[{"type":"thinking","thinking":"Now `marginTilde_convex`. This is the harder one. Let me look at what's needed.\n\nStatement:\n```lean\ntheorem marginTilde_convex (F : ℝ → ℝ) (hF : ContDiff ℝ 2 F)\n    (hmaj : ∀ T : ℝ, (Ncount 0 T : ℝ) ≤ F T)\n    (hψ : ∀ T : ℝ, Real.sqrt 20 + 1 ≤ T →\n      deriv (deriv F) T - deriv F T / (T - 1) < 0) :\n    ConvexOn ℝ (Set.Ici 20) (marginTilde F)\n```\n\nmarginTilde F n = S(n) − B̃(n) + 1 where:\n- S(n) = (1/2)n log n + (C1 − γ/2)n − 4\n- B̃(n) = 2F(T) + 4T·((2/3)C0hat T + 10.5 log(T+2) + 48) + 40·1.3352 + 2(e⁴+1)(7·Hhat T + 3.3) + IIMaj n + 1, with T = √n + 1.\n\nConvexity on [20, ∞). The D-057 proof: margiñ″ ≥ 0 via:\n- (S(n))″ = 1/(2n) > 0 (the (C1−γ/2)n and −4 are linear/constant → vanish).\n- Each B̃ component is concave (second derivative ≤ 0) via the chain-rule identity chain_rule_sqrt_shift: for φ(T(n)) with T(n) = √n+1: φ(T(n))″ = [φ″(T) − φ′(T)/(T−1)]/(4n) = ψ[φ](T)/(4n).\n  - 2F(T): ψ = the hψ hypothesis < 0 ✓ (factor 2).\n  - 4T·g(T) where g = (2/3)C0hat + 10.5 log(T+2) + 48: hmm — 4T·g(T) is NOT of the form φ(T(n)) directly... wait it is: φ(T) = 4T·g(T) composed with T(n). Its ψ-block: ψ[4Tg(T)] = (4Tg)″ − (4Tg)′/(T−1). D-057's 11-block table handles each piece.\n  - 2(e⁴+1)(7 Hhat T + 3.3): Hhat = 3C0hat² + 10.5 log(T+2) + 48.\n  - C0hat = 0.433 log(T+2) + 0.886 log log(T+2) + 2.9.\n  - IIMaj n: directly bounded (D-057: I_I term direct bound — its second derivative is tiny/negative?).\n\nThe full formalization of the 11-block table is a BIG job: requires computing second derivatives of log log, products, squares, and sign analysis on [√20+1, ∞). This is honestly a multi-hundred-line effort with real risk. The D-061 plan says tier (i) includes marginTilde_convex with \"the 11-block table\".\n\nHmm — but wait. Let me think about what's actually being claimed and whether there's a shortcut. The theorem is stated for ANY F with the ψ-hypothesis. The margin function is explicit. Convexity on [20,∞).\n\nAlternative characterization: ConvexOn ℝ (Set.Ici 20) f — for twice-differentiable f, `convexOn_of_deriv2_nonneg` exists in Mathlib: `convexOn_of_deriv2_nonneg (hfc : Convex ℝ s) (hf' : ContinuousOn f s) (hf'' : ∀ x ∈ interior s, DifferentiableAt ℝ (deriv f) x)?? Actually: `convexOn_of_deriv2_nonneg : Convex ℝ s → ContinuousOn f s → (∀ x ∈ interior s, DifferentiableAt ℝ f x)?? Let me recall: Mathlib has `convexOn_of_deriv2_nonneg (hfc : Convex ℝ s) (hf : ContinuousOn f s) (hf' : DifferentiableOn ℝ f (interior s)) (hf'' : ∀ x ∈ interior s, 0 ≤ deriv (deriv f) x) : ConvexOn ℝ s f`? Something like that — I believe `convexOn_of_deriv2_nonneg` exists with interior differentiability conditions. Set.Ici 20 interior = Ioi 20.\n\nSo the task: show deriv (deriv (marginTilde F)) n ≥ 0 for n > 20.\n\nmarginTilde F = S − B̃ + 1. Components:\n\n1. S(n) = ½n log n + c·n − 4: deriv = ½(log n + 1) + c; deriv² = 1/(2n) ✓.\n\n2. −2·(F ∘ T)(n) where T(n) = √n+1: second deriv = −2·ψ[F](T(n))/(4n) — via chain_rule_sqrt_shift (already proved!) with φ := F: (F∘T)″(n) = [F″(T) − F′(T)/√n]/(4n). And hψ says F″(T) − F′(T)/(T−1) < 0 for T ≥ √20+1; with T−1 = √n ✓ matches (chain_rule_sqrt_shift writes /Real.sqrt n). So −2·(...)/(4n) > 0 ✓.\n\n3. −4T·g(T) with g(T) = (2/3)C0hat T + 10.5 log(T+2) + 48: let φ(T) = 4T·g(T). ψ[φ](T) = φ″ − φ′/(T−1). Need ψ[φ](T) < 0 (then −ψ/(4n) > 0). Compute: φ = 4Tg; φ′ = 4g + 4Tg′; φ″ = 8g′ + 4Tg″. ψ = 8g′ + 4Tg″ − (4g + 4Tg′)/(T−1).