[{"type":"thinking","thinking":"Edit C1 compiles (identity + all ContDiffOn pieces + marginExpanded regularity). Commit, then Edit C2: the main theorem.\n\nThe main theorem proof:\n\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) := by\n  have hCDfull : ContDiffOn ℝ 2 (marginTilde F) (Set.Ioi 20) := by\n    rw [marginTilde_eq_expanded F]\n    exact contDiffOn_marginExpanded F hF\n  apply convexOn_of_deriv2_nonneg (convex_Ici 20)\n  · exact hCDfull.continuous.continuousOn.mono Ioi_subset_Ici_self\n  · rw [interior_Ici]\n    exact hCDfull.differentiableOn (by norm_num)\n  · rw [interior_Ici]\n    exact ((hCDfull.deriv_of_isOpen (m := 1) isOpen_Ioi (by norm_num))).differentiableOn\n      one_ne_zero\n  · intro n hn\n    rw [interior_Ici] at hn\n    have hn20 : 20 < n := hn\n    have hn0 : (0 : ℝ) < n := by linarith\n    -- value computation\n    have hval : deriv (deriv (marginTilde F)) n =\n        1 / (2 * n) - 2 * psiOf F (√n+1) / (4*n) − ... hmm\n```\n\nWait — hψ's form: `deriv (deriv F) T - deriv F T / (T - 1) < 0` = psiOf F T < 0 by def ✓.\n\nNow the value chain. Pieces and values:\n\n- S: marginS_deriv2 c hn0 : deriv (deriv (mS-lambda)) n = 1/(2n) where the lambda is `fun m => (1/2)*m*Real.log m + c*m - 4` with c = lagariasC1 − γ/2 ✓ matches the marginExpanded's first piece ✓.\n- mFblock: deriv (deriv (mFblock F)) n = (-2) * (deriv (deriv fun m => F (√m+1)) n) via deriv2_const_mul; then chain_rule_sqrt_shift F hF n hn0 : ... = (deriv (deriv F) (√n+1) − deriv F (√n+1)/√n)/(4n); then ← psiOf_sqrt_rel: = psiOf F (√n+1)/(4n). So value: (-2) * (psiOf F (√n+1)/(4n)).\n- mTL1b: deriv2_const_mul (-w) + chain_rule_sqrt_shift_on (fun t => t*log(t+2)) contDiffOn_TL1 n hn0 + psiOf_sqrt_rel + psiOf_T_mul_log_shift: value (-w)*(psiOf-TL1-form (√n+1)/(4n)).\n- ... same pattern per block (TL2, id, L1sq, L1L2, L2sq, L1, L2). For mIdb: the block is id?? mIdb m = (−w)*(√m+1) — the composition id∘T = T itself: deriv (deriv (fun m => (−w)*(√m+1))) n: via deriv2_const_mul on (fun m => √m+1): chain_rule_sqrt_shift_on id (contDiffOn_id...)?? — contDiffOn_id : ContDiffOn ℝ 2 id (Ioi (−1))?? — contDiffOn_id works for any set ✓. psiOf_id ✓.\n- mIIM: deriv2_const_mul (−1) + IIMaj facts — value (−1) * deriv (deriv IIMaj) n (kept as atom).\n- mK: deriv2_const_val.\n\nChain: write as successive deriv2_add applications with the diff-facts from the piece ContDiffOns. The bookkeeping — for each deriv2_add I need for BOTH sides (prefix, piece): DifferentiableAt (deriv ·) n + eventually-diff. Sources:\n- piece: hCDpiece (contDiffOn_mXxx) → diffAt-deriv: (hCDpiece.deriv_of_isOpen (m := 1) isOpen_Ioi (by norm_num)).differentiableOn one_ne_zero |>.differentiableAt (Ioi_mem_nhds hn20); eventual: filter_upwards [Ioi_mem_nhds hn20] (fun m hm => (hCDpiece.differentiableOn (by norm_num)).differentiableAt (Ioi_mem_nhds hm)).\n- prefix: same from the accumulated ContDiffOn (hCDacc_k := hCDacc_{k-1}.add hCDpiece).