[{"type":"thinking","thinking":"Confirmed:\n- `HasDerivAt.add_const : HasDerivAt (f · + c) f' x → HasDerivAt f f' x` — it's the mpr of `hasDerivAt_add_const_iff (c) : HasDerivAt (f · + c) f' x ↔ HasDerivAt f f' x` — so HasDerivAt.add_const REMOVES the const! Direction: mpr : HasDerivAt f f' x → HasDerivAt (f · + c) f' x — wait: alias ⟨_, HasDerivAt.add_const⟩ := hasDerivAt_add_const_iff — the iff's second component (mpr): Iff.mpr : RHS → LHS where LHS = HasDerivAt (f · + c) f' x, RHS = HasDerivAt f f' x. So HasDerivAt.add_const : HasDerivAt f f' x → HasDerivAt (f · + c) f' x — ADDS the const ✓ (takes h first? mpr of hasDerivAt_add_const_iff c — the (c) is the iff's explicit arg, so HasDerivAt.add_const (c)?? — alias preserves arg order: hasDerivAt_add_const_iff (c : F) : iff — .mpr applied: (hasDerivAt_add_const_iff c).mpr : HasDerivAt f f' x → HasDerivAt (f · + c) f' x. The alias HasDerivAt.add_const := fun c => (iff c).mpr — so HasDerivAt.add_const (c : F) (h : HasDerivAt f f' x)?? hmm — actually the alias ⟨_, X⟩ := thm means X := thm.2 = Iff.mpr — with c as a parameter: HasDerivAt.add_const c : HasDerivAt f f' x → HasDerivAt (f·+c) f' x. So usage: `(hasDerivAt_id' w₀).const_mul (5/2) |>.add_const??` hmm — as dot: h.add_const — if c comes first, dot-notation h.add_const puts h as the first EXPLICIT arg of matching type... Lean dot notation: h.add_const ⟨args⟩ inserts h at the first explicit arg of type HasDerivAt ... — if signature is add_const (c) (h'), then h.add_const discCenter-t would try to unify h with c : F ✗ then find h' slot ✓ (dot notation inserts into the first arg whose type matches the namespace head — HasDerivAt — so h goes to h' ✓ and c is explicit before... hmm dot notation: h.add_const X — h fills the first explicit HasDerivAt-typed arg, X fills remaining in order — if signature is (c : F) (h : HasDerivAt...) then h.add_const X: h → the HasDerivAt arg, X → c ✓ works regardless of order. So `(hasDerivAt_id' w₀).const_mul (5 / 2 : ℂ) |>.add_const (discCenter t)`? Hmm wait — const_mul first: HasDerivAt.const_mul (c) (h : HasDerivAt f f' x) : HasDerivAt (fun y => c * f y) (c * f') x? let me not verify — risk it and fix at compile. Actually simpler for hderiv_aff — use deriv_add_const/deriv_const_mul/deriv_id'' directly:\n\n```lean\n  have hderiv_aff : deriv (fun w : ℂ => discCenter t + (5 / 2 : ℂ) * w) w₀ = 5 / 2 := by\n    have h1 : (fun w : ℂ => discCenter t + (5 / 2 : ℂ) * w)\n        = fun w => (5 / 2 : ℂ) * w + discCenter t := funext fun w => by ring\n    rw [h1, deriv_add_const, deriv_const_mul (5 / 2 : ℂ) differentiableAt_id, deriv_id'']\n    ring\n```\nderiv_add_const (c) : deriv (fun y => f y + c) x = deriv f x ✓ (line ~111 in Deriv/Add). deriv_const_mul (c : 𝕜) (hd : DifferentiableAt 𝕜 f x) : deriv (fun y => c * f y) x = c * deriv f x ✓ standard. Then deriv (fun y => y)... wait after deriv_add_const: deriv (fun y => (5/2)*y) w₀; deriv_const_mul (5/2) differentiableAt_id — f := fun y => y? differentiableAt_id : DifferentiableAt 𝕜 id x — the function in deriv (fun y => (5/2)*y) is (fun y => y) — unify f := fun y => y vs id — deriv_const_mul's hd : DifferentiableAt 𝕜 f x — f from the LHS pattern deriv (fun y => c * f y) x — f := fun y => y — then hd must be DifferentiableAt ℂ (fun y => y) w₀ — differentiableAt_id : DifferentiableAt 𝕜 id x — id vs (fun y => y) — defeq ✓ elaboration of `differentiableAt_id` against the needed type — the needed type is a metavariable-directed... in rw [deriv_const_mul (5/2:ℂ) differentiableAt_id] — rw with the term: deriv_const_mul c hd : deriv (fun y => c * f y) x = c * deriv f x — f determined by hd's type: differentiableAt_id : DifferentiableAt ℂ id w₀ — f := id — so the rewrite pattern is deriv (fun y => (5/2) * id y) w₀ — goal has deriv (fun y => (5/2) * y) w₀ — id y vs y — syntactic mismatch ✗!! Use differentiableAt_id' : DifferentiableAt 𝕜 (fun x => x) x? exists: `differentiableAt_id' : DifferentiableAt 𝕜 (fun x => x) x` ✓ likely (paired with hasDerivAt_id'). Then f := fun y => y ✓ pattern deriv (fun y => (5/2)*(fun y => y) y)... — (fun y => y) applied — the LHS of deriv_const_mul is deriv (fun y => c * f y) x with f := (fun y => y): the pattern is deriv (fun y => c * (fun y => y) y) x — beta-redex vs goal's (5/2) * y — rw's matching is up to syntactic-with-some-defeq... rw uses `kabstract` which does NOT beta-reduce patterns by default... ugh. \n\nSIMPLEST ROBUST: prove hderiv_aff via HasDerivAt with explicit construction:\n```lean\n  have hderiv_aff : deriv (fun w : ℂ => discCenter t + (5 / 2 : ℂ) * w) w₀ = 5 / 2 := by\n    have hh : HasDerivAt (fun w : ℂ => (5 / 2 : ℂ) * w + discCenter t) ((5 / 2 : ℂ) * 1) w₀ :=\n      ((hasDerivAt_id' w₀).const_mul (5 / 2 : ℂ)).add_const (discCenter t)\n    ...