[{"type":"thinking","thinking":"`ContDiffOn.log (hf : ContDiffOn ℝ n f s) (hs : ∀ x ∈ s, f x ≠ 0) : ContDiffOn ℝ n (fun x => Real.log (f x)) s` ✓.\n\nNow write the assembly pieces. Per-block ContDiffOn on Ioi (−1):\n\n```lean\ntheorem contDiffOn_L1 : ContDiffOn ℝ 2 (fun t => Real.log (t + 2)) (Set.Ioi (-1)) :=\n  ((contDiffOn_id.add contDiffOn_const)).log (fun x hx => by\n    have : -1 < x := hx; have : (0:ℝ) < x + 2 := by linarith; positivity?? — goal (x+2) ≠ 0: ne_of_gt)\n```\ncontDiffOn_id : ContDiffOn 𝕜 n id s ✓; .add contDiffOn_const ✓. For the ≠ 0: fun x hx => ne_of_gt (by ... : 0 < x + 2). `by positivity` won't use hx; do: fun x hx => by have h : -1 < x := hx; exact ne_of_gt (by linarith : (0:ℝ) < x + 2).\n\nL2: ContDiffOn.log of contDiffOn_L1 with log(t+2) ≠ 0 on Ioi(−1): t+2 > 1 → log(t+2) > 0 ✓:\n  contDiffOn_L1.log (fun x hx => ne_of_gt (Real.log_pos (by have : -1 < x := hx; linarith : (1:ℝ) < x + 2)))\n\nL1²: contDiffOn_L1.pow 2?? ContDiffOn.pow : ContDiffOn ℝ n f s → ContDiffOn ℝ n (fun x => f x ^ k) s ✓ (exists? `ContDiffOn.pow` yes).\nL1L2: contDiffOn_L1.mul contDiffOn_L2.\nL2²: contDiffOn_L2.pow 2.\nTL1: contDiffOn_id.mul contDiffOn_L1.\nTL2: contDiffOn_id.mul contDiffOn_L2.\n\nWait — ContDiffOn.mul's conclusion: fun x => f x * g x ✓ matches TL1 := fun t => t * log(t+2) ✓.\n\nB. marginS_deriv2:\n```lean\ntheorem marginS_deriv2 (n : ℝ) (hn : 0 < n) (c : ℝ) :\n    deriv (deriv fun m => (1/2) * m * Real.log m + c * m - 4) n = 1 / (2 * n) := by\n  have h1 : ∀ᶠ m in 𝓝 n, HasDerivAt (fun m => (1/2) * m * Real.log m + c * m - 4)\n      ((1/2) * (Real.log m + 1) + c) m := by\n    filter_upwards [eventually_gt_nhds hn] with m hm\n    have hlog : HasDerivAt Real.log m⁻¹ m := Real.hasDerivAt_log (ne_of_gt hm)\n    have hml := (hasDerivAt_id' m).mul hlog  -- (m log m)' = 1*log m + m*m⁻¹\n    have h := ((hml.const_mul (1/2)).add ((hasDerivAt_id' m).const_mul c)).sub_const 4\n    -- massage: 1/2*(1*log m + m*m⁻¹) + c*1 = 1/2*(log m + 1) + c\n    have hm0 : m ≠ 0 := ne_of_gt hm\n    convert h using 2 <;> first | rfl | (field_simp [hm0]; ring)\n  have h2 : HasDerivAt (fun m => (1/2) * (Real.log m + 1) + c) ((1/2) * (1 / n)) n := by\n    have h := ((Real.hasDerivAt_log (ne_of_gt hn)).add_const 1).const_mul (1/2)\n    -- 1/2 * (m⁻¹) = 1/2 * (1/n)\n    convert h.add_const c using 2 <;> ring\n  exact deriv2_of_hasDerivAt_nhds h1 h2\n```\nsub_const: `HasDerivAt.sub_const (h) (c) : HasDerivAt (fun x => f x − c) f' x` ✓ exists (add_const's sibling). The const_mul on hml: fun m => (1/2)*(m*log m) — the goal's piece is (1/2)*m*log m = ((1/2)*m)*log m?? — parse: (1/2) * m * Real.log m = ((1/2)*m)*(log m) — DIFFERENT from (1/2)*(m*log m)! ring-equal but the HasDerivAt construction gives (1/2)*(m*log m). The function mismatch — convert using 2 with ring for function eq?? Function equality of lambdas — ring doesn't do funext. Restructure: build hml' : HasDerivAt (fun m => (1/2)*m*log m) ... directly: ((hasDerivAt_id' m).const_mul (1/2)).mul hlog : fun m => ((1/2)*m) * log m ✓ derivative: ((1/2)*1)*log m + ((1/2)*m)*m⁻¹ ✓ then .add (cm) .sub_const 4. Massage