[{"type":"thinking","thinking":"Block L2 compiles (formula + sign). Commit progress, then batch 2: L1², L1·L2, L2².\n\nBlock L1² (φ = (log(t+2))²):\n- D1 t = 2·L1·(t+2)⁻¹: HasDerivAt (fun t => (log(t+2))^2) (2*L1*(T+2)⁻¹) T: use `.pow 2`: HasDerivAt.pow : HasDerivAt f f' x → HasDerivAt (fun y => f y ^ 2) (2 * f x * f') x? For nat powers: `HasDerivAt.pow (n : ℕ) (hf : HasDerivAt f f' x) : HasDerivAt (fun y => f y ^ n) (n * f x ^ (n-1) * f') x` — for n=2: 2 * (L1)^1 * (T+2)⁻¹ ✓.\n- D2: d/dT [2·L1·(T+2)⁻¹] = 2[(T+2)⁻¹·(T+2)⁻¹ + L1·(−(T+2)⁻²)] = 2(1 − L1)/(T+2)². Via (const_mul 2).mul: u = log(t+2) [u' = (T+2)⁻¹], v = (t+2)⁻¹ [v' = −1/(T+2)²]: (u*v)' = u'v + uv'; then const_mul 2. D2 value: 2*((T+2)⁻¹*(T+2)⁻¹ + L1*(-1/(T+2)^2)) = 2(1−L1)/(T+2)² — massage with field_simp/ring.\n- ψ = 2(1−L1)/(T+2)² − 2L1(T+2)⁻¹/(T−1) → closed: 2(1−L1)/(T+2)² − 2L1/((T+2)(T−1)) ✓ field_simp.\n- Sign: L1 > 1 (from L1 ≥ 2): 1−L1 < 0 → first term < 0; second: 2L1/((T+2)(T−1)) > 0 → − < 0 ✓.\n\nBlock L1·L2 (φ = L1·L2):\n- D1 t = L2·(t+2)⁻¹ + L1·((t+2)·L1(t))⁻¹ — hmm L2's deriv: d/dt log(log(t+2)) = ((t+2)L1)⁻¹ — wait as computed: L2′ = (L1)⁻¹·(t+2)⁻¹ = ((t+2)·L1)⁻¹. D1 = L2·(t+2)⁻¹ + L1·((t+2)L1)⁻¹ = L2/(t+2) + 1/(t+2) = (L2+1)/(t+2) — massage.\n  Product rule: u = L1 (u' = (t+2)⁻¹), v = L2 (v' = ((t+2)·L1)⁻¹): D1 = u'·v + u·v' = (t+2)⁻¹·L2 + L1·((t+2)L1)⁻¹.\n- D2 = d/dT [(L2+1)·(T+2)⁻¹]: w = L2+1 (w' = ((T+2)L1)⁻¹), z = (T+2)⁻¹ (z' = −1/(T+2)²): D2 = w'z + wz' = ((T+2)L1)⁻¹(T+2)⁻¹ − (L2+1)/(T+2)² = [1/L1 − L2 − 1]/(T+2)² ✓ massage.\n  BUT D1 must be massaged to (L2+1)/(t+2) form FIRST so that D2's differentiation sees that function: deriv2_of_hasDerivAt_nhds takes D1 as a FUNCTION — I choose D1 := fun t => (Real.log (Real.log (t+2)) + 1) * (t+2)⁻¹ — then h1 (eventual HasDerivAt φ (D1 x)) needs the per-point product-rule + massage at every x — doable: inside filter_upwards, build the product rule at x and massage with field_simp there (x+2 ≠ 0, L1(x) ≠ 0 available from hx) ✓.\n- ψ: [1/L1 − L2 − 1]/(T+2)² − (L2+1)/((T+2)(T−1)).\n- Sign: 1/L1 ≤ 1/2 (L1 ≥ 2) < 1 + L2 (L2 ≥ 0.6931 → 1+L2 ≥ 1.6931): bracket 1/L1 − L2 − 1 ≤ 0.5 − 1.693 < 0 ✓; second term > 0 ✓. Needs L1 ≥ 2 and L2 ≥ 0.6931 — pass hypotheses (hL1 : 2 ≤ L1) (hL2 : 0.6931 ≤ L2) or take hT : √20+1 ≤ T and derive inside via a shared bounds lemma. I'll do the shared bounds lemma (block_range_bounds) and use it in all sign lemmas for blocks 5-9; blocks 10/11/2 already done with 1 < T only. Actually for uniformity, sign lemmas for L1², L1L2, L2² take {T} (hT : Real.sqrt 20 + 1 ≤ T) and use block_range_bounds.