[{"type":"thinking","thinking":"`sInf_partial_lower` compiles. Remaining pieces:\n- Piece 4: hB — oddHarm lower bound (as a standalone theorem).\n- Piece 5: the main assembly in sInfty_lower_bound.\n\nPiece 4:\n```lean\n/-- [PROVED session 011] oddHarm lower bound from the harmonic bounds:\n`oddHarm K ≥ log(2K+1) − ½log(K+1) + γ/2 − 1`. -/\ntheorem oddHarm_lower (K : ℕ) :\n    Real.log (2 * (K : ℝ) + 1) - Real.log ((K : ℝ) + 1) / 2 +\n        Real.eulerMascheroniConstant / 2 - 1 ≤ oddHarm K := by\n  rw [oddHarm_eq K]\n  have hbl := harmonic_lower (2 * K + 1) (by omega)\n  have hbu := harmonic_upper K\n  have hc : ((2 * K + 1 : ℕ) : ℝ) = 2 * (K : ℝ) + 1 := by push_cast; ring\n  rw [hc] at hbl\n  linarith\n```\nWait harmonic_lower (2K+1): log ↑(2K+1) + γ ≤ ↑(harmonic (2K+1)). rw [hc] at hbl → log (2↑K+1) + γ ≤ ... ✓. hbu : ↑(harmonic K) ≤ log (↑K+1) + γ ✓ matches goal's form ((K:ℝ)+1) ✓. linarith: goal: log(2K+1) − log(K+1)/2 + γ/2 − 1 ≤ H2K+1 − HK/2 − 1. From hbl: H2K+1 ≥ log(2K+1) + γ. From hbu: −HK/2 ≥ −(log(K+1)+γ)/2. Sum ✓ linarith.\n\nPiece 5: main theorem. Structure:\n\n```lean\ntheorem sInfty_lower_bound (n : ℕ) (hn : 20 ≤ n) : ... := by\n  have hn1 : 1 ≤ n := by omega\n  -- key: for every K, the bound reduces to a K-specialized real inequality\n  have hmain : ∀ K : ℕ,\n      (1 / 2) * (n : ℝ) * Real.log n +\n          (lagariasC1 - Real.eulerMascheroniConstant / 2) * n - 4 ≤\n        (1 - lagariasA1) * (n : ℝ) - 1 +\n          ((n : ℝ) * (Real.log (2 * (K : ℝ) + 1) - Real.log ((K : ℝ) + 1) / 2 +\n              Real.eulerMascheroniConstant / 2 - 1) - (K : ℝ)) →\n      (1 / 2) * (n : ℝ) * Real.log n +\n          (lagariasC1 - Real.eulerMascheroniConstant / 2) * n - 4 ≤\n        SInfty n := by\n    intro K hK\n    have h1 := sInf_partial_lower n K\n    have h2 := oddHarm_lower K\n    have h3 : (n : ℝ) * (Real.log (2 * (K : ℝ) + 1) - Real.log ((K : ℝ) + 1) / 2 +\n        Real.eulerMascheroniConstant / 2 - 1) ≤ (n : ℝ) * oddHarm K :=\n      mul_le_mul_of_nonneg_left h2 (by positivity)\n    have h4 : SInfty n = (1 - lagariasA1) * (n : ℝ) - 1 +\n        ∑' m : {m : ℕ // Odd m ∧ 3 ≤ m},\n          ((1 - 1 / (m : ℝ)) ^ n - 1 + (n : ℝ) / (m : ℝ)) := rfl\n    rw [h4]\n    linarith [h1, h3]\n  -- log identities\n  have h4pi : Real.log (4 * Real.pi) = 2 * Real.log 2 + Real.log Real.pi := by\n    rw [Real.log_mul (by norm_num : (4:ℝ) ≠ 0) Real.pi_ne_zero]\n    have e : Real.log (4:ℝ) = 2 * Real.log 2 := by\n      rw [show (4:ℝ) = 2^2 by norm_num, Real.log_pow]\n      norm_num\n    linarith [e]  -- hmm, log(4*π) = log 4 + log π; need = 2log2 + log π: rw [e]?? \n```\nHmm: after rw [log_mul], goal: log 4 + log π = 2 log 2 + log π. rw [e]?? e : log 4 = 2 log 2 — wait e is stated as Real.log 4 = 2 * Real.log 2 ✓ rw [e] → 2 log2 + log π = 2 log2 + log π → closed by rfl (rw auto-rfl) ✓. But careful `Real.log (4:ℝ)` — the def lagariasA1 has Real.log (4 * Real.pi); after log_mul → Real.log 4 + Real.log Real.pi ✓.