{
  "file_path": "D:\\Python\\Quantum\\research\\paper\\draft.tex",
  "old_string": "\\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}",
  "new_string": "\\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."
}

{
  "filePath": "D:\\Python\\Quantum\\research\\paper\\draft.tex",
  "oldString": "\\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}",
  "newString": "\\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.",
  "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{[authors TBD]}\n\\date{Started: 2026-08-17}\n\n\\begin{document}\n\\maketitle\n\n\\begin{abstract}\n[To be written. Placeholder: unconditional progress toward the Riemann\nHypothesis building on the recent two-thirds simplicity theorem.]\n\\end{abstract}\n\n\\section{Introduction}\n% State of the art: >= 2/3 simple on-line, >= 5/6 distinct (Anthropic, 2026).\n% Our contribution: [TBD from results ledger]\n\n\\section{Background: the Weil-form certificate method}\n% Summary of the inherited machinery: Weil explicit formula, Gabor\n% compression \\tilde G = P + Q, rank--trace inequality, Montgomery second\n% moment, window theory R(\\psi).\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 $\\zeta'/\\zeta$, 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) + 2\\log(T{+}2) + 2\\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$H(T) = 3C_0(T)^2 + 2\\log(T{+}2) + 2$.\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 $n \\ge 20$ (each component of $B$ is concave; the leading\nterm has second derivative $1/(2n)$), and\n$\\mathrm{margin}(5\\times 10^4) = 67\\,705 > 0$; hence\n$\\mathrm{margin}(n) > 0$ for all $n \\in [5\\times10^4, N_1]$\n(computed to $n = 4\\times10^5$ on a unit grid and to $N_1$ on a\nlogarithmic grid; $\\mathrm{margin}(N_1) = 2.46\\times 10^{26}$).\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 $N_0 = 5\\times 10^4$ by a factor $10$.\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\n\\appendix\n\\section{Numerical experiments}\n% Tables/figures generated from research/numerics/data/ CSVs.\n\n\\section{Formal verification}\n% Lean 4 skeleton status; axiom audit (propext, Classical.choice, Quot.sound only).\n\n\\bibliographystyle{plain}\n\\bibliography{references}\n\\end{document}\n",
  "structuredPatch": [
    {
      "oldStart": 131,
      "oldLines": 6,
      "newStart": 131,
      "newLines": 74,
      "lines": [
        " verifications.",
        " \\end{remark}",
        " ",
        "+\\subsection{The Suzuki operator line: exact factorization and the",
        "+spectral-measure equilibrium (R-003, D-040--D-046)}",
        "+",
        "+Suzuki \\cite{suzuki2026weil} constructs, for each $a > 0$, a",
        "+self-adjoint operator family $\\bar D_{a,\\theta}$ on model spaces",
        "+$\\mathcal H(T_a)$ built from the \\emph{screw function} $g$ of $\\zeta$",
        "+($\\hat g(z) = z^{-2}\\xi'(\\tfrac12-iz)/\\xi(\\tfrac12-iz)$), whose spectra",
        "+are the zeros of",
        "+$W(a,\\theta;z) = (z-i)V_+(z) + e^{i\\theta}(z+i)V_-(z)$, with $V_\\pm$",
        "+the Fourier transforms of deficiency vectors. His Corollary~1.6 states",
        "+that a limit formula $e^{\\phi(a,z)}W(a,\\theta(a);z) \\to",
        "+z^2\\xi(\\tfrac12-iz)/\\xi'(\\tfrac12-iz)$ (uniform on compacts, for some",
        "+choice of $\\theta(a)$, $\\phi$) would imply RH.",
        "+",
        "+\\begin{theorem}[R-003; modulo standard tempered-FT bookkeeping]",
        "+One has the exact factorization",
        "+$W(a,\\theta;z) = F(z)\\,M(a,\\theta;z)$ with",
        "+$F(z) = z^2\\xi(\\tfrac12-iz)/\\xi'(\\tfrac12-iz)$ and",
        "+$M = R_0 + R_\\tau$, where",
