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    "content": "contributes $\\le \\tfrac{\\sqrt n}{2}(e^{5/2}+1)\\cdot\\tfrac12$; the high\nsegment contracts geometrically (the repaired stepping ratio\n$q = e^{4/9}/2 < 1$, certified) and contributes $\\le 8.38$; and the\nright vertical is exponentially small, $\\le \\tfrac14 I_{\\mathrm{Maj}}(n)$.\nThe charge replacing the broken ``$3.3$'' is therefore\n$\\mathrm{DL}(n) = \\tfrac{\\sqrt n + 1}{4}(e^{5/2}+1) + 8.38 +\n\\tfrac14 I_{\\mathrm{Maj}}(n)$ per side, with the margin essentially\nunchanged (margins $13\\,160$ / $88\\,911$ / $2.458\\times10^{26}$ at\n$8\\times10^4$ / $10^5$ / $N_1$). We record for completeness that the\nper-side bound at $T = \\sqrt n$ appears to hold empirically by a strong\nhidden cancellation (E-032: true-integral to envelope ratio down to\n$5.6\\times10^{-10}$); whether that cancellation admits a proof is an\nopen question we do not rely on.\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",
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