\n```\nsame lambda-shape concerns. Honestly the cleanest: `fun_prop`-based HasDerivAt? There's `HasDerivAt` construction via `hasDerivAt_id'` + const_mul + add_const — the resulting function's lambda form: const_mul: HasDerivAt (fun x => c * f x)... with f := (fun x => x): fun x => c * (fun x => x) x — beta-redex AGAIN.\n\nAlternative — bypass entirely: deriv_const_add! `deriv_const_add (c : F) (hd : DifferentiableAt F f x) : deriv (fun y => c + f y) x = deriv f y x`?? hmm — there's `deriv_const_add (c : F) : deriv (fun y => c + f y) x = deriv f x`? I recall deriv_const_add (c) (hd)... exists: `theorem deriv_const_add (c : F) (hd : DifferentiableAt F f x) : deriv (fun y ↦ c + f y) x = deriv f x`? hmm not sure about hd requirement. Our function: fun w => discCenter t + (5/2)*w = const + (fun w => (5/2)*w): deriv_const_add (discCenter t) (hd : DifferentiableAt ℂ (fun w => (5/2)*w) w₀) → deriv (fun w => (5/2)*w) w₀. Then deriv_const_mul for that: fun w => (5/2)*w = fun w => (5/2)*id w?? — the pattern issue again: deriv_const_mul (c) (hd : DifferentiableAt 𝕜 f x) : deriv (fun y => c * f y) x — with the GOAL deriv (fun w => (5/2)*w) w₀: rw [deriv_const_mul] — rw with just the name: pattern deriv (fun y => ?c * ?f y) ?x — match against deriv (fun w => (5/2)*w) w₀: ?c := 5/2, ?f y := y ⟹ ?f := fun w => w — higher-order unification: ?f y =?= y — pattern unification ✓ ?f := fun y => y ✓ rw succeeds, side goal DifferentiableAt ℂ (fun w => w) w₀ — discharged by...? rw doesn't auto-discharge hd — deriv_const_mul has hd as a hypothesis — rw [deriv_const_mul] with the hypothesis as metavariable: creates side goal? rw [thm] where thm has hypotheses: they become NEW GOALS after the main one ✓ (rw can take such lemmas; the hypotheses become goals). Hmm — actually rw [lemma-with-hypotheses] fails with \"motive\" issues sometimes... it usually works and leaves side goals. Then `exact differentiableAt_id'`. And the final deriv (fun w => w) w₀ after deriv_const_mul rewrite: = (5/2) * deriv (fun w => w) w₀: rw [deriv_id'']: (5/2)*(fun _ => 1) w₀ = (5/2)*1... then ring/norm_num.\n\nHonestly, these micro-rewrites are unpredictable; the most robust is a one-liner via deriv's simp set:\n```lean\n    have hdiff : DifferentiableAt ℂ (fun w : ℂ => discCenter t + (5 / 2 : ℂ) * w) w₀ := by fun_prop\n    simp [deriv_const_add, deriv_const_mul, hdiff, deriv_id'']\n```\nHmm — let me just try variants at compile time. I'll write with the HasDerivAt approach using hasDerivAt_id' and adjust based on the error:\n\nattempt 1:\n```lean\n  have hderiv_aff : deriv (fun w : ℂ => discCenter t + (5 / 2 : ℂ) * w) w₀ = 5 / 2 := by\n    have hh : HasDerivAt (fun w : ℂ => discCenter t + (5 / 2 : ℂ) * w) (5 / 2 : ℂ) w₀ := by\n      have h1 := (hasDerivAt_id' w₀).const_mul (5 / 2 : ℂ)\n      -- h1 : HasDerivAt (fun x => (5/2) * x) ((5/2) * 1) w₀?? const_mul's value c*f'\n      have h2 := h1.const_add (discCenter t)  -- HasDerivAt.const_add? name!\n      ...\n```\nHasDerivAt.const_add (c) (h) : HasDerivAt (fun y => c + f y) ... — exists as alias of hasDerivAtFilter_const_add_iff presumably (HasDerivAt.const_add). Our form is discCenter t + (5/2)*w = const + f w with f w = (5/2)*w ✓ const_add fits directly!\n\nh1 := (hasDerivAt_id' w₀).const_mul (5/2:ℂ): HasDerivAt.const_mul (c : 𝕜) (hf : HasDerivAt f f' x) : HasDerivAt (fun y => c * f y) (c * f') y — dot: (hasDerivAt_id' w₀).const_mul (5/2:ℂ) : HasDerivAt (fun y => (5/2)*(fun x => x) y)?? — the body c * f y with f := fun x => x: c * (fun x => x) y — a redex (fun x => x) y — displayed/termed as (fun x => x) y — vs (5/2)*y target — rw/show bridging needed again. GRRR.\n\nOK — decisive robust approach: prove the HasDerivAt by fun_prop! `fun_prop` can prove HasDerivAt? fun_prop proves DifferentiableAt/Continuous/etc — and HasDerivAt? No — fun_prop doesn't compute the derivative VALUE. But there's `deriv` computation via `simp [deriv_add, deriv_const_mul, deriv_id'', ...]` with differentiability side conditions discharged by fun_prop... Actually the CLEANEST: use `deriv_add` with explicit differentiability:\n\n```lean\n  have hderiv_aff : deriv (fun w : ℂ => discCenter t + (5 / 2 : ℂ) * w) w₀ = 5 / 2 := by\n    rw [deriv_add (differentiableAt_const _) ((differentiableAt_id).const_mul _)]\n    rw [deriv_const, deriv_const_mul _ differentiableAt_id]\n    simp\n```\nHmm — deriv_add (hf : DifferentiableAt) (hg) : deriv (f + g) x = deriv f x + deriv g x — the function fun w => discCenter t + (5/2)*w = (fun _ => c) + (fun w => (5/2)*w) — the lambda (fun w => discCenter t + (5/2)*w) vs ((fun _ => c) + (fun w => (5/2)*w)) — Pi.add_apply defeq but syntactic match for rw [deriv_add ...] — deriv_add's LHS pattern: deriv (f + g) x — Hmm `deriv_add : deriv (f + g) x = ...