[{"type":"thinking","thinking":"User picked (a): Lean proofs, starting with sInfty_lower_bound. This is Session 011 now (or continuation of 010? Per protocol, new direction = new session; but the session 010 was just closed and committed. STATE.md says Session 011 next. We're still in the same conversation, but the archive protocol treats work blocks as sessions. I'll proceed as Session 011 and log at the end/breakpoints.)\n\nTask: prove `sInfty_lower_bound` in Lean. Recall the statement:\n\n```lean\ntheorem sInfty_lower_bound (n : ℕ) (hn : 20 ≤ n) :\n    (1 / 2) * n * Real.log n +\n        (lagariasC1 - Real.eulerMascheroniConstant / 2) * n - 4 ≤\n      SInfty n\n```\n\nwith\n```lean\ndef SInfty (n : ℕ) : ℝ :=\n  (1 - lagariasA1) * n - 1 +\n    tsum fun m : {m : ℕ // Odd m ∧ 3 ≤ m} =>\n      ((1 - 1 / (m : ℝ)) ^ n - 1 + (n : ℝ) / (m : ℝ))\n```\n\nThis is a REAL proof task and it's genuinely nontrivial in Lean:\n1. The tsum over odd m ≥ 3 — summability needs proof (term ~ n/(2m²) for large m... actually (1−1/m)^n − 1 + n/m ~ C(n,2)/m² for large m, summable since Σ1/m² converges... but the comparison depends on n; for each fixed n it's fine).\n2. The lower bound derivation: D-037's proof uses Bernoulli's inequality and explicit harmonic bounds. Translating to Lean is real work: split the tsum at M, bound the tail, etc.\n\nLet me recall the actual D-037 proof structure. I haven't read D-037's Step 1 proof in detail. The bound: S_∞(n) ≥ ½n log n + (C₁ − γ/2)n − 4 for n ≥ 20.\n\nProof idea (from the identity): S_∞(n) = (1−a₁)n − 1 + Σ_{m≥3 odd} f_n(m) with f_n(m) = (1−1/m)^n − 1 + n/m ≥ 0 by Bernoulli ((1−1/m)^n ≥ 1 − n/m). So each term is nonnegative! Then S_∞(n) ≥ (1−a₁)n − 1 + Σ_{m≥3 odd, m ≤ M} f_n(m) for any finite cutoff M (terms nonneg, assuming summability so tsum ≥ partial sums). Then choose M ~ something and bound the finite sum below via integral comparison to get ½n log n + ... \n\nThe standard approach: Σ_{m odd ≤ M} n/m = n·(½)(H-type sum over odds) ≈ (n/2)(log M + γ + log 2 − ...)... and the Σ[(1−1/m)^n − 1] part bounded below by −(#terms) etc. The precise D-037 derivation is in the archive — I should read D-037's Step 1 proof to mirror it.\n\nHonest assessment of difficulty in Lean: this is a multi-hour formalization even for an expert — tsum manipulations, summability, finite partial sums over odd numbers, log asymptotics with explicit constants, Bernoulli from Mathlib (`one_add_mul_le_pow`? Mathlib has Bernoulli: `one_sub_...`? There's `Real.one_add_mul_le_pow` (Bernoulli's inequality) — for (1+x)^n ≥ 1+nx with x ≥ −2? Mathlib: `one_add_mul_le_pow : 1 + n * a ≤ (1 + a)^n` for −2 ≤ a? Let me recall: Mathlib has `one_add_mul_le_pow` in Mathlib.Algebra.Order... something: `(1 : R) + n * a ≤ (1 + a) ^ n` requiring −2 ≤ a? There are versions. For our case a = −1/m, need 1 − n/m ≤ (1−1/m)^n — that's Bernoulli with a = −1/m ≥ −2 ✓.\n\nStrategy for this session: rather than attempting the full theorem in one giant leap, break into lemmas:\n- L1: summability of the odd-m family (needed for tsum ≥ partial sums via `sum_le_tsum`).\n- L2: pointwise Bernoulli nonnegativity of each term.\n- L3: for any finite cutoff, SInfty n ≥ (1−a₁)n − 1 + Σ_{odd m ≤ M} term.\n- L4: the finite-sum asymptotic bound (the real analytic content).\n\nAnd per the sub-stage rule (§8.5), each lemma proved = a clean committed state.\n\nAlso — should I prove `chain_rule_sqrt_shift` first instead? It's more mechanical (deriv of φ(√n+1)) — Mathlib has all the chain rule machinery. That might be a gentler first proof, building momentum. The user picked (a) with sInfty first per my description, but I can choose order within tier (i). Actually I recommended sInfty first; but a smarter order: chain_rule_sqrt_shift (quick win, validates our Lean workflow) → then sInfty_lower_bound pieces.\n\nPlan for this work block:\n1. Read D-037 Step 1 proof carefully.\n2. Prove chain_rule_sqrt_shift (warm-up).\n3. Attack sInfty_lower_bound in lemma stages.\n\nLet me set up todos and start. First read D-037's archimedean section.","signature":""},{"type":"text","text":"Going with (a) — Lean proofs. Session 011 begins. I'll work in small, committed stages, starting with a warm-up proof (`chain_rule_sqrt_shift` — mechanical, validates the workflow), then attacking `sInfty_lower_bound` lemma by lemma, mirroring the D-037 proof. First, let me study the archived proof."},{"type":"tool_use","id":"TodoWrite_147","name":"TodoWrite","input":{"todos":[{"content":"Read D-037 Step 1 proof (archimedean bound derivation)","status":"in_progress","activeForm":"Reading D-037 Step 1 proof"},{"content":"Prove chain_rule_sqrt_shift (warm-up, removes 1 sorry)","status":"pending","activeForm":"Proving chain_rule_sqrt_shift"},{"content":"Prove SInfty summability + Bernoulli nonnegativity lemmas","status":"pending","activeForm":"Proving summability and Bernoulli lemmas"},{"content":"Prove finite-cutoff lower bound for SInfty","status":"pending","activeForm":"Proving finite-cutoff bound"},{"content":"Prove finite-sum asymptotic estimate (main analytic content)","status":"pending","activeForm":"Proving finite-sum estimate"},{"content":"Assemble sInfty_lower_bound; PrintAxioms check; commit","status":"pending","activeForm":"Assembling and auditing sInfty_lower_bound"},{"content":"Session 011 end protocol","status":"pending","activeForm":"Running session end protocol"}]},"extra_content":null},{"type":"tool_use","id":"Bash_148","name":"Bash","input":{"command":"cd /d/Python/Quantum/research/derivations && grep -n \"Step 1\\|Bernoulli\\|harmonic\\|S_∞\\|SInfty\" D-037-finite-height-li-assembly.md | head -15","description":"Locate Step 1 proof in D-037"},"extra_content":null}]