{
  "file_path": "D:\\Python\\Quantum\\research\\numerics\\scripts\\db.py",
  "old_string": "def cmd_list(a) -> None:",
  "new_string": "def cmd_add_derivation(a) -> None:\n    con = connect()\n    con.execute(\n        \"\"\"INSERT OR REPLACE INTO derivations\n           (id, title, task, status, file, date) VALUES (?,?,?,?,?,?)\"\"\",\n        (a.id, a.title, a.task, a.status, a.file, date.today().isoformat()),\n    )\n    con.commit()\n    print(f\"logged derivation {a.id}\")\n\n\ndef cmd_add_graveyard(a) -> None:\n    con = connect()\n    con.execute(\n        \"\"\"INSERT OR REPLACE INTO graveyard\n           (id, title, track, obstruction, file, date) VALUES (?,?,?,?,?,?)\"\"\",\n        (a.id, a.title, a.track, a.obstruction, a.file, date.today().isoformat()),\n    )\n    con.commit()\n    print(f\"logged graveyard entry {a.id}\")\n\n\ndef cmd_list(a) -> None:"
}

{
  "filePath": "D:\\Python\\Quantum\\research\\paper\\draft.tex",
  "oldString": "with $R(\\psi_0) = 4/3$ for the triangle window 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 to $60$ digits in this project), giving the certified\nfractions $2 - R(\\psi)$: $0.6725$ of zeros simple and on the line,\n$0.83625$ distinct.",
  "newString": "with $R(\\psi_0) = 4/3$ 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.",
  "originalFile": "% ============================================================\n% NGUYEN-RH-1 \u2014 working paper draft (living document)\n% Grown incrementally from research/results/ (each R-# = verified part)\n% ============================================================\n\\documentclass[11pt]{article}\n\\usepackage{amsmath,amssymb,amsthm,mathtools}\n\\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue]{hyperref}\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{Toward the Riemann Hypothesis:\\\\\nWorking Draft --- Project NGUYEN-RH-1}\n\\author{Kimi K3\\thanks{Moonshot AI}\n\\and Claude Code\\thanks{Anthropic}\n\\and Dan T Nguyen\\thanks{ResearchAi}}\n\\date{Started: 2026-08-17; revised 2026-08-19}\n\n\\begin{document}\n\\maketitle\n\n\\begin{abstract}\nThree contributions, each carrying an explicit correctness tag.\n(1)~[proved, conditional on computation] An explicit finite-height\nLi-positivity theorem: the Keiper--Li coefficients satisfy\n$\\lambda_n > 0$ for all $1 \\le n \\le (T_0-1)^2 \\approx 9\\times10^{24}$,\nconditional on the interval-arithmetic verification of RH to height\n$T_0 = 3\\times10^{12}$ (Platt--Trudgian) and on finite Li-coefficient\ncomputations, with every constant traced (first assembled explicit\nchain; by Voros's sharpening this is the maximal content of the Li\nroute short of RH).\n(2)~[proved modulo itemized formalities] On Suzuki's operator line:\nan exact factorization $W = F\\cdot(R_0 + R_\\tau)$ of his\ncharacteristic function reducing his Corollary~1.6 to a\nWiener--Hopf tail-cancellation, and a Krein-space reduction of his\nlimit formula to a real-spectrum statement.\n(3)~[proved, conditional on RH] A valley theorem for the\nreproducing-kernel diagonal of the associated model spaces\n(plateau, floors, universal profile), explaining and certifying a\nspectral-measure analysis carried out numerically to $a = 12$: the\nkernel diagonal exhibits valleys exactly at the true zero ordinates\n(symbol $= 1 + {}$zero comb), positions lock quadratically, and the\nconsolidation obstruction is isolated as a generic Szeg\\H{o}\nequilibrium with residual $\\to Z/(2\\pi)$; all identified breaking\nchannels are mapped, with closures documented.\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 the output of a systematic research program that\nattacked RH along four tracks (certificate extension, spectral /\nHilbert--P\\'olya, positivity criteria, formal verification), closing\neach route either with a theorem or with a documented obstruction.