\n\nTo reduce repetition, note the accumulated prefix ContDiffOn chain is exactly what's inside contDiffOn_marginExpanded — but I need the PREFIXES, not just the whole. I'll build them inline in the chain.\n\nHonestly the cleanest write-up: a sequence of haves:\n\n```lean\n    -- diff-fact helper\n    have hdd : ∀ {P : ℝ → ℝ}, ContDiffOn ℝ 2 P (Set.Ioi 20) →\n        DifferentiableAt ℝ (deriv P) n := fun hCD =>\n      ((hCD.deriv_of_isOpen (m := 1) isOpen_Ioi (by norm_num)).differentiableOn\n        one_ne_zero).differentiableAt (Ioi_mem_nhds hn20)\n    have hev : ∀ {P : ℝ → ℝ}, ContDiffOn ℝ 2 P (Set.Ioi 20) →\n        (∀ᶠ m in 𝓝 n, DifferentiableAt ℝ P m) := fun hCD => by\n      filter_upwards [Ioi_mem_nhds hn20] with m hm\n      exact (hCD.differentiableOn (by norm_num)).differentiableAt (Ioi_mem_nhds hm)\n```\nhmm hev's proof: `fun hCD => by filter_upwards ...` — mixed term/tactic: `fun hCD => by ...` ✓ works.\n\nThen per piece values:\n\n```lean\n    have hTn : Real.sqrt 20 + 1 ≤ Real.sqrt n + 1 := by\n      have := Real.sqrt_le_sqrt (show (20:ℝ) ≤ n by linarith)\n      linarith\n    have vS : deriv (deriv fun m => (1/2)*m*Real.log m + (lagariasC1 - Real.eulerMascheroniConstant/2)*m - 4) n = 1/(2*n) :=\n      marginS_deriv2 _ hn0\n    have vF : deriv (deriv (mFblock F)) n = (-2) * (psiOf F (Real.sqrt n + 1) / (4 * n)) := by\n      unfold mFblock\n      rw [deriv2_const_mul (-2) (hdd (contDiffOn_mFblock F hF)) (hev (contDiffOn_mFblock F hF)),\n        chain_rule_sqrt_shift F hF n hn0, ← psiOf_sqrt_rel]\n    ...\n```\nCareful with deriv2_const_mul's conclusion: deriv (deriv (fun m => c * f m)) n = c * deriv (deriv f) n — with f := fun m => F (√m+1): the goal after unfold mFblock: deriv (deriv fun m => (-2) * F (√m+1)) n = ... — deriv2_const_mul (−2) hdd-facts hev-facts — but hdd/hev need ContDiffOn of f = fun m => F(√m+1) — NOT of mFblock! The contDiffOn_mFblock is for (−2)*(...). I need ContDiffOn of the UNWEIGHTED pieces! Hmm — the mX ContDiffOn lemmas are for the weighted pieces. deriv2_const_mul needs facts about the unweighted f. Options: (a) prove ContDiffOn for unweighted compositions separately (the .comp part without the const_mul — that's actually inside each contDiffOn_mX proof — factor out: contDiffOn_TL1c := contDiffOn_TL1.comp contDiffOn_sqrt_shift mapsTo_sqrt_shift (unweighted composition bundle). (b) state deriv2_const_mul variant taking facts about c*f directly?? No — the combinator needs deriv f differentiability.\n\nSo I need per-block UNWEIGHTED composition ContDiffOn: define:\ncontDiffOn_TL1c : ContDiffOn ℝ 2 ((fun t => t * Real.log (t + 2)) ∘ (fun m => Real.sqrt m + 1)) (Set.Ioi 20) := contDiffOn_TL1.comp contDiffOn_sqrt_shift mapsTo_sqrt_shift\n— but the piece mTL1b's body is ((fun t => t*log(t+2)) (√m+1)) — beta of the composition ✓ defeq; hdd/hev need ContDiffOn of P where P := fun m => (fun t => t*log(t+2)) (√m+1) — is `exact contDiffOn_TL1c` accepted (defeq composition vs application)? Should be (function composition reducible). And chain_rule_sqrt_shift_on (fun t => t*log(t+2)) contDiffOn_TL1 n hn0 : deriv (deriv fun n => (fun t => t*log(t+2)) (√n+1)) n — hmm the chain-rule lemma's function: `fun n => φ (√n+1)` with φ := (fun t => ...) — φ (√n+1) is the application ✓ matches.