\n\nReal.log_pow : Real.log (x^p) = p * Real.log x where p : ℕ cast: `Real.log_pow (x : ℝ) (n : ℕ) : log (x^n) = n * log x` ✓. So log(2^2) = ↑2 * log 2; norm_num turns ↑2 into 2 ✓.\n\nReal.pi_ne_zero : Real.pi ≠ 0 ✓ exists.\n\nh2pi similar.\n\nThen case split:\n```lean\n  rcases Nat.even_or_odd n with ⟨q, rfl⟩ | ⟨q, rfl⟩\n  · -- n = q + q\n    have hq10 : 10 ≤ q := by omega\n    have hQ : (10 : ℝ) ≤ (q : ℝ) := by exact_mod_cast hq10\n    have hq0 : (0 : ℝ) < (q : ℝ) := by linarith\n    refine hmain (q - 1) ?_\n    -- cast facts\n    have hK1 : 2 * ((q - 1 : ℕ) : ℝ) + 1 = 2 * (q : ℝ) - 1 := by\n      rw [Nat.cast_sub (by omega : 1 ≤ q), Nat.cast_one]; ring\n    have hK2 : ((q - 1 : ℕ) : ℝ) + 1 = (q : ℝ) := by\n      rw [Nat.cast_sub (by omega : 1 ≤ q), Nat.cast_one]; ring\n    have hK3 : ((q - 1 : ℕ) : ℝ) = (q : ℝ) - 1 := by\n      rw [Nat.cast_sub (by omega : 1 ≤ q), Nat.cast_one]\n    have hN : ((q + q : ℕ) : ℝ) = 2 * (q : ℝ) := by push_cast; ring\n    have hlogq : Real.log (q : ℝ) = Real.log (2 * (q : ℝ)) - Real.log 2 := by\n      have h := Real.log_mul (by norm_num : (2:ℝ) ≠ 0) hq0.ne'\n      linarith [h]\n    rw [hN, hK1, hK2, hK3, hlogq]\n```\nWAIT — the goal after refine hmain (q−1): `1/2*↑(q+q)*log ↑(q+q) + (C1−γ/2)*↑(q+q) − 4 ≤ (1−a1)*↑(q+q) − 1 + (↑(q+q)*(log(2*↑(q−1)+1) − log(↑(q−1)+1)/2 + γ/2 − 1) − ↑(q−1))`.\n\nhK2 : ↑(q−1) + 1 = ↑q — the goal has `Real.log (↑(q−1) + 1)` (from hmain's `Real.log ((K:ℝ)+1)` with K := q−1: `Real.log (↑(q-1)+1)`) ✓ rw [hK2] → log ↑q. Then hlogq: log ↑q → log(2↑q) − log2 ✓. hK1: 2*↑(q−1)+1 → 2↑q−1 ✓. hK3: ↑(q−1) → ↑q − 1 ✓. hN: ↑(q+q) → 2↑q ✓.\n\nOrder: rw [hN, hK1, hK2, hK3, hlogq] — after hK2, log ↑q appears; hlogq rewrites it ✓. But hlogq's LHS is `Real.log ↑q` — also `Real.log 2` appears... fine.\n\nThen unfold the defs and substitute log identities:\n```lean\n    unfold lagariasA1 lagariasC1\n    rw [h4pi, h2pi]\n```\nHmm — unfold inside the goal which contains both. Then:\n\n```lean\n    -- the γ, log π, log 2 pieces cancel exactly (D-037 §3 assembly)\n    have hE : (1 - (2 * Real.log 2 + Real.log Real.pi + Real.eulerMascheroniConstant) / 2) * (2 * (q:ℝ))\n        - 1 + (2 * (q:ℝ)) * (Real.log (2 * (q:ℝ) - 1) -\n          (Real.log (2 * (q:ℝ)) - Real.log 2) / 2 + Real.eulerMascheroniConstant / 2 - 1) -\n        ((q:ℝ) - 1) - ((1 / 2) * (2 * (q:ℝ)) * Real.log (2 * (q:ℝ)) +\n          ((Real.eulerMascheroniConstant - 1 - (Real.log 2 + Real.log Real.pi)) / 2 -\n            Real.eulerMascheroniConstant / 2) * (2 * (q:ℝ)) - 4) =\n        (2 * (q:ℝ)) * (Real.log (2 * (q:ℝ) - 1) - Real.log (2 * (q:ℝ))) + 4 := by\n      ring\n```\nThis is fragile — predicting the exact post-rewrite goal shape. Instead of hE on the whole inequality, move everything to one side via `suffices`? Or: use `le_sub_iff_add_le`-style normalization: Actually cleaner: state the goal as `LHS ≤ RHS`; apply `sub_nonneg.mp`? Let me do:\n\n```lean\n    rw [le_sub_iff]?? \n```\nSimplest robust: use linear arithmetic with the ring identity PROVIDED to linarith as a hypothesis — linarith doesn't do ring normalization of atoms... but `nlinarith`? The identity involves only linear combinations of atoms — linarith CAN cancel γ, log π, log 2 if I give it the needed atom-level facts. But the PRODUCT 2q*(log ...) — treat the products as atoms: h2q1 : (2*↑q)*(Real.log (2*↑q−1) − Real.log (2*↑q)) ≤ ... hmm products of 2↑q with log terms appear on both sides identically — as long as the SAME product terms appear, linarith treats them as atoms and cancels ✓.\n\nPlan: after all rewrites, goal: A ≤ B where A, B are linear combos of atoms {2↑q, ↑(q)... let me enumerate: variables: (↑q : ℝ) — hmm after hN, ↑(q+q) replaced by 2*↑q — but \"2*↑q\" is itself a product, not an atom! linarith normalizes polynomial arithmetic: 2*↑q is linear in atom ↑q ✓. Products like (2*↑q) * Real.log(...): linarith treats Real.log(...) as atom X, so 2*↑q*X is a PRODUCT of two atoms — linarith can't multiply BUT if the identical product 2*↑q*X appears on both sides, linarith's preprocessing treats each nonlinear monomial as a fresh atom consistently ✓ linarith does handle nonlinear monomials as atoms (it uses `ring_nf` normalization and abstracts nonlinear monomials). YES — linarith can close goals where nonlinear terms cancel identically.\n\nSo final step: provide the log-comparison fact in matching monomial form and call linarith... but the needed final inequality: 2↑q*(log(2↑q) − log(2↑q−1)) ≤ 2 (a nonlinear bound derived from hlogb1 by multiplication) — provide hlogb2 as I planned (proved via mul_le_mul_of_nonneg_left with explicit forms), then the goal reduces to linear-in-atoms: check: goal A ≤ B; B − A = 2↑q*(log(2↑q−1) − log(2↑q)) + 4 (computed earlier) — as monomial atoms: −2↑q*(log(2↑q) − log(2↑q−1)) + 4 ≥ 0 ⟸ hlogb2 : 2↑q*(log(2↑q) − log(2↑q−1)) ≤ 2 — but the monomials must MATCH: (2↑q)*(log(2↑q−1) − log(2↑q)) vs (2↑q)*(log(2↑q) − log(2↑q−1)) — as polynomials in atoms {↑q, log(2↑q), log(2↑q−1)}: first = 2↑q·L1 − 2↑q·L2, second = 2↑q·L2 − 2↑q·L1 — linarith's ring_nf normalizes both to monomial combos ✓ compatible.