[{"type":"thinking","thinking":"`riemannZeta_neg_two_mul {n : ℕ} (hn : 1 ≤ n) : riemannZeta (-2 * (n : ℂ)) = 0` exists in LiFEntire:473. For n = 1: riemannZeta (-2 * (1:ℂ)) = 0 — need −2·1 = −2: `by rw [riemannZeta_neg_two_mul (n := 1) le_rfl]; norm_num` or: have h := riemannZeta_neg_two_mul (n := 1) le_rfl; rw [mul_one] at h; exact h.\n\nLet me fix the sorry block and clean up that whole (−2)-forward direction. Also I should double check several other spots in the file that looked shaky:\n1. The `hr : (-2 : ℂ).re ∈ Set.uIcc z.re w.re` via `simp [Complex.neg_re, Complex.mul_re]` — the goal after rw [Set.uIcc_of_le hre, rectZ_re, rectW_re, Set.mem_Icc] is `−5/2 ≤ (−2:ℂ).re ∧ (−2:ℂ).re ≤ 2·√n`: (−2:ℂ).re = −2: simp should reduce to −5/2 ≤ −2 ∧ −2 ≤ 2√n: the first by norm_num, the second needs sqrt nonneg — simp can't prove −2 ≤ 2√n. Fix: constructor <;> [...; positivity-ish].\n2. The `h0` absurd case: `exact absurd (by rw [h0, zetaA_apply ...] at hzA; exact hzA) (by norm_num)` — zetaA 0 = (0−1)·ζ(0) = (−1)·(−1/2) = 1/2: hzA : zetaA 0 = 0: after rw [h0] at hzA... wait hzA : zetaA s = 0 and h0 : s = 0: rw [h0] at hzA → zetaA 0 = 0. zetaA_apply (0 ≠ 1): zetaA 0 = (0−1)·riemannZeta 0: rw in hzA: (0−1)·ζ(0) = 0. Then need to derive False: riemannZeta 0 = −1/2 (riemannZeta_zero ✓ used in LiMoments): (−1)·(−1/2) = 1/2 ≠ 0. So:\n```lean\n      · have h00 : zetaA 0 = 0 := by rwa [h0] at hzA\n        rw [zetaA_apply (by norm_num : (0 : ℂ) ≠ 1), riemannZeta_zero] at h00\n        norm_num at h00\n```\n3. The hm1' proof in the backward direction (−2n inside ⟹ m = 1): re = −2m ∈ [−5/2, 2√n]: −5/2 ≤ −2m ⟹ m ≤ 5/4 ⟹ m ≤ 1 (nat) + hm1 (1 ≤ m) ⟹ m = 1 ✓ my proof sketch has the right idea but the nlinarith/linarith chain needs care: hr.1 : −5/2 ≤ (−2·m:ℝ)?? wait hre2 : ((-2 * (m:ℂ)):ℂ).re = -2 * (m:ℝ) — hr : re ∈ Icc [−5/2, 2√n]: after rw [hre2] at hr: hr : −5/2 ≤ −2·m ∧ −2·m ≤ 2√n: from the first: 2m ≤ 5/2: m ≤ 5/4 ✓. My code: `have h1 : (m : ℝ) ≤ 5 / 4 := by have := hr.1; ...; nlinarith [this, hm0]` — hr.1 : −5/2 ≤ −2*(m:ℝ): −5/2 ≤ −2m ⟺ 2m ≤ 5/2 ⟺ m ≤ 5/4 ✓ linarith (not nlinarith — linear!): `linarith [hr.1]`? −5/2 ≤ −2·m ⟹ m ≤ 5/4: linear ✓. But hr.1 is after the hre2 rewrite: hr : (−2*(m:ℝ)) ∈ Set.Icc (−5/2) (2·√n)? Set.mem_Icc: hr.1 : −5/2 ≤ −2*(m:ℝ) ✓ linarith directly. My nlinarith usage is overkill but should work... Actually the code I wrote has `have h1 : (m : ℝ) ≤ 5 / 4 := by have := hr.1; have hm0 ...; nlinarith [this, hm0]` — `this` : −5/2 ≤ −2*(m:ℝ): nlinarith proves m ≤ 5/4 ✓ fine.\n   Then m ≤ 1: from m ≤ 5/4 < 2: Nat: (m:ℝ) < 2 → m < 2 → m ≤ 1 ✓.\n   Then rw [hm1'] makes s = −2·1: goal was membership in Sz = W ∪ {−2}: `right; simp` — s = −2·(1:ℂ): is −2·1 = −2 defeq for simp? (−2 * (1:ℂ)) = −2: mul_one ✓ simp closes: Finset.mem_singleton: −2*1 = −2 ✓.\n\nHmm wait — in the backward direction after obtain ⟨m, hm1, rfl⟩, s = −2·(m:ℂ) and the goal is `s ∈ W ∪ {−2}`: after rw [hm1']: goal `(−2 * ((1:ℕ):ℂ)) ∈ W ∪ {−2}`: `right; simp` — simp should show −2*1 = −2 then Finset.mem_singleton_self. Hmm — wait, actually my rw [hm1'] rewrites m := 1 — the ((1:ℕ):ℂ) cast: simp [Nat.cast_one, mul_one] ✓.