[{"type":"thinking","thinking":"shell_crit compiles. Now the final assembly theorem. Let me write it with the iterated filter splits and the three congruences.\n\n```lean\n/-- **The series circle bound** (ii′-5e-v-b capstone): on the γ-gap\ncircle |s| = R (k ≥ 4000), ‖seriesFun s‖ ≤ 2·lowZeroConst +\n2900·A₀·Ĉ₀·(log R)². -/\ntheorem seriesFun_norm_le_gapCircle {k : ℕ} (hk : 4000 ≤ k)\n    {R : ℝ} (hRlo : 4 * (k : ℝ) + 5 / 2 ≤ R) (hRhi : R ≤ 4 * (k : ℝ) + 3)\n    (hgap : ∀ ρ : ℂ, IsNontrivialZero ρ →\n      1 / (16 * C0hat (4 * (k : ℝ) + 9 / 2)) ≤ |ρ.im - R| ∨ 1 ≤ |ρ.im - R|)\n    {s : ℂ} (hs : ‖s‖ = R) :\n    ‖seriesFun s‖ ≤ 2 * lowZeroConst +\n      2900 * A0count * C0hat (4 * (k : ℝ) + 9 / 2) * (Real.log R) ^ 2 := by\n  classical\n  have hsz : ¬ IsNontrivialZero s :=\n    not_isNontrivialZero_on_gapCircle hk hRlo hgap hs\n  have hR : (16002 : ℝ) ≤ R := by\n    have h1 : (4000 : ℝ) ≤ (k : ℝ) := by exact_mod_cast hk\n    linarith [hRlo]\n  have hlogR : (9 : ℝ) ≤ Real.log R := nine_le_log_of_16002_le hR\n  have hlogRnn : (0 : ℝ) ≤ Real.log R := by linarith [hlogR]\n  have hĈ2 : (2 : ℝ) ≤ C0hat (4 * (k : ℝ) + 9 / 2) := two_le_C0hat (by\n    have hkR : (1 : ℝ) ≤ (k : ℝ) := by exact_mod_cast (by omega : 1 ≤ k)\n    linarith)\n  have hA0 := A0count_pos.le\n  set F := circleZeros R with hF\n  set g : {ρ : ℂ // IsNontrivialZero ρ} → ℝ := fun ρ => ‖seriesTerm s ρ‖ with hg\n  -- the five-way partition of the finite-part sum\n  set q1 : {ρ : ℂ // IsNontrivialZero ρ} → Prop := fun ρ => ‖(ρ : ℂ)‖ ≤ 3 with hq1\n  set q2 : {ρ : ℂ // IsNontrivialZero ρ} → Prop := fun ρ => ‖(ρ : ℂ)‖ ≤ R / 2 with hq2\n  set q3 : {ρ : ℂ // IsNontrivialZero ρ} → Prop := fun ρ => ‖(ρ : ℂ)‖ ≤ R - 2 with hq3\n  set q4 : {ρ : ℂ // IsNontrivialZero ρ} → Prop := fun ρ => ‖(ρ : ℂ)‖ < R + 2 with hq4\n  have hsplit1 : ∑ ρ ∈ F, g ρ\n      = ∑ ρ ∈ F.filter q1, g ρ + ∑ ρ ∈ F.filter (fun ρ => ¬ q1 ρ), g ρ :=\n    Finset.sum_filter_add_sum_filter_not F q1 g\n  ...\n```\nHmm wait — Finset.sum_filter_add_sum_filter_not (s) (p) (f) : ∑ x ∈ s, f x = ∑ x ∈ s.filter p, f x + ∑ x ∈ s.filter (fun x => ¬ p x), f x ✓.\n\nThe shell finsets: \n- shell_low's: F.filter (fun ρ : ST => ‖(ρ:ℂ)‖ ≤ 3) = F.filter q1 ✓ (q1 defined as that lambda ✓ defeq).\n- shell_inlo's: F.filter (fun ρ : ST => ¬‖(ρ:ℂ)‖ ≤ 3 ∧ ‖(ρ:ℂ)‖ ≤ R/2): from split of F.filter (¬q1) by q2: (F.filter (¬q1)).filter q2 = F.filter (fun ρ => ¬q1 ρ ∧ q2 ρ) (Finset.filter_filter) ✓ matches (¬‖ρ‖≤3 ∧ ‖ρ‖≤R/2) ✓ defeq.\n- shell_inhi's: F.filter (¬‖ρ‖ ≤ R/2 ∧ ‖ρ‖ ≤ R−2): the iterated piece is F.filter ((¬q1 ∧ ¬q2) ∧ q3) — congruence.\n- shell_crit's: F.filter (¬‖ρ‖ ≤ R−2 ∧ ‖ρ‖ < R+2): piece F.filter (((¬q1 ∧ ¬q2) ∧ ¬q3) ∧ q4) — congruence.\n- shell_up's: F.filter (¬‖ρ‖ < R+2): piece F.filter (((¬q1 ∧ ¬q2) ∧ ¬q3) ∧ ¬q4) — congruence.\n\nThe congruences (filter ext with the R-bounds):\n```\nhave hc3 : F.filter (fun ρ => (¬ q1 ρ ∧ ¬ q2 ρ) ∧ q3 ρ)\n    = F.filter (fun ρ => ¬ q2 ρ ∧ q3 ρ) := by\n  ext ρ\n  simp only [Finset.mem_filter, hq1, hq2, hq3]\n  constructor\n  · intro h; exact ⟨⟨h.2.1.2?...