← Session 36 — ii′-6-DL dog-leg + side bounds; Paper A review
thinking (1077 chars)
Key facts:
- `IIMaj (n : ℝ) : ℝ := 2 * (Real.sqrt n + 1) * (Real.exp (Real.sqrt n / 2) + 1) * 0.57 * 2 ^ (-(2 * Real.sqrt n - 2))` ✓ as I guessed (with the factor 2).
- `norm_logDeriv_zeta_le_iimZeta {n : ℕ} (hn : 5 ≤ n) {t : ℝ} : ‖logDeriv riemannZeta (((2 * Real.sqrt (n:ℝ)) : ℝ) + t * Complex.I)‖ ≤ 0.57 * 2 ^ (-(2*√n - 2))` — note the point form: `((2 * Real.sqrt (n:ℝ)) : ℝ) + t * Complex.I` — the `((2*√n) : ℝ)` is a ℝ-typed term coerced to ℂ: `↑(2 * √(n:ℝ)) + ↑t * I`.
- DL-4's kernel lemma point form: `(((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I)` ✓ same.
- `norm_one_add_one_div_le {s : ℂ} (hs : s ≠ 0) : ‖1 + 1/s‖ ≤ 1 + (s.re + 1)/Complex.normSq s` — hmm interesting — for s := w−1: ‖1 + 1/(w−1)‖ ≤ 1 + ((w−1).re + 1)/normSq (w−1) = 1 + (2√n − 1 + 1)/((2√n−1)² + t²) = 1 + (2√n)/((2√n−1)² + t²). Hmm — the recipe said (2·re(w−1)+1)/normSq = (4√n−1)/((2√n−1)²+t²) — different from norm_one_add_one_div_le's (re+1)/normSq = (2√n)/(...). Let me look at the full DL-4 proof to see the actual route — it's the template.
Let me read DL-4's full proof (lines 732-~830).
Templates located. Reading DL-4's kernel lemma proof in full — it's the direct template for the F-E variant.
tool: Read
input
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raw content_json (database row)
[{"type":"thinking","thinking":"Key facts:\n- `IIMaj (n : ℝ) : ℝ := 2 * (Real.sqrt n + 1) * (Real.exp (Real.sqrt n / 2) + 1) * 0.57 * 2 ^ (-(2 * Real.sqrt n - 2))` ✓ as I guessed (with the factor 2).\n- `norm_logDeriv_zeta_le_iimZeta {n : ℕ} (hn : 5 ≤ n) {t : ℝ} : ‖logDeriv riemannZeta (((2 * Real.sqrt (n:ℝ)) : ℝ) + t * Complex.I)‖ ≤ 0.57 * 2 ^ (-(2*√n - 2))` — note the point form: `((2 * Real.sqrt (n:ℝ)) : ℝ) + t * Complex.I` — the `((2*√n) : ℝ)` is a ℝ-typed term coerced to ℂ: `↑(2 * √(n:ℝ)) + ↑t * I`.\n- DL-4's kernel lemma point form: `(((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I)` ✓ same.\n- `norm_one_add_one_div_le {s : ℂ} (hs : s ≠ 0) : ‖1 + 1/s‖ ≤ 1 + (s.re + 1)/Complex.normSq s` — hmm interesting — for s := w−1: ‖1 + 1/(w−1)‖ ≤ 1 + ((w−1).re + 1)/normSq (w−1) = 1 + (2√n − 1 + 1)/((2√n−1)² + t²) = 1 + (2√n)/((2√n−1)² + t²). Hmm — the recipe said (2·re(w−1)+1)/normSq = (4√n−1)/((2√n−1)²+t²) — different from norm_one_add_one_div_le's (re+1)/normSq = (2√n)/(...). Let me look at the full DL-4 proof to see the actual route — it's the template.\n\nLet me read DL-4's full proof (lines 732-~830).","signature":""},{"type":"text","text":"Templates located. Reading DL-4's kernel lemma proof in full — it's the direct template for the F-E variant."},{"type":"tool_use","id":"Read_71","name":"Read","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean","offset":726,"limit":115},"extra_content":null}]