[{"type":"thinking","thinking":"`IIMaj_deriv2_lt` compiles. All pieces are now in place:\n\n- chain_rule_sqrt_shift_on / chain_rule_sqrt ✓\n- 8 ψ-blocks with formulas + signs ✓ (psiOf_id, psiOf_log_shift(_neg), psiOf_log_log_shift(_neg), psiOf_log_sq_shift(_neg), psiOf_log_mul_log_shift(_neg), psiOf_log_log_sq_shift(_neg), psiOf_T_mul_log_shift(_neg), psiOf_T_mul_log_log_shift(_neg))\n- block_range_bounds ✓\n- IIMaj_eq_psiTilde_sqrt, psiTilde_contDiff, deriv_psiTilde, deriv2_psiTilde, iim_u2_antitone, iimaj_bracket_bound, IIMaj_deriv2_lt ✓\n\nCommit now (crash safety — this was a big chunk), then the FINAL ASSEMBLY: marginTilde_convex.\n\nAssembly plan (the last piece):\nmarginTilde F n decomposition. Let me carefully expand marginTilde F:\n\nmarginTilde F n = S(n) − B̃(n) + 1 with\nS(n) = ½n log n + (C1−γ/2)n − 4\nB̃(n) = 2F(T) + 4T·((2/3)C0hat T + 10.5 L1 + 48) + 40·1.3352 + 2(e⁴+1)(7 Hhat T + 3.3) + IIMaj n + 1\n\nC0hat T = 0.433 L1 + 0.886 L2 + 2.9; Hhat T = 3 C0hat T² + 10.5 L1 + 48.\n\nExpand B̃ into blocks (weights):\n4T(2/3)C0hat: (8/3)(0.433 T·L1 + 0.886 T·L2 + 2.9 T) = 1.154666... T·L1 + 2.362666... T·L2 + 7.73333... T\n4T·10.5·L1 = 42 T·L1; 4T·48 = 192 T.\n2(e⁴+1)·7·Hhat: 14(e⁴+1)·[3 C0hat² + 10.5 L1 + 48] = 42(e⁴+1) C0hat² + 147(e⁴+1) L1 + 672(e⁴+1).\nC0hat² = (0.433 L1 + 0.886 L2 + 2.9)² = 0.187489 L1² + 0.784996 L2² + 8.41 + 2·0.433·0.886 L1L2 + 2·0.433·2.9 L1 + 2·0.886·2.9 L2 = 0.187489 L1² + 0.784996 L2² + 8.41 + 0.767276 L1L2 + 2.5114 L1 + 5.1388 L2.\n42(e⁴+1)·C0hat² = (e⁴+1)·[7.874538 L1² + 32.969832 L2² + 353.22 + 32.225592 L1L2 + 105.4788 L1 + 215.8296 L2].\nSo:\n- T·L1 weight: 1.154666... + 42 = 43.154666...: exact: (8/3)*0.433 = 3.464/3 = 1.15466666...; +42 → 43.15466666...\n- T·L2: (8/3)*0.886 = 7.088/3 = 2.36266666...\n- id(T): (8/3)*2.9 + 192 = 7.733333... + 192 = 199.733333...\n- L1²: (e⁴+1)·7.874538\n- L2²: (e⁴+1)·32.969832\n- L1L2: (e⁴+1)·32.225592\n- L1: (e⁴+1)·(105.4788 + 147) = (e⁴+1)·252.4788\n- L2: (e⁴+1)·215.8296\n- F: 2\n- constants: −4 (in S) + 40·1.3352 + 2(e⁴+1)(353.22... wait — the 8.41·42 = 353.22 inside 42(e⁴+1)C0hat², plus 672(e⁴+1) from 14(e⁴+1)·48, plus 2(e⁴+1)·3.3 = 6.6(e⁴+1), plus 40·1.3352 = 53.408, plus B̃'s +1, and margin's outer +1, and S's −4. All constants → K (don't care).\n\nSo marginTilde F n = [½n log n + (C1−γ/2)n − 4] − 2F(Tn) − Σ wᵢφᵢ(Tn) − IIMaj n − K'\nwhere the bracketed pieces and constants K' are CONSTANT in n.\n\nSecond derivative:\nmargiñ″(n) = 1/(2n) − 2·ψF(Tn)/(4n) − Σ wᵢψᵢ(Tn)/(4n) − IIMaj″(n)\n≥ 1/(2n) − IIMaj″(n) > 0   [since ψF < 0 (hψ), wᵢ > 0, ψᵢ < 0, so −2ψF/(4n) ≥ 0 and −Σwᵢψᵢ/(4n) ≥ 0]\n\nProof steps in Lean:\n1. Function identity: marginTilde F = fun n => (½n log n + c n − 4) − 2 F(√n+1) − [w-summed block terms] − IIMaj n + K — prove by unfolding marginTilde/C0hat/Hhat/IIMaj-defs and ring (with atoms L1(√n+1), L2, F, exp...). The weights: decimal fractions — ring normalizes ✓. But IIMaj appears UNEXPANDED in the identity (keep IIMaj n as a term) ✓ fine.