\n\nBut CAREFUL: will linarith's ring_nf see through the γ/log2/logπ/log(2↑q) atoms and the subtraction structure? Yes — atoms are non-algebraic subterms; ring_nf handles the rest.\n\nThe catch: hlogb2's statement must be in terms of atoms matching: Real.log (2*(q:ℝ)) and Real.log (2*(q:ℝ)−1) — these appear in the goal only if my rewrites produced exactly these arguments: goal has Real.log (2*↑q−1) ✓ (from hK1) and Real.log (2*↑q) (from hN inside log ↑(q+q) ✓). ✓\n\nSo hlogb2:\n```lean\n    have hpos : (0 : ℝ) < 2 * (q : ℝ) - 1 := by linarith\n    have hlogb1 : Real.log (2 * (q : ℝ)) - Real.log (2 * (q : ℝ) - 1) ≤\n        1 / (2 * (q : ℝ) - 1) := by\n      have h1 := Real.log_le_sub_one_of_pos\n        (x := (2 * (q : ℝ)) / (2 * (q : ℝ) - 1)) (div_pos (by linarith) hpos)\n      rw [Real.log_div (by linarith : (2:ℝ)*(q:ℝ) ≠ 0) hpos.ne'] at h1\n      have e : (2 * (q : ℝ)) / (2 * (q : ℝ) - 1) - 1 = 1 / (2 * (q : ℝ) - 1) := by\n        field_simp\n      linarith [h1, e]\n    have hlogb2 : (2 * (q : ℝ)) * (Real.log (2 * (q : ℝ)) - Real.log (2 * (q : ℝ) - 1)) ≤ 2 := by\n      have h2 := mul_le_mul_of_nonneg_left hlogb1 (by linarith : (0 : ℝ) ≤ 2 * (q : ℝ))\n      have e : (2 * (q : ℝ)) * (1 / (2 * (q : ℝ) - 1)) ≤ 2 := by\n        rw [mul_one_div, div_le_iff₀ hpos]\n        linarith\n      linarith [h2, e]\n    linarith [hlogb2]\n```\ne's `field_simp` — goal e: (2q)/(2q−1) − 1 = 1/(2q−1): field_simp clears denominators (needs hpos.ne' — field_simp uses hypotheses in context? It looks for ne facts via assumption... field_simp with hpos in context: it tries to discharge side conditions by `positivity`/assumption — hpos.ne' is derivable: provide explicitly: field_simp [hpos.ne']? field_simp takes simp args: field_simp [hpos.ne'] — hpos.ne' : 2↑q−1 ≠ 0 ✓. Then ring. Let me end e with `field_simp [hpos.ne']; ring`... wait field_simp might close or transform to 2q − (2q−1) = 1 → ring ✓. But if field_simp CLOSES it, ring errors no-goals — use `field_simp [hpos.ne']` alone if it closes... I'll write `field_simp [hpos.ne']; ring` and adjust based on error. Hmm, safer idiom: `rw [eq_div_iff hpos.ne']`? e : A/B − 1 = 1/B — messy. Just: have e : ... := by field_simp [hpos.ne']; ring — adjust if needed. Actually simplest bulletproof: \n```lean\n      have e : (2 * (q : ℝ)) / (2 * (q : ℝ) - 1) - 1 = 1 / (2 * (q : ℝ) - 1) := by\n        rw [div_sub' _ _ _ hpos.ne'??]