\n```\nHmm — with set-defined q1..q4, simp only [Finset.mem_filter, hq1, hq2, hq3] unfolds the predicates ✓. Then the membership forms: LHS: ρ ∈ F ∧ ((¬‖ρ‖≤3 ∧ ¬‖ρ‖≤R/2) ∧ ‖ρ‖≤R−2); RHS: ρ ∈ F ∧ (¬‖ρ‖≤R/2 ∧ ‖ρ‖≤R−2). Forward: drop. Backward: need ¬‖ρ‖≤3 from ¬‖ρ‖≤R/2: ‖ρ‖ > R/2 ≥ 8001 > 3: linarith [hR]. Structure:\n```\n  constructor\n  · rintro ⟨hF, ⟨h1, h2⟩, h3⟩\n    exact ⟨hF, h2, h3⟩\n  · rintro ⟨hF, h2, h3⟩\n    refine ⟨hF, ⟨?_, h2⟩, h3⟩\n    have hρ : R / 2 < ‖(ρ : ℂ)‖ := lt_of_not_ge h2\n    exact lt_of_not_ge... goal ¬‖ρ‖ ≤ 3: \n    · exact fun h => by linarith [hρ, hR]  -- hmm: ¬‖ρ‖≤3 ⟺ fun (h : ‖ρ‖≤3) => False: linarith\n```\n`exact fun h => by linarith [hρ, hR, h]` — or `exact not_le_of_gt (by linarith [hρ, hR] : (3:ℝ) < ‖(ρ:ℂ)‖)`. \n\nSimilarly hc4: LHS ((¬q1∧¬q2)∧¬q3)∧q4 → (¬q3 ∧ q4): drop: ¬q3 gives ‖ρ‖ > R−2 > 3 and > R/2 ✓ linarith.\nhc5: → ¬q4: drop: ¬q4 gives ‖ρ‖ ≥ R+2 > R−2, R/2, 3 ✓.\n\nThe assembling:\n```\n  have hsplit2 : ∑ ρ ∈ F.filter (fun ρ => ¬ q1 ρ), g ρ\n      = ∑ ρ ∈ (F.filter (fun ρ => ¬ q1 ρ)).filter q2, g ρ\n        + ∑ ρ ∈ (F.filter (fun ρ => ¬ q1 ρ)).filter (fun ρ => ¬ q2 ρ), g ρ :=\n    Finset.sum_filter_add_sum_filter_not _ q2 g\n  have hff2 : (F.filter (fun ρ => ¬ q1 ρ)).filter q2\n      = F.filter (fun ρ => ¬ q1 ρ ∧ q2 ρ) := Finset.filter_filter _ _ _\n```\nCareful — Finset.filter_filter : (s.filter p).filter q = s.filter (fun a => p a ∧ q a) — direction: filter q (filter p s) = filter (fun a => p a ∧ q a) s ✓. So (F.filter ¬q1).filter q2 = F.filter (fun ρ => (¬q1 ρ) ∧ q2 ρ) ✓.\n\nThis is getting verbose. Alternative: do the whole partition via a single big have using the 5-way split lemma I prove inline:\nActually, simplest robust structure — compute step by step with rw [hsplit1, hsplit2, ...] and congruences as rewrites:\n\n```\n  have hsplit : ∑ ρ ∈ F, g ρ\n      = (((∑ ρ ∈ F.filter q1, g ρ) + ∑ ρ ∈ F.filter (fun ρ => ¬q1 ρ ∧ q2 ρ), g ρ)\n        + ∑ ρ ∈ F.filter (fun ρ => ¬q2 ρ ∧ q3 ρ), g ρ)\n        + (∑ ρ ∈ F.filter (fun ρ => ¬q3 ρ ∧ q4 ρ), g ρ\n          + ∑ ρ ∈ F.filter (fun ρ => ¬ q4 ρ), g ρ) := ...\n```\nThe association/nesting: split1: S = S1 + T1 where T1 = ∑_{filter ¬q1}. split2: T1 = S2 + T2 where S2 = ∑_{filter (¬q1∧q2)} — wait (F.filter ¬q1).filter q2 = F.filter(¬q1 ∧ q2) ✓ and T2 = ∑_{(F.filter ¬q1).filter ¬q2} = ∑_{F.filter (¬q1∧¬q2)}. split3: T2 = S3 + T3, S3 = ∑_{(F.filter (¬q1∧¬q2)).filter q3} = ∑_{F.filter ((¬q1∧¬q2)∧q3)} =[hc3]= ∑_{F.filter (¬q2∧q3)}. T3 = ∑_{F.filter ((¬q1∧¬q2)∧¬q3)}. split4: T3 = S4 + S5: S4 = ∑_{filter (((¬q1∧¬q2)∧¬q3)∧q4)} =[hc4]= ∑_{F.filter (¬q3∧q4)}; S5 = ∑_{filter (((¬q1∧¬q2)∧¬q3)∧¬q4)} =[hc5]= ∑_{F.filter ¬q4}.\n\nThen S ≤ S1 + S2 + S3 + S4 + S5 ≤ 2·lowZeroConst + 60A₀L² + 60A₀L² + 2688ĈA₀L + 45A₀L² and the tail 91A₀L.