\n\nForm: I'll state:\nmarginTilde F = fun n => (1/2)*n*log n + (lagariasC1 − γ/2)*n − 2*(F (√n+1)) − (43.15466...)*(Tn L1 block) − ... − IIMaj n + K\n\nFor the deriv computations, group the blocks as COMPOSITIONS: φᵢ∘T where T = √n+1. The 8 block-functions:\nφ₂ := id\nφ₃ := fun t => Real.log (t+2)  [L1]\nφ₄ := fun t => Real.log (Real.log (t+2))  [L2]\nφ₅ := fun t => (Real.log (t+2))^2  [L1²]\nφ₆ := fun t => Real.log (t+2) * Real.log (Real.log (t+2))  [L1·L2]\nφ₇ := fun t => (Real.log (Real.log (t+2)))^2  [L2²]\nφ₈ := fun t => t * Real.log (t+2)  [T·L1]\nφ₉ := fun t => t * Real.log (Real.log (t+2))  [T·L2]\n\nmarginTilde F = fun n => S0 n − 2 F(Tn) − (w₂ φ₂(Tn) + w₃ φ₃(Tn) + ... + w₉ φ₉(Tn)) − IIMaj n + K\nwith S0 n = ½n log n + cn − 4.\n\nThe identity proof: unfold marginTilde C0hat Hhat; funext n; ring. The C0hat² expansion: 3*(0.433 L1 + 0.886 L2 + 2.9)^2 — ring handles ✓ (atoms: L1 := log(√n+1+2), L2 := log(log(...)), F, IIMaj n, exp 4). Note Hhat T = 3*C0hat T^2 + 10.5*log(T+2) + 48 — unfolds to the square ✓ ring expands ✓.\n\n2. Derivative of the whole: deriv (deriv (marginTilde F)) n — deriv through the sum via the identity + deriv_add/deriv_sub/deriv_const_mul chains... This is the nasty part: ~15 summands. Each piece's second derivative at n:\n- S0: deriv (deriv S0) n = 1/(2n): prove separately (lemma marginS_deriv2).\n- 2F∘T: chain_rule_sqrt_shift F hF n hn0 : = ψF-form... chain_rule_sqrt_shift gives (F″(Tn) − F′(Tn)/√n)/(4n) — with hψ's form F″(T) − F′(T)/(T−1): bridge √n = Tn−1: rw [show Real.sqrt n = (Real.sqrt n + 1) − 1 by ring] — then hψ Tn hTn gives < 0 ✓.\n- wᵢ·φᵢ∘T: chain_rule_sqrt_shift_on φᵢ hCᵢ n hn0 (needs ContDiffOn ℝ 2 φᵢ (Ioi (−1))!) — I HAVE NOT PROVEN THE ContDiffOn FOR THE BLOCKS! chain_rule_sqrt_shift_on requires ContDiffOn ℝ 2 φ (Ioi (−1)). I skipped that — the psiOf_* lemmas computed deriv-values via HasDerivAt, but chain_rule_sqrt_shift_on needs ContDiffOn. Need per-block ContDiffOn ℝ 2 on Ioi (−1):\n  - id: contDiffOn_id / (contDiff_id).contDiffOn ✓.\n  - log(t+2): ContDiffOn.log (contDiffOn_id.add contDiffOn_const)? — need (fun t => t+2) C^∞ on Ioi(−1) and ≠ 0 there: ContDiffOn.log (hf : ContDiffOn ℝ n f s) (hs : ∀ x ∈ s, f x ≠ 0)? Check name: `ContDiffOn.log` — grep. Then log log(t+2): ContDiffOn.log of the previous (needs log(t+2) ≠ 0 on Ioi(−1): t+2 > 1 → log > 0 ✓).\n  - squares/products: .mul/.pow.\n  - t*log, t*loglog: .mul with contDiffOn_id.\n  Each ~3-6 lines. 7 blocks (id covered) — ~40 lines.\n\n  ALTERNATIVE: bypass chain_rule_sqrt_shift_on for the explicit blocks: compute deriv (deriv (fun n => φᵢ(√n+1))) n DIRECTLY via deriv2_of_hasDerivAt_nhds with D1̃(n) := φᵢ′(√n+1)·(2√n)⁻¹ (chain rule) — requires per-point HasDerivAt of the composition + derivative of D1̃ — that's redoing chain_rule_sqrt_shift_on per block — MORE work. Prove the ContDiffOn's (cleaner).