[{"type":"thinking","thinking":"Errors:\n\n1. Line 3466 (hnear's separation step): `(one_div_le_one_div_of_le h2 hsep).trans_eq (one_div_one_div _)` — the expected final: `1/‖s−ρ‖ ≤ 32·Ĉ₀`: one_div_le_one_div_of_le (ha : 0 < a) (h : a ≤ b) : 1/b ≤ 1/a — with a := ‖s−ρ‖, b := 1/(32Ĉ₀)?? wait: hsep : 1/(32Ĉ₀) ≤ ‖s−ρ‖: one_div_le_one_div_of_le (hx : 0 < x) (hxy : x ≤ y) : 1/y ≤ 1/x — with x := 1/(32Ĉ₀) (positive h1) and y := ‖s−ρ‖: gives 1/‖s−ρ‖ ≤ 1/(1/(32Ĉ₀)) — I passed h2 (0 < ‖s−ρ‖) as the FIRST arg — WRONG: the signature: `one_div_le_one_div_of_le (ha : 0 < a) (hab : a ≤ b) : 1 / b ≤ 1 / a` — the FIRST arg is positivity of a (the SMALLER one). So: one_div_le_one_div_of_le h1 hsep : 1/‖s−ρ‖ ≤ 1/(1/(32Ĉ₀)) — then .trans_eq (one_div_one_div _ : 1/(1/x) = x) : 1/‖s−ρ‖ ≤ 32Ĉ₀ ✓. Fix: swap h2 → h1: `(one_div_le_one_div_of_le h1 hsep).trans_eq (one_div_one_div _)`.\n\n2. Line 3475 (hmax's h2): `abs_add_le _ _` gives |?a + ?b| ≤ |?a|+|?b| — expected |s.im + 1| ≤ |s.im| + 1: |(−1)| = 1 — the (1 : ℝ) is fine but abs_add_le _ _ unifies to |s.im + 1| ≤ |s.im| + |1| — the RHS |1| vs 1 mismatch: |1| = 1 needs abs_of_pos or norm_num. Fix: `have h1 : |s.im + 1| ≤ |s.im| + 1 := by have h := abs_add_le s.im (1 : ℝ); rwa [abs_one] at h`. abs_one : |1| = 1 ✓.\n   Also the hmax block has that monstrous by_cases' garbage I accidentally generated (the `.by_cases'` chain) — that's broken anyway. Let me rewrite hmax cleanly:\n   ```\n   have hmax : max |s.im - 1| |s.im + 1| ≤ |s.im| + 1 := by\n     apply max_le_iff.mpr\n     exact ⟨by\n       have h4 : s.im - 1 = s.im + (-1 : ℝ) := by ring\n       rw [h4]\n       have h := abs_add_le s.im (-1 : ℝ)\n       rwa [abs_neg, abs_one] at h,\n       by\n       have h := abs_add_le s.im (1 : ℝ)\n       rwa [abs_one] at h⟩\n   ```\n   Then use log(3·max+5) ≤ log(3(|t|+1)+5): Real.log_le_log (pos) (mul_le_mul_of_nonneg_left hmax (by norm_num)) — the hmax EQUALITY isn't needed; ≤ suffices and is 5 lines. Adjust the downstream: I had `rw [hmax]` — change to bound: log(3·max|t−1||t+1|+5) ≤ log(3(|t|+1)+5) via log_le_log + mul_le_mul + add_le_add_right.\n\n3. Line 3523 (hcap's 5c application): `logDeriv_riemannZeta_sub_nearZeros_le (t := s.im) ht10 rfl hhalf (...≤ 7/2) hζs` — the elaboration blew up: the `rfl` for hsim : s.im = t with t := s.im — should be fine... The error shows the 7/2 bound's proof being elaborated weirdly — `le_of_not_gt fun a => lt_irrefl (Eq.mp ...)` — the linarith inside produced garbage? The `(by linarith [hreg1.le] : s.re ≤ 7 / 2)` — hreg1 : ¬2 ≤ s.re: hreg1.le : s.re ≤ 2?? NO — `not_le.mp hreg1 : s.re < 2` — I wrote `hreg1.le` — that's le_of_not_ge hreg1 : s.re < 2 — .le gives s.re ≤ 2 ✓ then s.re ≤ 7/2 by linarith ✓. Hmm — but the error shows the application itself mismatched: maybe the ISSUE is `rfl : s.im = s.im` with the named arg (t := s.im) — hsim : s.im = t — t is assigned