\nEvery claim carries an explicit correctness tag, and every numerical\ninput is produced by a deterministic, archived script. The three\ntheorem-level contributions are:\n\n\\begin{enumerate}\n\\item \\textbf{An explicit finite-height Li theorem} (Section~3.1).\nConditional on the interval-arithmetic verification of RH to height\n$T_0 = 3\\times10^{12}$ \\cite{platt2021riemann} and on finite\nLi-coefficient computations, $\\lambda_n > 0$ for all\n$n \\le (T_0-1)^2 \\approx 9\\times10^{24}$, with all constants traced\nthrough Lagarias's contour argument \\cite{lagarias2007li}. By Voros's\nsharpening no infinite-range partial statement exists, so this is the\nmaximal content of the Li route short of RH; the explicit chain\nappears not to have been assembled before.\n\n\\item \\textbf{The Suzuki operator line, reduced} (Section~3.2).\nSuzuki's self-adjoint operator family $\\bar D_{a,\\theta}$\n\\cite{suzuki2026weil} realizes the Weil form through model spaces of a\n\\emph{screw function}; his Corollary~1.6 asserts that a certain limit\nformula would imply RH. We give an exact factorization of his\ncharacteristic function, $W = F\\,(R_0 + R_\\tau)$, isolating the open\ncontent of the limit formula as a finite-interval Wiener--Hopf\ntail-cancellation $(\\star)$ on the symbol $\\hat g = z^{-2}\\xi'/\\xi$,\nand a Krein-space reduction: the limit formula holds for some boundary\nflow $\\theta(a)$ if and only if the limit operator has real spectrum.\n\n\\item \\textbf{The valley theorem} (Section~3.3). Conditional on RH,\nthe reproducing-kernel diagonal $K_a(z,z)$ of the model spaces\n$\\mathcal H(T_a)$ is proved to exhibit a plateau $\\to 2a$ away from\nthe zeros and valleys of depth $\\log 2a + o(1)$ with a universal\nprofile at the zero ordinates. This certifies, and is certified by, a\nnumerical spectral-measure analysis to $a=12$: the operator's symbol\nis $1 + {}$the zero comb (the explicit formula in operator-symbol\nform), spectral atoms lock onto the true ordinates quadratically\n($0.65/a^2$), and the obstruction to ``consolidation'' of the measure\nonto the zeros is isolated as a generic Szeg\\H{o} equilibrium with\nresidual $\\to Z/(2\\pi)$ per window $[0,Z]$. A stress test with\nsynthetic off-line zero pairs shows the associated detector responds\nat three levels (bare form, pencil, tuned carrier).\n\\end{enumerate}\n\nSection~4 collects the obstruction analysis: routes that are closed,\nwith the obstruction identified in each case (six-wall ceiling map for\ncertificate extension; substitution symmetries; one-sided information\nin density routes; disjoint feasibility windows for the ground-state\nchannel). Appendix~A describes the numerical machinery;\nAppendix~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 triangle window 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 to $60$ digits in this project), giving the certified\nfractions $2 - R(\\psi)$: $0.6725$ of zeros simple and on the line,\n$0.83625$ distinct. A sharpness analysis locates the bandwidth-one\nceiling of the method at $\\approx 0.682$, and shows the obstructions\nto going further are structural (Section~4).\n\nTwo features of this machinery matter for the present paper. First,\nits only zero-side input is the Weil form itself, so any\nreformulation of the Weil form --- as an operator family (Suzuki's\nline, Section~3.2) or as a reproducing-kernel variational problem\n(Section~3.3) --- inherits the same positivity content. Second, the\ncertificate fails, as it must, for the Davenport--Heilbronn and\nEpstein zeta functions (which violate the RH-analogue); we use this\nas a standing sanity gate on every new criterion.\n\n\\section{Results}\n% Each new theorem/lemma imported from research/results/R-NNN-*.md\n% with its [PROVED]/[PROVED-CONDITIONAL] tag.\n\n\\subsection{An explicit finite-height Li-positivity 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] + o(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$H(T) = 3C_0(T)^2 + 10.5\\log(T{+}2) + 48$.