`? statement: deriv_add (hf : DifferentiableAt 𝕜 f x) (hg : DifferentiableAt 𝕜 g x) : deriv (f + g) x = deriv f x + deriv g x — pattern deriv (f + g) x vs goal deriv (fun w => c + (5/2)*w) w₀: is (fun w => c + (5/2)*w) syntactically (?f + ?g)? The lambda body c + (5/2)*w is HAdd.hAdd application — while (f + g) is Pi.instAdd application — `fun w => c + (5/2)*w` ≡ (fun _ => c) + (fun w => (5/2)*w) up to defeq (Pi.add_apply is definitional: (f+g) w = f w + g w by rfl ✓) — but rw's kabstract won't match without syntactic alignment. Use `show` to bridge:\n```lean\n    show deriv ((fun _ : ℂ => discCenter t) + fun w => (5 / 2 : ℂ) * w) w₀ = 5 / 2\n```\ndefeq ✓ (Pi.add_apply rfl). Then rw [deriv_add (differentiableAt_const _) ((differentiableAt_id).const_mul _)]: gives deriv (fun _ => c) w₀ + deriv (fun w => (5/2)*w) w₀ — wait second: deriv ((fun w => (5/2)*w))... deriv_const_mul (c) (hd): deriv (fun y => c * f y) x = c * deriv f x — pattern fun y => (5/2)*w — ?f y := y: HOU ✓: rw [deriv_const_mul (5/2:ℂ) differentiableAt_id] — hmm f := id from differentiableAt_id — pattern fun y => (5/2)*id y vs fun w => (5/2)*w — id y vs y mismatch ✗ again. Use differentiableAt_id' (f := fun x => x): pattern fun y => (5/2)*(fun x => x) y — redex vs (5/2)*w ✗ STILL mismatch. The redex problem!! rw [deriv_id''] expects deriv (fun x => x)... \n\nYou know what, the way to kill ALL these lambda-matching issues: `deriv_const_mul_field`? no. USE simp with the differentiability facts — simp handles redexes up to beta-eta:\n\n```lean\n  have hderiv_aff : deriv (fun w : ℂ => discCenter t + (5 / 2 : ℂ) * w) w₀ = 5 / 2 := by\n    have hd : DifferentiableAt ℂ (fun w : ℂ => (5 / 2 : ℂ) * w) w₀ :=\n      differentiableAt_id'.const_mul _\n    rw [show (fun w : ℂ => discCenter t + (5 / 2 : ℂ) * w)\n        = ((fun _ : ℂ => discCenter t) + fun w => (5 / 2 : ℂ) * w) from rfl]\n    rw [deriv_add (differentiableAt_const _) hd, deriv_const, deriv_const_mul _ differentiableAt_id',\n      deriv_id'']\n    simp\n```\n- differentiableAt_id' : DifferentiableAt 𝕜 (fun x => x) x ✓ (paired with hasDerivAt_id' — assume exists).\n- (differentiableAt_id').const_mul _ : DifferentiableAt ℂ (fun w => (5/2) * (fun x=>x) w)?? const_mul's result function: fun y => c * f y with f := fun x => x — the TYPE displays as DifferentiableAt ℂ (fun y => (5/2)* (fun x=>x) y) w₀ — my hd ascribes DifferentiableAt ℂ (fun w => (5/2)*w) w₀ — ascription checks defeq ✓ (beta) — `have hd : T := term` — term elaborated against T with defeq ✓ fine.\n- deriv_add (differentiableAt_const _) hd : deriv ((fun _ => c) + fun w => (5/2)*w) w₀ = deriv (fun _ => c) w₀ + deriv (fun w => (5/2)*w) w₀ — the hd's type must match DifferentiableAt ℂ (fun w => (5/2)*w) w₀ ✓ by ascription ✓. rw with this instance: pattern deriv (?f + ?g) ?x ✓ matches the shown form ✓.\n- deriv_const : deriv (fun _ => c) x = 0 ✓ (`deriv_const (c) : deriv (fun _ => c) x = 0` hmm — deriv_const (c : F) (x)? `deriv_const : deriv (fun _ => c) x = 0`? I think deriv_const (c : F) : deriv (fun _ => c) x = 0 with x implicit ✓).\n- deriv_const_mul (5/2:ℂ) differentiableAt_id' : deriv (fun y => (5/2) * (fun x => x) y) w₀ = (5/2) * deriv (fun x => x) w₀ — the pattern has the redex — goal has deriv (fun w => (5/2)*w) w₀ — MISMATCH ✗!! \n\nOK here's the thing — deriv_const_mul's LHS: `deriv (fun y => c * f y) x` — with f := (fun x => x), that's `fun y => c * (fun x => x) y` — when Lean PRINTS it might beta-reduce, but the TERM has the redex. rw matching: kabstract finds instances of the pattern in the goal — the pattern fun y => c * (fun x => x) y vs goal subterm fun w => (5/2)*w — kabstract with the pattern's head: lambda — compares bodies: c * ((fun x=>x) y) vs (5/2)*w — up to defeq? kabstract uses `AbstractMVars` with defeq at instances reducible-ish... actually kabstract DOES use defeq (isDefEq) for matching subterms! `rw` uses kabstract which by default matches up to reducible transparency... redexes: (fun x => x) y ≡ y by beta (reducibility-independent, always on) ✓ SO IT MATCHES. Hmm — actually I recall rw failing on beta-redex mismatches sometimes... kabstract's `AbstractMVars` abstracts occurrences found via `isDefEq` with default-ish settings in recent versions (they improved this). It MIGHT work. \n\nEnough — I'll write it with the show+rws and a `simp` finisher, and iterate at compile. Actually the SIMPLEST of all: skip the HasDerivAt/deriv-lemma mess:\n\n```lean\n  have hderiv_aff : deriv (fun w : ℂ => discCenter t + (5 / 2 : ℂ) * w) w₀ = 5 / 2 := by\n    have : HasDerivAt (fun w : ℂ => discCenter t + (5 / 2 : ℂ) * w) (5 / 2 : ℂ) w₀ := by\n      convert ((hasDerivAt_id' w₀).const_mul (5 / 2 : ℂ)).const_add (discCenter t) using 1\n      ring\n    exact this.deriv\n```\nconvert ... using 1: goal HasDerivAt (fun w => c + (5/2)*w) (5/2) w₀ vs provided HasDerivAt (fun y => (5/2)*(fun x=>x) y + ...) hmm const_add's value: c * f' — ((5/2)*1) — and function fun y => C + (5/2)*(fun x=>x) y... convert using 1: reduces to function-equality and value-equality: fun-eq: (fun w => c + (5/2)*w) = (fun y => C + (5/2)*(fun x=>x) y) — convert leaves this as a goal? using 1 gives congruence goals: the two functions equal + values equal — funext+ring... messy too.