the value: (1/2*1)*log m + (1/2*m)*m⁻¹ + c*1 = 1/2*(log m + 1) + c: field_simp [hm0]; ring ✓ — via convert h using 2 <;> first | rfl | (field_simp [...]; ring)? the function parts now match syntactically ✓ convert leaves only value goal ✓ field_simp;ring (or the instance-rfl issue — no sub here except sub_const 4 — hmm `(A + B) - 4` vs HasDerivAt ((hml').add hcm).sub_const 4 — function fun m => ((1/2)*m*log m + c*m) − 4 ✓ matches goal's (1/2)*m*log m + c*m − 4 ✓ parse ((1/2*m*log m) + (c*m)) − 4 ✓ same ✓. So convert h using 2 → value goal only → field_simp; ring. But watch the convert-instance issue from before (sub) — use <;> first | rfl | ring? field_simp needed for m⁻¹... write: convert h using 2; field_simp [hm0]; ring. If instance goals appear, add rfl-handling.\n\nC. combinators:\n```lean\ntheorem deriv2_add {f g : ℝ → ℝ} {n : ℝ}\n    (hf1 : DifferentiableAt ℝ (deriv f) n) (hg1 : DifferentiableAt ℝ (deriv g) n)\n    (hf2 : ∀ᶠ m in 𝓝 n, DifferentiableAt ℝ f m) (hg2 : ∀ᶠ m in 𝓝 n, DifferentiableAt ℝ g m) :\n    deriv (deriv (f + g)) n = deriv (deriv f) n + deriv (deriv g) n := by\n  have he : deriv (f + g) =ᶠ[𝓝 n] (fun m => deriv f m + deriv g m) := by\n    filter_upwards [hf2, hg2] with m hfm hgm\n    exact deriv_add hfm hgm\n  rw [(hf1.add hg1).congr_of_eventuallyEq?? ]\n```\ncareful: want deriv (deriv (f+g)) n = deriv (fun m => deriv f m + deriv g m) n = deriv (deriv f) n + deriv (deriv g) n. \nSecond step: deriv_add hf1 hg1 : deriv (deriv f + deriv g) n = deriv (deriv f) n + deriv (deriv g) n ✓.\nFirst step: EventuallyEq.deriv : f =ᶠ g → deriv f x = deriv g x — is it pointwise?? Earlier in psiOf_id I hit `EventuallyEq.deriv he : deriv (deriv id) =ᶠ[𝓝 T] deriv fun x => 1` — it gave =ᶠ! So EventuallyEq.deriv gives the eventuallyEq of derivs, NOT pointwise. The pointwise: `Filter.EventuallyEq.deriv_eq`? or `EventuallyEq.deriv` in newer Mathlib IS pointwise... it printed =ᶠ in the error earlier. Alternative: he.hasDerivAt?? — use: deriv (f+g) n... For pointwise: there is `Filter.EventuallyEq.deriv (h : f =ᶠ[𝓝 x] g) : deriv f x = deriv g x`?? The earlier print showed `deriv (deriv id) =ᶠ[𝓝 T] deriv fun x => 1` — hmm that was `he.deriv` where he : deriv id =ᶠ fun _ => 1 — so EventuallyEq.deriv maps =ᶠ to =ᶠ of derivs?! That contradicts my memory of a pointwise lemma. Maybe the pointwise one is `Filter.EventuallyEq.deriv_eq : f =ᶠ[𝓝 x] g → deriv f x = deriv g x`? Hmm — or the one I used in chain_rule_sqrt_shift: `hmul.congr_of_eventuallyEq hg1` — HasDerivAt congr (different). For pointwise deriv-eq: I'm now unsure of the name — options: EventuallyEq.deriv (gives =ᶠ), so pointwise deriv f x = deriv g x is NOT directly that... wait — actually maybe `EventuallyEq.deriv` DOES give pointwise and the earlier error print was showing something else... The print: \"EventuallyEq.deriv he is a proof of deriv (deriv id) =ᶠ[𝓝 T] deriv fun x => 1\" — nope, =ᶠ.\n\nSafe alternative: prove the pointwise equality via congr: have hev : deriv (f+g) =ᶠ[𝓝 n] (fun m => deriv f m + deriv g m); then `Filter.EventuallyEq.deriv_eq`?? — let me just grep for \"EventuallyEq.deriv\" variants.