\n\n3. Sum assembly: the second derivative of a sum/difference/scaled sum of ~12 functions. Use deriv_add etc. with differentiability witnesses at the point n, then deriv of the resulting deriv-function sum — with eventuallyEq technique... For 12 summands this is a big mechanical block. Structure to reduce pain: prove\n\nhave key : deriv (deriv (marginTilde F)) n = 1/(2*n) − 2*(ψFexpr)/(4*n) − Σ (wᵢψᵢexpr/(4n)) − deriv (deriv IIMaj) n\n\nvia: rw [marginTilde_identity]; then deriv (deriv (big lambda)) n: the big lambda is a sum — apply deriv-add/sub rules TWICE (once for the first deriv formula as a function — needs eventuallyEq — ugh, deriv_add only gives pointwise). The clean way: \n\nderiv (f+g) =ᶠ deriv f + deriv g near n (if both differentiable near n), then deriv (deriv (f+g)) n = deriv (deriv f + deriv g) n = deriv (deriv f) n + deriv (deriv g) n (deriv_add with DifferentiableAt (deriv f) n, (deriv g) n).\n\nPer piece we have differentiable-on-neighborhood (from the chain-rule proofs' internals — hg1 eventually). Rather than rebuild, define a tactic block: induction-free manual unfolding with `deriv_add`'s? It'll be long but mechanical. Alternative MUCH cleaner: state and prove a \"second derivative of sum\" combinator:\n\ntheorem deriv2_add {f g : ℝ → ℝ} {n : ℝ}\n    (hf : DifferentiableAt ℝ (deriv f) n) (hg : DifferentiableAt ℝ (deriv g) n)\n    (hf' : DifferentiableAt ℝ f n) (hg' : DifferentiableAt ℝ g n)\n    (hf'' : ∀ᶠ m in 𝓝 n, DifferentiableAt ℝ f m) (hg'' : ∀ᶠ m in 𝓝 n, DifferentiableAt ℝ g m) :\n    deriv (deriv (f + g)) n = deriv (deriv f) n + deriv (deriv g) n\nProof: deriv (f+g) =ᶠ[𝓝 n] (fun m => deriv f m + deriv g m) via filter_upwards [hf'', hg''] with m hfm hgm => deriv_add hfm hgm; then rw [EventuallyEq.deriv... hmm need pointwise: deriv (deriv (f+g)) n = deriv (fun m => deriv f m + deriv g m) n (congr_of_eventuallyEq.symm... ) = deriv (deriv f) n + deriv (deriv g) n via deriv_add hf hg.\n\nStill need per-piece: DifferentiableAt (deriv piece) n + eventually-differentiability — these come FOR FREE from the chain-rule applications? chain_rule_sqrt_shift_on's CONCLUSION gives deriv (deriv piece) n — implying deriv piece differentiable at n?? NOT extractable from the equation alone... but from the ContDiffOn hC: DifferentiableOn ℝ (deriv φ) (Ioi(−1)) → DifferentiableAt at Tn; and the composition's deriv: (φ∘T)' = (φ'∘T)·T' — differentiable at n since φ' differentiable at Tn and T smooth ✓. So per piece ~4-6 lines to get hf/hg-style facts. 12 pieces × 6 = 72 lines + combinators. Meh but doable.\n\nALTERNATIVELY — a slicker path avoiding sum-of-derivs entirely: prove the convexity inequality via HasDerivAt/deriv at the level of the FULL function using deriv2_of_hasDerivAt_nhds ONCE on marginTilde with D1 = full closed form and D2 = the lower-bound form? HasDerivAt of the full margin function: builds from per-piece HasDerivAt via .add/.sub/.mul combinators — one big term, no deriv_add chains needed! Then:\n\nh1 : ∀ᶠ m in 𝓝 n, HasDerivAt (marginTilde F) (D1 m) m — per-point from piece HasDerivAts.\nh2 : HasDerivAt D1 (D2val n) n.