s.im ✓ rfl should work... The error's elaborated garbage is from linarith's internals leaking (the ring normalization failed). The message is truncated — the mismatch might be hζs's TYPE: logDeriv_riemannZeta_sub_nearZeros_le's hs0 : riemannZeta s ≠ 0 ✓ hζs ✓. Or ht10 : 10 ≤ |s.im| — the capstone's ht : 10 ≤ |t| with t := s.im: 10 ≤ |s.im| ✓ ht10 ✓. Hmm — or hhalf : (1:ℝ)/2 ≤ s.re vs the capstone's hσlo : 1/2 ≤ s.re — (1:ℝ)/2 vs 1/2 — same term? `1/2` as ℝ: HDiv 1 2 ✓ same. The `(by linarith [hreg1.le] : s.re ≤ 7 / 2)` — maybe linarith failed and the garbage is its failure state (the \"lt_irrefl (Eq.mp (congrArg ...))\" IS linarith's internal error form when it fails). Why would s.re ≤ 7/2 fail from s.re < 2? `hreg1.le` — hreg1 : ¬ 2 ≤ s.re: `not_le.mp hreg1 : s.re < 2` — but I wrote `hreg1.le` — is `.le` valid on a Not? `lt_of_not_ge hreg1` is the way; `hreg1.le` might not elaborate (Not has no .le). THAT's likely it — le_of_lt? For ¬(2 ≤ s.re), `hreg1.le` — Dot-notation on a proof of ¬p: ¬p = (p → False): hreg1.le : ... doesn't exist — elaborator garbage. Fix: `(by linarith [lt_of_not_ge hreg1] : s.re ≤ 7 / 2)`. Also earlier `hreg1.le` appears in hcap's first branch AND `hreg1.le` in ht10's h1 pair `⟨hreg2, hreg1.le⟩` — same fix: lt_of_not_ge hreg1. And in hcap's second branch: `(by linarith [hhalf] : s.re ≤ 1 / 2)` — hhalf (pushed) : ¬1/2 ≤ s.re: lt_of_not_ge hhalf : s.re < 1/2: linarith gives ≤ ✓ — hhalf was pushed with `push Not at hhalf` giving s.re < 1/2 directly ✓ fine.\n\n   Hmm wait — in ht10's hre25: `have h1 : -5 / 2 ≤ s.re ∧ s.re ≤ 2 := ⟨hreg2, hreg1.le⟩` — hreg1.le same issue: use (lt_of_not_ge hreg1).le.\n\n4. The hcap's whole structure — the error mentions 3523 which is the 5c branch. Fix hreg1.le occurrences.\n\nLet me apply these fixes.","signature":""},{"type":"text","text":"Fixes: `one_div_le_one_div_of_le` arg order, `abs_one` for the |+1| forms, hmax simplified to ≤, and `hreg1.le` → `lt_of_not_ge`."},{"type":"tool_use","id":"Edit_205","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean","old_string":"              rw [show (zeroMult ρ : ℝ) / ‖s - ρ‖ = (zeroMult ρ : ℝ) * (1 / ‖s - ρ‖) from by ring]\n              apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)\n              have h1 : (0 : ℝ) < 1 / (32 * C0hat (4 * (k : ℝ) + 9 / 2)) := by positivity\n              have h2 : (0 : ℝ) < ‖s - ρ‖ := by linarith [hsep, h1]\n              exact (one_div_le_one_div_of_le h2 hsep).trans_eq (one_div_one_div _)","new_string":"              rw [show (zeroMult ρ : ℝ) / ‖s - ρ‖ = (zeroMult ρ : ℝ) * (1 / ‖s - ρ‖) from by ring]\n              apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)\n              have h1 : (0 : ℝ) < 1 / (32 * C0hat (4 * (k : ℝ) + 9 / 2)) := by positivity\n              exact (one_div_le_one_div_of_le h1 hsep).trans_eq (one_div_one_div _)"},"extra_content":null}]