\n\nBlock L2² (φ = L2²):\n- D1 = 2·L2·((t+2)L1)⁻¹ (via .pow 2 of L2-block's h1... L2's deriv at t: ((t+2)·L1)⁻¹ form — I have it in (log)⁻¹*(t+2)⁻¹ form; massage).\n- D2 = d/dT [2L2·((T+2)L1)⁻¹]: 2[L2′·((T+2)L1)⁻¹ + L2·(−((T+2)L1)⁻²·((T+2)L1)′)]: ((T+2)L1)′ = L1 + 1 (product rule: (t+2)'·L1 + (t+2)·L1' = L1 + (t+2)(t+2)⁻¹ = L1+1). L2′ = ((T+2)L1)⁻¹. So D2 = 2[((T+2)L1)⁻² − L2(L1+1)((T+2)L1)⁻²] = 2[1 − L2(L1+1)]/((T+2)L1)² ✓.\n- Sign: L2(L1+1) ≥ 0.6931·3 = 2.0793 > 1 ✓ first term < 0; second term > 0 ✓.\n\nThe per-point massage inside filter_upwards for D1 of L1L2 and L2² blocks: for L2²'s D1: fun t => 2·L2(t)·(((t+2)·L1(t))⁻¹) — per-point HasDerivAt: (L2's h1-instance).pow 2 gives HasDerivAt (fun t => (L2 t)^2) (2*(L2 x)^(1) * (L2' x)) x where L2' x = ((x+2)L1x)⁻¹ — but my h1 in the L2 block used D1 form (L1)⁻¹*(x+2)⁻¹ — the .pow multiplies whatever form — then massage to 2·L2·((x+2)L1)⁻¹ form per point via field_simp with the local ne facts ✓.\n\nTo reduce duplication, extract a reusable per-point lemma: hasDerivAt_log_log (x : ℝ) (hx : -1 < x) : HasDerivAt (fun t => Real.log (Real.log (t+2))) ((Real.log (x+2))⁻¹ * (x+2)⁻¹) x — standalone lemma! Then all blocks reuse it. Similarly hasDerivAt_log_shift (x) (hx : -2 < x). Let me refactor: add these two standalone lemmas before the block lemmas (the existing psiOf_log_shift/psiOf_log_log_shift can keep their inline versions or use them — minimal churn: just add the two standalone helpers and use in NEW blocks; leave existing code).\n\nWrite helpers + batch 2 (L1², L1L2, L2²) now. This is a big chunk; write carefully.\n\nFor L1²:\n```lean\ntheorem psiOf_log_sq_shift (T : ℝ) (hT : 1 < T) :\n    psiOf (fun t => Real.log (t + 2) ^ 2) T =\n      2 * (1 - Real.log (T + 2)) / (T + 2) ^ 2 -\n        2 * Real.log (T + 2) / ((T + 2) * (T - 1)) := by\n  have hT2 : (0 : ℝ) < T + 2 := by linarith\n  have h1 : ∀ᶠ x in 𝓝 T, HasDerivAt (fun t => Real.log (t + 2) ^ 2)\n      (2 * Real.log (x + 2) * (x + 2)⁻¹) x := by\n    filter_upwards [eventually_gt_nhds (by linarith : (-2 : ℝ) < T)] with x hx\n    have h := (hasDerivAt_log_shift x (by linarith)).pow 2  -- (log(t+2))^2, deriv 