[{"type":"thinking","thinking":"Now the hborder proof — rewrite the three classification branches with the key-disjunction pattern. The current structure per branch:\n1. nontrivial: rcases hs → 4 pieces (bottom im=−T, left re=−5/2, top im=T, right re=2√n).\n2. h0 (s = 0): same 4.\n3. −2n: same 4.\n\nReplace each `rcases hs with hb | hb | hb | hb` with the key-pattern. For the nontrivial case:\n```lean\n    rcases zeta_zero_classification hζ with hnt | h0 | hn2\n    · have key : s.im = z.im ∨ s.re = z.re ∨ s.im = w.im ∨ s.re = w.re := by\n        rw [RectangleBorder] at hs\n        simp only [Set.mem_union, Complex.mem_reProdIm, Set.mem_singleton_iff] at hs\n        tauto\n      rcases key with him | hre | him2 | hre2\n      · -- bottom: im = −T\n        rw [rectZ_im] at him\n        ... (the conj argument, him : s.im = −T)\n      · -- left: re = −5/2\n        rw [rectZ_re] at hre\n        rw [hre] at hnt\n        exact (by norm_num : ¬ ((0 : ℝ) < -5 / 2 ∧ -5 / 2 < 1)) ⟨hnt.2.1, hnt.2.2⟩\n      · -- top: im = T\n        rw [rectW_im] at him2\n        ... (direct pigeonhole)\n      · -- right: re = 2√n\n        rw [rectW_re] at hre2\n        exact riemannZeta_ne_zero_of_one_le_re (by rw [hre2]; linarith [...]) hζ\n```\n\nFor h0 (s = 0): the border Eqs: (0:ℂ).im/re = z.im/z.re/w.im/w.re: rewrite with rectZ_im/rectZ_re/rectW_im/rectW_re + Complex.zero_im/zero_re → contradictions (0 = −T, 0 = −5/2, 0 = T, 0 = 2√n).\n```lean\n    · rw [h0] at hs\n      have key : (0 : ℂ).im = z.im ∨ (0 : ℂ).re = z.re ∨ (0 : ℂ).im = w.im ∨\n          (0 : ℂ).re = w.re := by\n        rw [RectangleBorder] at hs\n        simp only [Set.mem_union, Complex.mem_reProdIm, Set.mem_singleton_iff] at hs\n        tauto\n      rcases key with him | hre | him2 | hre2\n      · rw [rectZ_im, Complex.zero_im] at him\n        exact absurd him (ne_of_gt (by linarith [hT1] : (0 : ℝ) > -T))\n      · rw [rectZ_re, Complex.zero_re] at hre\n        norm_num at hre\n      · rw [rectW_im, Complex.zero_im] at him2\n        exact absurd him2 (ne_of_gt (by linarith [hT1] : (0 : ℝ) < T))\n      · rw [rectW_re, Complex.zero_re] at hre2\n        have h0' : (0 : ℝ) ≤ Real.sqrt n := Real.sqrt_nonneg _\n        linarith [hre2, h0']\n```\nWait — in the h0 branch after rw [h0] at hs, hs is about (0:ℂ) — the `key` proof's `tauto` — the border membership of (0:ℂ) — the rw [h0] at hs replaced s with 0 in hs ✓ then key with s := (0:ℂ)... but my key statement has (0:ℂ).im/re — after rw [h0] at hs, the simp+tauto gives the key with (0:ℂ) ✓. Note: tauto's disjunction must match my key's form ✓.\n\nHmm — one issue: in h0 branch, after `rw [h0] at hs`, the key's statement mentions (0:ℂ) explicitly — but the tauto-generated disjunction is in terms of `(0:ℂ)` ✓ consistent.\n\nFor −2n: after obtain ⟨m, hm1, rfl⟩, s = −2m:\n```lean\n      obtain ⟨m, hm1, rfl⟩ := hn2\n      have key : ((-2 * (m : ℂ)) : ℂ).im = z.im ∨ ((-2 * (m : ℂ)) : ℂ).re = z.re ∨\n          ((-2 * (m : ℂ)) : ℂ).im = w.im ∨ ((-2 * (m : ℂ)) : ℂ).re = w.re := by\n        rw [RectangleBorder] at hs\n        simp only [Set.mem_union, Complex.mem_reProdIm, Set.mem_singleton_iff] at hs\n        tauto\n      have him0 : ((-2 * (m : ℂ)) : ℂ).im = 0 := by simp [Complex.mul_im]\n      have hre0 : ((-2 * (m : ℂ)) : ℂ).re = -2 * (m : ℝ) := by simp [Complex.mul_re, Complex.mul_im]\n      rcases key with him | hre | him2 | hre2\n      · rw [rectZ_im, him0] at him\n        exact absurd him (ne_of_gt (by linarith [hT1] : (0 : ℝ) > -T))\n      · rw [rectZ_re, hre0] at hre\n        -- hre : -2·m = -5/2\n        have hm : (m : ℝ) = 5 / 4 := by linarith\n        have hm5 : m = 1 := by\n          have h6 : (m : ℝ) < 2 := by linarith\n          have h7 : m < 2 := by exact_mod_cast h6\n          exact le_antisymm (Nat.lt_succ_iff.mp h7) hm1\n        rw [hm5] at hm\n        norm_num at hm\n      · rw [rectW_im, him0] at him2\n        exact absurd him2 (ne_of_gt (by linarith [hT1] : (0 : ℝ) < T))\n      · rw [rectW_re, hre0] at hre2\n        have hpos : (0 : ℝ) ≤ Real.sqrt n := Real.sqrt_nonneg _\n        have hm0 : (0 : ℝ) ≤ (m : ℝ) := Nat.cast_nonneg _\n        nlinarith [hre2, hpos, hm0]\n```\nWait — the left case: hre : (-2·m).re = z.re — after rw [rectZ_re] at hre: (−2·m).re = −5/2; rw [hre0] at hre — hre0 : ((-2·m)).re = −2·(m:ℝ): rewrites the LHS → −2m = −5/2 ✓ then m = 5/4 by linarith ✓.\n\nHmm — but the `rw [rectZ_re, hre0] at hre` — hre : ((-2*(m:ℂ)):ℂ).re = z.re: rw [rectZ_re] at hre → ... = −5/2; rw [hre0] at hre → −2*(m:ℝ) = −5/2 ✓.\n\nAnd him0/hre0: `by simp [Complex.mul_im]` / `by simp [Complex.mul_re, Complex.mul_im]` — re/im of (−2)·↑m: mul_re: (−2).re·↑m.re − (−2).im·↑m.im = −2·m.re − 0 = −2·m.re... simp should evaluate ((−2:ℂ).re = −2, .im = 0, (↑m).re = m, .im = 0) ✓.\n\nAlso — earlier lines: the nontrivial bottom case: him : s.im = −T (after rectZ_im): the conj argument uses him : s.im = -T ✓ and (conj s).im = −s.im = T: `Complex.conj_im` — my earlier code used `rw [Complex.conj_im, him]` ✓.\n\nLet me now rewrite the whole hborder block with these patterns.","signature":""},{"type":"text","text":"Now rewriting the whole hborder block with the key-disjunction pattern."},{"type":"tool_use","id":"Edit_282","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiAssembly.lean","old_string":"  -- the border is zero-free\n  have hborder : ∀ s ∈ RectangleBorder z w, riemannZeta s ≠ 0 := by\n    intro s hs hζ\n    rcases zeta_zero_classification hζ with hnt | h0 | hn2\n    · -- nontrivial: re ∈ (0,1); the only border pieces with re in (0,1) are top/bottom\n      rcases hs with hb | hb | hb | hb\n      · -- bottom: im = −T; conj ρ has im = T in the window — pigeonhole contradiction\n        have him : s.im = -T := by\n          have := hb.2\n          rw [rectZ_im] at this\n          exact Set.mem_singleton_iff.mp this\n        have hconj := hTwide ((starRingEnd ℂ) s) (isNontrivialZero_conj hnt)\n          (by rw [Complex.conj_im, him]; linarith [hTn])\n          (by rw [Complex.conj_im, him]; linarith [hTn1])\n        have hpos : (0 : ℝ) < 1 / (9 * C0hat (Real.sqrt n + 2)) := by\n          have h := two_le_C0hat (by linarith [hn1sqrt] : (1 : ℝ) ≤ Real.sqrt n + 2)\n          exact one_div_pos.mpr (by linarith)\n        have hzero : |((starRingEnd ℂ) s).im - T| = 0 := by\n          rw [Complex.conj_im, him, sub_self, abs_zero]\n        linarith [hconj, hpos, hzero]\n      · -- left: re = −5/2, not in (0,1)\n        have hre' : s.re = -5 / 2 := by\n          have := hb.1\n          rw [rectZ_re] at this\n          exact Set.mem_singleton_iff.mp this\n        rw [hre'] at hnt\n        exact (by norm_num : ¬ ((0 : ℝ) < -5 / 2 ∧ -5 / 2 < 1)) ⟨hnt.2.1, hnt.2.2⟩\n      · -- top: im = T; pigeonhole contradiction directly\n        have him : s.im = T := by\n          have := hb.2\n          rw [rectW_im] at this\n          exact Set.mem_singleton_iff.mp this\n        have hdir := hTwide