← Session 18 — Lean tier (ii): finite-height proof + certified base points
thinking (5130 chars)
Now write the paper edits:
**Edit 1 — §3.1 Step 4 (line 247-255):** replace with the FAlt/two-point form + footnote trail.
Current:
```
\emph{Step 4 (margin).}
$\mathrm{margin}(n) := \tfrac12 n\log n + (C_1 - \gamma/2)n - 4 - B(n) + 1$
is convex for $n \ge 20$ (proved: chain-rule reduction to eleven
one-variable sign conditions, each discharged elementarily — D-057),
and $\mathrm{margin}(8\times 10^4) = 16\,020 > 0$ with positive
derivative; hence
$\mathrm{margin}(n) > 0$ for all $n \in [8\times10^4, N_1]$
(also computed to $n = 4\times10^5$ on a unit grid and to $N_1$ on a
logarithmic grid; $\mathrm{margin}(N_1) = 2.458\times 10^{26}$).
```
New:
```
\emph{Step 4 (margin).}
$\mathrm{margin}(n) := \tfrac12 n\log n + (C_1 - \gamma/2)n - 4 - B(n) + 1$
is convex for $n \ge 20$ (proved: chain-rule reduction to eleven
one-variable sign conditions, each discharged elementarily — D-057),
and $\mathrm{margin}(8\times 10^4) = 15\,764 > 0$ with
$\mathrm{margin}(10^5) > \mathrm{margin}(8\times10^4)$; by convexity
(secant slopes increase) hence
$\mathrm{margin}(n) > 0$ for all $n \in [10^5, N_1]$\footnote{The
margin value at $8\times10^4$ has a documented trail: $16\,020$ with
the Trudgian $F + S_b$ zero-count majorant (E-028); $15\,764$ with the
mildly enlarged majorant consumed by the machine verification, in
which a negative-weight term of $F + S_b$ is dropped so that every
convexity block has provably negative $\psi$ (the true majorant's
$\psi$ crosses zero near $T = 16$), at a traced cost of
$\approx 0.9\sqrt n$, i.e.\ $256$ of the $16\,020$; and $\ge 9\,960$
as the certified rational interval lower bound that the machine
discharge actually consumes.}
(also computed to $n = 4\times10^5$ on a unit grid and to $N_1$ on a
logarithmic grid; $\mathrm{margin}(N_1) = 2.458\times 10^{26}$).
```
Hmm — careful about what "the machine verification" consumes: the certified lower bound 8001 (336000−328000+1)! My D-062 §5 said ≥ 9 960 but the FINAL Lean targets were 336000/328000 → margin ≥ 8001. Let me recompute the trail number for the footnote: the Lean-certified lower bound is margiñ(8×10⁴) ≥ 336000 − 328000 + 1 = 8001. Earlier "9 960" was the pre-tuning D-062 estimate. The footnote should say ≥ 8 001 (what the Lean proof certifies) or "> 8 000". Let me use the exact: 336000 − 328000 + 1 = 8001. So footnote: "≥ 8\,001 as the certified rational interval lower bound consumed by the machine discharge (NguyenRH1, \texttt{marginTilde\_FAlt\_80000\_pos})."
Also fix D-062 §5's "9 960" → keep as is (it was the plan-stage estimate) but note the executed values? D-062 says "margin ≥ 9 960 > 0 ✓ (hbase)" — the executed Lean targets give 8001. Slight mismatch between plan and execution — update D-062 §5 with a one-line erratum/note. Also journal.
Also Step 5's overlap: "The ranges overlap N₀ = 8×10⁴ by a factor 6.25" — with the 10⁵ split: the margin regime starts at 10⁵ and finite verification covers to 5×10⁵, so overlap factor 5. Update Step 5's last sentence: "The ranges overlap the split point $10^5$ by a factor $5$." Wait — the pen-and-paper N₀ = 8×10⁴ with 6.25× overlap (5×10⁵/8×10⁴ = 6.25) is the E-028 story. The Lean assembly splits at 10⁵ (overlap 5). For consistency with "the bound the Lean theorem actually consumes": change to "The ranges overlap the $n = 10^5$ split by a factor $5$ (and the adopted crossover $N_0 = 8\times10^4$ by $6.25$)."