\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 $\\mathrm{margin}(8\\times 10^4) = 16\\,020 > 0$ with positive\nderivative; hence\n$\\mathrm{margin}(n) > 0$ for all $n \\in [8\\times10^4, N_1]$\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 $N_0 = 8\\times 10^4$ by a factor $6.25$.\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\\subsection{The Suzuki operator line: exact factorization and the\nspectral-measure equilibrium (R-003, D-040--D-046)}\n\nSuzuki \\cite{suzuki2026weil} constructs, for each $a > 0$, a\nself-adjoint operator family $\\bar D_{a,\\theta}$ on model spaces\n$\\mathcal H(T_a)$ built from the \\emph{screw function} $g$ of $\\zeta$\n($\\hat g(z) = z^{-2}\\xi'(\\tfrac12-iz)/\\xi(\\tfrac12-iz)$), whose spectra\nare the zeros of\n$W(a,\\theta;z) = (z-i)V_+(z) + e^{i\\theta}(z+i)V_-(z)$, with $V_\\pm$\nthe Fourier transforms of deficiency vectors. His Corollary~1.6 states\nthat a limit formula $e^{\\phi(a,z)}W(a,\\theta(a);z) \\to\nz^2\\xi(\\tfrac12-iz)/\\xi'(\\tfrac12-iz)$ (uniform on compacts, for some\nchoice of $\\theta(a)$, $\\phi$) would imply RH.\n\n\\begin{theorem}[R-003; modulo standard tempered-FT bookkeeping]\nOne has the exact factorization\n$W(a,\\theta;z) = F(z)\\,M(a,\\theta;z)$ with\n$F(z) = z^2\\xi(\\tfrac12-iz)/\\xi'(\\tfrac12-iz)$ and\n$M = R_0 + R_\\tau$, where\n$R_0 = 2e^{a+i\\theta/2}\\sin(az - \\theta/2) + o(e^a)$ is an explicit\nsine skeleton and $R_\\tau$ is the out-of-interval tail transform.\nHence the limit formula is equivalent to the \\emph{tail-cancellation\nstatement} $(\\star)$: $R_\\tau$ cancels the skeleton leaving a zero-free\nremainder \u2014 a finite-interval Wiener--Hopf remainder estimate on the\nsymbol $\\hat g$, whose factorization is gated by the zero locations.\n\\end{theorem}\n\nWe studied the spectral (Clark/de~Branges) measures\n$\\mu_a = \\sum_k w_k\\,\\delta_{z_k}$ of $\\bar D_{a,\\pi}$ numerically to\n$a = 12$ ($w_k = 1/K_a(z_k,z_k)$, $K_a$ the model-space reproducing\nkernel; Galerkin machinery validated against the zero-side explicit\nformula to $4.5\\times10^{-6}$). The picture that emerges\n(D-040--D-045):\n\\begin{enumerate}\n\\item \\emph{The metric knows the zeros.} $K_a(z,z)$ has deep narrow\nvalleys exactly at the true ordinates $\\gamma$ (depth\n$\\Delta(a) \\approx \\log a + 0.64$, width $\\sim 1.5/a$), while\n$K_a(z,z) \\to 2a$ (the free Paley--Wiener diagonal) on the plateau:\nthe operator's symbol is \\emph{1 + the zero comb} \u2014 the explicit\nformula in operator-symbol form.\n\\item \\emph{Positions lock and masses are right.} Each $\\gamma$ has a\nspectral atom within $\\approx 0.65/a^2$ (quadratic locking, correcting\nan earlier wrong-direction metric), carrying mass $\\approx m_\\gamma = 1$.\n\\item \\emph{The equilibrium.} The residual mass over the excess\n(lattice) atoms tends to $Z/(2\\pi)$ per window $[0,Z]$, not to zero:\nthe family equilibrates exactly at the consolidation boundary\n(generic Szeg\\H{o} bulk behavior). At fixed $\\theta$ this is structural\n($\\theta$-independence is measured: residual spread $\\le 2.2\\%$ over\nthe full extension family).\n\\item \\emph{The breaking mechanism.} A shift $\\lambda \\to \\lambda_a^-$\n(bottom of the Weil-form spectrum) pins the valley floors (masses stay\n$1$) while the plateau response diverges \u2014 the unique channel that\nkills the residual. It is machinery-blocked: $\\lambda_a$ decays\nsuperexponentially ($0.414$ at $a{=}0.25$ to $<3\\times10^{-12}$ at\n$a{=}2$) below any achievable discretization floor at zero-resolving\n$a$ (Graveyard G-004).\n\\end{enumerate}\n\nOn the operator-theoretic side (D-041): the limit operator $D$ exists\nunconditionally as a $J$-self-adjoint operator in a Krein space\n(negative index = number of off-line zero pairs, Bombieri), with\n$\\mathrm{spec}(D) = \\Gamma$; and deficiency-element convergence is\nequivalent to strong resolvent convergence\n$\\bar D_{a,\\theta(a)} \\to D$ (Theorem D1), giving the reduction:\nthe limit formula (1.12) holds for some $\\theta(a), \\phi$ if and only\nif $\\mathrm{spec}(D) \\subset \\mathbb R$. The real-spectrum persistence\nstep is exactly RH (the paywall), as expected.