2*L1^(1)*(x+2)⁻¹\n    rwa [pow_one] at h  -- 2 * L1 ^ 1 * (x+2)⁻¹ → 2*L1*(x+2)⁻¹ ✓\n  have h2 : HasDerivAt (fun t => 2 * Real.log (t + 2) * (t + 2)⁻¹)\n      (2 * (1 - Real.log (T + 2)) / (T + 2) ^ 2) T := by\n    have hu := hasDerivAt_log_shift T (by linarith : -2 < T)  -- u' = (T+2)⁻¹\n    have hv : HasDerivAt (fun t => (t + 2)⁻¹) (-1 / (T + 2) ^ 2) T := by\n      have h := ((hasDerivAt_id' T).add_const 2).inv (by linarith : (T + 2) ≠ 0)\n      have e : (-(1 : ℝ)) / (T + 2) ^ 2 = -1 / (T + 2) ^ 2 := by ring\n      rwa [e] at h\n    have h := (hu.mul hv).const_mul 2  -- 2 * (u'v + uv')\n    have hT2' : (T + 2) ≠ 0 := by linarith\n    have e : 2 * ((T + 2)⁻¹ * (T + 2)⁻¹ + Real.log (T + 2) * (-1 / (T + 2) ^ 2)) =\n        2 * (1 - Real.log (T + 2)) / (T + 2) ^ 2 := by\n      field_simp [hT2']; ring\n    rwa [e] at h\n  have hd2 := deriv2_of_hasDerivAt_nhds h1 h2\n  have hd1 : deriv (fun t => Real.log (t + 2) ^ 2) T = 2 * Real.log (T + 2) * (T + 2)⁻¹ :=\n    h1.self_of_nhds.deriv\n  unfold psiOf\n  rw [hd2, hd1]\n  have hT1 : (T - 1) ≠ 0 := by linarith\n  have hT2' : (T + 2) ≠ 0 := by linarith\n  field_simp [hT1, hT2']\n  ring\n```\nHasDerivAt.pow signature: `HasDerivAt.pow (hf : HasDerivAt f f' x) (n : ℕ)?? or (n) first?` — in Mathlib: `theorem HasDerivAt.pow (n : ℕ) (hf : HasDerivAt f f' x) : HasDerivAt (fun y => f y ^ n) ((n : 𝕜) * f x ^ (n - 1) * f') x`?? or protected theorem HasDerivAt.pow (hf) (n)? Order matters. The usage `(hasDerivAt_log_shift x hx).pow 2` — if pow takes n first as explicit, dot notation puts hf in the hf slot ✓ (dot notation fills the first explicit arg of matching type). Derivative: ↑2 * f x ^ (2-1) * f' = 2 * L1^1 * (T+2)⁻¹ — need cast ↑2 and pow 2-1=1: massage: the e-rewrite: 2 * L1^1 * (x+2)⁻¹: rwa [pow_one] at h — h's deriv is (↑(2:ℕ)) * L1^1 * (x+2)⁻¹: cast: Nat.cast 2 = 2 — push_cast? Just: have e : (↑(2:ℕ)) * L1^1 * (x+2)⁻¹ = 2*L1*(x+2)⁻¹ := by norm_num?? hmm — write: rw [pow_one] at h; then h : ... ((2:ℕ):ℝ) * L1 * (x+2)⁻¹ — the cast (↑2): norm_num1? Use: rwa [pow_one, Nat.cast_ofNat] at h? Nat.cast_ofNat : ((OfNat.ofNat n : ℕ) : α) = OfNat.ofNat n ✓ simp lemma. Try rwa [pow_one, Nat.cast_ofNat] at h — or simp only [pow_one, Nat.cast_ofNat] at h; exact h. Also the exponent in HasDerivAt.pow's result: f x ^ (n - 1) with n=2: f x ^ (2-1) = f x ^ 1: 2-1 as ℕ = 1 ✓ defeq; pow_one rewrites L1^1 → L1 ✓.