\nThen deriv (deriv (marginTilde F)) n = D2val n, and D2val n = 1/(2n) − ... − IIMaj″(n) ≥ 1/(2n) − IIMaj″(n)?? — wait I need the VALUE = formula, then the inequality. D2val would be the exact second derivative: 1/(2n) − 2ψF/(4n) − Σwᵢψᵢ/(4n) − IIMaj″-closed. Then margiñ″ ≥ 1/(2n) − IIMaj″ > 0 via the sign facts.\n\nThe per-piece HasDerivAt for the compositions φᵢ(√m+1): from the block per-point HasDerivAt (hasDerivAt_log_shift etc. are for φᵢ at POINT x) composed with T(m) = √m+1: HasDerivAt.comp — gives φᵢ′(Tm)·(2√m)⁻¹. For F: DifferentiableAt ℝ F (Tm) from hF.differentiable → hasDerivAt.\n\nThe D1's derivative (h2): D1(m) = S0'(m) − 2F'(Tm)·(2√m)⁻¹ − Σwᵢφᵢ'(Tm)·(2√m)⁻¹ − IIMaj-D1(m)... \n\nWAIT — this is exactly redoing the chain-rule proofs piecewise. The chain-rule lemmas ALREADY package this. Better to use them + the deriv2_add/sub/const_mul combinators. The per-piece differentiability facts needed for the combinators:\n\nFor piece P(m) = φᵢ(Tm): \n- DifferentiableAt ℝ P m for m near n: φ differentiable at Tm (from ContDiffOn) + T smooth ✓ via DifferentiableAt.comp.\n- DifferentiableAt ℝ (deriv P) n: deriv P =ᶠ (fun m => φᵢ'(Tm)·(2√m)⁻¹) near n (from the chain-rule hg1-style eventuallyEq — which I can re-derive via deriv_comp per point); the RHS function differentiable at n: φᵢ' differentiable at Tn (from ContDiffOn.deriv_of_isOpen) ✓ chain.\n\nHmm — honestly, the cleanest packaging: prove for EACH piece a lemma `deriv2_piece : deriv (deriv P) n = value n ∧ DifferentiableAt ℝ (deriv P) n ∧ (∀ᶠ m in 𝓝 n, DifferentiableAt ℝ P m)` — the last two are needed for the combinators. Actually the combinators (deriv2_add) need only (DifferentiableAt (deriv f) n) + (eventually DifferentiableAt f)... Let me simplify the combinator needs:\n\nderiv2_add needs: hf' : DifferentiableAt ℝ (deriv f) n; hg' : DifferentiableAt ℝ (deriv g) n; and the eventuallyEq deriv (f+g) =ᶠ deriv f + deriv g: needs DifferentiableAt f m ∧ g m eventually.\n\nSo per piece: (i) DifferentiableAt ℝ (deriv P) n, (ii) ∀ᶠ m, DifferentiableAt ℝ P m. And the VALUE deriv (deriv P) n = ... from chain-rule.\n\nFor (i): from chain-rule's value formula? No — independently: deriv P =ᶠ Q near n where Q smooth-at-n: deriv P differentiable at n ⟸ (EventuallyEq.differentiableAt_iff? `Filter.EventuallyEq.differentiableAt_iff : f =ᶠ[𝓝 x] g → (DifferentiableAt ℝ f x ↔ DifferentiableAt ℝ g x)` ✓ exists). So (i) via the eventuallyEq + Q differentiable (from HasDerivAt Q value at n — hasDerivAt→differentiableAt ✓). The eventuallyEq deriv P =ᶠ Q: per point m near n: deriv P m = Q m via deriv_comp etc. — this is the hg1 argument again per piece... FINE — but I can extract it ONCE per piece type since the chain-rule lemma's proof had it... can't extract from the lemma.