\n\nLet me define a small bundle per block inline in the proof instead of 7 more top-level lemmas. In the main proof, per block:\n```lean\n    have hCDTL1c : ContDiffOn ℝ 2 (fun m => (fun t => t * Real.log (t + 2)) (Real.sqrt m + 1)) (Set.Ioi 20) :=\n      contDiffOn_TL1.comp contDiffOn_sqrt_shift mapsTo_sqrt_shift\n```\n— the comp's resulting function is (fun t => ...) ∘ (fun m => √m+1) — defeq to the application-lambda ✓ exact.\n\nThis is getting to be a LOT of haves (7 blocks × (hCD + value)). The value per block:\n```lean\n    have vTL1 : deriv (deriv fun m => (fun t => t * Real.log (t + 2)) (Real.sqrt m + 1)) n =\n        psiOf (fun t => t * Real.log (t + 2)) (Real.sqrt n + 1) / (4 * n) := by\n      rw [chain_rule_sqrt_shift_on (fun t => t * Real.log (t + 2)) contDiffOn_TL1 n hn0,\n        ← psiOf_sqrt_rel]\n```\nThen mTL1b's value: deriv (deriv mTL1b) n = (−w) * vTL1-value: deriv2_const_mul (−w) (hdd hCDTL1c) (hev hCDTL1c) gives deriv (deriv (fun m => (−w)*((fun t => t*log(t+2)) (√m+1)))) n = (−w) * deriv (deriv (fun m => (fun t => t*log(t+2))(√m+1))) n ✓ then rw [vTL1]. And mTL1b ≡ that lambda (unfold mTL1b ✓ defeq — the body IS that ✓ syntactically? mTL1b's def body: (-((8/3)*0.433 + 42)) * ((fun t => t * Real.log (t + 2)) (Real.sqrt m + 1)) ✓ EXACTLY ✓).\n\nThen the accumulation: unfold marginExpanded in the goal; the nested lambda; 11 deriv2_add's with prefix ContDiffOns. Prefix ContDiffOn list mirrors contDiffOn_marginExpanded's internals — I'll rebuild inline:\nhCD1 : ContDiffOn ℝ 2 (fun n => mS-lambda n + mFblock F n) (Ioi 20) := (contDiffOn_mS _).add (contDiffOn_mFblock F hF)\nhCD2 := hCD1.add contDiffOn_mTL1b\n... through hCD11 (all pieces) — 11 haves (~11 lines using previous).\nThen:\ne1 := deriv2_add (hdd (contDiffOn_mS _)) (hdd (contDiffOn_mFblock F hF)) (hev ...) (hev ...) : deriv (deriv (fun m => (mS m + mFblock F m))) n = ...\ne2 := deriv2_add (hdd hCD1) (hdd contDiffOn_mTL1b) (hev hCD1) (hev contDiffOn_mTL1b)\n...\ne11 := deriv2_add (hdd hCD10) (hdd contDiffOn_const) (hev hCD10) (hev contDiffOn_const)\n\nWait — the mK piece: its function in marginExpanded is the CONSTANT mK — as `fun n => mK`?? marginExpanded's body ends with `+ mK` — as a function of n that's fun _ => mK — deriv2_add with g := fun _ => mK: facts: DifferentiableAt (deriv (fun _ => mK)) n: deriv_const → const fun → differentiableAt_const ✓ (not from ContDiffOn — contDiffOn_const works too: hdd contDiffOn_const — the hdd helper expects ContDiffOn ℝ 2 P — contDiffOn_const : ContDiffOn 𝕜 n fun _ => c ✓ for any n including 2 ✓).\n\nThen the final goal: 0 ≤ deriv (deriv (marginExpanded-lambda)) n — rw [e11]? The e's rewrite stepwise: e11 : deriv (deriv (prefix10 + mK)) n = deriv (deriv prefix10) n + deriv (deriv (fun _ => mK)) n — rw [e11, deriv2_const_val] → ...