\n\n4. The h0 case in hborder (s = 0 on the border): after rw [h0] at hs, the border cases: bottom im = −T: hb.2 : (0:ℂ).im ∈ {z.im}: `Set.mem_singleton_iff.mp hb.2` — hb.2 is the im-membership of the ×ℂ: (0:ℂ).im ∈ {−T}? Wait — the bottom piece is [[z.re, w.re]] ×ℂ {z.im}: membership: ⟨re ∈ [[..]], im ∈ {z.im}⟩: hb.2 : im ∈ {z.im} = z.im: rw [rectZ_im] at this: (0:ℂ).im = −T: (0:ℂ).im = 0: `rw [rectZ_im, Complex.zero_im]` — hmm the membership gives (0:ℂ).im = z.im... my code: `Set.mem_singleton_iff.mp hb.2` : (0:ℂ).im = z.im? mem_singleton_iff : a ∈ {b} ↔ a = b: (0:ℂ).im = z.im — z.im = −T (rectZ_im): so 0 = −T: the absurd: ne_of_gt (0 < T)?? 0 = −T with T > 0: −T ≠ 0: the equation is (0:ℂ).im = z.im i.e. 0 = −T: absurd needs 0 ≠ −T: `by rw [rectZ_im, Complex.zero_im]; exact ne_of_gt (by linarith : (0:ℝ) < T)` — produces 0 ≠ −T: hmm ne_of_gt h : a ≠ b where h : a > b? ne_of_gt : a > b → a ≠ b: with h : 0 < T: gives 0 ≠ T — but the equation is 0 = −T! Mismatch: need 0 ≠ −T: ne_of_gt (by linarith : (0:ℝ) < T) : (0:ℝ) ≠ T... the target after rw [rectZ_im, Complex.zero_im]: `¬ ((0:ℂ).im = -T)` → `¬ (0 = -T)`: 0 ≠ −T. ne_of_gt (h : 0 < T) : 0 ≠ T — wrong sign. Use: `by rw [rectZ_im, Complex.zero_im]; exact (by linarith [hT1] : (0:ℝ) ≠ -T)`? or ne_of_lt (by linarith : (-T) < 0): −T ≠ 0: the eq is 0 = −T: (0 ≠ −T) = (-T ≠ 0).symm: `exact (ne_of_lt (by linarith [hT1] : (-(T:ℝ)) < 0)).symm`. Hmm — but wait, the membership structure: `Set.mem_singleton_iff.mp hb.2` — hb.2 : (0:ℂ).im ∈ {z.im}?: the mem_singleton gives (0:ℂ).im = z.im: after rw [rectZ_im] at that: (0:ℂ).im = −T; then absurd (that) (proof that (0:ℂ).im ≠ −T): `rw [Complex.zero_im]`: (0:ℝ) ≠ −T ✓. My original code had `by rw [rectZ_im, Complex.zero_im]; exact ne_of_gt ...` — the rw happens in the GOAL of the absurd's second argument: goal `¬ ((0:ℂ).im = z.im)`?? — hmm no: `exact absurd (Set.mem_singleton_iff.mp hb.2) (by rw [rectZ_im, Complex.zero_im]; exact ne_of_gt ...)` — the `by rw` block proves `¬ ((0:ℂ).im = z.im)` — wait no: absurd (h : a) (h' : ¬a). Here a := ((0:ℂ).im = z.im)?? mem_singleton_iff.mp hb.2 : (0:ℂ).im = z.im — hmm wait, is the membership (0:ℂ).im ∈ {z.im} giving (0:ℂ).im = z.im? mem_singleton_iff : a ∈ {b} ↔ a = b ✓ so (0:ℂ).im = z.im. And z.im — I need to rw [rectZ_im] INSIDE the proof of ¬((0:ℂ).im = z.im): the by-block's goal: ¬ ((0:ℂ).im = z.im): rw [rectZ_im]: ¬((0:ℂ).im = −T): rw [Complex.zero_im]: ¬((0:ℝ) = −T): exact (ne_of_lt ...).symm or... ne_of_gt (h : a > b) : a ≠ b: 0 > −T: 0 ≠ −T ✓ — `ne_of_gt (by linarith [hT1] : (0:ℝ) > -T)` : (0:ℝ) ≠ −T ✓✓ — my original `ne_of_gt (by linarith : (0 : ℝ) < T)` gives 0 ≠ T — WRONG (need 0 ≠ −T). Fix to (0:ℝ) > -T.\n\nHmm wait — but the equation after rw [Complex.zero_im] is ¬((0:ℝ) = −T) i.e. 0 ≠ −T ✓ ne_of_gt (0 > −T) ✓.