\n\nActually — recall the chain_rule_sqrt_shift proof: it used (hmul.congr_of_eventuallyEq hg1).deriv — HasDerivAt of the explicit function + congr → deriv of the original = explicit's value at that point ✓ — that's the pattern: to compute deriv (deriv (f+g)) n: exhibit HasDerivAt (fun m => deriv f m + deriv g m) V n := hf1.add hg1 |>.hasDerivAt; then congr_of_eventuallyEq with he.symm?? congr_of_eventuallyEq (h : HasDerivAt f' ...) (h1 : f' =ᶠ g')... — the signature: HasDerivAt.congr_of_eventuallyEq (h : HasDerivAt f f' x) (h₁ : f =ᶠ[𝓝 x] g) : HasDerivAt g f' x — so (hf1.add hg1).hasDerivAt : HasDerivAt (fun m => deriv f m + deriv g m) V n; congr with he.symm : (fun m => deriv f m + deriv g m) =ᶠ deriv (f+g)?? he : deriv (f+g) =ᶠ (fun m => ...) — he.symm ✓. Gives HasDerivAt (deriv (f+g)) V n → .deriv : deriv (deriv (f+g)) n = V ✓. So:\n\n  have he : deriv (f + g) =ᶠ[𝓝 n] (fun m => deriv f m + deriv g m) := ...\n  have h2 : HasDerivAt (fun m => deriv f m + deriv g m) (deriv (deriv f) n + deriv (deriv g) n) n :=\n    (hf1.add hg1).hasDerivAt\n  exact (h2.congr_of_eventuallyEq he.symm).deriv\n\nderiv2_sub, deriv2_const_mul (deriv (c * f) n = c * deriv f n: deriv_const_mul c hf1' : deriv (fun x => c * f x) n = c * deriv f n ✓ exists `deriv_const_mul`; for the combinator: similar structure with DifferentiableAt (c • deriv f)... deriv (fun m => c * f m) =ᶠ fun m => c * deriv f m (deriv_const_mul at each point: `deriv_const_mul c (hfm.differentiableAt?? — deriv_const_mul (c : ℝ) (hf : DifferentiableAt ℝ f x) : deriv (fun x => c * f x) x = c * deriv f x` ✓). HasDerivAt (fun m => c*deriv f m) (c*V) := hf1.const_mul c |>.hasDerivAt.\n\nAnd deriv2_neg for subtraction via add of neg or sub directly.\n\nD. marginTilde_identity: as analyzed; write with explicit weight decimals:\nw_TL1 = 43.154666666...? — decimal: (8/3)*0.433 + 42: 8*0.433 = 3.464; /3 = 1.1546666...; non-terminating! ring handles fractions as rationals — but I must WRITE the weight. Use the fractional form instead: don't pre-combine — in the identity, keep the weights in UNCOMBINED form and let ring prove: i.e. the identity RHS uses separate terms with the ORIGINAL constants: e.g. (8/3)*0.433 * (TL1-block) + 42*(TL1-block) — but then deriv assembly has more pieces... Alternatively pick the block decomposition where each block FUNCTION appears once with an ugly exact rational weight: 43.15466666666666... no — must be exact. Write as fractions: (8*0.433/3 + 42) — as an arithmetic expression inside the identity: the identity is proved by ring so any presentation works; for the deriv VALUE step, the weight multiplies the block — deriv_const_mul with c = that expression ✓ fine. So use weights as arithmetic expressions: w1 := ((8/3)*0.433 + 42), w2 := ((8/3)*0.886), w3 := ((8/3)*2.9 + 192), w4 := (42*(0.433^2))*(Real.exp 4 + 1), w5 := (42*(2*0.433*0.886))*(e⁴+1), w6 := (42*(0.886^2))*(e⁴+1), w7 := (42*(2*0.433*2.9) + 147)*(e⁴+1), w8 := (42*(2*0.886*2.9))*(e⁴+1). Hmm wait recompute: C0hat² coefficients: 3C0hat²·14(e⁴+1) = 42(e⁴+1)C0hat²: L1²: 42·0.433² = 42·0.187489 = 7.874538 ✓ w4 = 7.874538·(e⁴+1) — as expression 42*0.433^2*(e⁴+1) ✓; L1L2: 42·(2·0.433·0.886) = 42·0.767276 = 32.225592 ✓; L2²: 42·0.886² = 32.969832 ✓; L1: 42·(2·0.433·2.9) + 147 = 42·2.5114 + 147 = 105.4788+147 = 252.4788 ✓ times (e⁴+1); L2: 42·(2·0.886·2.9) = 42·5.1388 = 215.8296 ✓ times (e⁴+1); const: 42·2.9²·(e⁴+1) + 672(e⁴+1) + 6.6(e⁴+1) + 40·1.3352 + 1 − ... — fold into K.