\n   g(T) = (2/3)(0.433 log(T+2) + 0.886 log log(T+2) + 2.9) + 10.5 log(T+2) + 48.\n   Let L = log(T+2), ℓ = log L = log log(T+2). g = a₁L + a₂ℓ + c with a₁ = (2/3)(0.433) + 10.5 = 0.288667 + 10.5 = 10.788667, a₂ = (2/3)(0.886) = 0.590667, c = (2/3)(2.9) + 48 = 1.93333 + 48 = 49.93333.\n   g′ = a₁/(T+2) + a₂/(L(T+2)) = (a₁ + a₂/L)/(T+2).\n   g″ = −a₁/(T+2)² + a₂·d/dT[1/(L(T+2))] = −a₁/(T+2)² − a₂(1/L + 1/L²)/(T+2)²... let me compute: d/dT [1/(L(T+2))] = −[L′(T+2) + L]/(L(T+2))² = −[1 + L]/(L²(T+2)²)? Hmm: L′ = 1/(T+2). d/dT (L(T+2)) = 1 + L. So d/dT[1/(L(T+2))] = −(1+L)/(L²(T+2)²).\n   So g″ = −a₁/(T+2)² − a₂(1+L)/(L²(T+2)²) < 0.\n   ψ[φ] = 8g′ + 4Tg″ − (4g + 4Tg′)/(T−1).\n   Sign analysis: 8g′ = 8(a₁ + a₂/L)/(T+2) > 0 but small ~ 8a₁/T; 4Tg″ ~ −4a₁T/(T+2)² ~ −4a₁/T; −(4g)/(T−1) ~ −4a₁L/T dominates (negative, log factor); −4Tg′/(T−1) ~ −4a₁/T. Sum ~ −4a₁L/T + (8−4−4)a₁/T + ... = −4a₁L/T + lower order < 0 for large T. At T = √20+1 ≈ 5.47: need numeric check. D-057's 11-block table did this rigorously.\n\n   This is where it gets heavy: rigorous sign proofs of such expressions on a half-line, requiring log bounds, etc. The 11-block table approach decomposes into elementary blocks whose signs are verifiable.\n\n4. −2(e⁴+1)(7 Hhat T + 3.3): Hhat = 3C0hat² + 10.5L + 48. φ(T) = 7·Hhat(T)·const... ψ[3C0hat² + ...]: C0hat² has derivative 2C0hat·C0hat′ — sign analysis with squares — heavier.\n\n5. −IIMaj n: IIMaj(n) = 2(√n+1)(e^{√n/2}+1)·0.57·2^{−(2√n−2)}. D-057 treats I_I via a \"direct bound\" (not the ψ formalism) — showing |IIMaj″| tiny or IIMaj″ ≤ 0 directly? Let me read D-057 §3 to see the actual treatment.\n\nHonestly, formalizing the FULL marginTilde_convex with rigorous sign tables is a large project — likely 500+ lines of Lean with delicate derivative computations of log log functions and square terms. This could eat the whole session (and context). \n\nLet me check D-057 for the exact block decomposition to size this properly. Also check E-029 (sympy table).\n\nGiven the session's already substantial progress (sInfty proved + committed), a reasonable approach: assess marginTilde_convex carefully; if it's as heavy as it looks, do a sub-stage: prove the S″ + F-block part and decompose the rest into cleanly-stated helper lemmas with sorries? NO — sorries inside a \"proved\" theorem reintroduce sorryAx; the honest approach is either full proof or leave the theorem as skeleton.\n\nAlternative: prove a REDUCED version? No — don't weaken statements (audit rule).\n\nLet me read D-057 first to see the 11-block table and decide. It's important to understand what \"ψ-block sign hypothesis\" covers: the theorem's hψ is ONLY about F. The other blocks (4T·g, Hhat, IIMaj) must be handled by direct computation inside the proof — that's the 11-block table content.\n\nLet me read D-057.","signature":""},{"type":"text","text":"Now `marginTilde_convex` — the heavy one. Let me read D-057's 11-block table to size the formalization."},{"type":"tool_use","id":"Bash_83","name":"Bash","input":{"command":"wc -l /d/Python/Quantum/research/derivations/D-057-margin-convexity.md 2>/dev/null || ls /d/Python/Quantum/research/derivations/ | grep D-057","description":"Locate D-057"},"extra_content":null}]