+ 0; then e10 etc. Then substitute all piece values (vS, vF, vTL1-weighted...). Then the final inequality:\n\nmargiñ″(n) = 1/(2n) + (−2)·psiOf F Tn/(4n) + Σ (−wᵢ)·psiOfᵢ Tn/(4n) + (−1)·IIMaj″(n) + 0\n≥ 1/(2n) + 0 + ... + 0 − IIMaj″(n)   [−2ψF ≥ 0 since ψF < 0 (hψ Tn hTn); −wᵢψᵢ ≥ 0 since wᵢ > 0, ψᵢ < 0 (psiOf_*_neg hTn)]\n> 0   [IIMaj″(n) < 1/(2n) by IIMaj_deriv2_lt]\n\nThe linarith at the end needs the pieces: h1 : 0 < -(2 * psiOf F Tn / (4*n))... provide: hF1 : 0 ≤ -2 * (psiOf F Tn / (4*n)) : from hψ Tn hTn : psiOf F Tn < 0 (unfold psiOf at hψ's result? hψ gives deriv (deriv F) T − deriv F T/(T−1) < 0 — psiOf F T unfolds to that ✓ have hψn : psiOf F (√n+1) < 0 := hψ _ hTn — type: hψ's conclusion is the unfolded form — psiOf is a def — `hψ _ hTn : deriv (deriv F) (√n+1) − deriv F (√n+1)/((√n+1)−1) < 0` — and I need psiOf F (√n+1) < 0 — defeq ✓ exact hψ _ hTn (with psiOf reducible def — defeq ✓).\nThen per block: psiOf_T_mul_log_shift_neg (hTn) : psiOf (TL1-block) (√n+1) < 0 ✓ (statement form matches the value's psiOf argument ✓).\nPer-block weighted: 0 ≤ (-w) * (psiOf .. / (4*n)): from psiOf < 0 and −w < 0... wait the value is (−w)·(ψᵢ/(4n)) with w > 0: (−w) < 0, ψᵢ < 0 → product > 0 ✓. nlinarith or mul_pos_of_neg_of_neg / (4n) > 0. Provide per block: have hi : 0 ≤ (-wᵢ) * (psiOf φᵢ Tn / (4*n)) := by have h1 := psiOf_..._neg hTn; have h2 : (0:ℝ) < 4*n := by positivity?? (n > 0 ✓ linarith); positivity-ish: (−wᵢ)*(ψᵢ/(4n)) ≥ 0 ⟸ (−wᵢ) ≤ 0... write: have : (0:ℝ) ≤ (-wᵢ) * (psiOf φᵢ Tn / (4*n)) := by\n  have hp : psiOf φᵢ (√n+1) < 0 := psiOf_..._neg hTn\n  have hw : (-wᵢ : ℝ) < 0 := by norm_num\n  have hn4 : (0 : ℝ) < 4 * n := by linarith\n  have h3 : (0:ℝ) < psiOf φᵢ (√n+1) / (4*n) := by... no — ψᵢ/(4n) < 0: div_neg_of_neg_of_pos hp hn4; then (−wᵢ)·(neg) ≥ 0: mul_nonneg_of_nonpos_of_nonpos hw.le (div_neg_of_neg_of_pos hp hn4).le ✓ mul_nonneg_of_nonpos_of_nonpos ✓ exists.\nAnd the weights −wᵢ < 0 via norm_num (wᵢ are positive decimals/expressions with exp 4 + 1 > 0: −(42*0.433^2*(exp 4 + 1)) < 0: norm_num? exp 4 + 1 > 0 needs (Real.exp_pos 4): hw : (-(42*0.433^2*(Real.exp 4 + 1)) : ℝ) < 0 := by have := Real.exp_pos 4; nlinarith?? — hmm: 42*0.433^2*(exp4+1) > 0: mul_pos (by norm_num) (by linarith [Real.exp_pos 4])... nlinarith [Real.exp_pos 4] should manage: goal -(X) < 0 with X = 42*0.433^2*(E+1): nlinarith needs X > 0: E+1 > 1 > 0 ✓ nlinarith [Real.exp_pos 4]. Try per-weight.\n\nThis is big but writable. The final linarith: margiñ″ = 1/(2n) + (sum of ≥0 terms) − IIMaj″(n): 1/(2n) − IIMaj″(n) > 0 by IIMaj_deriv2_lt hn20.le; total ≥ that + 0s > 0 ✓ linarith [iimlt, hF1, h1b, ..., h8b].\n\nOne issue: the accumulated value expression's FORM — after rw [e1..e11, deriv2_const_val, vS, vF-weighted, ...], the goal 0 ≤ [big sum] — the big sum's exact term order/association — linarith normalizes ✓ atoms: psiOf F Tn, psiOf φᵢ Tn (each a def-application atom), deriv (deriv IIMaj) n, n, √n... — linarith treats them as atoms ✓ with the sign facts provided ✓.