\n\nFINAL DECISION — brute force via `HasDerivAt` constructor from simpler parts using exact combinators is getting silly. Use the calculus `fun_prop`+`deriv` simp route which is designed for this:\n\n```lean\n  have hderiv_aff : deriv (fun w : ℂ => discCenter t + (5 / 2 : ℂ) * w) w₀ = 5 / 2 := by\n    rw [deriv_add_const]\n    rw [deriv_const_mul (5 / 2 : ℂ) differentiableAt_id']\n    rw [deriv_id'']\n    simp\n```\n- deriv_add_const (c) : deriv (fun y ↦ f y + c) x = deriv f x — goal's function fun w => discCenter t + (5/2)*w is NOT of the form (fun y => f y + c) (const is on the LEFT) — so first convert: our function = fun w => (5/2)*w + discCenter t (comm): \n```lean\n    have h1 : (fun w : ℂ => discCenter t + (5 / 2 : ℂ) * w)\n        = fun w => (5 / 2 : ℂ) * w + discCenter t := funext fun w => by ring\n    rw [h1, deriv_add_const, deriv_const_mul (5 / 2 : ℂ) differentiableAt_id', deriv_id'']\n    simp\n```\nderiv_const_mul (c) (hd) : deriv (fun y => c * f y) x = c * deriv f x — with hd := differentiableAt_id' : DifferentiableAt ℂ (fun x => x) w₀ — f := fun x => x — the rw-pattern: deriv (fun y => (5/2) * (fun x => x) y) w₀ — goal (after deriv_add_const): deriv (fun w => (5/2)*w) w₀ — kabstract defeq match (beta) — SHOULD work (rw does abstract up to defeq — I've seen deriv_const_mul used in this project? journal gotchas don't mention trouble). deriv_id'' : deriv (fun x => x) = fun _ => 1 — rw: pattern `deriv fun x => x` (not applied!) — goal subterm deriv (fun x => x) w₀ — deriv_id'' as a rewrite: it's deriv (fun x => x) = fun _ => 1 — rewriting the FUNCTION — the goal contains deriv (fun x => x) applied to w₀ — rw [deriv_id''] rewrites `deriv (fun x => x)` subterm → (fun _ => 1) — then goal (fun _ => 1) w₀ = ... — beta-redex — then `simp` or the final (5/2)*((fun _ => 1) w₀) = 5/2: simp beta-reduces + ring: use `simp` then if needed. Actually after all rws the goal: (5/2) * ((fun _ : ℂ => 1) w₀) + 0 = 5/2 hmm order: deriv_add_const: deriv (f+c) = deriv f: goal deriv (fun w => (5/2)*w) w₀ = 5/2 — wait deriv_add_const (c : F) : deriv (fun y ↦ f y + c) x = deriv f x — rw replaces deriv (fun w => (5/2)*w + c) w₀ with deriv (fun w => (5/2)*w) w₀ ✓ goal: deriv (fun w => (5/2)*w) w₀ = 5/2. deriv_const_mul (5/2) differentiableAt_id': rewrites to (5/2) * deriv (fun x => x) w₀ = 5/2. deriv_id'': (5/2) * (fun _ => 1) w₀ = 5/2: simp: beta + mul_one ✓ — but the base: (fun _ => 1) w₀ — simp only [] beta? plain `simp` handles ✓. If deriv_id'' doesn't apply (form: deriv (fun x : 𝕜 => x) = fun _ => 1 — as a rewrite rule L→R on the FUNCTION deriv (fun x => x) — the goal has deriv (fun x => x) w₀ — the subterm deriv (fun x => x) appears ✓ rw works).\n\nOK — good enough; iterate at compile. Similarly for hd in hlogP: deriv (fun w : ℂ => w - ρ) w₀ = 1:\n```lean\n      have hd : deriv (fun w : ℂ => w - ρ) w₀ = 1 := by\n        rw [deriv_sub_const, deriv_id'']\n```\nderiv_sub_const (c) : deriv (fun y ↦ f y − c) x = deriv f x ✓ (confirmed exists line 414) — then deriv (fun w => w) w₀: deriv_id'' matches (fun w => w) ≡ (fun x => x) alpha ✓ — hmm deriv_id'' is about `fun x : 𝕜 => x` — alpha-equivalent ✓ — then (fun _ => 1) w₀ = 1 — rfl-close by rw ✓ or trailing simp. Add `; rfl` hmm rw closes if the goal becomes rfl after all rewrites — after deriv_id'' goal is (5/2)... no for hd: goal deriv (fun w => w−ρ) w₀ = 1: rw [deriv_sub_const]: deriv (fun w => w) w₀ = 1; rw [deriv_id'']: (fun _ => 1) w₀ = 1 — defeq-rfl — rw closes goals that become rfl (up to reducible defeq incl beta ✓). OK.\n\nOne more name check: differentiableAt_id' exists? and deriv_add_const's exact form `deriv_add_const (c : F) : deriv (fun y ↦ f y + c) x = deriv f x` — from the file listing above line ~111: `theorem deriv_add_const (c : F) : deriv (fun y ↦ f y + c) x = deriv f x` ✓ (I saw deriv_add_const in the sed output: \"theorem deriv_add_const (c : F) : deriv (fun y ↦ f y + c) x = deriv f x\" hmm the sed output showed derivWithin_add_const and deriv_add_const — yes line ~111: deriv_add_const exists ✓).