{
  "file_path": "D:\\Python\\Quantum\\research\\correspondence\\lagarias-F-D-package\\README.md",
  "content": "# Attachment package: the (6.12) finding\n\nCompanion material for the letter to Prof. Lagarias of 2026-08-24.\n\n## Contents\n\n| file | what it is |\n|------|-----------|\n| `finding-note.pdf` | **The finding note (4 pp, compiled)** \u2014 the claim, the issue (e/2 > 1 per-step ratio), the numerical evidence tables, the mechanism (peak at \u03c3 \u2248 0.36\u221an, height ~e^{0.066\u221an}), the repair verification (T = c\u221an), the assessment, the questions, reproduction instructions |\n| `FINDING-NOTE.md` | the same note in Markdown (source of the PDF) |\n| `finding-note.tex` | LaTeX source of the PDF (pdflatex, two passes) |\n| `E-031-right-tail-stepping.py` | **the mpmath script** producing: (V1) the product profiles and peak tracking across n \u2208 {10\u00b3, 10\u2074, 10\u2075}, (V2) the right-tail envelope integrals at n \u2208 {10\u2074, 10\u2075}, (V3) the product log-profiles at T = c\u221an, c \u2208 {1, 1.5, 2} |\n| `evidence.csv` | the evidence tables in tidy long form (all numbers in the note) |\n\n## Reproduce\n\n```\npython E-031-right-tail-stepping.py --force\n```\n\nRequires only Python 3 + mpmath (dps 25, deterministic). Writes\n`E-031-right-tail-stepping.csv` + a JSON sidecar with the check verdicts\n(V1 peak tracking, V2 envelope ~5.9\u00d710\u00b9\u2070 at n = 10\u2075, V3 repair\nmonotone-decrease at c = 1.5 and c = 2 \u2014 all checks should pass).\n\n## One-paragraph summary\n\nIn the proof of Theorem 6.1 of \"Li coefficients for automorphic\nL-functions\" (Lagarias, arXiv:math/0404394), the right part of the\nhorizontal segments is bounded by a unit-step argument whose two\ningredients compose to a per-step ratio \u2264 e/2 \u2248 1.359 > 1 (zeta factor\n\u2264 1/2 via (6.10), kernel factor \u2264 e via (6.12)). The product\n|k_n(s)\u00b7L\u2032/L(s)| therefore *grows* from \u03c3 = 3 to \u03c3 \u2248 0.36\u221an, peaking at\n~e^{0.066\u221an} (measured ~1.6\u00d710\u2079 at n = 10\u2075; right-tail envelope\nintegral ~5.9\u00d710\u00b9\u2070) instead of decreasing geometrically to give\n|I_II| = O(\u221an). The issue is confined to the proof of the horizontal\nbound; the theorem's conclusion is consistent with all measurements and\nlikely stands (the two horizontals are conjugate sides and the\nintegrand's phase winds hundreds of radians). The repair T = c\u221an with\nc > 1.2012 makes the per-step ratio < 1 and restores the argument\nverbatim (verified numerically).\n"
}

{
  "type": "create",
  "filePath": "D:\\Python\\Quantum\\research\\correspondence\\lagarias-F-D-package\\README.md",
  "content": "# Attachment package: the (6.12) finding\n\nCompanion material for the letter to Prof. Lagarias of 2026-08-24.\n\n## Contents\n\n| file | what it is |\n|------|-----------|\n| `finding-note.pdf` | **The finding note (4 pp, compiled)** \u2014 the claim, the issue (e/2 > 1 per-step ratio), the numerical evidence tables, the mechanism (peak at \u03c3 \u2248 0.36\u221an, height ~e^{0.066\u221an}), the repair verification (T = c\u221an), the assessment, the questions, reproduction instructions |\n| `FINDING-NOTE.md` | the same note in Markdown (source of the PDF) |\n| `finding-note.tex` | LaTeX source of the PDF (pdflatex, two passes) |\n| `E-031-right-tail-stepping.py` | **the mpmath script** producing: (V1) the product profiles and peak tracking across n \u2208 {10\u00b3, 10\u2074, 10\u2075}, (V2) the right-tail envelope integrals at n \u2208 {10\u2074, 10\u2075}, (V3) the product log-profiles at T = c\u221an, c \u2208 {1, 1.5, 2} |\n| `evidence.csv` | the evidence tables in tidy long form (all numbers in the note) |\n\n## Reproduce\n\n```\npython E-031-right-tail-stepping.py --force\n```\n\nRequires only Python 3 + mpmath (dps 25, deterministic). Writes\n`E-031-right-tail-stepping.csv` + a JSON sidecar with the check verdicts\n(V1 peak tracking, V2 envelope ~5.9\u00d710\u00b9\u2070 at n = 10\u2075, V3 repair\nmonotone-decrease at c = 1.5 and c = 2 \u2014 all checks should pass).\n\n## One-paragraph summary\n\nIn the proof of Theorem 6.1 of \"Li coefficients for automorphic\nL-functions\" (Lagarias, arXiv:math/0404394), the right part of the\nhorizontal segments is bounded by a unit-step argument whose two\ningredients compose to a per-step ratio \u2264 e/2 \u2248 1.359 > 1 (zeta factor\n\u2264 1/2 via (6.10), kernel factor \u2264 e via (6.12)). The product\n|k_n(s)\u00b7L\u2032/L(s)| therefore *grows* from \u03c3 = 3 to \u03c3 \u2248 0.36\u221an, peaking at\n~e^{0.066\u221an} (measured ~1.6\u00d710\u2079 at n = 10\u2075; right-tail envelope\nintegral ~5.9\u00d710\u00b9\u2070) instead of decreasing geometrically to give\n|I_II| = O(\u221an). The issue is confined to the proof of the horizontal\nbound; the theorem's conclusion is consistent with all measurements and\nlikely stands (the two horizontals are conjugate sides and the\nintegrand's phase winds hundreds of radians). The repair T = c\u221an with\nc > 1.2012 makes the per-step ratio < 1 and restores the argument\nverbatim (verified numerically).\n",
  "structuredPatch": [],
  "originalFile": null
}