\n\nAlso for vF: value = (-2) * (psiOf F (√n+1)/(4*n)) — sign: −2·ψF/(4n) ≥ 0 from ψF < 0 ✓ mul_nonneg_of_nonpos_of_nonpos (by norm_num : (−2:ℝ) ≤ 0) (div_neg_of_neg_of_pos hψn hn4).le.\n\nNow write Edit C2. It's long; write carefully. The per-block pattern ×7 (TL1, TL2, id, L1sq, L1L2, L2sq, L1, L2 — that's 8 blocks: TL1, TL2, id, L1sq, L1L2, L2sq, L1, L2) + S + F + IIM + K = 12 pieces ✓ matches marginExpanded's 12 summands (mS, mFblock, mTL1b, mTL2b, mIdb, mL1sqb, mL1L2b, mL2sqb, mL1b, mL2b, mIIM, mK).\n\nBlock table (piece, weight-def, block φ, contDiffOn_block, psiOf value lemma, psiOf_neg lemma):\n1. mTL1b, w=((8/3)*0.433 + 42), φ=fun t => t*log(t+2), contDiffOn_TL1, psiOf_T_mul_log_shift, psiOf_T_mul_log_shift_neg\n2. mTL2b, w=((8/3)*0.886), φ=fun t => t*log(log(t+2)), contDiffOn_TL2, psiOf_T_mul_log_log_shift, _neg\n3. mIdb, w=((8/3)*2.9 + 192), φ=id, contDiffOn_id, psiOf_id (gives eq + <0 in one), — note psiOf_id's hypothesis is 1 < T ✓ (hTn gives √20+1 ≤ Tn → 1 < Tn ✓ linarith). The value: psiOf id Tn = −1/(Tn−1) ✓ and < 0 ✓ from the same lemma (conjunction).\n4. mL1sqb, w=(42*0.433^2*(e⁴+1)), φ=fun t => log(t+2)^2, contDiffOn_L1sq, psiOf_log_sq_shift, _neg\n5. mL1L2b, w=(42*(2*0.433*0.886)*(e⁴+1)), φ=fun t => log(t+2)*log(log(t+2)), contDiffOn_L1L2, psiOf_log_mul_log_shift, _neg\n6. mL2sqb, w=(42*0.886^2*(e⁴+1)), φ=fun t => log(log(t+2))^2, contDiffOn_L2sq, psiOf_log_log_sq_shift, _neg\n7. mL1b, w=((42*(2*0.433*2.9) + 147)*(e⁴+1)), φ=fun t => log(t+2), contDiffOn_L1, psiOf_log_shift, psiOf_log_shift_neg\n8. mL2b, w=(42*(2*0.886*2.9)*(e⁴+1)), φ=fun t => log(log(t+2)), contDiffOn_L2, psiOf_log_log_shift, _neg\n\nThe deriv2_const_mul weight: the piece def mTL1b has the weight written as (-((8/3)*0.433 + 42)) — deriv2_const_mul with c := (-((8/3)*0.433 + 42)) — must match the piece body syntactically after unfold ✓.\n\nFor the final expression I want terms like (-w)*(psiOf φ Tn/(4*n)) — nonneg facts per block.\n\nThe chain: 11 deriv2_add applications. Writing now — this will be a big edit (~200 lines). Let me be very careful with the structure. To reduce risk of association mismatches in the final linarith, after the e-rewrites the goal is `0 ≤ deriv (deriv (lambda)) n` — the e11 rewrite turns deriv (deriv (full-lambda)) n into the sum — then rw all piece values — then linarith closes. If some rw pattern fails (association/mismatch in how deriv2_add's LHS unifies with the goal's lambda), errors will guide.\n\nActually — a subtlety: after `unfold marginExpanded` in the goal, the lambda is `fun n => (((...(mS n + mFblock F n) + mTL1b n) + ...) + mK)`?? marginExpanded's body as I defined: `(((((((((((S n + mFblock F n) + mTL1b n) + mTL2b n) + mIdb n) + mL1sqb n) + mL1L2b n) + mL2sqb n) + mL1b n) + mL2b n) + mIIM n) + mK` ✓ left-nested ✓. deriv2_add unifies f := the prefix lambda, g := mK-piece-lambda (fun _ => mK? — the last term is `+ mK` — a CONSTANT TERM not applied to n!! The lambda body: ... + mIIM n) + mK — mK is ℝ, not (P n)! So the outermost sum: (fun n => prefix n) + ... hmm the body is `X n + mK` where mK : ℝ — as a function: fun n => X n + mK — deriv2_add with g := fun _ => mK: unification: (fun n => X n + mK) =?