\n\nThe identity:\nmarginTilde F = fun n => (1/2)*n*log n + (lagariasC1 − γ/2)*n − 2*F(√n+1)\n  − ((8/3)*0.433 + 42) * ((√n+1) * log(√n+1+2))\n  − ((8/3)*0.886) * ((√n+1) * log(log(√n+1+2)))\n  − ((8/3)*2.9 + 192) * (√n+1)\n  − (42*0.433^2) * (e⁴+1) * (log(√n+1+2))^2\n  − (42*(2*0.433*0.886)) * (e⁴+1) * (log(√n+1+2) * log(log(√n+1+2)))\n  − (42*0.886^2) * (e⁴+1) * (log(log(√n+1+2)))^2\n  − (42*(2*0.433*2.9) + 147) * (e⁴+1) * log(√n+1+2)\n  − (42*(2*0.886*2.9)) * (e⁴+1) * log(log(√n+1+2))\n  − IIMaj n + K\nwith K = −4 − 40*1.3352 − 2*(e⁴+1)*(7*(3*2.9^2 + 48) + 3.3) − 1 + 1 hmm: let me recompute K: margin = S − B̃ + 1; S = ½n log n + cn − 4; B̃ = [2F + 4T(...) + 40·1.3352 + 2(e⁴+1)(7Hhat+3.3) + IIMaj + 1]. So margin = ½n log n + cn − 4 − 2F − 4T(...) − 53.408 − 2(e⁴+1)(7Hhat+3.3) − IIMaj − 1 + 1. The 7Hhat's constant part: 7·(3·8.41 + 48) = 7·(25.23+48) = 7·73.23 = 512.61; +3.3 = 515.91; ×2(e⁴+1) = 1031.82(e⁴+1). K = −4 − 53.408 − 1031.82(e⁴+1) − 1 + 1 = −58.408 − 1031.82(e⁴+1). In expression form: K := (-4 - 40*1.3352 - 2*(Real.exp 4 + 1)*(7*(3*2.9^2 + 48) + 3.3) - 1 + 1) — let ring sort it.\n\nThe identity proof: funext n; unfold marginTilde C0hat Hhat; ring. Concern: the `let T := √n+1` in marginTilde's def — unfold should expose (zeta). ring with atoms: log(√n+1+2), log(log(...)), F(√n+1), exp(√n/2)... IIMaj n — IIMaj stays FOLDED (don't unfold IIMaj — keep as atom ✓ don't include in unfold list). exp 4 atom ✓.\n\nBut CAREFUL: ring needs the LHS unfolded to the same atom expressions: marginTilde's B̃ has `2 * F T + 4 * T * ((2/3) * C0hat T + 10.5 * Real.log (T + 2) + 48) + 40 * 1.3352 + 2 * (Real.exp 4 + 1) * (7 * Hhat T + 3.3) + IIMaj n + 1` — C0hat T = 0.433*log(T+2) + 0.886*log(log(T+2)) + 2.9; Hhat T = 3*C0hat T^2 + 10.5*log(T+2) + 48. ring expands C0hat T^2 ✓. The identity I write must match ring's normalization — ring will verify either way ✓.\n\nE. marginTilde ContDiffOn ℝ 2 (Ioi 20): via the identity + per-piece ContDiffOn (Ioi 20) + add/sub/const_mul/mul:\n- S0: (contDiffOn_id.mul (contDiffOn_log' ...)) — n log n on Ioi 20: ContDiffOn.mul contDiffOn_id (ContDiffOn.log contDiffOn_id (fun x hx => ne_of_gt (by ...: 0 < x))) ✓ plus const_mul/linear/const.\n- F∘T: hF.contDiffOn.comp (T ContDiffOn Ioi 20, maps into univ): ContDiffOn.comp (contDiffOn...) — for F global: (contDiffOn_univ... use hF.contDiffOn.comp (hT, mapsTo). T = √n+1 on Ioi 20: Real.contDiffOn_sqrt?? sqrt is C^∞ on Ioi 0: `Real.contDiffOn_sqrt : ContDiffOn ℝ n Real.sqrt (Ioi 0)`? Hmm — sqrt ContDiffOn {0}ᶜ: `Real.contDiffOn_sqrt : ContDiffOn ℝ n Real.sqrt {0}ᶜ`? I recall `contDiffOn_sqrt : ContDiffOn ℝ n Real.sqrt {0}ᶜ` in Mathlib.Analysis.SpecialFunctions.Sqrt or Pow — check. Then .add contDiffOn_const for +1.