s hnt (by rw [him]; exact hTn) (by rw [him]; linarith [hTn1])\n        have hpos : (0 : ℝ) < 1 / (9 * C0hat (Real.sqrt n + 2)) := by\n          have h := two_le_C0hat (by linarith [hn1sqrt] : (1 : ℝ) ≤ Real.sqrt n + 2)\n          exact one_div_pos.mpr (by linarith)\n        have hzero : |s.im - T| = 0 := by rw [him, sub_self, abs_zero]\n        linarith [hdir, hpos, hzero]\n      · -- right: re = 2√n ≥ 2 > 1\n        have hre' : s.re = 2 * Real.sqrt n := by\n          have := hb.1\n          rw [rectW_re] at this\n          exact Set.mem_singleton_iff.mp this\n        exact riemannZeta_ne_zero_of_one_le_re (by rw [hre']; linarith [Real.one_le_sqrt.mpr (by exact_mod_cast hn : (1 : ℝ) ≤ n)]) hζ\n    · -- s = 0: not on the border (im 0 = ±T needs T = 0; re 0 = −5/2 / 2√n false)\n      rw [h0] at hs\n      rcases hs with hb | hb | hb | hb\n      · exact absurd (Set.mem_singleton_iff.mp hb.2) (by rw [rectZ_im, Complex.zero_im]; exact ne_of_gt (by linarith : (0 : ℝ) < T))\n      · exact absurd (Set.mem_singleton_iff.mp hb.1) (by rw [rectZ_re, Complex.zero_re]; norm_num)\n      · exact absurd (Set.mem_singleton_iff.mp hb.2) (by rw [rectW_im, Complex.zero_im]; exact ne_of_gt (by linarith : (0 : ℝ) < T))\n      · exact absurd (Set.mem_singleton_iff.mp hb.1) (by rw [rectW_re, Complex.zero_re]; norm_num [Real.sqrt_nonneg (n : ℝ)])\n    · -- s = −2n, n ≥ 1: re = −2n; border re-pieces are −5/2 and 2√n; im = 0 ≠ ±T\n      obtain ⟨m, hm1, rfl⟩ := hn2\n      have him0 : ((-2 * (m : ℂ)) : ℂ).im = 0 := by\n        simp [Complex.neg_re, Complex.neg_im, Complex.mul_re, Complex.mul_im]\n      have hre0 : ((-2 * (m : ℂ)) : ℂ).re = -2 * (m : ℝ) := by\n        simp [Complex.mul_re, Complex.mul_im]\n      rcases hs with hb | hb | hb | hb\n      · have h := Set.mem_singleton_iff.mp hb.2\n        rw [rectZ_im, him0] at h\n        exact absurd h (ne_of_gt (by linarith [hT1] : (0 : ℝ) > -T))\n      · have hre' : z.re = -2 * (m : ℝ) := by\n          have h := Set.mem_singleton_iff.mp hb.1\n          rw [hre0] at h\n          exact h.symm\n        have hzr : z.re = -5 / 2 := rectZ_re T\n        rw [hzr] at hre'\n        have hm : (m : ℝ) = 5 / 4 := by linarith\n        have hm5 : m = 1 := by\n          have h6 : (m : ℝ) < 2 := by linarith\n          have h7 : m < 2 := by exact_mod_cast h6\n          exact le_antisymm (Nat.lt_succ_iff.mp h7) hm1\n        rw [hm5] at hm\n        norm_num at hm\n      · have h := Set.mem_singleton_iff.mp hb.2\n        rw [rectW_im, him0] at h\n        exact absurd h (ne_of_gt (by linarith [hT1] : (0 : ℝ) < T))\n      · have hre' : w.re = -2 * (m : ℝ) := by\n          have h := Set.mem_singleton_iff.mp hb.1\n          rw [hre0] at h\n          exact h.symm\n        have hwr : w.re = 2 * Real.sqrt n := rectW_re n T\n        rw [hwr] at hre'\n        have hpos : (0 : ℝ) ≤ Real.sqrt n := Real.sqrt_nonneg _\n        have hm0 : (0 : ℝ) ≤ (m : ℝ) := Nat.cast_nonneg _\n        nlinarith [hre', hpos, hm0]","new_string":"  -- the border is zero-free\n  have hborder : ∀ s ∈ RectangleBorder z w, riemannZeta s ≠ 0 := by\n    intro s hs hζ\n    have key : s.im = z.im ∨ s.re = z.re ∨ s.im = w.im ∨ s.re = w.re := by\n      rw [RectangleBorder] at hs\n      simp only [Set.mem_union, Complex.mem_reProdIm, Set.mem_singleton_iff] at hs\n      tauto\n    rcases zeta_zero_classification hζ with hnt | h0 | hn2\n    · -- nontrivial: re ∈ (0,1); border pieces with re in (0,1) are top/bottom\n      rcases