\n\nHmm — actually the shape might be ((n:𝕜)) with n:ℕ cast: (↑2) * L1^(2-1) * f'. 2-1 evaluates to 1 ✓ (kernel computes), so pretty-print may show L1 ^ 1 or L1 — pow_one might be needed ✓ planned.\n\nhasDerivAt_log_shift helper:\n```lean\ntheorem hasDerivAt_log_shift (x : ℝ) (hx : -2 < x) :\n    HasDerivAt (fun t => Real.log (t + 2)) (x + 2)⁻¹ x := by\n  have h := (Real.hasDerivAt_log (show x + 2 ≠ 0 by linarith)).comp x\n    ((hasDerivAt_id' x).add_const 2)\n  rwa [mul_one] at h\n\ntheorem hasDerivAt_log_log (x : ℝ) (hx : -1 < x) :\n    HasDerivAt (fun t => Real.log (Real.log (t + 2)))\n      ((Real.log (x + 2))⁻¹ * (x + 2)⁻¹) x := by\n  have hin := hasDerivAt_log_shift x (by linarith)\n  have hout : HasDerivAt Real.log (Real.log (x + 2))⁻¹ (Real.log (x + 2)) :=\n    Real.hasDerivAt_log (ne_of_gt (Real.log_pos (by linarith)))\n  exact hout.comp x hin\n```\n\nFor L1·L2:\n```lean\ntheorem psiOf_log_mul_log_shift (T : ℝ) (hT : 1 < T) :\n    psiOf (fun t => Real.log (t + 2) * Real.log (Real.log (t + 2))) T =\n      ((Real.log (T + 2))⁻¹ - Real.log (Real.log (T + 2)) - 1) / (T + 2) ^ 2 -\n        (Real.log (Real.log (T + 2)) + 1) / ((T + 2) * (T - 1)) := by\n  have hT2 : (0 : ℝ) < T + 2 := by linarith\n  have hL1 : (0 : ℝ) < Real.log (T + 2) := Real.log_pos (by linarith)\n  have h1 : ∀ᶠ x in 𝓝 T, HasDerivAt\n      (fun t => Real.log (t + 2) * Real.log (Real.log (t + 2)))\n      ((Real.log (Real.log (x + 2)) + 1) * (x + 2)⁻¹) x := by\n    filter_upwards [eventually_gt_nhds (by linarith : (-1 : ℝ) < T)] with x hx\n    have hu := hasDerivAt_log_shift x (by linarith)\n    have hv := hasDerivAt_log_log x hx\n    have h := hu.mul hv\n    have hx2 : (x + 2) ≠ 0 := by linarith\n    have hxL : Real.log (x + 2) ≠ 0 := ne_of_gt (Real.log_pos (by linarith))\n    have e : (x + 2)⁻¹ * Real.log (Real.log (x + 2)) +\n        Real.log (x + 2) * ((Real.log (x + 2))⁻¹ * (x + 2)⁻¹) =\n        (Real.log (Real.log (x + 2)) + 1) * (x + 2)⁻¹ := by\n      field_simp [hx2, hxL]; ring\n    rwa [e] at h\n  have h2 : HasDerivAt (fun t => (Real.log (Real.log (t + 2)) + 1) * (t + 2)⁻¹)\n      (((Real.log (T + 2))⁻¹ - Real.log (Real.log (T + 2)) - 1) / (T + 2) ^ 2) T := by\n    have hw : HasDerivAt (fun t => Real.log (Real.log (t + 2)) + 1)\n        ((Real.log (T + 2))⁻¹ * (T + 2)⁻¹) T :=\n      (hasDerivAt_log_log T (by linarith)).add_const 1\n    