmoment-ladder certificates cap below $2/3$ at every order (G-002).\n\\item \\emph{Multiple-zero bounds as chain input} (G-001): the\n$s_2/p$ substitution symmetry makes depth$\\to$0 off-line pairs\nindistinguishable from double zeros in all two-moment data.\n\\item \\emph{Density theorems $\\to$ de~Bruijn--Newman $\\Lambda$}\n(G-003): one-sided information \u2014 density theorems lower-bound the\non-line fraction while $\\Lambda$ requires an upper bound on the\ncomplex fraction.\n\\item \\emph{$\\xi^{(k)}$-certificate routes} (T4/T4b, D-022, D-031):\nthe strength mechanism (a perfect-square node in the diagonal law)\ndoes not transfer to $\\xi$, and the $s_2/p$ symmetry caps every\nderivative class below $100\\%$.\n\\end{itemize}\n\n\\appendix\n\\section{Numerical experiments}\n% Tables/figures generated from research/numerics/data/ CSVs.\n\nAll experiments are deterministic scripts with CSV+JSON sidecars in the\nproject archive (\\texttt{research/numerics/}); checks as stated.\n\n\\subsection{Li-coefficient machinery (E-011/E-015/E-021/E-023)}\n\\begin{itemize}\n\\item $\\lambda_n > 0$ verified for $n \\le 3000$ from the arithmetic\nside (Bombieri--Lagarias Thm~2, zero-free; \\textsc{flint} Stieltjes +\nArb $\\zeta$, 1000--1450-digit working precision), cross-validated\nagainst the zero-side computation (agreement $4.8\\times10^{-25}$\nrelative) and $\\lambda_1$ agreeing with its closed form to better\nthan $10^{-28}$ relative.\n\\item Exact archimedean part: $S_\\infty(n) = \\tfrac12 n\\log n + C_1 n\n- \\tfrac14 - \\tfrac{1}{24n} + o(1/n)$ measured to $n = 5000$\n($c_0 = -1/4$ identified exactly; residuals match $-1/4 - 1/(24n)$ to\nall printed digits), and $|S_f(n)| \\le 13.52$ for $n \\le 5000$\n(max at $n = 3298$) vs the traced worst-case $\\sim 2280$ there \u2014\nthe contour constants are $\\sim 170\\times$ loose, as documented.\n\\item Margin assembly (E-028, fully traced constants $A = 10.5$,\n$B_0 = 48$): crossover at $n = 75\\,451$; adopted $N_0 = 8\\times10^4$\nwith margin $16\\,020$ and positive derivative ($15\\,764$ with the\nformal development's majorant --- Step-4 footnote); convexity proved\n(D-057) and grid-confirmed to $N_1 \\approx 9\\times10^{24}$;\n$\\mathrm{margin}(N_1) = 2.458\\times10^{26}$. (First-estimate\nconstants $2/2$: crossover $32\\,871$; measured $0.9/1.0$: $30\\,217$.)\n\\end{itemize}\n\n\\section{Formal verification}\n\nThe inherited density theorem \\cite{anthropic2026riemann} is formally\nverified in Lean~4 (repository \\texttt{zeta-23-lean}, tag v1.0); this\nproject performed a full \\texttt{lake} rebuild (9010 jobs) and a live\n\\texttt{\\#print axioms} audit of all comparator statements, confirming\nthe published axiom profile verbatim: no axioms beyond \\texttt{propext},\n\\texttt{Classical.choice}, and \\texttt{Quot.sound}, with \\texttt{sorry}\noccurring only in the trusted challenge-statement stubs.\n\nThe new result of this paper is formalized in the companion\npackage \\texttt{NguyenRH1} (same toolchain, path-linked to the vendored\n\\texttt{zeta-23-lean} and its \\textsc{Mathlib} build; no new axioms are\nintroduced anywhere). The first tier is complete: the archimedean lower\nbound for $S_\\infty(n)$ and the margin convexity are machine-proved\n(\\texttt{sInfty\\_lower\\_bound}, \\texttt{marginTilde\\_convex}, and the\nchain-rule identity \\texttt{chain\\_rule\\_sqrt\\_shift}), each with axiom\nprofile exactly the three standard axioms above. A design note on the\nconvexity proof: the pen-and-paper argument uses eleven one-variable\nsign conditions, but the Lean development proves an eight-block\n$\\psi$-table plus a separate analytic bound for the $I_I$ term\n($\\widetilde B_4$); the divergence from the eleven-row table is that\n$\\widetilde B_4$ is not concave, so it is excluded from the $\\psi$\nmachinery and bounded directly. That direct bound is itself a worked\nexample of this project's measured-to-traced conversion: the grid\ncheck of E-029 (maximum $2n\\,|\\widetilde B_4''(n)| = 0.189$ on\n$[20, 10^4]$, with the decay argument covering the tail) is superseded\nby a fully analytic factor-monotonicity proof of the sufficient\ninequality $\\widetilde\\psi''(s) - \\widetilde\\psi'(s)/s \\le 1.316 < 2$\nfor $s \\ge \\sqrt{20}$, i.e.\\ $\\widetilde B_4''(n) < 1/(2n)$ for all\n$n \\ge 20$ (\\texttt{iimaj\\_bracket\\_bound},\n\\texttt{IIMaj\\_deriv2\\_lt}).\n\nThe 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.\n\n\\bibliographystyle{plain}\n\\bibliography{references}\n\\end{document}\n",
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