\n- blocks φᵢ∘T: contDiffOn_Lx.comp (T, mapsTo Ioi 20 → Ioi (−1)) ✓ T(x) = √x+1 ≥ √20+1 > −1 ✓ mapsTo.\n- IIMaj: psiTilde ContDiff global → IIMaj = psiTilde∘√: contDiffOn of psiTilde.contDiffOn.comp(sqrt...) ✓ or (psiTilde_contDiff).contDiffOn.comp...\n- Combine ~15 summands via ContDiffOn.sub/add chains (const weights via contDiffOn_const.mul or .const_mul?).\n\nE is another ~60 lines. Then:\nhf : ContinuousOn (marginTilde F) (Ici 20) := (contDiffOn on Ioi 20).continuousOn.mono (Ioi_subset_Ici_self)?? — ContinuousOn on Ici: from ContDiffOn on Ioi 20 — continuousOn on Ioi ⊆ Ici ✓ .mono.\nhf' : DifferentiableOn ℝ (marginTilde F) (interior (Ici 20)): interior_Ici → Ioi 20: contDiffOn.differentiableOn (n≠0) ✓.\nhf'' : DifferentiableOn ℝ (deriv (marginTilde F)) (Ioi 20): contDiffOn.deriv_of_isOpen isOpen_Ioi (m+1≤n) → ContDiffOn ℝ 1 (deriv f) (Ioi 20) → .differentiableOn ✓.\n\nF. value+inequality per n ∈ Ioi 20:\nderiv (deriv (marginTilde F)) n = [compute via combinators + chain-rules + marginS + IIMaj] = 1/(2n) − 2ψF/(4n) − Σwᵢψᵢ/(4n) − IIMaj″(n) ≥ 1/(2n) − IIMaj″(n) > 0.\n\nThe value equality via deriv2_add/sub/const_mul combinators — each combinator application needs the per-piece (DifferentiableAt (deriv piece) n) + (eventually DifferentiableAt piece) — from per-piece ContDiffOn (Ioi 20):\n- DifferentiableAt ℝ (deriv pᵢ) n: per-piece ContDiffOn ℝ 2 (Ioi 20) → .deriv_of_isOpen isOpen_Ioi (1+1≤2) → ContDiffOn ℝ 1 (deriv pᵢ) → .differentiableOn (1≠0) → .differentiableAt (Ioi_mem_nhds (n > 20)) ✓.\n- eventually DifferentiableAt pᵢ: (contDiffOn.differentiableOn).differentiableAt eventually ✓: ∀ᶠ m in 𝓝 n, DifferentiableAt pᵢ m: from DifferentiableOn pᵢ (Ioi 20) and Ioi 20 ∈ 𝓝 n: filter_upwards [Ioi_mem_nhds h20] with m hm => (hCD.differentiableOn ..).differentiableAt (Ioi_mem_nhds hm) ✓.\n\nPer-piece ContDiffOn on Ioi 20 (needed by BOTH E and F): prove per piece a bundle lemma? E needs ContDiffOn of the WHOLE — which composes the same per-piece facts. So make per-piece ContDiffOn lemmas on Ioi 20 (10 pieces: S0, F∘T, 8 blocks, IIMaj, const) — the blocks: contDiffOn_Lx.comp (hT, hmaps). And T: contDiffOn_T : ContDiffOn ℝ 2 (fun n => √n + 1) (Ioi 20)?? — Ioi 20 ⊆ {0}ᶜ ✓ from Real.contDiffOn_sqrt on {0}ᶜ: (Real.contDiffOn_sqrt.mono (subset)) .add contDiffOn_const — or directly on Ioi 20 via sqrt contDiffOn (Ioi 0) mono. Check Real.contDiffOn_sqrt name.\n\nThis is a lot of pieces but all mechanical now. Let me check Real.contDiffOn_sqrt + ContDiffOn.comp signature, then write everything in 2-3 edits.","signature":""},{"type":"text","text":"Checking the last two API names for the assembly."},{"type":"tool_use","id":"Bash_262","name":"Bash","input":{"command":"grep -rn \"contDiffOn_sqrt\\|theorem ContDiffOn.comp \\|theorem ContDiff.comp \" /d/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/Analysis/SpecialFunctions/Sqrt.lean /d/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/Analysis/Calculus/ContDiff/Operations.lean 2>/dev/null | head -8","description":"Check sqrt ContDiff and ContDiffOn.comp"},"extra_content":null}]