...\n```\nmeh. Try field_simp; iterate.\n\nWait — actually there's subtlety in hlogb1's h1: Real.log_le_sub_one_of_pos gives log(x) ≤ x − 1. x = 2q/(2q−1). rw [Real.log_div] at h1: log(2q) − log(2q−1) ≤ 2q/(2q−1) − 1 ✓. Then linarith [h1, e] ✓.\n\nBUT — div_pos (by linarith : (0:ℝ) < 2*(q:ℝ)) hpos ✓.\n\nNow — the (1/2) * ↑(q+q) * Real.log ↑(q+q) — after rw [hN]: (1/2)*(2*↑q)*log(2*↑q) ✓ atom log(2↑q) ✓ matches hlogb2's atoms ✓.\n\nOne more concern: after `refine hmain (q - 1) ?_`, does the goal's LHS (theorem statement instance) syntactically match what hmain produces? hmain's first hypothesis is `(1/2)*↑n*log ↑n + (lagariasC1 − γ/2)*↑n − 4 ≤ ...` with n replaced by q+q ✓ exactly the goal ✓ refine unifies.\n\nAlso NOTE: hmain refers to theorem-goal LHS — I must state hmain with casts written as in the goal: goal has `(1 / 2) * ↑n * Real.log ↑n` where the statement wrote `(1 / 2) * n * Real.log n` — elaborated with ↑n ✓. In hmain I write `(1 / 2) * (n : ℝ) * Real.log (n : ℝ)` ✓ same.\n\nAnd SInfty n — h4 : SInfty n = (1−a1)*↑n − 1 + ∑' ... := rfl ✓ (def unfold).\n\nIn hmain's proof: rw [h4]; linarith [h1, h3]: goal: LHS ≤ (1−a1)*↑n − 1 + tsum. h1 : ↑n*oddHarm K − ↑K ≤ tsum. h3 : ↑n*(logstuff) ≤ ↑n*oddHarm K. hK (hypothesis): LHS ≤ (1−a1)*↑n − 1 + (↑n*(logstuff) − ↑K). linarith: tsum atom T; h1: A ≤ T where A := ↑n*oddHarm − ↑K; h3: B ≤ ↑n*oddHarm i.e. B − ↑K ≤ A... goal: LHS ≤ (1−a1)n − 1 + T: from hK: LHS ≤ (1−a1)n − 1 + (B − ↑K)... wait hmain's hypothesis has (↑n*(logstuff) − ↑K) and h3 gives ↑n*logstuff ≤ ↑n*oddHarm so (B − K) ≤ (oddHarm·n − K) = A ≤ T ✓ linarith chains ✓.\n\npositivity for 0 ≤ ↑n ✓.\n\nOdd branch:\n```lean\n  · -- n = 2 * q + 1\n    have hq10 : 10 ≤ q := by omega\n    have hQ : (10 : ℝ) ≤ (q : ℝ) := by exact_mod_cast hq10\n    have hq0 : (0 : ℝ) < (q : ℝ) := by linarith\n    refine hmain q ?_\n    have hN : ((2 * q + 1 : ℕ) : ℝ) = 2 * (q : ℝ) + 1 := by push_cast; ring\n    have hK1 : 2 * (q : ℝ) + 1 = 2 * (q : ℝ) + 1 := rfl  -- K = q: log (2*↑q+1) vs log ↑(2q+1)\n```\nHmm — the goal after refine hmain q: terms: log(2*↑q+1), log(↑q+1), ↑(q), and ↑(2q+1) from the theorem LHS / hmain's (1−a1)*↑n. Also hmain's log(2*↑K+1) with K=q: 2*↑q+1. The LHS has log ↑(2q+1). rw [hN] turns ↑(2q+1) → 2↑q+1 everywhere ✓ then log ↑(2q+1) → log(2↑q+1) ✓ matches.