= f + g — f := fun n => X n, g := fun _ => mK — pointwise: f n + g n = X n + mK ✓ g n = mK ✓ unify g := fun _ => mK ✓ should work (higher-order pattern).\n\nSimilarly mIIM n ✓ piece.\n\nLet me also double check deriv2_add usage for e1: the FULL lambda's outermost structure: (prefix11 + mK) — the FIRST deriv2_add I apply should split the OUTERMOST: f := prefix11 (all but mK), g := fun _ => mK. Then rw [that] → deriv prefix11 + deriv const. Then next split prefix11 = prefix10 + mIIM: etc. So apply e11 first (outermost), then e10... in rw order: rw [e_outer, deriv2_const_val, e_next, ...]. The deriv2_add applications need the prefix ContDiffOns hCD10, hCD9, ... — build them as haves (hCD1..hCD10 incremental).\n\nOK writing. For hdd/hev helpers as local haves with {P} implicit — hdd hCD : DifferentiableAt ℝ (deriv P) n — P implicit from hCD's type ✓.\n\nOne more: the mS piece — its lambda in marginExpanded is anonymous: `fun m => (1/2)*m*Real.log m + (lagariasC1 − γ/2)*m − 4` ✓ marginS_deriv2's statement matches ✓ and contDiffOn_mS (c := lagariasC1 − γ/2) ✓.\n\nFor the F-piece: mFblock F — unfold mFblock in the value lemma? The value: deriv (deriv (mFblock F)) n — the accumulation rewrites need this exact form. deriv2_const_mul gives deriv (deriv fun m => (−2)*F(√m+1)) n — mFblock F ≡ that (unfold ✓). To bridge: `have vF : deriv (deriv (mFblock F)) n = (-2) * (psiOf F (√n+1)/(4*n)) := by unfold mFblock; rw [deriv2_const_mul ..., chain_rule_sqrt_shift F hF n hn0, ← psiOf_sqrt_rel]` — after unfold, goal: deriv (deriv fun m => (−2)*F(√m+1)) n = (−2)*(psiOf F (√n+1)/(4*n)): rw [deriv2_const_mul (−2) hdd-f hev-f] → goal: (−2) * deriv (deriv fun m => F(√m+1)) n = (−2)*(psiOf ...): rw [chain_rule_sqrt_shift F hF n hn0] → (−2)*((deriv (deriv F)(√n+1) − deriv F(√n+1)/√n)/(4*n)) = ...: rw [← psiOf_sqrt_rel] → matches ✓ closed by rfl (rw auto).\n\nThe deriv2_const_mul facts: hdd/hev for f := fun m => F (√m+1): need ContDiffOn ℝ 2 (fun m => F (√m+1)) (Ioi 20): (hF.contDiffOn.comp contDiffOn_sqrt_shift (Set.mapsTo_univ _ _)) ✓ — defeq with the lambda ✓ (composition). Have it as hCDFc.\n\nFor blocks: hCDTL1c etc. as above (composition, defeq to application lambda — `exact` should unify since ContDiffOn.comp produces (g ∘ f) — and the stated type uses the application lambda — defeq ✓).\n\nFor mIdb: piece (−w)*(√m+1) — f := fun m => √m+1 = id ∘ (√m+1)?? — chain_rule_sqrt_shift_on id contDiffOn_id n hn0 : deriv (deriv fun n => id (√n+1)) n = ... — id (√n+1) vs (√n+1): defeq ✓. psiOf_id Tn hT : psiOf id Tn = −1/(Tn−1) ∧ psiOf id Tn < 0 ✓ (hT : 1 < Tn).