\n\nHmm — alternative: get (i) and (ii) DIRECTLY from a ContDiffOn statement for the COMPOSITION: if ψ := φ∘T is ContDiffOn ℝ 2 on a neighborhood of n, then DifferentiableAt (deriv ψ) n ✓ (from contDiffOn.deriv_of_isOpen + interior) and DifferentiableAt ψ m nearby ✓. So: prove per piece ContDiffOn ℝ 2 P (Ioi 0) — via ContDiffOn.comp of the block's ContDiffOn (Ioi(−1)) with T's ContDiffOn (Ioi 0 → maps into Ioi(−1) ✓ T(m) = √m+1 > 1 > −1 ✓). Then ALL of (i), (ii), AND the chain-rule need (chain_rule_sqrt_shift_on's hφ is on φ not P — hmm chain_rule_sqrt_shift_on takes hφ : ContDiffOn ℝ 2 φ (Ioi(−1)) ✓ same ContDiffOn fact reused for BOTH purposes ✓).\n\nSo the ONLY missing infrastructure: per-block ContDiffOn ℝ 2 φᵢ (Ioi (−1)) [7 blocks: L1, L2, L1², L1L2, L2², TL1, TL2 — id free]. Check ContDiffOn.log exists.\n\nThen the assembly:\n- marginTilde_identity (function eq by ring)\n- marginS pieces\n- per-piece ContDiffOn + psiOf values + signs\n- sum machinery\n- final: deriv (deriv (marginTilde F)) n ≥ 1/(2n) − IIMaj″(n) > 0... wait — need = then ≥. The equality: margiñ″(n) = 1/(2n) − 2ψF/(4n) − Σwᵢψᵢ/(4n) − IIMaj″(n) — via combinators + chain-rule values + psiOf values.\n- Then hf'', hf' for convexOn_of_deriv2_nonneg: DifferentiableOn f (interior (Ici 20)) = Ioi 20: marginTilde differentiableOn Ioi 20: pieces differentiable at each n > 20 (T smooth, blocks on Ioi(−1), F global, IIMaj global, S0) — via the sum structure ✓. deriv f differentiableOn: from the per-piece DifferentiableAt (deriv piece) n at every n > 20 + deriv-add chain... OR: use the h1/h2 (HasDerivAt of full D1) approach at each n > 20?? — the sum-combinators at each point give deriv f = explicit sum formula on Ioi 20, and that formula is differentiableOn (Ioi 20) by the same per-piece facts (now for D1-pieces) — DifferentiableOn.congr. \n\nThat's the full plan — sizable (~250-350 lines). The per-piece packaging will dominate. Let me define a structure to make it mechanical:\n\nFor each block φᵢ define pᵢ : ℝ → ℝ := fun n => φᵢ (√n + 1) and prove a bundle:\ntheorem piece_i (n : ℝ) (hn : 20 ≤ n) :\n    deriv (deriv pᵢ) n = ψᵢ-closed n ∧ DifferentiableAt ℝ (deriv pᵢ) n\n\nHmm the ∧-bundle per piece... Actually — wait. There's a MUCH simpler observation for hf' and hf'' (the convexOn_of_deriv2_nonneg hypotheses): they're DifferentiableOn conditions on f and deriv f over Ioi 20. If I prove marginTilde F is ContDiffOn ℝ 2 on Ioi 20 (as a whole!), then:\n- ContinuousOn: contDiffOn.continuousOn ✓\n- DifferentiableOn f: contDiffOn.differentiableOn (n≠0) ✓\n- DifferentiableOn (deriv f): contDiffOn.deriv_of_isOpen (Ioi open) → ContDiffOn ℝ 1 (deriv f) (Ioi 20) → differentiableOn ✓\n\nContDiffOn ℝ 2 (marginTilde F) (Ioi 20): from the identity (marginTilde = explicit sum), each summand ContDiffOn: S0 (n log n — ContDiffOn on Ioi 20: log smooth for n > 0 ✓), F∘T (comp), blocks (comp), IIMaj (global ContDiff → On), constants. ContDiffOn.comp (block) (T smooth Ioi 20, maps into Ioi(−1)) ✓. Sum: ContDiffOn.add/sub ✓. So ONE ContDiffOn proof (~30-50 lines) handles ALL the regularity! And for the VALUES (deriv (deriv f) n), use the combinators deriv2_add etc. — which need per-piece DifferentiableAt (deriv piece) n — obtainable from the SAME per-piece ContDiffOn (deriv_of_isOpen + mem). OK — the per-piece ContDiffOn does triple duty. \n\nThen:\n- convexity: convexOn_of_deriv2_nonneg (convex_Ici 20) hf hf' hf'' hf''_nonneg.