[{"type":"thinking","thinking":"`iimaj_bracket_bound` COMPILES (with the 800k heartbeat allowance). Now the IIMaj_deriv2_lt glue lemma, then the final assembly.\n\nIIMaj_deriv2_lt:\n```lean\n/-- [PROVED session 011] IIMaj″(n) < 1/(2n) for n ≥ 20 (D-057 §3's\nmax 2n·|B̃₄″| < 1, proved analytically). -/\ntheorem IIMaj_deriv2_lt {n : ℝ} (hn : 20 ≤ n) :\n    deriv (deriv IIMaj) n < 1 / (2 * n) := by\n  have hn0 : (0 : ℝ) < n := by linarith\n  have hId : IIMaj = fun m => psiTilde (Real.sqrt m) := funext fun m => IIMaj_eq_psiTilde_sqrt m\n  rw [hId, chain_rule_sqrt psiTilde psiTilde_contDiff n hn0, deriv_psiTilde, deriv2_psiTilde]\n  have hs : Real.sqrt 20 ≤ Real.sqrt n := Real.sqrt_le_sqrt hn\n  have hb := iimaj_bracket_bound hs\n  -- goal: (D2(√n) − D1(√n)/√n)/(4n) < 1/(2n)\n  have h4n : (0 : ℝ) < 4 * n := by linarith\n  have hs0 : Real.sqrt n ≠ 0 := Real.sqrt_ne_zero'.2 hn0\n  have hbridge : (deriv (deriv psiTilde) (Real.sqrt n) - deriv psiTilde (Real.sqrt n) / Real.sqrt n) / (4 * n) < 1 / (2 * n) := by\n    rw [div_lt_div_iff_of_pos?? ...]\n  ...\n```\nAfter the rewrites (deriv_psiTilde, deriv2_psiTilde), the goal's expression IS the iimaj_bracket_bound's LHS divided by (4n): hb : [4.56*(2*(−a eA−b eB) + (s+1)*(a²eA+b²eB)) − (4.56*((eA+eB)+(s+1)*(−a eA−b eB)))/s] < 2 with s := √n — the goal: (that)/(4n) < 1/(2n). Since 4n > 0: (X)/(4n) < 1/(2n) ⟺ X/(4n) < 2/(4n) ⟸ X < 2: div_lt_div_of... for STRICT: gcongr? or: 1/(2n) = 2/(4n) (field_simp) then (div_lt_div_iff_of_pos_right? unknown) — do: have e : (1:ℝ)/(2*n) = 2/(4*n) := by field_simp [hn0.ne']; ring?? field_simp closes maybe; then rw [e]; gcongr? For div_lt: `div_lt_div_of_lt_left`? Name risk. Alternative: mul: X/(4n) < 2/(4n) ⟺ X < 2 by (div_lt_div_iff_of_pos_right h4n)? or mul_lt_mul_right: X*(4n)⁻¹ < 2*(4n)⁻¹ ⟸ X < 2: mul_lt_mul_of_nonneg_right?? no — for STRICT: mul_lt_mul_of_pos_right (X < 2) (inv_pos.mpr h4n)... wait: X/(4n) = X*(4n)⁻¹: strict mono of (· * c) for c > 0: (mul_lt_mul_right (inv_pos.mpr h4n)).mpr hb : X * (4n)⁻¹ < 2 * (4n)⁻¹ ✓ `mul_lt_mul_right (hc : 0 < c) : a * c < b * c ↔ a < b` ✓ exists I'm confident. And X/(4n) = X*(4n)⁻¹: div_eq_mul_inv ✓; 1/(2n) = 2/(4n) = 2*(4n)⁻¹: div_eq_mul_inv + field: 1/(2n) = 2/(4n): field_simp; ring?? 1/(2n) = 2/(4n) ⟺ 4n = 2*(2n) ✓ field_simp [hn0.ne'] closes or leaves ring-goal. Then rewrite goal to X*(4n)⁻¹ < 2*(4n)⁻¹ via div_eq_mul_inv rewrites.\n\nCareful: does the goal after the chain_rule_sqrt rw have the bracket as the bracket? chain_rule_sqrt gives (deriv (deriv psiTilde) (√n) − deriv psiTilde (√n)/√n)/(4n); rw [deriv_psiTilde, deriv2_psiTilde] substitutes the closed forms ✓ then it's hb's LHS / (4n) where hb's LHS is the unfolded iimA/iimB expression — BUT hb (iimaj_bracket_bound) is stated with iimA/iimB FOLDED and the goal after rw [deriv_psiTilde, deriv2_psiTilde] also has iimA/iimB folded (deriv_psiTilde/deriv2_psiTilde statements use iimA/iimB ✓). ✓ match.\n\nAlso — hs : √20 ≤ √n: Real.sqrt_le_sqrt hn ✓ (hn : 20 ≤ n — hmm hn : (20:ℝ) ≤ n ✓ sqrt_le_sqrt ✓).\n\nWrite it, build, commit. Then assembly.","signature":""},{"type":"text","text":"Bracket bound compiles. Adding the `IIMaj_deriv2_lt` glue lemma."},{"type":"tool_use","id":"Edit_250","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean","old_string":"/-- [SKELETON] Margin convexity (D-057, draft §3.1 Step 4): for any","new_string":"/-- [PROVED session 011] IIMaj″(n) < 1/(2n) for n ≥ 20 (the analytic\nversion of D-057 §3's \"max 2n·|B̃₄″| = 0.189 < 1\"). -/\ntheorem IIMaj_deriv2_lt {n : ℝ} (hn : 20 ≤ n) :\n    deriv (deriv IIMaj) n < 1 / (2 * n) := by\n  have hn0 : (0 : ℝ) < n := by linarith\n  have hId : IIMaj = fun m => psiTilde (Real.sqrt m) :=\n    funext fun m => IIMaj_eq_psiTilde_sqrt m\n  rw [hId, chain_rule_sqrt psiTilde psiTilde_contDiff n hn0, deriv_psiTilde,\n    deriv2_psiTilde]\n  have hb := iimaj_bracket_bound (Real.sqrt_le_sqrt hn)\n  have h4n : (0 : ℝ) < 4 * n := by linarith\n  have e : (1 : ℝ) / (2 * n) = 2 / (4 * n) := by\n    field_simp [hn0.ne']\n    ring\n  rw [e]\n  have e2 : (2 : ℝ) / (4 * n) = 2 * (4 * n)⁻¹ := by rw [div_eq_mul_inv]\n  rw [e2, div_eq_mul_inv]\n  exact (mul_lt_mul_right (inv_pos.mpr h4n)).mpr hb\n\n/-- [SKELETON] Margin convexity (D-057, draft §3.1 Step 4): for any"},"extra_content":null}]