have hz : HasDerivAt (fun t => (t + 2)⁻¹) (-1 / (T + 2) ^ 2) T := by\n      have h := ((hasDerivAt_id' T).add_const 2).inv (by linarith : (T + 2) ≠ 0)\n      have e : (-(1 : ℝ)) / (T + 2) ^ 2 = -1 / (T + 2) ^ 2 := by ring\n      rwa [e] at h\n    have h := hw.mul hz\n    have hT2' : (T + 2) ≠ 0 := by linarith\n    have hL1' : Real.log (T + 2) ≠ 0 := ne_of_gt hL1\n    have e : ((Real.log (T + 2))⁻¹ * (T + 2)⁻¹) * (T + 2)⁻¹ +\n        (Real.log (Real.log (T + 2)) + 1) * (-1 / (T + 2) ^ 2) =\n        ((Real.log (T + 2))⁻¹ - Real.log (Real.log (T + 2)) - 1) / (T + 2) ^ 2 := by\n      field_simp [hT2', hL1']; ring\n    rwa [e] at h\n  ...assembly same pattern\n```\nhw's derivative: (hasDerivAt_log_log T _).add_const 1 : HasDerivAt (fun t => L2(t) + 1) ((L1)⁻¹*(T+2)⁻¹) T ✓ (add_const on value ✓ that's the right add_const usage).\n\nFor L2²:\n```lean\ntheorem psiOf_log_log_sq_shift (T : ℝ) (hT : 1 < T) :\n    psiOf (fun t => Real.log (Real.log (t + 2)) ^ 2) T =\n      2 * (1 - Real.log (Real.log (T + 2)) * (Real.log (T + 2) + 1)) /\n          ((T + 2) * Real.log (T + 2)) ^ 2 -\n        2 * Real.log (Real.log (T + 2)) / ((T + 2) * Real.log (T + 2) * (T - 1)) := by\n  h1: per-point: (hasDerivAt_log_log x hx).pow 2 → 2*L2^1*((L1x)⁻¹*(x+2)⁻¹): massage to 2*L2*(((x+2)*L1x)⁻¹)?? — I want D1 in the form fun t => 2 * L2(t) * ((t+2)*L1(t))⁻¹ for D2 differentiation:\n  D1 t := 2 * Real.log (Real.log (t+2)) * ((t + 2) * Real.log (t + 2))⁻¹\n  per-point massage: 2 * L2x ^ 1 * ((L1x)⁻¹ * (x+2)⁻¹) = 2*L2x*((x+2)*L1x)⁻¹: field_simp [hx2, hxL]: ((x+2)*L1)⁻¹ = (x+2)⁻¹*L1⁻¹ (mul_inv) — field_simp normalizes ✓ or by rw [mul_inv]: (L1)⁻¹*(x+2)⁻¹ = ((x+2)*L1)⁻¹?? mul_inv : (a*b)⁻¹ = a⁻¹*b⁻¹ → (x+2)⁻¹ * L1⁻¹ = ((x+2)*L1)⁻¹ by (mul_inv).symm: e : (Real.log (x+2))⁻¹ * (x+2)⁻¹ = ((x+2)*Real.log (x+2))⁻¹ := by rw [mul_inv]; ring?? mul_inv rw gives (x+2)⁻¹ * L1⁻¹ on LHS of mul_inv ((x+2)*L1)⁻¹ = (x+2)⁻¹*L1⁻¹ — so (mul_inv).symm : (x+2)⁻¹*L1⁻¹ = ((x+2)*L1)⁻¹: e := by rw [mul_inv, mul_comm ((x+2)⁻¹) ((Real.log (x+2))⁻¹]]: after rw [mul_inv] the goal RHS ((x+2)*L1)⁻¹ becomes (x+2)⁻¹*L1⁻¹; goal: L1⁻¹*(x+2)⁻¹ = (x+2)⁻¹*L1⁻¹: ring ✓.