\n\nNow assemble the main theorem text fully. Remaining pieces after the deriv identities:\n\n```lean\n  -- J_B from LogOfAnalyticFunction\n  obtain ⟨J_B, hJ_an, hJ0, hJderiv, hJre⟩ :=\n    LogOfAnalyticFunction (r := 3 / 5) (R := rC ht) (B := Cf (rC ht) (scaledZeta t))\n      (by norm_num) (by linarith [rC_ge ht])\n      ((Cf_collZeros_analytic ht).mono (Metric.closedBall_subset_closedBall (by\n        have h1 := rC_lt_one ht\n        linarith [rC_pos ht])))\n      (Cf_collZeros_ne_zero ht)\n  set Mbound : ℝ := Real.log ((9 / 4) * |t| ^ 2 / gapD t ^ collMultSum ht) with hMdef\n  have hMpos : 0 < Mbound := by\n    rw [hMdef]\n    apply Real.log_pos\n    have hδM : gapD t ^ collMultSum ht ≤ 1 :=\n      pow_le_one₀ (gapD_pos t).le ((gapD_le t).trans (by norm_num))\n    have hδM0 : 0 < gapD t ^ collMultSum ht := pow_pos (gapD_pos t) _\n    have hX1 : (1:ℝ) ≤ (9 / 4) * |t| ^ 2 := by\n      have hsq : (10:ℝ) ^ 2 ≤ |t| ^ 2 := (sq_le_sq₀ (by norm_num) (abs_nonneg t)).mpr ht\n      nlinarith\n    rw [le_div_iff₀ hδM0, one_mul]\n    exact hδM.trans hX1\n  have hReJ : ∀ z ∈ Metric.ball 0 (rC ht), (J_B z).re ≤ Mbound := by\n    intro z hz\n    have hzC : z ∈ Metric.closedBall 0 (rC ht) := Metric.ball_subset_closedBall hz\n    have h1 := hJre z hz\n    have hBz := Cf_collZeros_norm_le_ball ht hzC\n    have hBz0 := Cf_collZeros_ne_zero ht hzC\n    have hlogle : Real.log ‖Cf (rC ht) (scaledZeta t) z‖ ≤ Mbound :=\n      Real.log_le_log (norm_pos_iff.mpr hBz0) hBz\n    have hcen := Cf_collZeros_log_norm_zero_nonneg ht\n    rw [hMdef] at hlogle  -- Mbound def\n    rw [← h1]\n    linarith [hlogle, hcen]\n```\nhmm — hMdef : Mbound = Real.log(...): `set ... with hMdef` — hlogle : Real.log ‖B z‖ ≤ Mbound already (Mbound via set-folded in the statement? I wrote the have with Mbound directly ✓ since set replaced occurrences — but the RHS of hBz is the expanded form; Real.log_le_log ... hBz : Real.log ‖B z‖ ≤ Real.log((9/4)|t|²/δ^M) — with set active, does that display as Mbound? set folds occurrences in the GOAL at set-time, not in later terms. The `have hlogle : ... ≤ Mbound := Real.log_le_log ... hBz` — the ascribed type ≤ Mbound vs the term's ≤ Real.log(...) — defeq via hMdef (Mbound is a let/defined local): set introduces Mbound as a local DEFINITION (fvar with value) — defeq ✓ exact-elaboration ✓ fine.\n\nThen h1 : Real.log ‖B z‖ − Real.log ‖B 0‖ = (J_B z).re; goal (J_B z).re ≤ Mbound: rw [← h1] — rewrites (J_B z).re → the log-diff: goal log‖B z‖ − log‖B 0‖ ≤ Mbound: linarith [hlogle (log‖B z‖ ≤ Mbound), hcen (0 ≤ log‖B 0‖)] ✓.\n\nBC:\n```lean\n  have hBC : ‖deriv J_B w₀‖ ≤ 16 * Mbound * (rC ht) ^ 2 / (rC ht - 3 / 5) ^ 3 :=\n    BorelCaratheodoryDeriv hMpos (by norm_num) (by linarith [rC_ge ht])\n      hJ_an.analyticOn hJ0 hReJ (by\n        rw [Metric.mem_closedBall, dist_zero_right]\n        exact hw₀norm)\n  have h450 : 16 * Mbound * (rC ht) ^ 2 / (rC ht - 3 / 5) ^ 3 ≤ 450 * Mbound := by\n    have hrc1 := rC_ge ht\n    have hrc2 := rC_le ht\n    have hrc0 := rC_pos ht\n    have h1 : (rC ht) ^ 2 ≤ (24 / 25 : ℝ) ^ 2 :=\n      (sq_le_sq₀ hrc0.le (by norm_num)).mpr hrc2\n    have h2 : (8 / 25 : ℝ) ^ 3 ≤ (rC ht - 3 / 5) ^ 3 :=\n      pow_le_pow_left₀ (by norm_num) (by linarith) 3\n    rw [div_le_iff₀ (pow_pos (by linarith : (0:ℝ) < rC ht - 3 / 5) 3)]\n    nlinarith [hMpos.le, h1, h2, mul_nonneg hMpos.le (sub_nonneg.mpr h1),\n      mul_nonneg hMpos.le (sub_nonneg.mpr h2)]\n```\nGoal after rw: 16*Mbound*rC² ≤ 450*Mbound*(rC−3/5)³: nlinarith hints as planned ✓.\n\nderiv identity at w₀:\n```lean\n  have hderivJ : deriv J_B w₀\n      = deriv (Cf (rC ht) (scaledZeta t)) w₀ / Cf (rC ht) (scaledZeta t) w₀ :=\n    hJderiv w₀ (by rw [Metric.mem_closedBall, dist_zero_right]; exact hw₀norm)\n  have hderivB : deriv (Cf (rC ht) (scaledZeta t)) w₀\n      = deriv (fun w => scaledZeta t w / collProd ht w) w₀ := hB_eq.deriv\n  have hBval : Cf (rC ht) (scaledZeta t) w₀ = scaledZeta t w₀ / collProd ht w₀ :=\n    hB_eq.eq_of_nhds\n  have hlogFP : logDeriv (fun w => scaledZeta t w / collProd ht w) w₀\n      = (5 / 2 : ℂ) * logDeriv riemannZeta s\n        - ∑ ρ ∈ collZeros ht, (analyticOrderNatAt (scaledZeta t) ρ : ℂ) / (w₀ - ρ) := by\n    rw [hlogB, hlogF, hlogP]\n  have hkey : deriv J_B w₀ = (5 / 2 : ℂ) * logDeriv riemannZeta s\n      - ∑ ρ ∈ collZeros ht, (analyticOrderNatAt (scaledZeta t) ρ : ℂ) / (w₀ - ρ) := by\n    rw [hderivJ, hderivB, hBval]\n    exact hlogFP\n```\nhkey's last: goal deriv (F/P) w₀ / (F/P) w₀ = ... — exact hlogFP : logDeriv (F/P) w₀ = ... — logDeriv f x ≡ deriv f x / f x definitionally ✓ exact works via defeq (delta logDeriv — logDeriv is a def `logDeriv f := fun x => deriv f x / f x`? then logDeriv (F/P) w₀ ≡ deriv (F/P) w₀/(F/P) w₀ by delta+beta ✓).