← Session 35 — ii′-6-R: Lagarias 6.1 right-tail re-proof (E-032/D-073)
thinking (1808 chars)
All key signatures confirmed:
- `kFun_shift_norm_le_exp_four_add_one (hσ : s.re ≤ 3) (hT : n ≤ s.im²) : ‖kFun n (s+1)‖ ≤ e⁴+1` — NOTE: it's stated for `kFun n (s + 1)` with s.re ≤ 3! So at the point w = σ + iT, apply with s := w − 1, s.re = σ − 1 ≤ 3 ⟺ σ ≤ 4 — hmm. Wait: to bound kFun at the point p with re p ≤ 3... the lemma bounds kFun n (s+1) where s.re ≤ 3 and n ≤ s.im². So for the point p = σ + iT: set s = p − 1 = (σ−1) + iT: s.re = σ−1 ≤ 3 ⟺ σ ≤ 4; s.im = T, n ≤ T² ✓. So kFun n p ≤ e⁴+1 for re p ≤ 4 — covers the mid segment [−5/2, 3] ✓ (that's why it's "shift" — the shift by 1 is how the D-036 statement normalizes it).
- `kFun_shift_norm_le_exp_sqrt_half_add_one (hn : 0 < n) (hσ : 2√n ≤ s.re) : ‖kFun n (s+1)‖ ≤ e^{√n/2}+1` — for I_I at p = 2√n + it: s = p − 1, s.re = 2√n − 1 ≥ 2√n?? FALSE — s.re = 2√n − 1 < 2√n. Hmm! So for the right vertical at re p = 2√n, applying with s = p − 1 gives s.re = 2√n − 1, which does NOT satisfy 2√n ≤ s.re.
Hmm. But DL-4's `kFun_right_vertical_norm_le_exp_sqrt_half_add_one` — let me check its statement: "for n ≥ 16 and ..." at LiDogLeg:727. It's probably stated directly at the point 2√n + it with the (2√n−1) denominator absorbed: (1+1/(2√n−1))ⁿ ≤ e^{n/(2√n−1)} and n/(2√n−1) ≤ √n/2 for... n/(2√n−1) ≤ √n/2 ⟺ 2n ≤ √n(2√n−1) = 2n − √n ⟺ 0 ≤ −√n false. So the bound e^{n/(2√n−1)} ≤ e^{√n/2} is FALSE. It must be e^{√n/2+δ}. Let me check DL-4's exact statement — it says "Kernel at the far-right vertical (D-074 §8): for n ≥ 16 and ..." and gives ≤ exp(√n/2) + 1. How? |s/(s−1)| ≤ 1 + 1/|s−1|, |s−1| ≥ |s|−1 ≥ 2√n − 1. n/(2√n−1): for n = 16: 16/7 = 2.2857 vs √16/2 = 2. So 2.2857 > 2 — the exp bound e^{√n/2} fails at n=16?? Unless the lemma uses |s−1| ≥ ... hmm wait: maybe it's stated with 2√n + it for t with |t| ≥ ... no. Let me just read DL-4's lemma.
Checking DL-4's right-vertical kernel lemma (the `(s+1)`-shift signature looks off by one for the I_I application).
tool: Read
input
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raw content_json (database row)
[{"type":"thinking","thinking":"All key signatures confirmed:\n- `kFun_shift_norm_le_exp_four_add_one (hσ : s.re ≤ 3) (hT : n ≤ s.im²) : ‖kFun n (s+1)‖ ≤ e⁴+1` — NOTE: it's stated for `kFun n (s + 1)` with s.re ≤ 3! So at the point w = σ + iT, apply with s := w − 1, s.re = σ − 1 ≤ 3 ⟺ σ ≤ 4 — hmm. Wait: to bound kFun at the point p with re p ≤ 3... the lemma bounds kFun n (s+1) where s.re ≤ 3 and n ≤ s.im². So for the point p = σ + iT: set s = p − 1 = (σ−1) + iT: s.re = σ−1 ≤ 3 ⟺ σ ≤ 4; s.im = T, n ≤ T² ✓. So kFun n p ≤ e⁴+1 for re p ≤ 4 — covers the mid segment [−5/2, 3] ✓ (that's why it's \"shift\" — the shift by 1 is how the D-036 statement normalizes it).\n- `kFun_shift_norm_le_exp_sqrt_half_add_one (hn : 0 < n) (hσ : 2√n ≤ s.re) : ‖kFun n (s+1)‖ ≤ e^{√n/2}+1` — for I_I at p = 2√n + it: s = p − 1, s.re = 2√n − 1 ≥ 2√n?? FALSE — s.re = 2√n − 1 < 2√n. Hmm! So for the right vertical at re p = 2√n, applying with s = p − 1 gives s.re = 2√n − 1, which does NOT satisfy 2√n ≤ s.re. \n\nHmm. But DL-4's `kFun_right_vertical_norm_le_exp_sqrt_half_add_one` — let me check its statement: \"for n ≥ 16 and ...\" at LiDogLeg:727. It's probably stated directly at the point 2√n + it with the (2√n−1) denominator absorbed: (1+1/(2√n−1))ⁿ ≤ e^{n/(2√n−1)} and n/(2√n−1) ≤ √n/2 for... n/(2√n−1) ≤ √n/2 ⟺ 2n ≤ √n(2√n−1) = 2n − √n ⟺ 0 ≤ −√n false. So the bound e^{n/(2√n−1)} ≤ e^{√n/2} is FALSE. It must be e^{√n/2+δ}. Let me check DL-4's exact statement — it says \"Kernel at the far-right vertical (D-074 §8): for n ≥ 16 and ...\" and gives ≤ exp(√n/2) + 1. How? |s/(s−1)| ≤ 1 + 1/|s−1|, |s−1| ≥ |s|−1 ≥ 2√n − 1. n/(2√n−1): for n = 16: 16/7 = 2.2857 vs √16/2 = 2. So 2.2857 > 2 — the exp bound e^{√n/2} fails at n=16?? Unless the lemma uses |s−1| ≥ ... hmm wait: maybe it's stated with 2√n + it for t with |t| ≥ ... no. Let me just read DL-4's lemma.","signature":""},{"type":"text","text":"Checking DL-4's right-vertical kernel lemma (the `(s+1)`-shift signature looks off by one for the I_I application)."},{"type":"tool_use","id":"Read_28","name":"Read","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean","offset":727,"limit":100},"extra_content":null}]