\n\nThe shell applications: shell_low hs hsz hR — its finset: (circleZeros R).filter (fun ρ : ST => ‖(ρ:ℂ)‖ ≤ 3) — F.filter q1: is F.filter q1 DEFEQ to that? F := circleZeros R (set alias), q1 := (fun ρ => ‖(ρ:ℂ)‖ ≤ 3) (set alias) — after `set`, occurrences are replaced by the aliases: shell_low's statement has the unfolded forms... When I `have h1 := shell_low hs hsz hR`, h1's type displays with circleZeros R and the lambda — matching F.filter q1 requires unfolding the set aliases. `set ... with hF` introduces hF : F = circleZeros R — I can `rw [hF] at ...` hmm — set replaces occurrences of circleZeros R in the GOAL by F; hypotheses h1 keep circleZeros R. Then ∑ ρ ∈ F.filter q1, g ρ vs h1's ∑ ρ ∈ (circleZeros R).filter (fun ρ => ‖(ρ:ℂ)‖ ≤ 3), ‖seriesTerm s ρ‖: to use h1 as a bound for the first piece, I need the pieces stated in the shell's unfolded form. Simplest: DON'T use set for F/g/q1..q4 — write everything unfolded (verbose but no mismatch). The g alias: shell lemmas give bounds on ∑ ‖seriesTerm s ρ‖ directly — keep g := fun ρ => ‖seriesTerm s ρ‖ as a plain lambda everywhere (no set).\n\nOK let me write it fully unfolded. The predicates as plain lambdas with pinned binder types. The sum_filter_add_sum_filter_not applications with explicit p:\n\n```\n  have hsplit1 : ∑ ρ ∈ circleZeros R, (fun ρ => ‖seriesTerm s ρ‖) ρ\n      = ∑ ρ ∈ (circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} => ‖(ρ : ℂ)‖ ≤ 3),\n          (fun ρ => ‖seriesTerm s ρ‖) ρ\n        + ∑ ρ ∈ (circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n            ¬ ‖(ρ : ℂ)‖ ≤ 3), (fun ρ => ‖seriesTerm s ρ‖) ρ :=\n    Finset.sum_filter_add_sum_filter_not _ _ _\n```\nHmm — the summand in seriesFun_norm_le_split is `‖seriesTerm s ρ‖` (an application, not a lambda-applied). tsum_split_finset gave: ∑ ρ ∈ circleZeros R, ‖seriesTerm s ρ‖ — the elaborated form is ∑ ρ ∈ circleZeros R, (fun ρ => ‖seriesTerm s ρ‖) ρ — the display is the application. sum_filter_add_sum_filter_not _ _ _ should unify (the p gets inferred from... the goal — since I'm ascribing hsplit1's type, unification pins p := (fun ρ : ST => ‖ρ‖ ≤ 3) ✓).\n\nThen the shell bound applications: shell_low hs hsz hR : ∑ ρ ∈ (circleZeros R).filter (fun ρ : ST => ‖(ρ:ℂ)‖ ≤ 3), ‖seriesTerm s ρ‖ ≤ 2·lowZeroConst — the summand there is ‖seriesTerm s ρ‖ — application vs (fun ρ => ...)ρ — defeq (beta) ✓ linarith can bridge? For the final linarith I'll pass the shell bounds as haves; linarith sees the sum terms as atoms — the atoms must be SYNTACTICALLY equal for linarith to match! The beta-redex vs reduced application: linarith normalizes via... linarith uses syntactic atom matching up to reducible defeq — beta-redexes might not match! Banked gotcha from session 031: \"state split goals with `show` in reduced form\" — the summand beta-redex issue. To be safe: beta-reduce the pieces with `show` or `simp only []`... The cleanest: after establishing hsplit (the partition identity with the pieces in whatever form), convert with `conv` to beta-reduce... Alternatively state hsplit with the summand ALREADY as application: sum_filter_add_sum_filter_not's