\n- hf''_nonneg : ∀ n ∈ Ioi 20, 0 ≤ deriv (deriv (marginTilde F)) n — via the value computation + inequality.\n\nValue computation via combinators: I still need per-piece deriv-values + per-piece DifferentiableAt (deriv piece) n for the deriv2_add combinator. With per-piece ContDiffOn (Ioi 20):\n- DifferentiableAt ℝ (deriv pᵢ) n: from ContDiffOn ℝ 2 → deriv ContDiffOn ℝ 1 (Ioi 20 open) → differentiableOn → differentiableAt (Ioi_mem_nhds) ✓.\n- eventually DifferentiableAt pᵢ: from differentiableOn + nhds ✓.\n\nCombinators: deriv2_add, deriv2_sub, deriv2_const_mul, plus values from chain_rule_sqrt_shift_on (blocks, F — wait F is global ContDiff: chain_rule_sqrt_shift F hF — the GLOBAL version ✓ already proved) and marginS_deriv2 and IIMaj_deriv2 (need deriv (deriv IIMaj) n VALUE — from chain_rule_sqrt psiTilde... via hId: deriv (deriv IIMaj) n = [D2(√n) − D1(√n)/√n]/(4n) — hmm for the inequality I only need IIMaj″(n) < 1/(2n) (IIMaj_deriv2_lt) and margiñ″ = [rest] − IIMaj″: so I need the VALUE equality including IIMaj″ — from chain_rule_sqrt as in IIMaj_deriv2_lt ✓ reuse.\n\nmarginS_deriv2: S0 n = ½n log n + c n − 4: deriv2 = 1/(2n): via deriv2_of_hasDerivAt_nhds: D1 m = ½(log m + 1) + c; D2 = ½·(1/m): HasDerivAt (fun m => (1/2)*m*log m + c*m − 4) (D1 m) per point (m > 0): (log m)' = 1/m: (½(m log m))' = ½(log m + 1) ✓; (c m)' = c ✓. D1' = ½·(1/m): (½(log m + 1) + c)' = ½·(1/m) ✓.\n\nOK — executing. Order:\nA. Per-block ContDiffOn lemmas (7).\nB. marginS_deriv2.\nC. deriv2 combinators (add, sub, const_mul, neg maybe).\nD. marginTilde_identity.\nE. marginTilde_contDiffOn (regularity package) — optional but simplifies convexOn_of_deriv2_nonneg hypotheses.\nF. marginTilde_convex proof: value equality + inequality + regularity.\n\nThis is the final big push. Write A+B+C first (one edit), build, then D+E, then F.\n\nFor A: check ContDiffOn.log name.","signature":""},{"type":"text","text":"All IIMaj pieces compile. Committing this milestone, then checking `ContDiffOn.log` for the final assembly."},{"type":"tool_use","id":"Bash_261","name":"Bash","input":{"command":"cd /d/Python/Quantum && git add research/lean/NguyenRH1/NguyenRH1/LiSkeleton.lean && git -c user.name=\"NGUYEN-RH-1\" -c user.email=\"research@local\" commit -m \"Session 011: IIMaj bound complete — iim_u2_antitone + iimaj_bracket_bound (factor-monotone, ≤1.316<2) + IIMaj_deriv2_lt (IIMaj″ < 1/(2n) for n≥20)\" && grep -rn \"theorem ContDiffOn.log\\|protected theorem ContDiffOn.log\" /d/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean /d/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/Analysis/SpecialFunctions/Log/ContDiff.lean 2>/dev/null | head -4","description":"Commit IIMaj bound; find ContDiffOn.log"},"extra_content":null}]