[{"type":"thinking","thinking":"Oops — `(by norm_num : (1 : ℝ) < 1)` is false! The hc argument for FAlt's domain c = 1 should be `hc : c < 1` — but c = 1 is NOT < 1. Problem!\n\nWait: chain_rule_sqrt_shift_on requires c < 1 so that Ioi c ∈ 𝓝 (√m+1) via √m+1 ≥ 1 > c. For FAlt the domain is Ioi 1 — c = 1 fails the strict inequality: at m → 0⁺, √m+1 → 1, so Ioi 1 is NOT a neighborhood of √m+1 when m = ... wait we apply at n with hn : 0 < n, so √n + 1 > 1 strictly, and Ioi 1 IS a neighborhood of √n+1 for n > 0. The issue is only my lemma's hmem: ∀ m > 0, Ioi c ∈ 𝓝 (√m+1) — for c = 1: need 1 < √m+1 ⟺ m > 0 ✓ TRUE for all m > 0! Ioi_mem_nhds needs c < √m + 1: 1 < √m+1 ⟺ 0 < √m ⟺ m > 0 ✓. So actually hc should be `c ≤ 1`! For c = 1: √m+1 > 1 ⟺ m > 0 = hm ✓. For c < 1: √m+1 ≥ 1 > c ✓. So the condition c ≤ 1 works with a slightly adjusted hmem proof: Ioi_mem_nhds (show c < √m+1): cases c = 1: need 1 < √m+1 ⟺ 0 < √m: Real.sqrt_pos.mpr hm. General: by_cases... simplest: have : c ≤ 1 → c < Real.sqrt m + 1 := fun hc => lt_of_le_of_lt hc (by have := Real.sqrt_pos.mpr hm; linarith). Wait lt_of_le_of_lt : c ≤ 1 → 1 < √m+1 → c < √m+1 ✓ where 1 < √m+1 ⟺ 0 < √m from sqrt_pos.mpr hm ✓.\n\nSo change hc : c < 1 to hc : c ≤ 1, and hmem becomes:\n```\nexact Ioi_mem_nhds (lt_of_le_of_lt hc (by have h := Real.sqrt_pos.mpr hm; linarith))\n```\nHmm wait, for the ORIGINAL c = −1 the proof used `by linarith [Real.sqrt_nonneg m]` — the new proof with sqrt_pos works for all c ≤ 1 including −1 ✓.\n\nThen vF call: hc = (le_refl 1). Fix.","signature":""},{"type":"text","text":"Correction — `c = 1` needs `≤`: for `n > 0`, `√n + 1 > 1` strictly, so `hc : c ≤ 1` suffices. Fixing."},{"type":"tool_use","id":"Edit_39","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean","old_string":"/-- [PROVED session 011; generalized session 012 to any c < 1]\nLocalized variant of `chain_rule_sqrt_shift`: φ only needs to be C² on\n(c, ∞) with c < 1 (all D-057 blocks: L1-singularities at T = −2,\nL2-singularities at T = −1, so c = −1 suffices for them; the explicit\nTrudgian majorant FAlt of session 012 has log-singularities at 0 and 1,\nso it uses c = 1). -/\ntheorem chain_rule_sqrt_shift_on (φ : ℝ → ℝ) {c : ℝ} (hc : c < 1)\n    (hφ : ContDiffOn ℝ 2 φ (Set.Ioi c)) (n : ℝ) (hn : 0 < n) :\n    deriv (deriv fun n => φ (Real.sqrt n + 1)) n =\n      (deriv (deriv φ) (Real.sqrt n + 1) -\n        deriv φ (Real.sqrt n + 1) / Real.sqrt n) / (4 * n) := by\n  have hφd : DifferentiableOn ℝ φ (Set.Ioi c) :=\n    hφ.differentiableOn (by norm_num)\n  have hφdd : DifferentiableOn ℝ (deriv φ) (Set.Ioi c) :=\n    (hφ.deriv_of_isOpen (m := 1) isOpen_Ioi (by norm_num)).differentiableOn\n      one_ne_zero\n  have hmem : ∀ m : ℝ, 0 < m → Set.Ioi c ∈ 𝓝 (Real.sqrt m + 1) := by\n    intro m hm\n    exact Ioi_mem_nhds (by linarith [Real.sqrt_nonneg m])","new_string":"/-- [PROVED session 011; generalized session 012 to any c ≤ 1]\nLocalized variant of `chain_rule_sqrt_shift`: φ only needs to be C² on\n(c, ∞) with c ≤ 1 (all D-057 blocks: L1-singularities at T = −2,\nL2-singularities at T = −1, so c = −1 suffices for them; the explicit\nTrudgian majorant FAlt of session 012 has log-singularities at 0 and 1,\nso it uses c = 1). -/\ntheorem chain_rule_sqrt_shift_on (φ : ℝ → ℝ) {c : ℝ} (hc : c ≤ 1)\n    (hφ : ContDiffOn ℝ 2 φ (Set.Ioi c)) (n : ℝ) (hn : 0 < n) :\n    deriv (deriv fun n => φ (Real.sqrt n + 1)) n =\n      (deriv (deriv φ) (Real.sqrt n + 1) -\n        deriv φ (Real.sqrt n + 1) / Real.sqrt n) / (4 * n) := by\n  have hφd : DifferentiableOn ℝ φ (Set.Ioi c) :=\n    hφ.differentiableOn (by norm_num)\n  have hφdd : DifferentiableOn ℝ (deriv φ) (Set.Ioi c) :=\n    (hφ.deriv_of_isOpen (m := 1) isOpen_Ioi (by norm_num)).differentiableOn\n      one_ne_zero\n  have hmem : ∀ m : ℝ, 0 < m → Set.Ioi c ∈ 𝓝 (Real.sqrt m + 1) := by\n    intro m hm\n    have h1 : (1 : ℝ) < Real.sqrt m + 1 := by\n      have h := Real.sqrt_pos.mpr hm\n      linarith\n    exact Ioi_mem_nhds (lt_of_le_of_lt hc h1)"},"extra_content":null}]