conclusion has f applied — with f := fun ρ => ‖seriesTerm s ρ‖ the terms are (fun ρ => ...) ρ — beta-redexes. If instead I apply sum_filter_add_sum_filter_not with f := (fun ρ => ‖seriesTerm s ρ‖) — same thing. The Finset.sum notation: ∑ ρ ∈ s, f ρ — the pretty printer shows ‖seriesTerm s ρ‖ when f is the lambda (it eta-reduces? The sum is over `(fun ρ => ‖seriesTerm s ρ‖)` — displayed as ∑ ρ ∈ s, ‖seriesTerm s ρ‖). Shell lemmas' sums are over the same lambda — same atoms ✓. I think it's fine since both come from the same syntactic source. Just be consistent: everywhere use the summand `‖seriesTerm s ρ‖` and let Lean unify.\n\nFor hsplit's pieces after filter_filter rewrites — the conjunction lambdas must match the shells' pinned lambdas exactly:\n- inlo: (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} => ¬ ‖(ρ : ℂ)‖ ≤ 3 ∧ ‖(ρ : ℂ)‖ ≤ R / 2) ✓ copied from shell_inlo.\n- inhi: (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} => ¬ ‖(ρ : ℂ)‖ ≤ R / 2 ∧ ‖(ρ : ℂ)‖ ≤ R - 2) ✓.\n- crit: (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} => ¬ ‖(ρ : ℂ)‖ ≤ R - 2 ∧ ‖(ρ : ℂ)‖ < R + 2) ✓.\n- up: (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} => ¬ ‖(ρ : ℂ)‖ < R + 2) ✓.\n\nThe congruences hc3/hc4/hc5 rewrite the iterated pieces to these. The intermediate forms from filter_filter: (fun ρ => (fun ... ¬q1-ish) ρ ∧ ...) — hmm filter_filter gives filter (fun a => p a ∧ q a) with p,q the lambdas — the resulting lambda is (fun ρ => ¬‖(ρ:ℂ)‖ ≤ 3 ∧ ‖(ρ:ℂ)‖ ≤ R/2) — with the SAME binder type as the pieces? p := (fun ρ : ST => ¬‖(ρ:ℂ)‖ ≤ 3) — binder pinned ✓ then the conjunction lambda's binder : ST ✓ and the body ¬‖(ρ:ℂ)‖ ≤ 3 ∧ ‖(ρ:ℂ)‖ ≤ R/2 — should match shell_inlo's lambda syntactically if I write the filters with identical text. ✓\n\nNow write the assembly:\n\n```lean\n  -- shells\n  have hsh1 := shell_low hs hsz hR\n  have hsh2 := shell_inlo hs hsz hR\n  have hsh3 := shell_inhi hs hsz hR\n  have hsh4 := shell_crit hk hRlo hgap hs hsz\n  have hsh5 := shell_up hs hsz hR\n  have htail := series_tail_le hs hR\n  have hsplit0 := seriesFun_norm_le_split (s := s) (R := R)\n  -- the partition identity\n  have hsplit1 : ... := Finset.sum_filter_add_sum_filter_not (circleZeros R)\n    (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} => ‖(ρ : ℂ)‖ ≤ 3) (fun ρ => ‖seriesTerm s ρ‖)\n  have hsplit2 : ... (on the ¬q1 piece by q2)\n  have hsplit3 : ... \n  have hsplit4 : ...\n  have hff2 : ((circleZeros R).filter (¬q1)).filter q2 = (circleZeros R).filter (¬q1 ∧ q2) := Finset.filter_filter _ _ _\n  ...\n  have hc3 : (circleZeros R).filter (fun ρ : ST => (¬‖(ρ:ℂ)‖ ≤ 3 ∧ ¬‖(ρ:ℂ)‖ ≤ R/2) ∧ ‖(ρ:ℂ)‖ ≤ R−2)\n      = (circleZeros R).filter (fun ρ : ST => ¬‖(ρ:ℂ)‖ ≤ R/2 ∧ ‖(ρ:ℂ)‖ ≤ R−2) := by ext...\n  have hc4 : ... = crit's\n  have hc5 : ... = up's\n```\n\nThen chain everything with linarith [hsplit0, htail, hsh1..5, hsplit1..4, hff2.., hc3..5, hfin].