\n\nhζeq:\n```lean\n  have hζeq : logDeriv riemannZeta s\n      = (2 / 5 : ℂ) * deriv J_B w₀\n        + ∑ ρ ∈ collZeros ht, (analyticOrderNatAt (scaledZeta t) ρ : ℂ) /\n            (s - (discCenter t + (5 / 2 : ℂ) * ρ)) := by\n    have h1 : (5 / 2 : ℂ) * logDeriv riemannZeta s\n        = deriv J_B w₀ + ∑ ρ ∈ collZeros ht, (analyticOrderNatAt (scaledZeta t) ρ : ℂ) / (w₀ - ρ) := by\n      rw [hkey]\n      ring\n    calc logDeriv riemannZeta s\n        = (2 / 5 : ℂ) * ((5 / 2 : ℂ) * logDeriv riemannZeta s) := by ring\n      _ = (2 / 5 : ℂ) * (deriv J_B w₀ + ∑ ρ ∈ collZeros ht,\n            (analyticOrderNatAt (scaledZeta t) ρ : ℂ) / (w₀ - ρ)) := by rw [h1]\n      _ = (2 / 5 : ℂ) * deriv J_B w₀ + ∑ ρ ∈ collZeros ht,\n            (analyticOrderNatAt (scaledZeta t) ρ : ℂ) /\n              (s - (discCenter t + (5 / 2 : ℂ) * ρ)) := by\n          rw [mul_add, Finset.mul_sum]\n          apply congrArg ((2 / 5 : ℂ) * deriv J_B w₀ + ·)\n          apply Finset.sum_congr rfl\n          intro ρ hρ\n          have hX : (w₀ - ρ) ≠ 0 :=\n            sub_ne_zero.mpr (fun h => hFw₀ (by rw [h]; exact ((mem_collZeros ht).mp hρ).2))\n          have hden : s - (discCenter t + (5 / 2 : ℂ) * ρ) = (5 / 2 : ℂ) * (w₀ - ρ) := by\n            rw [hw₀def]; ring\n          rw [mul_div_assoc, hden]\n          field_simp [hX, (by norm_num : (5 / 2 : ℂ) ≠ 0)]\n          ring\n```\nHmm — h1's proof: goal (5/2)·logDeriv ζ s = deriv J_B w₀ + Σ; rw [hkey] rewrites deriv J_B w₀ → (5/2)L − Σ: goal (5/2)L = ((5/2)L − Σ) + Σ: ring ✓.\n\nThe congrArg application: `apply congrArg ((2 / 5 : ℂ) * deriv J_B w₀ + ·)` — the `·` in a term — congrArg (fun X => A + X)? congrArg (f) (h : a = b) : f a = f b — apply: goal A + X = A + Y — unify f ?a = A + X... f := fun X => A + X, ?a := X ✓ new goal X = Y ✓.\n\nWait — after rw [mul_add, Finset.mul_sum]: goal: (2/5)*deriv J_B w₀ + ∑ ρ, (2/5)*((m_ρ:ℂ)/(w₀−ρ)) = (2/5)*deriv J_B w₀ + ∑ ρ, (m_ρ:ℂ)/(s−aff ρ). congrArg ✓ then sum_congr per-term: (2/5)*((m:ℂ)/(w₀−ρ)) = (m:ℂ)/(s−affρ). rw [mul_div_assoc]: ((2/5)*m)/(w₀−ρ) = ...; rw [hden]: RHS denominator s−affρ → (5/2)(w₀−ρ): goal ((2/5)*m)/(w₀−ρ) = m/((5/2)*(w₀−ρ)): field_simp [hX, h5]: cross-mult: ((2/5)*m)*((5/2)*(w₀−ρ)) = m*(w₀−ρ): ring ✓ — field_simp might produce a different normal form; ring closes ✓.\n\nPush-forward and split:\n```lean\n  set img := (collZeros ht).image (fun w : ℂ => discCenter t + (5 / 2 : ℂ) * w) with himg\n  have hsum_img : ∑ ρ ∈ img, (zeroMult ρ : ℂ) / (s - ρ)\n      = ∑ ρ ∈ collZeros ht, (analyticOrderNatAt (scaledZeta t) ρ : ℂ) /\n          (s - (discCenter t + (5 / 2 : ℂ) * ρ)) := by\n    rw [himg, Finset.sum_image (affMap_injOn _)]\n    apply Finset.sum_congr rfl\n    intro ρ hρ\n    rw [zeroMult_affMap ht hρ]\n  have hsplit : ∑ ρ ∈ img, (zeroMult ρ : ℂ) / (s - ρ)\n      = ∑ ρ ∈ nearZeros t, (zeroMult ρ : ℂ) / (s - ρ)\n        + ∑ ρ ∈ img \\ nearZeros t, (zeroMult ρ : ℂ) / (s - ρ) := by\n    have hsub : nearZeros t ⊆ img := by rw [himg]; exact nearZeros_subset_image ht\n    rw [← Finset.sum_sdiff hsub]  -- hmm sum_sdiff: does it exist as to_additive of prod_sdiff?\n```\n`Finset.prod_sdiff [DecidableEq ι] (h : s₁ ⊆ s₂) : (∏ x ∈ s₂ \\ s₁, f x) * ∏ x ∈ s₁, f x = ∏ x ∈ s₂, f x` with @[to_additive] — so `Finset.sum_sdiff (h : s₁ ⊆ s₂) : (∑ x ∈ s₂ \\ s₁, f x) + ∑ x ∈ s₁, f x = ∑ x ∈ s₂, f x` EXISTS via to_additive ✓ (my earlier grep just didn't match because it's generated). So:\n```lean\n    have h := Finset.sum_sdiff hsub (f := fun ρ => (zeroMult ρ : ℂ) / (s - ρ))\n    -- h : (∑ extras) + (∑ near) = ∑ img\n    linarith?? no — complex sums. Use: \n    rw [eq_sub_iff_add_eq]... \n```\nCleaner: goal Σ_img = Σ_near + Σ_extras: from h : Σ_extras + Σ_near = Σ_img: `rw [add_comm] at h; exact h.symm` ✓:\n```lean\n    rw [add_comm] at h\n    exact h.symm\n```\nhmm h : Σ_extras + Σ_near = Σ_img; add_comm at h: hmm rw [add_comm] at h rewrites the FIRST + in h: Σ_near + Σ_extras = Σ_img ✓; .symm : Σ_img = Σ_near + Σ_extras ✓ exact.\n\nWait — hsub after `set img`: nearZeros_subset_image ht : nearZeros t ⊆ (collZeros ht).image aff — with img set-bound, rw [himg] at... the goal nearZeros t ⊆ img: rw [himg]: himg : img := ... — set's rewrite: himg : img = (collZeros ht).image ... — rw [himg] replaces img with the image ✓ then exact.\n\nFinal assembly:\n```lean\n  have hfinal : logDeriv riemannZeta s - ∑ ρ ∈ nearZeros t, (zeroMult ρ : ℂ) / (s - ρ)\n      = (2 / 5 : ℂ) * deriv J_B w₀ + ∑ ρ ∈ img \\ nearZeros t, (zeroMult ρ : ℂ) / (s - ρ) := by\n    rw [hζeq, ← hsum_img, hsplit]\n    ring\n  rw [hfinal]\n  -- the extras norm bound\n  have hextras : ‖∑ ρ ∈ img \\ nearZeros t, (zeroMult ρ : ℂ) / (s - ρ)‖\n      ≤ Real.log ((9 / 4) * |t| ^ 2) / Real.log (33 / 32) := by\n    -- from sdiff_sum machinery + sum_image_zeroMult_le\n    have h1 : ‖∑ ρ ∈ img \\ nearZeros t, (zeroMult ρ : ℂ) / (s - ρ)‖\n        ≤ ∑ ρ ∈ img \\ nearZeros t, (zeroMult ρ : ℝ) := by\n      refine (norm_sum_le _ _).trans (Finset.sum_le_sum fun ρ hρ => ?_)\n      have hρimg : ρ ∈ img := (Finset.mem_sdiff.mp hρ).1\n      have hρnotnear : ρ ∉ nearZeros t := (Finset.mem_sdiff.mp hρ).2\n      rw [himg] at hρimg\n      obtain ⟨w, hw, rfl⟩ := Finset.mem_image.mp hρimg\n      ... (per-term bound as planned) ...\n    refine h1.trans ?