← Session 35 — ii′-6-R: Lagarias 6.1 right-tail re-proof (E-032/D-073)
thinking (1487 chars)
Zero errors. DL-4 compiles. Now: full build + audit extension + E-034 note (VRightBound charged IIMaj/2 in Lean vs E-034's IIMaj/4 — hmm — wait, actually: the Lean V_right bound is (U−T)·(e^{√n/2}+1)·2^{−(2√n−2)}·dirichletC2 with dirichletC2 ≤ 1. The E-034 fold used VRightBound = IIMaj/4 = 0.57·((√n+1)/2)(e+1)2^{−X}. The Lean-provable bound (with (U−T) ≤ (√n+1)/2 in DL-5) gives ((√n+1)/2)(e+1)2^{−X}·1 — which is BIGGER than IIMaj/4 (0.57×-smaller). So the fold's VRightBound slot should be charged at the Lean-provable ((√n+1)/2)(e+1)2^{−X} = IIMaj/(4·0.57) ≈ 0.4386·IIMaj — which is ≤ IIMaj/2 ✓. So charge VRightBound := IIMaj/2 in the fold for honesty. The E-034 script has VRightBound = IIMaj/4 — update it to IIMaj/2 with a comment (fold indifferent: IIMaj ~1e-105). Re-run E-034 at that (fast fold-only? the script re-runs quadrature ~6 min — run detached).
Actually — hmm, let me double-check the DL-5 accounting: my Lean V_right bound (with the U−T explicit) is (U−T)·(e^{√n/2}+1)·(2^{−(2√n−2)}·dirichletC2). With U−T = T/2 ≤ (√n+1)/2 and dirichletC2 ≤ 1: ≤ ((√n+1)/2)·(e^{√n/2}+1)·2^{−(2√n−2)}. And IIMaj/2 = ((√n+1))·(e^{√n/2}+1)·0.57·2^{−(2√n−2)}. Ratio: ((√n+1)/2)·1 vs (√n+1)·0.57: (1/2)·1 = 0.5 vs 0.57: 0.5 < 0.57 ✓ so ((√n+1)/2)(e+1)2^{−X} ≤ IIMaj/2 ✓✓. Good — VRightBound := IIMaj/2 is provable. (Even better: the Lean bound is ~0.44·IIMaj, between IIMaj/4 and IIMaj/2.)
Update E-034's VRightBound to IIMaj/2 + comment, rerun, then commit DL-4 with audit extension.
Zero errors — DL-4 compiles. Now build, audit, and re-validate E-034 at the Lean-provable VRightBound = IIMaj/2 (the dirichletC2 ≤ 1 charge makes IIMaj/4 unprovable; IIMaj/2 fits with slack).
tool: Bash
input
{
"command": "cd D:/Python/Quantum/research/lean/NguyenRH1 && export PATH=\"$HOME/.elan/bin:$PATH\" && lake build NguyenRH1 2>&1 | tail -2",
"description": "Full root build with DL-4",
"timeout": 600000
}
tool: Edit
input
{
"file_path": "D:\\Python\\Quantum\\research\\numerics\\scripts\\E-034-dogleg-validation.py",
"old_string": "def VRightBound(n):\n return IIMaj(n) / 4",
"new_string": "def VRightBound(n):\n # LEAN-PROVABLE charge IIMaj/2 (DL-4, vIntegral_Jlag_right_norm_le:\n # (U-T)*(e^{sqrt(n)/2}+1)*2^{-(2 sqrt(n)-2)}*dirichletC2 with U-T <=\n # (sqrt(n)+1)/2 and dirichletC2 <= 1 gives <= IIMaj/2 \u2014 the tight IIMaj/4\n # would need dirichletC2 <= 0.57; the fold is indifferent: IIMaj ~ 1e-105).\n return IIMaj(n) / 2"
}
raw content_json (database row)
[{"type":"thinking","thinking":"Zero errors. DL-4 compiles. Now: full build + audit extension + E-034 note (VRightBound charged IIMaj/2 in Lean vs E-034's IIMaj/4 — hmm — wait, actually: the Lean V_right bound is (U−T)·(e^{√n/2}+1)·2^{−(2√n−2)}·dirichletC2 with dirichletC2 ≤ 1. The E-034 fold used VRightBound = IIMaj/4 = 0.57·((√n+1)/2)(e+1)2^{−X}. The Lean-provable bound (with (U−T) ≤ (√n+1)/2 in DL-5) gives ((√n+1)/2)(e+1)2^{−X}·1 — which is BIGGER than IIMaj/4 (0.57×-smaller). So the fold's VRightBound slot should be charged at the Lean-provable ((√n+1)/2)(e+1)2^{−X} = IIMaj/(4·0.57) ≈ 0.4386·IIMaj — which is ≤ IIMaj/2 ✓. So charge VRightBound := IIMaj/2 in the fold for honesty. The E-034 script has VRightBound = IIMaj/4 — update it to IIMaj/2 with a comment (fold indifferent: IIMaj ~1e-105). Re-run E-034 at that (fast fold-only? the script re-runs quadrature ~6 min — run detached).\n\nActually — hmm, let me double-check the DL-5 accounting: my Lean V_right bound (with the U−T explicit) is (U−T)·(e^{√n/2}+1)·(2^{−(2√n−2)}·dirichletC2). With U−T = T/2 ≤ (√n+1)/2 and dirichletC2 ≤ 1: ≤ ((√n+1)/2)·(e^{√n/2}+1)·2^{−(2√n−2)}. And IIMaj/2 = ((√n+1))·(e^{√n/2}+1)·0.57·2^{−(2√n−2)}. Ratio: ((√n+1)/2)·1 vs (√n+1)·0.57: (1/2)·1 = 0.5 vs 0.57: 0.5 < 0.57 ✓ so ((√n+1)/2)(e+1)2^{−X} ≤ IIMaj/2 ✓✓. Good — VRightBound := IIMaj/2 is provable. (Even better: the Lean bound is ~0.44·IIMaj, between IIMaj/4 and IIMaj/2.)\n\nUpdate E-034's VRightBound to IIMaj/2 + comment, rerun, then commit DL-4 with audit extension.","signature":""},{"type":"text","text":"Zero errors — DL-4 compiles. Now build, audit, and re-validate E-034 at the Lean-provable VRightBound = IIMaj/2 (the dirichletC2 ≤ 1 charge makes IIMaj/4 unprovable; IIMaj/2 fits with slack)."},{"type":"tool_use","id":"Bash_365","name":"Bash","input":{"command":"cd D:/Python/Quantum/research/lean/NguyenRH1 && export PATH=\"$HOME/.elan/bin:$PATH\" && lake build NguyenRH1 2>&1 | tail -2","description":"Full root build with DL-4","timeout":600000},"extra_content":null},{"type":"tool_use","id":"Edit_366","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\numerics\\scripts\\E-034-dogleg-validation.py","old_string":"def VRightBound(n):\n return IIMaj(n) / 4","new_string":"def VRightBound(n):\n # LEAN-PROVABLE charge IIMaj/2 (DL-4, vIntegral_Jlag_right_norm_le:\n # (U-T)*(e^{sqrt(n)/2}+1)*2^{-(2 sqrt(n)-2)}*dirichletC2 with U-T <=\n # (sqrt(n)+1)/2 and dirichletC2 <= 1 gives <= IIMaj/2 — the tight IIMaj/4\n # would need dirichletC2 <= 0.57; the fold is indifferent: IIMaj ~ 1e-105).\n return IIMaj(n) / 2"},"extra_content":null}]