\n\n\\subsection{The valley theorem (D-048--D-050)}\n\nThe spectral-measure picture above rests on a theorem about the\nreproducing-kernel diagonal $K_a(z,z) = \\langle e_z, T_a^{-1}e_z\\rangle$\nof the model spaces $\\mathcal H(T_a)$.\n\n\\begin{theorem}[valley law; conditional on RH]\nAssume RH. For $a \\ge 1$, $z \\in \\mathbb R$:\n\\begin{enumerate}\n\\item \\emph{(plateau)} $K_a(z,z) \\le 2a$ exactly, and\n$K_a(z,z) \\ge 2a(1 - C'(z,\\delta)/a)$ for\n$\\mathrm{dist}(z,\\Gamma) \\ge \\delta$, with explicit $C'$;\n\\item \\emph{(floors)} at each zero $\\gamma$,\n$K_a(\\gamma,\\gamma) \\le 1/m_\\gamma$ exactly; and if $\\gamma$ has no\nother zero within distance $1$,\n\\[\n1 - \\tfrac{1}{2a} - C\\tfrac{1+\\log\\gamma}{a^2}\n\\;\\le\\; K_a(\\gamma,\\gamma),\n\\]\nwith explicit $C$; in general the same rate holds up to an explicit\nlocal-configuration constant (a Gram-determinant factor and an\nannulus count $E(\\gamma,a)$, zero generically) via node-killing\nprojection test functions (D-058);\n\\item \\emph{(profile)} near such a zero, $K_a$ has the universal lower\nenvelope $1/(\\mathrm{sinc}^2((z-\\gamma)a) + 1/(2a) + D_1/(4a^2))$.\n\\end{enumerate}\nIn particular the valley depth $\\Delta(a) = \\log 2a + o(1)$.\n\\end{theorem}\n\n\\begin{proof}[Proof idea (D-049, D-058)]\nThe variational identity\n$K_a(z,z) = \\sup_v |\\hat v(z)|^2/(\\|v\\|^2 + Q_W^a(v))$ (the sup over\nthe zero-side form domain, which contains the flat bumps \u2014 Lemma FD)\nwith the Weil explicit formula $Q_W^a(v) = \\sum_\\gamma m_\\gamma|\\hat\nv(\\gamma)|^2$ (the pure comb under RH). Plateau upper bound:\n$T_a \\ge I$. Floor: single-term domination above; below, the flat\nbump for isolated zeros, and in general the projection of $e_\\gamma$\noff the span of the nearby evaluation vectors (which vanishes exactly\nat the neighboring zeros). The combinatorial input is Trudgian's\nzero-density bound.\n\\end{proof}\n\n\\begin{remark}[the detector content; D-050, E-027]\nThe detector has three levels (measured, E-027; all four cases\n$a \\in \\{4,8\\}$, two heights): (L1) the bare Weil form is indefinite\nfor \\emph{any} off-line deviation $\\delta > 0$ (strict Schwarz on the\ntwo evaluation functionals; no threshold), with tiny magnitude near\n$\\delta = 0$; (L2) the pencil $T_a = I + B_a$ loses positivity at a\nmagnitude-set onset (measured $\\delta \\approx 0.11$ at $a=4$,\n$0.038$ at $a=8$); (L3) the tuned flat-carrier quotient turns negative\nexactly at $\\delta = \\pi/(2a)$ (proved one-direction; measured to\n$0.9\\%$). Hence the measured valley picture (floors $\\approx 1$ at\nthe true ordinates, plateau $\\to 2a$) is a numerical verification of\nWeil positivity in the $\\mathcal H(T_a)$ metric to scale $a = 12$,\nand an off-line pair produces negative form directions localized at\nits height.\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-004 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\\item \\emph{The $\\lambda \\to \\lambda_a^-$ ground-state channel}\n(G-004, D-046): the feasibility windows are disjoint \u2014 ground-state\nresolution requires $a \\lesssim 1.3$ (superexponential decay of\n$\\lambda_a$) while zero resolution requires $a \\gtrsim 4$.\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 $\\lambda_1$ to 102 digits against\n$\\gamma/2$-type closed forms.\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; 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\\subsection{Spectral-measure program (E-016--E-025)}\n\\begin{itemize}\n\\item Galerkin realization of Suzuki's Weil-form operator validated\nagainst the zero-side explicit formula to $4.5\\times10^{-6}$;\nzero-location by sign-change + Brent refinement; winding-number\ndiagnostics for off-axis zeros.