_\n    refine (Finset.sum_le_sum_of_subset_of_nonneg Finset.sdiff_subset\n      (fun i _ _ => Nat.cast_nonneg _)).trans ?_\n    rw [himg]\n    exact sum_image_zeroMult_le ht\n```\nWait — the per-term: after obtain rfl, goal: ‖(zeroMult (aff w) : ℂ)/(s − aff w)‖ ≤ (zeroMult (aff w) : ℝ). norm_div, norm_natCast: (m:ℝ)/‖s − aff w‖ ≤ m: need ‖s − aff w‖ ≥ 1: via himge (1 ≤ |(aff w).im − t|) — from hρnotnear: ρ = aff w ∉ nearZeros: with hρdisc (aff w ∈ zeroDiscFinset): mem_nearZeros would give |im−t| < 1 — so ¬ ⟹ 1 ≤ |im − t| ✓ (not_lt.mp). hρdisc: as planned (needs the div_eq_zero_iff resolve — ρ ∈ zeroDiscFinset: mem_zeroDiscFinset : ρ ∈ zeroDiscFinset t ↔ ρ ∈ closedBall ∧ ζ ρ = 0 ✓).\n\nnorm_natCast name: RCLike.norm_natCast — hmm in the RCLike/Basic.lean:654 it's inside `namespace RCLike`? The grep showed `theorem norm_natCast (n : ℕ) : ‖(n : K)‖ = n` — likely inside namespace RCLike ⟹ RCLike.norm_natCast ✓. Try that; fallback Complex.norm_natCast.\n\nThen:\n```lean\n  calc ‖(2 / 5 : ℂ) * deriv J_B w₀ + ∑ ρ ∈ img \\ nearZeros t, (zeroMult ρ : ℂ) / (s - ρ)‖\n      ≤ ‖(2 / 5 : ℂ) * deriv J_B w₀‖\n        + ‖∑ ρ ∈ img \\ nearZeros t, (zeroMult ρ : ℂ) / (s - ρ)‖ := norm_add_le _ _\n    _ ≤ (2 / 5) * (450 * Mbound) + Real.log ((9 / 4) * |t| ^ 2) / Real.log (33 / 32) := by\n        have h25 : ‖(2 / 5 : ℂ)‖ = 2 / 5 := by\n          rw [show (2 / 5 : ℂ) = ((2 / 5 : ℝ) : ℂ) by norm_num, Complex.norm_real,\n            Real.norm_eq_abs, abs_of_nonneg (by norm_num : (0:ℝ) ≤ 2 / 5)]\n        rw [norm_mul, h25]\n        exact add_le_add\n          (mul_le_mul_of_nonneg_left (hBC.trans h450) (by norm_num : (0:ℝ) ≤ 2 / 5)) hextras\n    _ ≤ 100000 * (Real.log |t|) ^ 2 := by\n        have h1 : Mbound ≤ 420 * (Real.log |t|) ^ 2 := by\n          rw [hMdef]\n          exact Mbound_le ht\n        have h2 := jensen_bound_elementary ht\n        have hL := two_le_log_abs_t ht\n        have h42 : 33 + 66 * Real.log |t| ≤ 42 * (Real.log |t|) ^ 2 := by\n          have hprod : (0:ℝ) ≤ (Real.log |t| - 2) * (42 * Real.log |t| + 17) :=\n            mul_nonneg (sub_nonneg.mpr hL) (by nlinarith [hL])\n          nlinarith [hprod, hL]\n        nlinarith [h1, h2, h42, hL, sq_nonneg (Real.log |t|)]\n```\nThe last nlinarith: goal (2/5)*(450*Mbound) + Jensen ≤ 100000L² with h1 : Mbound ≤ 420L², h2 : Jensen ≤ 33+66L, h42 : 33+66L ≤ 42L²: (2/5)*450*Mbound = 180*Mbound ≤ 75600L²; Jensen ≤ 42L²; total ≤ 75642L² ≤ 100000L²: nlinarith [h1, h2, h42, sq_nonneg]: 180*h1 + h42 + 24358*L²: coefficients 180 (on h1), 1 (h42), and 24358*(L² ≥ 0): nlinarith multiplies hypotheses by positive constants: 180*h1 ✓, h42 ✓, and needs (100000−75642)L² = 24358L² ≥ 0 — from sq_nonneg L times 24358 ✓. Should find it. Hmm — (2/5)*(450*Mbound): nlinarith normalizes the numeral arithmetic ✓ (2/5)*450 = 180 exactly ✓.\n\nWait — h2 is used but with the chain h2.trans h42 — nlinarith gets both, fine (it can chain linearly: Jensen ≤ 33+66L ≤ 42L² — nlinarith treats 33+66L as linear in L and 42L² quadratic — with h42 providing the bridge ✓).\n\nAlso — one important check: `hw₀norm` : ‖w₀‖ ≤ 3/5 — used for closedBall(0, 3/5) membership (hJderiv/hBC) AND closedBall(0,1) (hF_an: hw₀norm.trans (by norm_num : 3/5 ≤ 1)) ✓.\n\nAnd hBC's membership arg: w₀ ∈ closedBall 0 (3/5) ✓.\n\nI think there's one more subtlety: in hlogB, logDeriv_div's f := scaledZeta t (a def!) — logDeriv_div (x) (hf : f x ≠ 0)... — the function in logDeriv (fun w => scaledZeta t w / collProd ht w) w₀: f := fun w => scaledZeta t w? The lemma: logDeriv (fun z => f z / g z) x with f z := scaledZeta t z — unification f := scaledZeta t ✓ (the lambda fun w => scaledZeta t w is eta of scaledZeta t ✓ handled by unification with eta ✓ generally OK). hlogB's conclusion: logDeriv (scaledZeta t) w₀ − logDeriv (collProd ht) w₀ — then in hlogFP I rw [hlogB, hlogF, hlogP]: hlogF : logDeriv (scaledZeta t) w₀ = ... — hlogB's RHS contains logDeriv (scaledZeta t) w₀ ✓ matches hlogF's LHS ✓.\n\nAlso hlogF's statement: logDeriv (scaledZeta t) w₀ = (5/2)*logDeriv ζ s — inside its proof, `rw [hFeq]` — hFeq : scaledZeta t = fun w => (ζ∘aff) w / c — the goal: logDeriv (scaledZeta t) w₀ = (5/2)*logDeriv ζ s — rw [hFeq]: logDeriv (fun w => (ζ∘aff) w / c) w₀ = ... ✓ then logDeriv_div with f := (ζ∘aff), g := fun _ => c... logDeriv_div (x) (hf : f x ≠ 0) — f must match: (fun w => (ζ∘aff) w / c) = (fun z => f z / g z) with f := ζ∘aff? f z := (ζ∘aff) z ✓ f := riemannZeta ∘ aff ✓. hf : (ζ∘aff) w₀ ≠ 0 = hga ✓. hg : (fun _ => c) w₀ = c ≠ 0 = hζs0 ✓. hdf : DifferentiableAt ℂ (ζ∘aff) w₀ := hζs_diff'.comp w₀ haff_diff — DifferentiableAt.comp (x) (hf : DifferentiableAt f (g x)) (hg : DifferentiableAt g x) : DifferentiableAt (f ∘ g) x — signature `DifferentiableAt.comp (x : 𝕜) (hf : DifferentiableAt 𝕜' f (g x)) (hg : DifferentiableAt 𝕜 g x) : DifferentiableAt 𝕜' (f ∘ g) x` ✓ hζs_diff' : DifferentiableAt ℂ ζ (discCenter t + (5/2)*w₀) — needed at ((fun w => discCenter t + (5/2)*w) w₀) — the applied form — defeq beta ✓ unification should manage (the g in comp unifies with aff lambda, then g w₀ ≡ aff-expr by beta ✓).