\n\nHmm — linarith with EQUALITIES of finset sums: the partition gives sum-equalities (S = S1 + T1 etc.); the shells give inequalities on the pieces; linarith combines linear ✓ as long as the atoms align. The pieces after rewriting via hff2/hc3/hc4/hc5 must be TEXTUALLY the shells' sums. Use rw to normalize the pieces in the split chain:\n\nActually cleanest: build the full partition as one big have by successive rw:\n```\n  have hpart : ∑ ρ ∈ circleZeros R, ‖seriesTerm s ρ‖\n      = (((∑ ρ ∈ (circleZeros R).filter (low), ‖seriesTerm s ρ‖\n        + ∑ ρ ∈ (circleZeros R).filter (inlo), ‖seriesTerm s ρ‖)\n        + ∑ ρ ∈ (circleZeros R).filter (inhi), ‖seriesTerm s ρ‖)\n        + ∑ ρ ∈ (circleZeros R).filter (crit), ‖seriesTerm s ρ‖)\n        + ∑ ρ ∈ (circleZeros R).filter (up), ‖seriesTerm s ρ‖ := by\n    rw [Finset.sum_filter_add_sum_filter_not (circleZeros R) (low) _,\n      Finset.sum_filter_add_sum_filter_not _ (fun ρ : ST => ‖(ρ:ℂ)‖ ≤ R/2) _,\n      Finset.sum_filter_add_sum_filter_not _ (fun ρ : ST => ‖(ρ:ℂ)‖ ≤ R−2) _,\n      Finset.sum_filter_add_sum_filter_not _ (fun ρ : ST => ‖(ρ:ℂ)‖ < R+2) _]\n    -- now the pieces are in filter-of-filter form; normalize with filter_filter + congruences\n    rw [Finset.filter_filter, Finset.filter_filter, Finset.filter_filter, ...]\n    -- then congruence rewrites hc3 hc4 hc5\n    rw [hc3, hc4, hc5]\n```\nThe rewriting chain in a big goal — fragile (the filter predicates in the rw's must match). Alternative: establish the sum-chain via explicit haves and finish with linarith. That's more verbose but each step checkable. Let me do the haves + linarith approach:\n\n```\n  have hsplit1 := Finset.sum_filter_add_sum_filter_not (circleZeros R)\n    (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} => ‖(ρ : ℂ)‖ ≤ 3) (fun ρ => ‖seriesTerm s ρ‖)\n  -- hsplit1 : ∑ F = ∑_{F.filter q1} + ∑_{F.filter (¬q1)}\n  have hsplit2 := Finset.sum_filter_add_sum_filter_not\n    ((circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} => ¬ ‖(ρ : ℂ)‖ ≤ 3))\n    (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} => ‖(ρ : ℂ)‖ ≤ R / 2) (fun ρ => ‖seriesTerm s ρ‖)\n  -- T1 = ∑_{(F.filter ¬q1).filter q2} + ∑_{(F.filter ¬q1).filter ¬q2}\n  have hff2 : ((circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} => ¬ ‖(ρ : ℂ)‖ ≤ 3)).filter\n        (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} => ‖(ρ : ℂ)‖ ≤ R / 2)\n      = (circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n          ¬ ‖(ρ : ℂ)‖ ≤ 3 ∧ ‖(ρ : ℂ)‖ ≤ R / 2) := Finset.filter_filter _ _ _\n  rw [hff2] at hsplit2\n  have hff2' : ((circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} => ¬ ‖(ρ : ℂ)‖ ≤ 3)).filter\n        (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} => ¬ ‖(ρ : ℂ)‖ ≤ R / 2)\n      = (circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n          ¬ ‖(ρ : ℂ)‖ ≤ 3 ∧ ¬ ‖(ρ : ℂ)‖ ≤ R / 2) := Finset.filter_filter _ _ _\n  rw [hff2'] at hsplit2\n  have hsplit3 := Finset.sum_filter_add_sum_filter_not\n    ((circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n      ¬ ‖(ρ : ℂ)‖ ≤ 3 ∧ ¬ ‖(ρ : ℂ)‖ ≤ R / 2))\n    (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} => ‖(ρ : ℂ)‖ ≤ R - 2) (fun ρ => ‖seriesTerm s ρ‖)\n  have hff3 : (...).filter (q3) = (circleZeros R).filter (fun ρ : ST => (¬‖ρ‖≤3 ∧ ¬‖ρ‖≤R/2) ∧ ‖ρ‖≤R−2) :=\n    Finset.filter_filter _ _ _\n  rw [hff3] at hsplit3\n  have hff3' : (...).filter (¬q3) = ... (fun ρ : ST => (¬‖ρ‖≤3 ∧ ¬‖ρ‖≤R/2) ∧ ¬‖ρ‖≤R−2) := Finset.filter_filter _ _ _\n  rw [hff3'] at hsplit3\n  have hsplit4 := Finset.sum_filter_add_sum_filter_not\n    ((circleZeros R).filter (fun ρ : ST => (¬‖ρ‖≤3 ∧ ¬‖ρ‖≤R/2) ∧ ¬‖ρ‖≤R−2))\n    (fun ρ : ST => ‖(ρ:ℂ)‖ < R+2) _\n  have hff4 : ... = filter (((¬q1∧¬q2)∧¬q3)∧q4) := Finset.filter_filter _ _ _\n  have hff4' : ... = filter (((¬q1∧¬q2)∧¬q3)∧¬q4) := Finset.filter_filter _ _ _\n  rw [hff4, hff4'] at hsplit4\n  have hc3 : (circleZeros R).filter (fun ρ : ST => (¬‖(ρ:ℂ)‖ ≤ 3 ∧ ¬‖(ρ:ℂ)‖ ≤ R/2) ∧ ‖(ρ:ℂ)‖ ≤ R−2)\n      = (circleZeros R).filter (fun ρ : ST => ¬‖(ρ:ℂ)‖ ≤ R/2 ∧ ‖(ρ:ℂ)‖ ≤ R−2) := by\n    ext ρ\n    simp only [Finset.mem_filter]\n    constructor\n    · rintro ⟨hF, ⟨-, h2⟩, h3⟩\n      exact ⟨hF, h2, h3⟩\n    · rintro ⟨hF, h2, h3⟩\n      refine ⟨⟨hF, ⟨?_, h2⟩⟩, h3⟩  -- hmm the anonymous constructor nesting\n      ...\n```\nCareful with the And-nesting in the membership: filter pred (¬q1∧¬q2)∧q3: mem ↔ ρ ∈ F ∧ ((¬q1 ρ ∧ ¬q2 ρ) ∧ q3 ρ): rintro ⟨hF, ⟨⟨h1, h2⟩, h3⟩⟩ vs q2∧q3-form: ρ ∈ F ∧ (¬q2 ρ ∧ q3 ρ): rintro ⟨hF, ⟨h2, h3⟩⟩. Forward: ⟨⟨-, h2⟩, h3⟩ hmm: ⟨hF, ⟨⟨h1, h2⟩, h3⟩⟩ → ⟨hF, ⟨h2, h3⟩⟩ ✓. Backward: ⟨hF, ⟨h2, h3⟩⟩ → ⟨hF, ⟨⟨(proof ¬q1), h2⟩, h3⟩⟩. ¬q1 ρ = ¬‖ρ‖ ≤ 3: from h2 : ¬‖ρ‖ ≤ R/2: ‖ρ‖ > R/2 ≥ 8001 > 3: `fun h => by linarith [lt_of_not_ge h2, hR, h]` — or `not_le_of_gt (by linarith [lt_of_not_ge h2, hR] : (3:ℝ) < ‖(ρ:ℂ)‖)`.\n\nThe λ binder types in the conjunction lambdas: filter_filter produces (fun a => p a ∧ q a) with p,q's binders pinned ST — the resulting lambda binder : ST ✓ and body `(¬‖(a:ℂ)‖ ≤ 3) ∧ (¬‖(a:ℂ)‖ ≤ R/2) ∧ ...` hmm — filter_filter: filter p (filter q s) = filter (fun a => q a ∧ p a): the q,p lambdas APPLIED to a: q a ∧ p a — NOT the unfolded ¬‖a‖≤3 ∧ ‖a‖≤R/2 unless q,p were exactly those lambdas (then q a = (¬‖(a:ℂ)‖≤3) after beta). Syntactically the filter predicate becomes (fun a => q a ∧ p a) — with q,p the pinned lambdas: q a beta-reduces to ¬‖(a:ℂ)‖≤3 — the shell lambdas have the SAME reduced body but the filter in hff2's RHS I stated as (fun ρ : ST => ¬‖(ρ:ℂ)‖ ≤ 3 ∧ ‖(ρ:ℂ)‖ ≤ R/2) — is (fun a => q a ∧ p a) SYNTACTICALLY (fun ρ => ¬‖ρ‖≤3 ∧ ‖ρ‖≤R/2)? q a = (fun ρ : ST => ¬‖(ρ:ℂ)‖ ≤ 3) a — a beta-redex! So filter_filter's RHS predicate is (fun a => (¬‖(a:ℂ)‖ ≤ 3) ∧ (‖(a:ℂ)‖ ≤ R/2)) after beta — Lean's filter_filter statement: `theorem Finset.filter_filter (p q : α → Prop) [DecidablePred p] [DecidablePred q] (s : Finset α) : (s.filter p).filter q = s.filter fun a => p a ∧ q a` — the RHS has the lambda (fun a => p a ∧ q a) — with p,q the actual lambdas — beta-redex applications p