**Edit 2 — numerics bullet (line 475-479):** "adopted N₀ = 8×10⁴ with margin 16 020 and positive derivative" — this is the E-028 numerics report, factually correct as numerics. The mathematician's complaint was about main-text/appendix consistency. The numerics bullet can keep 16 020 IF the trail is visible — add "(16\,020 with the Trudgian $F+S_b$ majorant; 15\,764 with the formal-development majorant — see the Step-4 footnote)". Minimal edit: append a clause.
**Edit 3 — appendix formal-status paragraph:** update the five-hypotheses paragraph to three hypotheses (the discharge landed):
Current new paragraph says: "machine-proved from five named hypotheses: the two computational-perimeter inputs ..., two elementary evaluations of the explicit margin function (... a finite interval-arithmetic computation), and a single named analytic hypothesis SfBoundProp ..."
Replace with:
"The theorem \texttt{lambda\_pos\_finite\_height} is machine-proved from three named hypotheses: the two computational-perimeter inputs of Theorem~\ref{thm:finite-height-li} (Platt--Trudgian; the Li verifications) and a single named analytic hypothesis \texttt{SfBoundProp}: ... The two elementary margin evaluations ($\mathrm{margin}(8\cdot10^4) > 0$ and $\mathrm{margin}(10^5) > \mathrm{margin}(8\cdot10^4)$) are discharged in the development by certified rational interval bounds on $\log$/$\exp$/$\sqrt{\cdot}$ (\texttt{LiBasePoints.lean}: \texttt{marginTilde\_FAlt\_80000\_pos}, \texttt{marginTilde\_FAlt\_80000\_lt\_100000}; the concluding three-hypothesis form is \texttt{lambda\_pos\_finite\_height\_verified})."
Also the tail of that paragraph about the majorant — keep the FAlt note (still accurate).
Let me now make these three edits.
Making the three paper edits: §3.1 Step 4 with the footnote trail, the numerics bullet, and the appendix paragraph (now three hypotheses — the discharge landed).
tool: Edit
input
{
"file_path": "D:\\Python\\Quantum\\research\\paper\\draft.tex",
"old_string": "\\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 $6.25$.",
"new_string": "\\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\nthe mildly enlarged majorant consumed by the machine verification ---\na negative-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 $6.25$)."
}
raw content_json (database row)
[{"type":"thinking","thinking":"Now write the paper edits:\n\n**Edit 1 — §3.1 Step 4 (line 247-255):** replace with the FAlt/two-point form + footnote trail.\n\nCurrent:\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 — 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\nNew:\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 — D-057),\nand $\\mathrm{margin}(8\\times 10^4) = 15\\,764 > 0$ with\n$\\mathrm{margin}(10^5) > \\mathrm{margin}(8\\times10^4)$; 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, in\nwhich a negative-weight term of $F + S_b$ is dropped so that every\nconvexity block has provably negative $\\psi$ (the true majorant's\n$\\psi$ crosses zero near $T = 16$), at a traced cost of\n$\\approx 0.9\\sqrt n$, i.e.\\ $256$ of the $16\\,020$; and $\\ge 9\\,960$\nas the certified rational interval lower bound that the machine\ndischarge actually consumes.} \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\nHmm — careful about what \"the machine verification\" consumes: the certified lower bound 8001 (336000−328000+1)! My D-062 §5 said ≥ 9 960 but the FINAL Lean targets were 336000/328000 → margin ≥ 8001. Let me recompute the trail number for the footnote: the Lean-certified lower bound is margiñ(8×10⁴) ≥ 336000 − 328000 + 1 = 8001. Earlier \"9 960\" was the pre-tuning D-062 estimate. The footnote should say ≥ 8 001 (what the Lean proof certifies) or \"> 8 000\". Let me use the exact: 336000 − 328000 + 1 = 8001. So footnote: \"≥ 8\\,001 as the certified rational interval lower bound consumed by the machine discharge (NguyenRH1, \\texttt{marginTilde\\_FAlt\\_80000\\_pos}).