\n\nThe hlogcomp: logDeriv_comp hζs_diff' haff_diff : logDeriv (ζ ∘ aff) w₀ = logDeriv ζ (aff w₀) * deriv aff w₀ ✓.\n\nAfter rw [hlogcomp, hlogconst, sub_zero, hderiv_aff]: goal: logDeriv ζ (aff-applied w₀) * (5/2) = (5/2)*logDeriv ζ s — the aff-applied: ((fun w : ℂ => discCenter t + (5/2)*w) w₀) — beta-redex in the goal: `show logDeriv riemannZeta (discCenter t + (5 / 2 : ℂ) * w₀) * (5 / 2 : ℂ) = (5 / 2 : ℂ) * logDeriv riemannZeta s` — defeq beta ✓ — then rw [haff]: logDeriv ζ s * (5/2) = (5/2)*logDeriv ζ s: ring ✓. Hmm — but will `rw [hlogcomp]` work — hlogcomp's LHS: logDeriv (ζ ∘ aff) w₀ vs goal (after hFeq + logDeriv_div rw): logDeriv (ζ∘aff) w₀ − logDeriv (fun _ => c) w₀ — the term logDeriv (ζ∘aff) w₀ ✓ syntactic ✓.\n\nOK — also `show` in hζs_diff' : haff.symm ▸ hζs_diff: haff : aff-expr = s; haff.symm : s = aff-expr; ▸ rewrites s → aff-expr in hζs_diff's type: DifferentiableAt ℂ ζ (aff-expr) ✓ — the ▸ term: `haff.symm ▸ hζs_diff` — ascribed type given ✓ should elaborate (▸ as term with explicit target type works).\n\nOne more: `Real.log_le_iff_le_exp` — check name: it's `Real.log_le_iff_le_exp (hx : 0 < x) : log x ≤ y ↔ x ≤ exp y` — I used it in log_9_4_le_one and it compiled ✓ (chunk 3 green). Good.\n\nNow write the whole main theorem + hextras lemma. Actually — I planned hextras inline (h1 inside the final). Fine.\n\nAlso need: `Finset.mem_sdiff`, `Finset.sdiff_subset` ✓ standard. `not_lt.mp` ✓.\n\nLet me also prepare hρdisc's proof inline (used in hextras):\n```lean\n      have hρdisc : discCenter t + (5 / 2 : ℂ) * w ∈ zeroDiscFinset t := by\n        rw [mem_zeroDiscFinset]\n        constructor\n        · rw [Metric.mem_closedBall, Complex.dist_eq, add_sub_cancel_left, norm_mul]\n          have h25 : ‖(5 / 2 : ℂ)‖ = 5 / 2 := by\n            rw [show (5 / 2 : ℂ) = ((5 / 2 : ℝ) : ℂ) by norm_num, Complex.norm_real,\n              Real.norm_eq_abs, abs_of_nonneg (by norm_num : (0:ℝ) ≤ 5 / 2)]\n          rw [h25]\n          have hwle := ((mem_collZeros ht).mp hw).1\n          calc (5 / 2 : ℝ) * ‖w‖ ≤ (5 / 2) * rC ht :=\n                mul_le_mul_of_nonneg_left hwle (by norm_num)\n            _ = collRadius ht := by unfold rC; ring\n            _ ≤ 5 / 2 := (collRadius_le ht).trans (by norm_num)\n        · have hFw : scaledZeta t w = 0 := ((mem_collZeros ht).mp hw).2\n          unfold scaledZeta at hFw\n          exact (div_eq_zero_iff.mp hFw).resolve_right hζs0\n```\nCareful — hζs0 : riemannZeta (discCenter t) ≠ 0 — the div_eq_zero_iff.mp hFw : riemannZeta (discCenter t + (5/2)*w) = 0 ∨ riemannZeta (discCenter t) = 0 — .resolve_right hζs0 : riemannZeta (discCenter t + (5/2)*w) = 0 ✓ — and the goal (mem_zeroDiscFinset's second): riemannZeta (discCenter t + (5/2)*w) = 0 ✓ exactly. But wait — unfold scaledZeta at hFw: scaledZeta t w = riemannZeta (discCenter t + (5/2)*w) / riemannZeta (discCenter t) — the `unfold scaledZeta` — scaledZeta t w — unfold rewrites to the def body applied: (fun t w => ...) t w → after beta: the division ✓.\n\ndiv_eq_zero_iff : a / b = 0 ↔ a = 0 ∨ b = 0 ✓ (in GroupWithZero — `div_eq_zero_iff : a / b = 0 ↔ a = 0 ∨ b = 0` ✓).\n\nMetric.mem_closedBall : x ∈ closedBall c r ↔ dist x c ≤ r ✓; Complex.dist_eq : dist a b = ‖a − b‖ ✓ (or just dist_eq_norm). In the existing file they used `Metric.mem_closedBall, Complex.dist_eq` ✓ same pattern.\n\nadd_sub_cancel_left : a + b − a = b ✓ pattern (discCenter t + (5/2)*w) − discCenter t ✓.\n\nOK — writing the final block now. It's long; I'll write it as ONE edit and compile. Also mark todos.","signature":""},{"type":"text","text":"§10 green. Now the main theorem (§11) — the J_B/BC assembly."},{"type":"tool_use","id":"TodoWrite_97","name":"TodoWrite","input":{"todos":[{"content":"Read D-067 §6 + journal-023 + LiDisc.lean + LiContour.lean (context)","status":"completed","activeForm":"Reading context files"},{"content":"R-gap selection: scaled exists_avoiding_midpoint on [2.3, 2.4]","status":"completed","activeForm":"Building R-gap selection"},{"content":"g := ζ/∏(s−ρₖ)^mₖ analytic + nonvanishing on closed ball","status":"completed","activeForm":"Building g analytic/nonvanishing"},{"content":"Bounds: |g| outer circle + −log|g(s₀)|","status":"completed","activeForm":"Proving g bounds"},{"content":"BC assembly → loose R-bound theorem (main theorem)","status":"in_progress","activeForm":"Assembling Borel–Carathéodory"},{"content":"Build green + axiom audit + commit sub-stage","status":"pending","activeForm":"Building and auditing"}]},"extra_content":null}]