\n\nFor mIIM: deriv (deriv (mIIM)) n = (−1)*deriv (deriv IIMaj) n: deriv2_const_mul (−1) with facts for IIMaj: ContDiffOn ℝ 2 IIMaj (Ioi 20) — from contDiffOn_mIIM? No — mIIM is the WEIGHTED one; need unweighted IIMaj ContDiffOn — the hcomp inside contDiffOn_mIIM is psiTilde∘√ — and IIMaj ≡ fun m => psiTilde(√m) via hId: hCDIIM : ContDiffOn ℝ 2 IIMaj (Ioi 20) := by rw [hId]; exact (psiTilde_contDiff.contDiffOn).comp hsqrt' (mapsTo_univ) — with hsqrt' := contDiffOn_id.sqrt ... — build inline.\n\nmK piece: facts: contDiffOn_const works with hdd/hev helpers ✓ (they take ContDiffOn ℝ 2 P — contDiffOn_const : ContDiffOn 𝕜 n (fun _ => c) for ANY n ✓ n=2 ✓).\n\nFinal linarith assembly. Nonneg term facts:\n- hnn1 : 0 ≤ -2 * (psiOf F (√n+1) / (4*n)) := mul_nonneg_of_nonpos_of_nonpos (by norm_num) (div_neg_of_neg_of_pos hψn hn4).le — wait div_neg_of_neg_of_pos gives ψ/(4n) < 0 → .le ✓ then (−2)·(neg) ≥ 0: mul_nonneg_of_nonpos_of_nonpos (a ≤ 0) (b ≤ 0) : 0 ≤ a*b ✓ signature (ha : a ≤ 0) (hb : b ≤ 0) ✓.\n- per block: 0 ≤ (-wᵢ) * (psiOf φᵢ (√n+1) / (4*n)) := mul_nonneg_of_nonpos_of_nonpos hw.le (div_neg_of_neg_of_pos hp hn4).le with hw : (-wᵢ:ℝ) ≤ 0 := by norm_num / for exp-weights: (−(42*0.433^2*(exp 4 + 1))) ≤ 0: have := Real.exp_pos 4; nlinarith?? — try: by positivity? positivity can't prove ≤ 0 of negative... `by have h := Real.exp_pos 4; nlinarith [h]`? For the L1sq weight: −(42*0.433²·(E+1)): need 42·0.433²·(E+1) ≥ 0: nonneg: E+1 > 0 ✓ → neg ≤ 0: nlinarith [Real.exp_pos 4] — nlinarith with products of constants — should fold ✓. Let me write hw via: by have hE := Real.exp_pos 4; nlinarith [hE].\n\nAlternatively avoid per-weight sign fuss: the value term is (−wᵢ)·(ψᵢ/(4n)) — show ≥ 0 via: since ψᵢ < 0: ψᵢ/(4n) < 0; and (−wᵢ) < 0: product ≥ 0 ✓ — same thing.\n\nThe IIM term: value (−1)*deriv (deriv IIMaj) n — in the final sum. IIMaj_deriv2_lt hn20.le : deriv (deriv IIMaj) n < 1/(2*n) ✓.\n\nFinal: 0 ≤ 1/(2n) + hnn1-terms + (−1)*IIMaj″ + 0 — linarith [IIMaj_deriv2_lt, hnn1, hnn-blocks...]: goal 0 ≤ (1/(2n)) + Σ(nonneg) − IIMaj″: from IIMaj″ < 1/(2n): (1/(2n)) − IIMaj″ > 0 ✓ linarith with the nonneg facts ✓.\n\nCareful: after all the rw's, the goal's expression — the sum's exact shape (associations) — linarith normalizes ✓.\n\nLet me now also double-check `hψ`'s usage: the theorem gives hψ : ∀ T, √20+1 ≤ T → deriv (deriv F) T − deriv F T/(T−1) < 0. I need psiOf F (√n+1) < 0: psiOf F T unfolds to exactly that ✓ so `have hψn : psiOf F (√n+1) < 0 := hψ _ hTn` — type mismatch? hψ's conclusion vs psiOf application — psiOf is a def: `psiOf F (√n+1)` — the goal type psiOf F (√n+1) < 0 vs hψ's type deriv (deriv F) (√n+1) − ... < 0 — defeq via unfold — exact should accept (defeq at default transparency — psiOf is a regular def, unfolds ✓).\n\nWriting Edit C2 now — big.","signature":""},{"type":"text","text":"Edit C1 compiles. Committing, then Edit C2 — the main theorem."},{"type":"tool_use","id":"Bash_302","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: marginTilde_eq_expanded identity (ring-verified) + full ContDiffOn regularity of marginExpanded on Ioi 20\" && git log --oneline -1","description":"Commit identity and regularity"},"extra_content":null}]