\n\\item Clark/de~Branges weights $1/K_a(z_k,z_k)$ via Schur-reduced\nbordered Toeplitz solves (validated to $3.6\\times10^{-14}$ against\ndense bordered solves; $h$-refinement $1.1\\times10^{-3}$).\n\\item Valley observable: $\\Delta(a) = 2.16, 2.51, 2.74, 2.87, 2.94$\nfor $a = 4,6,8,10,12$; plateau $K \\to 2a$; floors $\\approx 1.04$;\nmass within $\\pm0.1$ of true zeros $63\\% \\to 72.5\\%$; positional\nlocking $\\approx 0.65/a^2$; residual $\\to Z/(2\\pi)$.\n\\item $\\theta$-independence (E-024): residual spread $\\le 2.2\\%$ over\n$7$ boundary conditions $\\times$ $3$ values of $a$.\n\\item $\\lambda$-flow (E-025): discretization floor documented\n(pencil bottom crosses zero at $a \\approx 5.2$ for $h = 0.01$,\n$O(h)$ artifact); channel response measured at $a=4$\n($\\Delta\\colon 2.16 \\to 6.99$, residual $48 \\to 11$).\n\\item Detector stress test (E-027): a synthetic off-line zero pair\n(mid-gap heights $23.0$, $51.4$; $a \\in \\{4,8\\}$; Legendre--Galerkin\nzero-side form, $N > \\gamma^* a$) confirms the three-level detector\npicture: bare-form negativity onset immediate ($\\sim 10^{13}\\times$\ncontrol floor at the smallest $\\delta$), pencil positivity loss at\n$\\delta \\approx 0.11$ ($a=4$) / $0.038$ ($a=8$), tuned-carrier\nthreshold at $\\pi/(2a)$ to $\\le 0.9\\%$; negative directions\nlocalized at the pair (contrast $3.4$--$6.1$).\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); a\nstatic audit performed in this project confirmed the published axiom\nprofile: no real axioms beyond \\texttt{propext},\n\\texttt{Classical.choice}, and \\texttt{Quot.sound}, with\n\\texttt{sorry} occurring only in the trusted challenge-statement stubs\n(full \\texttt{lake} rebuild deferred).\n\nThe new results of this paper are \\emph{not yet} formalized. In order\nof increasing formalization cost, the targets are: (i) the elementary\narchimedean lower bound for $S_\\infty(n)$ and the margin convexity\n(eleven one-variable sign conditions, each elementary --- suited to a\n\\texttt{positivity}/interval-arithmetic tactic pipeline); (ii) the\ntraced contour bound for $|S_f(n)|$ (requires explicit $N(T)$, $S(T)$,\nand partial-fraction bounds for $\\zeta'/\\zeta$ --- a nontrivial but\nfinite library task); (iii) the valley theorem (requires a formal\nmodel-space/reproducing-kernel infrastructure that does not currently\nexist in \\textsc{Mathlib}); (iv) the Wiener--Hopf reduction\n(Section~3.2), whose remaining steps are open mathematics rather than\nformalization work. The finite-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} would leave 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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        " \\iint |u-v|\\,\\psi(u)\\psi(v)\\,du\\,dv}{\\bigl(\\int \\psi\\bigr)^2}",
        " \\qquad (\\psi \\in C([-\\tfrac12,\\tfrac12]) \\text{ even}, \\ \\psi > 0),",
        " \\]",
        "-with $R(\\psi_0) = 4/3$ for the triangle window and",
        "+with $R(\\psi_0) = 4/3$ for the flat window",
        "+$\\psi_0 = \\mathbf 1_{[-1/2,1/2]}$ and",
        " $R(\\psi_{MT}) = \\tfrac12 + \\tfrac{1}{\\sqrt2}\\cot\\tfrac{1}{\\sqrt2}",
        " = 1.32749\\ldots$ for the Montgomery--Taylor window (both verified",
        "-symbolically to $60$ digits in this project), giving the certified",
        "-fractions $2 - R(\\psi)$: $0.6725$ of zeros simple and on the line,",
        "-$0.83625$ distinct. A sharpness analysis locates the bandwidth-one",
        "+symbolically and against $60$-digit reference values in this project,",
        "+E-001), giving the certified fractions $2 - R(\\psi)$: $0.6725$ of",
        "+zeros simple and on the line, $0.83625$ distinct. A sharpness analysis locates the bandwidth-one",
        " ceiling of the method at $\\approx 0.682$, and shows the obstructions",
        " to going further are structural (Section~4).",
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