\n\nlog(↑q+1): need = log((N+1)/2) = log(2↑q+2) − log 2: hlogq1 : Real.log ((q:ℝ)+1) = Real.log (2*(q:ℝ)+2) − Real.log 2:\n```lean\n    have hlogq1 : Real.log ((q : ℝ) + 1) = Real.log (2 * (q : ℝ) + 2) - Real.log 2 := by\n      have h1 : (2 * (q : ℝ) + 2) = 2 * ((q : ℝ) + 1) := by ring\n      have h2 := Real.log_mul (by norm_num : (2:ℝ) ≠ 0) (by positivity : ((q:ℝ)+1) ≠ 0)\n      -- h2 : log (2*(↑q+1)) = log 2 + log (↑q+1)\n      rw [← h1] at h2  -- h2 : log (2↑q+2) = log 2 + log (↑q+1)\n      linarith [h2]\n```\npositivity for (↑q+1) ≠ 0: positivity proves 0 < x goals; for ≠ goals positivity works too (ne of pos) ✓.\n\nThen final: goal after rw [hN, hlogq1]; unfold; rw [h4pi, h2pi]:\natoms: log(2↑q+1), log(2↑q+2), log 2, log π, γ, ↑q.\nB − A = (N/2)(log N − log(N+1)) + 7/2 with N := 2↑q+1, N+1 := 2↑q+2 ✓ atoms match.\nBound: hlogb2' : ((2*(q:ℝ)+1)/2) * (Real.log (2*(q:ℝ)+2) − Real.log (2*(q:ℝ)+1)) ≤ 1/2:\nfrom log((N+1)/N) ≤ 1/N:\n```lean\n    have hpos : (0 : ℝ) < 2 * (q : ℝ) + 1 := by linarith\n    have hlogb1 : Real.log (2 * (q : ℝ) + 2) - Real.log (2 * (q : ℝ) + 1) ≤\n        1 / (2 * (q : ℝ) + 1) := by\n      have h1 := Real.log_le_sub_one_of_pos\n        (x := (2 * (q : ℝ) + 2) / (2 * (q : ℝ) + 1)) (div_pos (by linarith) hpos)\n      rw [Real.log_div (by linarith : (2:ℝ)*(q:ℝ)+2 ≠ 0) hpos.ne'] at h1\n      have e : (2 * (q : ℝ) + 2) / (2 * (q : ℝ) + 1) - 1 = 1 / (2 * (q : ℝ) + 1) := by\n        field_simp [hpos.ne']\n        ring\n      linarith [h1, e]\n    have hlogb2 : ((2 * (q : ℝ) + 1) / 2) *\n        (Real.log (2 * (q : ℝ) + 2) - Real.log (2 * (q : ℝ) + 1)) ≤ 1 / 2 := by\n      have h2 := mul_le_mul_of_nonneg_left hlogb1 (by positivity : (0:ℝ) ≤ (2*(q:ℝ)+1)/2)\n      have e : ((2 * (q : ℝ) + 1) / 2) * (1 / (2 * (q : ℝ) + 1)) = 1 / 2 := by\n        field_simp [hpos.ne']\n      linarith [h2, e]\n    linarith [hlogb2]\n```\ne: ((2q+1)/2)*(1/(2q+1)) = 1/2: field_simp [hpos.ne'] — clears: (2q+1)*... might close directly; if it leaves goal, ring; add `; ring` — if field_simp closes, ring errors \"no goals\" — grr. Use: `by rw [div_mul_div_comm, mul_one_div, div_div]`? overthinking. Alternative: `by field_simp [hpos.ne']` alone — if it doesn't close, error says unsolved goals; then add ring. I'll write `field_simp [hpos.ne']; ring` and if \"no goals\" remove ring. Even-branch e similar.\n\nHmm — actually a cleaner uniform way for these tiny field identities: `by rw [eq_div_iff hpos.ne']`? For e (even): (2q)/(2q−1) − 1 = 1/(2q−1): rewrite as: have e : (2*(q:ℝ))/(2*(q:ℝ)−1) = 1 + 1/(2*(q:ℝ)−1) := by rw [div_add_one?? no... \n\nKeep field_simp approach, iterate on errors.