a, q a. When I rw [hff2] at hsplit2 where hff2's RHS is my hand-written (fun ρ : ST => ¬‖(ρ:ℂ)‖ ≤ 3 ∧ ‖(ρ:ℂ)‖ ≤ R/2) — the rw [hff2] rewrites LHS-of-hff2 to RHS-of-hff2 in hsplit2 — hsplit2's piece is (F.filter ¬q1).filter q2 — matching hff2's LHS — the rw works by matching hff2's LHS pattern against hsplit2 ✓ replacing with hff2's RHS (my clean form) ✓. So the rw produces my clean lambda regardless of the redex form ✓ — as long as hff2's LHS matches hsplit2's piece textually. hff2 := Finset.filter_filter _ _ _ — its type: (s.filter ?p).filter ?q = s.filter (fun a => ?p a ∧ ?q a) — ascription matches LHS ✓.\n\nBUT: the rw [hff2] at hsplit2 — rw rewrites instances of hff2's LHS — the piece in hsplit2 — ✓. However — the DecidablePred instances: filter_filter needs [DecidablePred p] [DecidablePred q] — hsplit2's filters carry the classical instances (classical in scope) — unification handles ✓ hopefully.\n\nAlso NOTE: hsplit2 as I stated it is Finset.sum_filter_add_sum_filter_not applied to (F.filter ¬q1) with p := q2: conclusion: ∑_{F.filter ¬q1} f = ∑_{(F.filter ¬q1).filter q2} f + ∑_{(F.filter ¬q1).filter (fun ρ => ¬ q2 ρ)} f — the second piece's predicate: (fun ρ : ST => ¬ ‖(ρ:ℂ)‖ ≤ R/2) ✓.\n\nOK — this is going to be long but mechanical. One more consideration: the final linarith combination needs hsplit0 (‖seriesFun‖ ≤ S + tail), hsplit1..4 (S = ..., T1 = ..., T2 = ..., T3 = S4+S5), the congruences applied, shells, htail, and the final constant absorption hfin (the nlinarith). The pieces' atoms: after all the rws at hsplit2/3/4, and hc3/hc4/hc5 — the congruences rewrite the SUM atoms: hc3 : filter(...) = filter(inhi-form) — rw [hc3] at hsplit3: hsplit3's S3-piece becomes the inhi-form sum ✓.\n\nThen: S = S1 + T1, T1 = S2 + T2, T2 = S3 + T3, T3 = S4 + S5 with S1..S5 exactly the shell sums (after rws) ⟹ S = S1+S2+S3+S4+S5 by linarith (equalities between sum atoms ✓).\n\nThen ‖seriesFun‖ ≤ S + tail ≤ (2·lowZeroConst + 60A₀L² + 60A₀L² + 2688ĈA₀L + 45A₀L²) + 91A₀L and hfin absorbs ≤ 2900A₀ĈL².\n\nhfin: 2*lowZeroConst + 60A₀L²+60A₀L²+2688ĈA₀L+45A₀L² + 91A₀L ≤ 2*lowZeroConst + 2900*A₀*Ĉ*L²: \nnlinarith [hĈ2, hlogR, hA0, hL2, lowZeroConst_nonneg] — hmm the L ≤ L² fact: hL2 : Real.log R ≤ (Real.log R)^2 := by nlinarith [hlogR] — wait that itself is nonlinear: L² ≥ L ⟸ L ≥ 9: nlinarith [hlogR]: goal L ≤ L² ⟺ 0 ≤ L(L−1): L ≥ 9 ⟹ L(L−1) ≥ 0: nlinarith does (L)·(L−1) hmm — nlinarith [hlogR] should: product of (L−9) and L: L² −9L ≥ 0 ⟹ L² ≥ 9L ≥ L ✓ via pairs. Actually: L ≤ L² ⟸ L² − L = L(L−1) ≥ 0: nlinarith considers L·L ≥ 9L (from hlogR multiplied by L ≥ 0?) — nlinarith multiplies hypotheses: hlogR·(L) hmm. nlinarith [hlogR] : proves L ≤ L^2? It needs: L² − L ≥ 0: try: (L − 9)·L ≥ 0 (from hlogR and L ≥ 0): nlinarith knows pow_two_nonneg? The term L^2 − L: with 9 ≤ L: L^2 ≥ 9L ≥ L... nlinarith proves: L^2 − L = L·(L−1): hmm it needs (L−1) ≥ 0 and L ≥ 0: products: hlogR (9 ≤ L) with itself: 81 ≤ L²; then L² ≥ 81 ≥ L ✓ since L ≥ 9 