\"\n\nAlso fix D-062 §5's \"9 960\" → keep as is (it was the plan-stage estimate) but note the executed values? D-062 says \"margin ≥ 9 960 > 0 ✓ (hbase)\" — the executed Lean targets give 8001. Slight mismatch between plan and execution — update D-062 §5 with a one-line erratum/note. Also journal.\n\nAlso Step 5's overlap: \"The ranges overlap N₀ = 8×10⁴ by a factor 6.25\" — with the 10⁵ split: the margin regime starts at 10⁵ and finite verification covers to 5×10⁵, so overlap factor 5. Update Step 5's last sentence: \"The ranges overlap the split point $10^5$ by a factor $5$.\" Wait — the pen-and-paper N₀ = 8×10⁴ with 6.25× overlap (5×10⁵/8×10⁴ = 6.25) is the E-028 story. The Lean assembly splits at 10⁵ (overlap 5). For consistency with \"the bound the Lean theorem actually consumes\": change to \"The ranges overlap the $n = 10^5$ split by a factor $5$ (and the adopted crossover $N_0 = 8\\times10^4$ by $6.25$).\"\n\n**Edit 2 — numerics bullet (line 475-479):** \"adopted N₀ = 8×10⁴ with margin 16 020 and positive derivative\" — this is the E-028 numerics report, factually correct as numerics. The mathematician's complaint was about main-text/appendix consistency. The numerics bullet can keep 16 020 IF the trail is visible — add \"(16\\,020 with the Trudgian $F+S_b$ majorant; 15\\,764 with the formal-development majorant — see the Step-4 footnote)\". Minimal edit: append a clause.\n\n**Edit 3 — appendix formal-status paragraph:** update the five-hypotheses paragraph to three hypotheses (the discharge landed):\n\nCurrent new paragraph says: \"machine-proved from five named hypotheses: the two computational-perimeter inputs ..., two elementary evaluations of the explicit margin function (... a finite interval-arithmetic computation), and a single named analytic hypothesis SfBoundProp ...\"\n\nReplace with:\n\"The theorem \\texttt{lambda\\_pos\\_finite\\_height} is machine-proved from three named hypotheses: the two computational-perimeter inputs of Theorem~\\ref{thm:finite-height-li} (Platt--Trudgian; the Li verifications) and a single named analytic hypothesis \\texttt{SfBoundProp}: ... The two elementary margin evaluations ($\\mathrm{margin}(8\\cdot10^4) > 0$ and $\\mathrm{margin}(10^5) > \\mathrm{margin}(8\\cdot10^4)$) are discharged in the development by certified rational interval bounds on $\\log$/$\\exp$/$\\sqrt{\\cdot}$ (\\texttt{LiBasePoints.lean}: \\texttt{marginTilde\\_FAlt\\_80000\\_pos}, \\texttt{marginTilde\\_FAlt\\_80000\\_lt\\_100000}; the concluding three-hypothesis form is \\texttt{lambda\\_pos\\_finite\\_height\\_verified}).\"\n\nAlso the tail of that paragraph about the majorant — keep the FAlt note (still accurate).\n\nLet me now make these three edits.","signature":""},{"type":"text","text":"Making the three paper edits: §3.1 Step 4 with the footnote trail, the numerics bullet, and the appendix paragraph (now three hypotheses — the discharge landed)."},{"type":"tool_use","id":"Edit_175","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\paper\\draft.tex","old_string":"\\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 — 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 $6.25$.","new_string":"\\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 — 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\nthe mildly enlarged majorant consumed by the machine verification ---\na negative-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 $6.25$)."},"extra_content":null}]