[{"type":"thinking","thinking":"The user wants to continue with 6-c, starting with 6-c-ii. Per STATE.md, the recipe for 6-c-ii is banked:\n\n**6-c-ii RECIPE (I_III, left vertical σ = −5/2; target statement):**\n`vIntegral_Jlag_left_norm_le (hZC) (hZC2) (hNLZ) (hMZ : MZeroProp) {n} (hn : 100000 ≤ n) {T} (hTn : √n ≤ T) (hTn1 : T ≤ √n+1) : ‖VIntegral (Jlag n) (-5/2) (-T) T‖ ≤ 4*T*((2/3)*Z2hat T + 10.5*log(T+2) + 48) + 40*1.3352`.\n\nSteps:\n1. VIntegral_norm + norm_integral_le_integral_norm (−T ≤ T): ‖·‖ ≤ ∫_{−T}^{T} g, g t := ‖Jlag n (↑(-5/2:ℝ)+↑t·I)‖. Integrability of g: Jlag continuous on the path (kFun_differentiableAt (s≠1 via re=−5/2) .continuousAt + my continuousAt_logDeriv_zeta (zeta_ne_zero_left_vertical) → .mul/.neg) composed with the path map u ↦ ↑x+↑u·I (fun_prop) → .norm → ContinuousOn.intervalIntegrable.\n2. Split ∫_{−T}^{T} = ∫_{−T}^{−10} + ∫_{−10}^{10} + ∫_{10}^{T} (integral_add_adjacent_intervals ×2). The left piece: ∫_{−T}^{−10} g = ∫_{10}^{T} g — g is EVEN (jlag_left_vertical_norm_even) + integral_comp_neg (∫_a^b f(−x) = ∫_{−b}^{−a} f).\n3. Middle: per t ∈ Icc(−10,10): g t ≤ 2·1.3352 (Jlag_norm_eq + kFun_norm_le_two (re ≤ 1/2) + hMZ t (|t| ≤ 10); CAST BRIDGE: MZeroProp's point is (-5/2:ℂ)+↑t·I, mine is ↑(-5/2:ℝ)+↑t·I — `(-5/2:ℂ) = ↑(-5/2:ℝ)` by push_cast; ring). Then integral_mono_on + integral_const: ≤ 20·(2·1.3352) = 40·1.3352.\n4. Tails: per t ∈ Icc(10,T): g t ≤ K := 2*((2/3)*Z2hat T + 10.5*log(T+2) + 48). Via Jlag_norm_eq + kFun_norm_le_two + ‖logDeriv‖ ≤ ‖logDeriv − Σnear‖ + ‖Σnear‖:\n   - capstone logDeriv_riemannZeta_sub_nearZeros_le_tight hZC hZC2 hNLZ (ht : 10 ≤ t) (hσlo/hσhi : re = −5/2, by simp) (him : s.im = t, by simp) (hζ : zeta_ne_zero_left_vertical t) ⟹ ‖·−Σ‖ ≤ 10.5·log(|t|+2)+48 = 10.5·log(t+2)+48 (abs_of_nonneg) ≤ 10.5·log(T+2)+48 (Real.log_le_log, t ≤ T).\n   - ‖Σnear‖ ≤ Σ ‖mult/(s−ρ)‖ ≤ (2/5)·Σ mult ≤ (2/5)·Z2hat t ≤ (2/3)·Z2hat T: per ρ: ‖s−ρ‖ ≥ |re(s−ρ)| = 5/2+ρ.re ≥ 5/2 (abs_re_le_norm; re(s−ρ) = −5/2−ρ.re, abs_of_neg) ⟹ 1/‖s−ρ‖ ≤ 2/5; Σ mult ≤ Z2hat t (nearZerosSub_mult_sum_le hZC2); Z2hat_mono (1 ≤ t ≤ T) + 2/5 ≤ 2/3 + Z2hat T ≥ 0.\n   Then integral_mono_on + integral_const: ∫_{10}^{T} g ≤ (T−10)·K.\n5. Assemble: ∫ ≤ 2·(T−10)·K + 40·1.3352 ≤ 2T·K + 40·1.3352 = 4T·((2/3)Z2hat T + 10.5 log(T+2) + 48) + 40·1.3352 (K ≥ 0 since Z2hat T ≥ 0 for T ≥ 1: log(T+2) ≥ 1, loglog ≥ 0).\n\nFirst, I need to look at the existing LiSideBounds.lean to see what scaffolding exists (Z2hat_mono, nearZerosSub_mult_sum_le, continuousAt/On_logDeriv_zeta, Jlag_norm_eq, zeta_ne_zero_left_vertical, zeta_ne_zero_horizontal, rectangleIntegral_Jlag_eq, VIntegral_norm, HIntegral_split, jlag_left_vertical_norm_even).\n\nLet me set up a task list and read the file. Let me first read LiSideBounds.lean to see what was built in 6-c-i.\n\nThis is a big Lean task. Let me plan:\n1. Read LiSideBounds.lean (the 6-c-i scaffolding).\n2. Check the exact signatures of key lemmas: VIntegral_norm, Jlag_norm_eq, kFun_norm_le_two, zeta_ne_zero_left_vertical, jlag_left_vertical_norm_even, nearZerosSub_mult_sum_le, Z2hat_mono, logDeriv_riemannZeta_sub_nearZeros_le_tight, MZeroProp def.\n3. Write the 6-c-ii proof per the recipe.\n4. Build it with lake env lean on the single file, then lake build the module.\n5. Commit per sub-stage protocol.\n\nLet me start. I should use TodoWrite for the task tracking. Then read the file.\n\nThe file is at research/lean/NguyenRH1/NguyenRH1/LiSideBounds.lean presumably. Let me read it.","signature":""},{"type":"text","text":"Resuming ii′-6-c. Setting up the sub-stage plan and reading the 6-c-i scaffolding first."},{"type":"tool_use","id":"TodoWrite_2","name":"TodoWrite","input":{"todos":[{"content":"6-c-ii: I_III left-vertical bound vIntegral_Jlag_left_norm_le","status":"in_progress","activeForm":"Proving 6-c-ii I_III left-vertical bound"},{"content":"6-c-iii: I_I right-vertical bound (F-E form)","status":"pending","activeForm":"Proving 6-c-iii I_I bound"},{"content":"6-c-iv: mid-horizontal segment machine","status":"pending","activeForm":"Proving 6-c-iv mid-horizontal machine"},{"content":"6-c-v: horizontal assembly (mid + DL-5 tails)","status":"pending","activeForm":"Assembling 6-c-v horizontal bound"},{"content":"6-d: fold ⟹ sfBoundProp (then commit, update archive)","status":"pending","activeForm":"Proving 6-d fold to sfBoundProp"}]},"extra_content":null},{"type":"tool_use","id":"Bash_3","name":"Bash","input":{"command":"ls \"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\\" && wc -l \"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean\"","description":"List Lean package files and LiSideBounds line count"},"extra_content":null}]