\n\nSame pattern for the top side (im = T): 0 = T: ne_of_gt (0 < T) gives 0 ≠ T ✓ correct there... my code has `ne_of_gt (by linarith : (0 : ℝ) < T)` for the T-case: equation 0 = T: 0 ≠ T ✓ OK. And for the −2n cases: same patterns.\n\nAlso the left-border for the −2n case: re = −2m = −5/2 ⟹ m = 5/4 ✗: my proof: `have hm5 : m = 1 := ...` — wait that's WRONG: re = −5/2 means 2m = 5/2: m = 5/4 — NOT a natural! The contradiction: 2m = 5/2 has no natural solution: m ≤ 1 (from 2m ≤ 5/2 → m ≤ 5/4 < 2) and m ≥ 1 ⟹ m = 1 ⟹ 2·1 = 5/2 ⟹ 2 = 5/2 contradiction. My code derives m = 1 then rw [hm5] at this (this : (m:ℝ) = 5/4): gives (1:ℝ) = 5/4: norm_num at this → False ✓. OK that works but convoluted; fine.\n\nBut wait — there's an issue with `have : (-2 : ℝ) * (m : ℝ) = -5 / 2 := by have h := congrArg Complex.re hre'; simp [...] at h; rw [rectZ_re] at h; exact_mod_cast h` — hre' : ((-2*(m:ℂ)):ℂ).re = −5/2?? — hre' came from `Set.mem_singleton_iff.mp hb.1` : ((-2*(m:ℂ)):ℂ).re = z.re: rw [rectZ_re] at that gives re = −5/2 directly. My code does congrArg Complex.re hre' — but hre' is ALREADY about .re! Let me recheck my code:\n\n```lean\n      · have hre' : ((-2 * (m : ℂ)) : ℂ).re = -5 / 2 := Set.mem_singleton_iff.mp hb.1\n```\nhb.1 : ((-2*(m:ℂ)):ℂ).re ∈ {z.re}: mp gives re = z.re = −5/2 after... no wait — mp gives ((-2·m)).re = z.re: need rw [rectZ_re] at it: `Set.mem_singleton_iff.mp hb.1 : ((-2*(m:ℂ)):ℂ).re = z.re` — so hre' as written is WRONG (it's = z.re, not = −5/2 directly). Fix: `have hre' : ((-2 * (m : ℂ)) : ℂ).re = -5 / 2 := by have h := Set.mem_singleton_iff.mp hb.1; rwa [rectZ_re] at h`. Then ((-2·m)).re = −5/2: compute re of −2·m: simp [Complex.mul_re, ...]: −2·m. Then (2·m:ℝ) = 5/2.\n\nHmm — also `simp [Complex.mul_re, Complex.neg_re] at h` — re((−2:ℂ)*m) = (−2).re·m.re − (−2).im·m.im = −2·m − 0 = −2·m ✓ simp should give it. My code then does the cast chain. Let me simplify: \n```lean\n      · have hre' : ((-2 * (m : ℂ)) : ℂ).re = -5 / 2 := by\n          have h := Set.mem_singleton_iff.mp hb.1\n          rwa [rectZ_re] at h\n        have h2 : (-2 : ℝ) * (m : ℝ) = -5 / 2 := by\n          have h := hre'\n          simp [Complex.mul_re, Complex.mul_im, Complex.ofReal_re] at h\n          ... \n```\nHmm — ((-2 * (m:ℂ)):ℂ).re: the expression is (-2 * (m:ℂ)) where −2 : ℂ and (m:ℂ) = ↑(m:ℝ)? No — (m : ℂ) is Nat.cast: Complex.mul_re: re(a·b) = a.re·b.re − a.im·b.im: (−2).re = −2, (−2).im = 0, (↑m).re = ↑m, (↑m).im = 0: = −2·↑m. simp [Complex.mul_re] at hre' turns hre' into −2·(m:ℝ) = −5/2 (with the Nat.cast_re simp lemma) ✓. Then:\n```lean\n        have hm : (m : ℝ) = 5 / 4 := by\n          have h := hre'\n          simp only [Complex.mul_re, Complex.mul_im, Complex.neg_re, Complex.neg_im,\n            Complex.ofNat_re, Complex.ofNat_im, Nat.cast_re, Nat.cast_im] at h\n          linarith\n```\nThis is getting messy. Simpler approach for both re-cases of −2n: compute the re directly as a have:\n```lean\n      have hre2m : ((-2 * (m : ℂ)) : ℂ).re = -2 * (m : ℝ) := by\n        simp [Complex.mul_re, Complex.mul_im]\n```\nre(−2·m) = (−2).re·m.re − (−2).im·m.im = −2·m.re − 0 = −2·m ✓ simp [Complex.mul_re] alone might leave it as (−2).re * (m:ℂ).re: add the ofNat re lemmas: Complex.ofNat_re? The atoms: (−2:ℂ).re — simp evaluates to −2 via... `Complex.ofNat_re`? or `show ((-2:ℂ).re = -2) from rfl`? (−2:ℂ).re should compute to −2 definitionally (OfNat ofNat's re). Try `by simp [Complex.mul_re, Complex.mul_im]`.\n\nThis file has several shaky spots. Rather than perfecting it in my head, let me fix the obvious issues (the sorry, the hre' pattern, the ne_of_gt sign, the h0 absurd case, the (−2).re uIcc) and iterate on the build. Let me rewrite the problematic parts.","signature":""},{"type":"text","text":"`riemannZeta_neg_two_mul` exists (LiFEntire). Fixing the sorry and the shaky spots before the first build."},{"type":"tool_use","id":"Edit_265","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiAssembly.lean","old_string":"      · refine ⟨⟨?_, ?_⟩, ?_⟩\n        · have hr : (-2 : ℂ).re ∈ Set.uIcc z.re w.re := by\n            rw [Set.uIcc_of_le hre, rectZ_re, rectW_re, Set.mem_Icc]\n            simp [Complex.neg_re, Complex.mul_re]\n          exact hr\n        · have hi : (-2 : ℂ).im ∈ Set.uIcc z.im w.im := by\n            rw [Set.uIcc_of_le him, rectZ_im, rectW_im, Set.mem_Icc]\n            simp [Complex.neg_re, Complex.neg_im, Complex.mul_re, Complex.mul_im]\n            constructor <;> linarith [hT1]\n          exact hi\n        · have h2ne : (-2 : ℂ) ≠ 1 := by norm_num\n          rw [zetaA_apply h2ne]\n          have hζ2 : riemannZeta (-2 : ℂ) = 0 := by\n            have h := RvM.completedRiemannZeta_eq_zero_iff.mp (by\n              have h2zeros : riemannZeta (-2 : ℂ) = 0 := by\n                rw [riemannZeta_neg_two]\n                exact Real.Gamma_ne_zero_of_zpow_neg ?_  -- placeholder\n              exact h2zeros)\n            sorry\n          rw [hζ2, mul_zero]","new_string":"      · refine ⟨⟨?_, ?_⟩, ?_⟩\n        · have hr : (-2 : ℂ).re ∈ Set.uIcc z.re w.re := by\n            rw [Set.uIcc_of_le hre, rectZ_re, rectW_re, Set.mem_Icc]\n            have h2re : (-2 : ℂ).re = -2 := by simp\n            rw [h2re]\n            constructor\n            · norm_num\n            · have h0' : (0 : ℝ) ≤ Real.sqrt n := Real.sqrt_nonneg _\n              linarith\n          exact hr\n        · have hi : (-2 : ℂ).im ∈ Set.uIcc z.im w.im := by\n            rw [Set.uIcc_of_le him, rectZ_im, rectW_im, Set.mem_Icc]\n            have h2im : (-2 : ℂ).im = 0 := by simp\n            rw [h2im]\n            constructor <;> linarith [hT1]\n          exact hi\n        · have h2ne : (-2 : ℂ) ≠ 1 := by norm_num\n          rw [zetaA_apply h2ne]\n          have hζ2 : riemannZeta (-2 : ℂ) = 0 := by\n            have h := riemannZeta_neg_two_mul (n := 1) le_rfl\n            rw [show (-2 : ℂ) * ((1 : ℕ) : ℂ) = -2 by push_cast; ring] at h\n            exact h\n          rw [hζ2, mul_zero]"},"extra_content":null}]