\n  h2: D1' : w = 2*L2 (w' = 2*((L1)⁻¹*(T+2)⁻¹)), z = ((t+2)*L1)⁻¹: z' = −((t+2)*L1)⁻² * ((t+2)*L1)': ((t+2)*L1)' : product: (id+2).mul (log_shift): 1*L1 + (T+2)*(T+2)⁻¹ = L1 + 1 (massage mul_inv_cancel₀). So z' = −(L1+1)/((T+2)*L1)² (via .inv: HasDerivAt ((t+2)*L1)⁻¹ (−(L1+1)/((T+2)L1)^2) T: .inv of the product-HasDerivAt gives −(L1+1)/((T+2)*L1)^2 ✓ then (fun t => 2*L2 t) — hmm D1 = w * z with w = 2*L2: actually D1 t = 2*L2(t)*z(t): (const_mul 2 of L2).mul z: w' = 2*(L2') = 2*((L1)⁻¹*(T+2)⁻¹): (hasDerivAt_log_log T _).const_mul 2.\n  D2 = w'z + wz' = 2*((L1)⁻¹*(T+2)⁻¹)*((T+2)L1)⁻¹ + 2*L2*(−(L1+1)/((T+2)L1)²)\n     = 2/((T+2)²L1²) − 2L2(L1+1)/((T+2)L1)² = 2[1 − L2(L1+1)]/((T+2)L1)² ✓ massage field_simp.\n```\nThe z (=(t+2)*L1)⁻¹) inverse derivation: \n  hz_inner : HasDerivAt (fun t => (t+2)*Real.log (t+2)) (Real.log (T+2) + 1) T := by\n    have h := ((hasDerivAt_id' T).add_const 2).mul (hasDerivAt_log_shift T (by linarith))\n    -- 1 * L1 + (T+2) * (T+2)⁻¹\n    have e : (1:ℝ) * Real.log (T+2) + (T+2)*(T+2)⁻¹ = Real.log (T+2) + 1 := by\n      rw [mul_inv_cancel₀ (by linarith : (T+2) ≠ 0)]; ring\n    rwa [e] at h\n  hz : HasDerivAt (fun t => ((t+2)*Real.log (t+2))⁻¹) (-(Real.log (T+2)+1)/((T+2)*Real.log (T+2))^2) T :=\n    hz_inner.inv (mul_ne_zero hT2' hL1')\n✓ .inv gives −(L1+1)/((T+2)L1)² directly ✓.\n\nOK — also the sign lemmas:\n\npsiOf_log_sq_shift_neg (hT : √20+1 ≤ T): from block_range_bounds: L1 ≥ 2 > 1: 1−L1 < 0: first term 2(1−L1)/(T+2)² < 0 (div_neg_of_neg_of_pos? (1−L1) < 0, denom > 0: 2*(neg)/pos < 0 ✓); second term > 0: 2L1/((T+2)(T−1)) > 0 → − < 0: sum: first + (−second): e-factor: 2(1−L1)/(T+2)² − 2L1/((T+2)(T−1)) = 2(1−L1)/(T+2)² + (−(2L1/((T+2)(T−1)))) — linarith with: t1 : 2*(1−L1)/(T+2)^2 < 0 (div_neg... prove via: 2*(1−L1) < 0, (T+2)^2 > 0 → div_neg_of_neg_of_pos : a < 0 → 0 < b → a/b < 0 ✓), t2 : 0 < 2*L1/((T+2)*(T−1)) (div_pos), then linarith [t1, t2]?? goal: A − B < 0 with A < 0, B > 0: A − B < 0 ✓ linarith — the earlier linarith failure was atom-mismatch; here t1's A is literally `2*(1−L1)/(T+2)^2` and goal has exactly that ✓ match.\n\npsiOf_log_mul_log_shift_neg: bracket: (L1)⁻¹ − L2 − 1: L1 ≥ 2 → (L1)⁻¹ ≤ 1/2 (inv_le_inv_of_le₀? `one_div_le_one_div_of_le (h : 0 < a) : a ≤ b → 1/b ≤ 1/a` ✓); L2 ≥ 0.6931: bracket ≤ 0.5 − 0.6931 − 1 < 0 → first term < 0 (div_neg_of_neg_of_pos); second > 0 ✓ linarith.\n\npsiOf_log_log_sq_shift_neg: L2(L1+1) ≥ 0.6931·(2+1) = 2.0793 > 1: 1 − L2(L1+1) < 0: first term < 0; second: 2L2/((T+2)L1(T−1)) > 0 ✓ linarith.