        "+$R_0 = 2e^{a+i\\theta/2}\\sin(az - \\theta/2) + o(e^a)$ is an explicit",
        "+sine skeleton and $R_\\tau$ is the out-of-interval tail transform.",
        "+Hence the limit formula is equivalent to the \\emph{tail-cancellation",
        "+statement} $(\\star)$: $R_\\tau$ cancels the skeleton leaving a zero-free",
        "+remainder \u2014 a finite-interval Wiener--Hopf remainder estimate on the",
        "+symbol $\\hat g$, whose factorization is gated by the zero locations.",
        "+\\end{theorem}",
        "+",
        "+We studied the spectral (Clark/de~Branges) measures",
        "+$\\mu_a = \\sum_k w_k\\,\\delta_{z_k}$ of $\\bar D_{a,\\pi}$ numerically to",
        "+$a = 12$ ($w_k = 1/K_a(z_k,z_k)$, $K_a$ the model-space reproducing",
        "+kernel; Galerkin machinery validated against the zero-side explicit",
        "+formula to $4.5\\times10^{-6}$). The picture that emerges",
        "+(D-040--D-045):",
        "+\\begin{enumerate}",
        "+\\item \\emph{The metric knows the zeros.} $K_a(z,z)$ has deep narrow",
        "+valleys exactly at the true ordinates $\\gamma$ (depth",
        "+$\\Delta(a) \\approx \\log a + 0.64$, width $\\sim 1.5/a$), while",
        "+$K_a(z,z) \\to 2a$ (the free Paley--Wiener diagonal) on the plateau:",
        "+the operator's symbol is \\emph{1 + the zero comb} \u2014 the explicit",
        "+formula in operator-symbol form.",
        "+\\item \\emph{Positions lock and masses are right.} Each $\\gamma$ has a",
        "+spectral atom within $\\approx 0.65/a^2$ (quadratic locking, correcting",
        "+an earlier wrong-direction metric), carrying mass $\\approx m_\\gamma = 1$.",
        "+\\item \\emph{The equilibrium.} The residual mass over the excess",
        "+(lattice) atoms tends to $Z/(2\\pi)$ per window $[0,Z]$, not to zero:",
        "+the family equilibrates exactly at the consolidation boundary",
        "+(generic Szeg\\H{o} bulk behavior). At fixed $\\theta$ this is structural",
        "+($\\theta$-independence is measured: residual spread $\\le 2.2\\%$ over",
        "+the full extension family).",
        "+\\item \\emph{The breaking mechanism.} A shift $\\lambda \\to \\lambda_a^-$",
        "+(bottom of the Weil-form spectrum) pins the valley floors (masses stay",
        "+$1$) while the plateau response diverges \u2014 the unique channel that",
        "+kills the residual. It is machinery-blocked: $\\lambda_a$ decays",
        "+superexponentially ($0.414$ at $a{=}0.25$ to $<3\\times10^{-12}$ at",
        "+$a{=}2$) below any achievable discretization floor at zero-resolving",
        "+$a$ (Graveyard G-004).",
        "+\\end{enumerate}",
        "+",
        "+On the operator-theoretic side (D-041): the limit operator $D$ exists",
        "+unconditionally as a $J$-self-adjoint operator in a Krein space",
        "+(negative index = number of off-line zero pairs, Bombieri), with",
        "+$\\mathrm{spec}(D) = \\Gamma$; and deficiency-element convergence is",
        "+equivalent to strong resolvent convergence",
        "+$\\bar D_{a,\\theta(a)} \\to D$ (Theorem D1), giving the reduction:",
        "+the limit formula (1.12) holds for some $\\theta(a), \\phi$ if and only",
        "+if $\\mathrm{spec}(D) \\subset \\mathbb R$. The real-spectrum persistence",
        "+step is exactly RH (the paywall), as expected.",
        "+",
        " \\section{Obstruction analysis}",
        " % Documented dead ends from research/graveyard/ where they",
        " % delimit the method (contributes to success criterion 4)."
      ]
    }
  ],
  "userModified": false,
  "replaceAll": false
}