[{"type":"thinking","thinking":"Audit CLEAN: all 13 decls, 3 standard axioms only, 0 sorryAx.\n\nNow commit 6-c-ii per the crash-safe sub-stage protocol. Files to stage:\n- research/lean/NguyenRH1/NguyenRH1/LiSideBounds.lean (6-c-ii)\n- research/lean/NguyenRH1/NguyenRH1.lean (root import)\n- research/lean/NguyenRH1/AuditSession041.lean\n- research/lean/NguyenRH1/audit-session041-axioms.txt\n\nNOT the correspondence files (user's own work) or Scratch.lean (user's scratch) or the paper build artifacts.\n\nGit identity: pass per-command (-c user.name=\"NGUYEN-RH-1\" -c user.email=\"research@local\").\n\nCommit message style from history: \"ii′-6-c-i: LiSideBounds.lean scaffolding GREEN (build 8770 jobs)\". So: \"ii′-6-c-ii: I_III left-vertical bound vIntegral_Jlag_left_norm_le PROVED (build 8774 jobs, audit clean)\". Include key details: the parse-capture gotcha banked, T ≥ 316, even symmetry, MZeroProp short piece.\n\nThen continue to 6-c-iii (I_I, F-E form). Per the todo list, after committing, proceed to 6-c-iii.\n\nActually — the session protocol says commit per sub-stage. The user asked to \"continue with the rest of the 6-c, starting with 6-c-ii\" — so I should proceed through 6-c-iii, 6-c-iv, 6-c-v (and 6-d if all goes well). This is a long stretch of Lean work. Let me commit 6-c-ii now, then proceed to 6-c-iii.","signature":""},{"type":"text","text":"Audit clean — 13 decls, 3 standard axioms, 0 sorryAx. Committing 6-c-ii per the sub-stage protocol."},{"type":"tool_use","id":"Bash_68","name":"Bash","input":{"command":"cd \"D:/Python/Quantum\" && git add research/lean/NguyenRH1/NguyenRH1/LiSideBounds.lean research/lean/NguyenRH1/NguyenRH1.lean research/lean/NguyenRH1/AuditSession041.lean research/lean/NguyenRH1/audit-session041-axioms.txt && git -c user.name=\"NGUYEN-RH-1\" -c user.email=\"research@local\" commit -m \"$(cat <<'EOF'\nii′-6-c-ii: I_III left-vertical bound vIntegral_Jlag_left_norm_le PROVED (build 8774 jobs, audit clean)\n\nLiSideBounds.lean §2: ‖VIntegral (Jlag n) (-5/2) (-T) T‖ ≤\n4*T*((2/3)*Z2hat T + 10.5*log(T+2) + 48) + 40*1.3352 (n ≥ 10⁵ so T ≥ 316).\nKernel ≤ 2 (kFun_norm_le_two); tails t ∈ [10,T] via the 5f tight R-bound\ncapstone + the near count ≤ (2/5)·Z₂(t) (‖s−ρ‖ ≥ 5/2 from re = −5/2);\nmiddle |t| ≤ 10 via the perimeter MZeroProp (cast bridge ↑(-5/2:ℝ) =\n(-5/2:ℂ)); left tail = right tail by the even symmetry +\nintervalIntegral.integral_comp_neg. Root-imported (NguyenRH1.lean);\nAuditSession041: 13 decls (§1 scaffolding + §2), 3 standard axioms only,\n0 sorryAx.\n\nBanked gotchas: `∫ x in a..b, body` captures trailing `+`/`-` terms into\nthe integrand (precedence) — parenthesize every integral summand;\nContinuousAt.comp needs (g :=) (f :=) (x :=) pinned (arg-first\nelaboration misdecomposes the point); norm_le_insert' : ‖a‖ ≤ ‖b‖+‖a−b‖\n(norm_le_insert has the args swapped); rw [hre] closes `-5/2 ≤ -5/2` by\nauto-rfl le_refl (le_of_eq hre.symm is deterministic); integral_comp_neg\ntakes the integrand f EXPLICITLY; Continuous.norm' instance mismatch —\nuse continuous_norm.comp.\nEOF\n)\" && git log --oneline -1","description":"Commit 6-c-ii with audit trail"},"extra_content":null}]