key with him | hre | him2 | hre2\n      · -- bottom: im = −T; conj ρ has im = T in the window — pigeonhole contradiction\n        rw [rectZ_im] at him\n        have hconj := hTwide ((starRingEnd ℂ) s) (isNontrivialZero_conj hnt)\n          (by rw [Complex.conj_im, him]; linarith [hTn])\n          (by rw [Complex.conj_im, him]; linarith [hTn1])\n        have hpos : (0 : ℝ) < 1 / (9 * C0hat (Real.sqrt n + 2)) := by\n          have h := two_le_C0hat (by linarith [hn1sqrt] : (1 : ℝ) ≤ Real.sqrt n + 2)\n          exact one_div_pos.mpr (by linarith)\n        have hzero : |((starRingEnd ℂ) s).im - T| = 0 := by\n          rw [Complex.conj_im, him, sub_self, abs_zero]\n        linarith [hconj, hpos, hzero]\n      · -- left: re = −5/2, not in (0,1)\n        rw [rectZ_re] at hre\n        rw [hre] at hnt\n        exact (by norm_num : ¬ ((0 : ℝ) < -5 / 2 ∧ -5 / 2 < 1)) ⟨hnt.2.1, hnt.2.2⟩\n      · -- top: im = T; pigeonhole contradiction directly\n        rw [rectW_im] at him2\n        have hdir := hTwide s hnt (by rw [him2]; exact hTn) (by rw [him2]; linarith [hTn1])\n        have hpos : (0 : ℝ) < 1 / (9 * C0hat (Real.sqrt n + 2)) := by\n          have h := two_le_C0hat (by linarith [hn1sqrt] : (1 : ℝ) ≤ Real.sqrt n + 2)\n          exact one_div_pos.mpr (by linarith)\n        have hzero : |s.im - T| = 0 := by rw [him2, sub_self, abs_zero]\n        linarith [hdir, hpos, hzero]\n      · -- right: re = 2√n ≥ 2 > 1\n        rw [rectW_re] at hre2\n        exact riemannZeta_ne_zero_of_one_le_re (by rw [hre2];\n          linarith [Real.one_le_sqrt.mpr (by exact_mod_cast hn : (1 : ℝ) ≤ n)]) hζ\n    · -- s = 0: not on the border (im 0 = ±T needs T = 0; re 0 = −5/2 / 2√n false)\n      rw [h0] at hs\n      rw [h0] at key\n      rcases key with him | hre | him2 | hre2\n      · rw [rectZ_im, Complex.zero_im] at him\n        exact absurd him (ne_of_gt (by linarith [hT1] : (0 : ℝ) > -T))\n      · rw [rectZ_re, Complex.zero_re] at hre\n        norm_num at hre\n      · rw [rectW_im, Complex.zero_im] at him2\n        exact absurd him2 (ne_of_gt (by linarith [hT1] : (0 : ℝ) < T))\n      · rw [rectW_re, Complex.zero_re] at hre2\n        have h0' : (0 : ℝ) ≤ Real.sqrt n := Real.sqrt_nonneg _\n        linarith [hre2, h0']\n    · -- s = −2n, n ≥ 1: re = −2n; border re-pieces are −5/2 and 2√n; im = 0 ≠ ±T\n      obtain ⟨m, hm1, rfl⟩ := hn2\n      rw [show ((-2 : ℂ) * ((m : ℝ) : ℂ)) = (-2 : ℂ) * (m : ℂ) from by push_cast; ring] at key ⊢\n      have him0 : ((-2 : ℂ) * (m : ℂ)).im = 0 := by\n        simp [Complex.mul_im]\n      have hre0 : ((-2 : ℂ) * (m : ℂ)).re = -2 * (m : ℝ) := by\n        simp [Complex.mul_re, Complex.mul_im]\n      rcases key with him | hre | him2 | hre2\n      · rw [rectZ_im, him0] at him\n        exact absurd him (ne_of_gt (by linarith [hT1] : (0 : ℝ) > -T))\n      · rw [rectZ_re, hre0] at hre\n        have hm : (m : ℝ) = 5 / 4 := by linarith\n        have hm5 : m = 1 := by\n          have h6 : (m : ℝ) < 2 := by linarith\n          have h7 : m < 2 := by exact_mod_cast h6\n          exact le_antisymm (Nat.lt_succ_iff.mp h7) hm1\n        rw [hm5] at hm\n        norm_num at hm\n      · rw [rectW_im, him0] at him2\n        exact absurd him2 (ne_of_gt (by linarith [hT1] : (0 : ℝ) < T))\n      · rw [rectW_re, hre0] at hre2\n        have hpos : (0 : ℝ) ≤ Real.sqrt n := Real.sqrt_nonneg _\n        have hm0 : (0 : ℝ) ≤ (m : ℝ) := Nat.cast_nonneg _\n        nlinarith [hre2, hpos, hm0]"},"extra_content":null}]