[{"type":"thinking","thinking":"Interesting — the git log says Session 011 already exists: \"Session 011: chain_rule_sqrt_shift PROVED (first sorry eliminated, 9→8)\". And the file already has chain_rule_sqrt_shift proved. So a previous (crashed?) session already did part of this task. Wait — the git log at conversation start shows commit 53cc122 \"Session 011: chain_rule_sqrt_shift PROVED\". But STATE.md wasn't updated (it still says session 010 complete, next action session 011). Hmm, actually the commit exists, meaning a previous session-011 instance proved chain_rule_sqrt_shift and committed. But STATE.md wasn't updated — possibly crash happened mid-session, or the commit was made before updating STATE.md.\n\nSo current state: 9 skeleton theorems → 8 sorries remain. The user picked (a): Lean proof tier (i). Remaining in tier (i): sInfty_lower_bound and marginTilde_convex.\n\nPlan: work on sInfty_lower_bound first. Let me look at the math. This is a substantial task. Let me think about what's provable in Lean.\n\nsInfty_lower_bound: S_∞(n) ≥ ½ n log n + (C₁ − γ/2) n − 4 for n ≥ 20, where\n\nS_∞(n) = (1−a₁)n − 1 + Σ_{m ≥ 3 odd} [(1−1/m)ⁿ − 1 + n/m]\n\nwith a₁ = (log 4π + γ)/2, C₁ = (γ − 1 − log 2π)/2.\n\nThe proof per D-037 is \"elementary (Bernoulli + explicit harmonic bounds)\". This requires summing the series over odd m ≥ 3. The tsum over {m : ℕ // Odd m ∧ 3 ≤ m}.\n\nThis is a genuinely hard formalization: we need a lower bound on Σ_{m≥3 odd} [(1−1/m)ⁿ − 1 + n/m].\n\nLet me recall the mathematics. The terms: (1−1/m)ⁿ − 1 + n/m ≥ 0 by Bernoulli (since (1−1/m)ⁿ ≥ 1 − n/m). So the sum is nonneg term-by-term. To get a lower bound of ½ n log n + O(n), we need the sum to be large.\n\nStandard approach: bound (1−1/m)ⁿ ≥ e^{−n/m}·(1 − n/m²) or similar... Actually for lower bounds, (1−1/m)ⁿ ≥ e^{−n/m}(1 − n/m²)? Hmm. Let me think of the D-037 derivation. Let me read D-037 to see the exact proof strategy used for sInfty_lower_bound.\n\nAlso marginTilde_convex is much harder (ConvexOn with second derivatives of a complicated function).\n\nThe honest approach: read D-037 §(archimedean lower bound) and D-056/D-057 for the proof details, then plan the Lean proof.\n\nGiven the difficulty, I should enter plan mode? The user already chose option (a), which is well-specified: prove the elementary Lean tier. The task is clear enough — implement proofs. I don't need plan approval; the user picked this directly. I'll proceed but should manage with todos.\n\nLet me first read D-037's Step 1 (archimedean lower bound proof) to get the exact elementary argument.","signature":""},{"type":"text","text":"A previous Session 011 instance already proved `chain_rule_sqrt_shift` (commit `53cc122`, 9→8 sorries) before your crash — that work is committed and safe. Remaining in tier (i): `sInfty_lower_bound` and `marginTilde_convex`.\n\nLet me pull up the underlying mathematics from D-037/D-057 to plan the proofs."},{"type":"tool_use","id":"Grep_5","name":"Grep","input":{"pattern":"S_∞|sInfty|lower bound|Step 1","path":"D:\\Python\\Quantum\\research\\derivations\\D-037-finite-height-li.md","output_mode":"content","-n":true,"head_limit":40},"extra_content":null}]