\n\nblock_range_bounds:\n```lean\ntheorem block_range_bounds {T : ℝ} (hT : Real.sqrt 20 + 1 ≤ T) :\n    (5.4721 : ℝ) ≤ T ∧ 2 ≤ Real.log (T + 2) ∧\n      (0.6931 : ℝ) ≤ Real.log (Real.log (T + 2)) := by\n  have hs : (4.4721 : ℝ) ≤ Real.sqrt 20 := by\n    rw [Real.le_sqrt (by norm_num : (0:ℝ) ≤ 4.4721)]\n    norm_num\n  have hT5 : (5.4721 : ℝ) ≤ T := by linarith\n  have hT2 : (7.4721 : ℝ) ≤ T + 2 := by linarith\n  have hL1 : (2:ℝ) ≤ Real.log (T + 2) := by\n    have he2 : Real.exp 2 ≤ T + 2 := by\n      have hd := Real.exp_one_lt_d9\n      have e : Real.exp 2 = Real.exp 1 * Real.exp 1 := by\n        rw [← Real.exp_add]; norm_num\n      rw [e]\n      calc Real.exp 1 * Real.exp 1 ≤ 2.7182818286 * 2.7182818286 := by\n            apply mul_le_mul (le_of_lt hd) (le_of_lt hd) (by positivity) (Real.exp_pos 1).le\n        _ ≤ 7.4721 := by norm_num\n        _ ≤ T + 2 := hT2\n    have h2 := Real.log_le_log (Real.exp_pos 2) he2\n    rwa [Real.log_exp] at h2\n  have hL2 : (0.6931:ℝ) ≤ Real.log (Real.log (T + 2)) := by\n    have h2 := Real.log_le_log (by norm_num : (0:ℝ) < 2) hL1\n    have h3 := Real.log_two_gt_d9\n    linarith\n  exact ⟨hT5, hL1, hL2⟩\n```\nCheck names: Real.le_sqrt — signature: `Real.le_sqrt (hx : 0 ≤ x) : x ≤ sqrt y ↔ x ^ 2 ≤ y`? or with hy? Might be `Real.le_sqrt (hx : 0 ≤ x) (hy : 0 ≤ y) : x ≤ sqrt y ↔ x ^ 2 ≤ y`. I'll write `rw [Real.le_sqrt (by norm_num)]` and if arity wrong, adjust. Real.log_le_log : `Real.log_le_log (h₀ : 0 < x) (h₁ : x ≤ y) : log x ≤ log y`? Hmm — might be `Real.log_le_log_iff (hy : 0 < y) : log x ≤ log y ↔ x ≤ y` or `Real.log_le_log_of_le`... grep needed. exp_one_lt_d9 ✓ exists (used in EulerMascheroni file). Real.log_two_gt_d9 ✓ exists. Real.exp_add ✓. Real.log_exp ✓.\n\nLet me grep Real.log_le_log quickly before writing.","signature":""},{"type":"text","text":"Block L2 compiles. Quick check of two lemma names I'll need for the bounds lemma, then batch 2."},{"type":"tool_use","id":"Bash_123","name":"Bash","input":{"command":"grep -rn \"theorem log_le_log \\|theorem log_le_log(\" /d/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | head -4; grep -rn \"theorem le_sqrt \\|theorem le_sqrt(\" /d/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/Analysis/SpecialFunctions/Sqrt.lean /d/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/Analysis/SpecialFunctions/Pow/*.lean 2>/dev/null | head -4","description":"Check log_le_log and le_sqrt signatures"},"extra_content":null}]