\n\nOne thing to double check — the odd branch's final linarith: B − A = (N/2)(log N − log(N+1)) + 7/2. hlogb2 : (N/2)(log(N+1) − log N) ≤ 1/2. So B − A ≥ −1/2 + 7/2 = 3 > 0 ✓ linarith closes with monomial atom consistency: B − A expanded in monomials {↑q·log(2↑q+1), ↑q·log(2↑q+2), log(2↑q+1), log(2↑q+2), ↑q, log 2, ...}. Wait — also non-monomial products: (1/2)*↑(2q+1)*log ↑(2q+1) → after hN: (1/2)*(2↑q+1)*log(2↑q+1) — expands to ↑q·log(2↑q+1) + (1/2)log(2↑q+1) ✓ ring_nf handles. And (2↑q+1)*(log(2↑q+1) − (log(2↑q+2)−log2)/2 + γ/2 − 1) — products of (2↑q+1) with atoms ✓ ring_nf normalizes to monomials ✓ linarith consistent ✓.\n\nEven branch check: B − A = 2↑q*(log(2↑q−1) − log(2↑q)) + 4 — earlier computed D = N(log(N−1) − log N) + 4 with N = 2↑q ✓. hlogb2 : 2↑q*(log(2↑q) − log(2↑q−1)) ≤ 2 → B − A ≥ 2 > 0 ✓.\n\nAlso don't forget: `Real.log n` in the goal — after rcases rfl, n := q+q: `Real.log ↑(q+q)` ✓ hN rewrites.\n\n`unfold lagariasA1 lagariasC1` — unfold replaces occurrences of the CONSTANT lagariasA1 with its definition ✓ then rw [h4pi, h2pi] rewrite Real.log (4*π) and Real.log (2*π) ✓.\n\nWait — one problem: hmain's hypothesis and conclusion both contain lagariasA1/lagariasC1 — after refine hmain (q−1) ?_, the remaining goal contains them ✓ unfold works on goal.\n\nAlso h2pi/h4pi must be before rcases (shared) ✓.\n\nAnother check — `Real.log_mul`: signature `Real.log_mul (hx : x ≠ 0) (hy : y ≠ 0) : Real.log (x * y) = Real.log x + Real.log y` ✓.\n\nFor h4pi: rw [Real.log_mul ...] rewrites Real.log (4 * Real.pi) in the GOAL — but h4pi is itself a have with goal `Real.log (4*π) = 2log2 + logπ` — inside its own proof, rw [Real.log_mul] on goal ✓.\n\nNow write piece 4 (oddHarm_lower) and piece 5 (main). Let me write piece 4 first + build, then piece 5 even branch + build, then odd branch + build. Small steps.","signature":""},{"type":"text","text":"`sInf_partial_lower` compiles. Piece 4: the `oddHarm` lower bound from the harmonic bounds."},{"type":"tool_use","id":"Edit_71","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean","old_string":"/-- [SKELETON] Archimedean lower bound (draft §3.1 Step 1):","new_string":"/-- [PROVED session 011] oddHarm lower bound (D-037 §3): from H1/H2,\n`oddHarm K ≥ log(2K+1) − ½log(K+1) + γ/2 − 1`. -/\ntheorem oddHarm_lower (K : ℕ) :\n    Real.log (2 * (K : ℝ) + 1) - Real.log ((K : ℝ) + 1) / 2 +\n        Real.eulerMascheroniConstant / 2 - 1 ≤ oddHarm K := by\n  rw [oddHarm_eq K]\n  have hbl := harmonic_lower (2 * K + 1) (by omega)\n  have hbu := harmonic_upper K\n  have hc : ((2 * K + 1 : ℕ) : ℝ) = 2 * (K : ℝ) + 1 := by push_cast; ring\n  rw [hc] at hbl\n  linarith\n\n/-- [SKELETON] Archimedean lower bound (draft §3.1 Step 1):"},"extra_content":null}]