← Session 30 — ii′-5e squeeze capstone + 5f perimeter repair
thinking (1412 chars)
Good. `C0hat t = 0.433 * log(t+2) + 0.886 * log(log(t+2)) + 2.9`.
For Ĉ₀(R+2) ≤ 5·log R (R ≥ 16002, so log R ≥ 9):
- log(R+4) ≤ 2·log R? log(R+4) ≤ log(R²) = 2 log R needs R+4 ≤ R² — true for R ≥ 3. But careful: log(R+4) — actually C0hat (R+2) uses log((R+2)+2) = log(R+4). And log(log(R+4)) ≤ log(2 log R) = log 2 + log log R ≤ 1 + log R... since log R ≥ 9, log log R ≤ log R (log x ≤ x). So:
- 0.433·log(R+4) ≤ 0.433·2·log R = 0.866·log R.
- 0.886·log(log(R+4)) ≤ 0.886·log(2·log R). Now 2·log R ≤ (log R)² for log R ≥ 2 ⟹ log(2 log R) ≤ 2·log log R ≤ 2·log R... hmm log(log R) ≤ log R since log x ≤ x. So 0.886·log(log(R+4)) ≤ 0.886·2·log R? Wait: log(log(R+4)) ≤ log(2·log R) (from log(R+4) ≤ 2 log R and log monotone, need 2 log R > 0 ✓). Then log(2·log R) = log 2 + log(log R) ≤ 1 + log R ≤ 2·log R... since log R ≥ 9 ≥ 1. OK so ≤ 2·log R, giving 0.886·2·log R = 1.772·log R.
- 2.9 ≤ 2.9·log R /9 ≤ ... 2.9 ≤ 0.33·log R (log R ≥ 9 ⟹ 2.9 ≤ 2.9/9·log R ≤ 0.33·log R).
Total: (0.866 + 1.772 + 0.33)·log R ≈ 2.97·log R ≤ 5·log R. ✓ Comfortable.
So I'll prove `C0hat_add_two_le : C0hat (R + 2) ≤ 5 * Real.log R` for (9 ≤ log R) or (R ≥ some bound). The exact decomposition: C0hat (R+2) = 0.433·log(R+4) + 0.886·log(log(R+4)) + 2.9.
Now `IsNontrivialZero` — it's not defined in our package; it's from Zeta23 presumably. Let me check: LiSeries.lean line 55 uses it without definition... grep in Zeta23.
C0hat is the honest majorant `0.433·log(t+2) + 0.886·log log(t+2) + 2.9`. Now checking `IsNontrivialZero` (Zeta23) and a few more lemma names.
tool: Bash
input
{
"command": "grep -rn \"def IsNontrivialZero\" D:/Python/Quantum/zeta-23-lean/Zeta23/ | head -2; grep -rn \"IsNontrivialZero\" D:/Python/Quantum/research/lean/NguyenRH1/NguyenRH1/LiSeries.lean | grep \"theorem\\|lemma\" | head -8; echo ===; grep -rn \"theorem analyticAt_differentiableAt\\|AnalyticAt.differentiableAt\" D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/Analysis/Analytic/Basic.lean | head -3",
"description": "Find IsNontrivialZero definition and AnalyticAt.differentiableAt"
}
raw content_json (database row)
[{"type":"thinking","thinking":"Good. `C0hat t = 0.433 * log(t+2) + 0.886 * log(log(t+2)) + 2.9`.\n\nFor Ĉ₀(R+2) ≤ 5·log R (R ≥ 16002, so log R ≥ 9):\n- log(R+4) ≤ 2·log R? log(R+4) ≤ log(R²) = 2 log R needs R+4 ≤ R² — true for R ≥ 3. But careful: log(R+4) — actually C0hat (R+2) uses log((R+2)+2) = log(R+4). And log(log(R+4)) ≤ log(2 log R) = log 2 + log log R ≤ 1 + log R... since log R ≥ 9, log log R ≤ log R (log x ≤ x). So:\n - 0.433·log(R+4) ≤ 0.433·2·log R = 0.866·log R.\n - 0.886·log(log(R+4)) ≤ 0.886·log(2·log R). Now 2·log R ≤ (log R)² for log R ≥ 2 ⟹ log(2 log R) ≤ 2·log log R ≤ 2·log R... hmm log(log R) ≤ log R since log x ≤ x. So 0.886·log(log(R+4)) ≤ 0.886·2·log R? Wait: log(log(R+4)) ≤ log(2·log R) (from log(R+4) ≤ 2 log R and log monotone, need 2 log R > 0 ✓). Then log(2·log R) = log 2 + log(log R) ≤ 1 + log R ≤ 2·log R... since log R ≥ 9 ≥ 1. OK so ≤ 2·log R, giving 0.886·2·log R = 1.772·log R.\n - 2.9 ≤ 2.9·log R /9 ≤ ... 2.9 ≤ 0.33·log R (log R ≥ 9 ⟹ 2.9 ≤ 2.9/9·log R ≤ 0.33·log R).\n Total: (0.866 + 1.772 + 0.33)·log R ≈ 2.97·log R ≤ 5·log R. ✓ Comfortable.\n\nSo I'll prove `C0hat_add_two_le : C0hat (R + 2) ≤ 5 * Real.log R` for (9 ≤ log R) or (R ≥ some bound). The exact decomposition: C0hat (R+2) = 0.433·log(R+4) + 0.886·log(log(R+4)) + 2.9.\n\nNow `IsNontrivialZero` — it's not defined in our package; it's from Zeta23 presumably. Let me check: LiSeries.lean line 55 uses it without definition... grep in Zeta23.","signature":""},{"type":"text","text":"C0hat is the honest majorant `0.433·log(t+2) + 0.886·log log(t+2) + 2.9`. Now checking `IsNontrivialZero` (Zeta23) and a few more lemma names."},{"type":"tool_use","id":"Bash_20","name":"Bash","input":{"command":"grep -rn \"def IsNontrivialZero\" D:/Python/Quantum/zeta-23-lean/Zeta23/ | head -2; grep -rn \"IsNontrivialZero\" D:/Python/Quantum/research/lean/NguyenRH1/NguyenRH1/LiSeries.lean | grep \"theorem\\|lemma\" | head -8; echo ===; grep -rn \"theorem analyticAt_differentiableAt\\|AnalyticAt.differentiableAt\" D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/Analysis/Analytic/Basic.lean | head -3","description":"Find IsNontrivialZero definition and AnalyticAt.differentiableAt"},"extra_content":null}]