and 81 ≥ 9... L² ≥ 81 and L ≤ ... no: L ≥ 9 and L² ≥ 81 doesn't give L² ≥ L directly... it does: L² ≥ 81 ≥ L ⟸ L ≤ 81? NO L could be 100. Hmm: L² ≥ L ⟺ L ≥ 1 (L ≥ 0): L ≥ 9 ≥ 1 ✓: L·L ≥ 1·L = L: mul_le_mul_of_nonneg_left (1 ≤ L) (L ≥ 0)... nlinarith: L²−L ≥ 0 ⟸ (L−1)·L ≥ 0: nlinarith considers products of pairs among [hlogR] and atoms: (L−9)(L) — from hlogR: L−9 ≥ 0 and L ≥ ... nlinarith might need (L)(L−1) — the pair (L−1 ≥ 0) isn't a hypothesis but (L ≥ 9) implies it — nlinarith multiplies (hlogR : 9 ≤ L) by (hlogR): 81 ≤ L²: gives L² ≥ 81: and needs L² ≥ L: L² ≥ 81 and L ≥ 9 ⟹ L² ≥ 81 ≥ ... no: L ≥ 9 ⟹ L² ≥ 9L (L·L ≥ 9·L needs L ≥ 0 — the product (L)·(hlogR): nlinarith multiplies hypotheses by atoms: hlogR·L: 9L ≤ L² ✓ then L² ≥ 9L ≥ L ⟸ 9L ≥ L ⟸ L ≥ 0 ✓. So nlinarith [hlogR] proves L ≤ L² ✓ (it does multiply hypotheses by variables).\n\nThen the absorption nlinarith [hĈ2, hlogR, hA0, hL2]:\ngoal: 165A₀L² + 2688ĈA₀L + 91A₀L ≤ 2900A₀ĈL².\nAs computed: suffices: 2688ĈA₀L ≤ 2688ĈA₀L² (ĈA₀·(L²−L) ≥ 0) and 256A₀L² ≤ 212·2A₀L² ≤ 212ĈA₀L² hmm: 165+91 = 256: 256A₀L² ≤ 256·(Ĉ/2)A₀L² = 128ĈA₀L² (Ĉ ≥ 2) and 2688+128 = 2816 ≤ 2900 ✓. The nlinarith products: (Ĉ−2)·(A₀L²) ≥ 0; (L²−L)·(A₀) ≥ 0; (L²−L)·(ĈA₀) ≥ 0: nlinarith [hĈ2, hL2, hA0, mul_nonneg hA0 (pow_two_nonneg (Real.log R)), mul_nonneg (mul_nonneg hĈ2'... ] — Ĉ ≥ 0 from hĈ2: give hints: hnn1 : 0 ≤ A0count * (Real.log R)^2; hnn2 : 0 ≤ C0hat(...) * A0count. nlinarith [hĈ2, hlogR, hA0, hL2, hnn1, hnn2]. The key products: (Ĉ−2)·(A₀L²) — nlinarith multiplies (hĈ2 : 2 ≤ Ĉ) by hnn1 ✓; (L²−L)·(A₀) — hL2·... nlinarith multiplies hL2 by hA0?? It multiplies PAIRS of hypotheses (and their negations): hL2·hA0: A₀·(L²−L) ≥ 0 ✓. And 2688ĈA₀L ≤ 2688ĈA₀L²: ĈA₀·(L²−L): hL2·(ĈA₀): product of hL2 with hnn2 ✓. OK it should find it. If not, I'll decompose manually.\n\nWait — also the 2·lowZeroConst appears on both sides: linarith cancels ✓.\n\nLet me now write the assembly. It's long; write carefully.","signature":""},{"type":"text","text":"shell_crit compiles. Now the final assembly theorem with the iterated filter-split partition."},{"type":"tool_use","id":"Edit_114","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean","old_string":"end NguyenRH1\n\nend","new_string":"/-- **The series circle bound** (ii′-5e-v-b chunk 3 capstone): on the\nγ-gap circle |s| = R (k ≥ 4000), ‖seriesFun s‖ ≤ 2·lowZeroConst +\n2900·A₀·Ĉ₀(4k+9/2)·(log R)². The ONE tsum split at {‖ρ‖ < 2R+1}, the\nfive-way shell partition of the finite part (iterated filter splits;\nthe redundant ¬q1/¬q2 conjuncts dropped via the range implications),\nthe shell bounds (low 2·lowZeroConst; inlo/inhi 60·A₀L²; crit\n2688·Ĉ₀A₀L; up 45·A₀L²), and the far tail ≤ 91·A₀L. -/\ntheorem seriesFun_norm_le_gapCircle {k : ℕ} (hk : 4000 ≤ k)\n    {R : ℝ} (hRlo : 4 * (k : ℝ) + 5 / 2 ≤ R) (hRhi : R ≤ 4 * (k : ℝ) + 3)\n    (hgap : ∀ ρ : ℂ, IsNontrivialZero ρ →\n      1 / (16 * C0hat (4 * (k : ℝ) + 9 / 2)) ≤ |ρ.im - R| ∨ 1 ≤ |ρ.im - R|)\n    {s : ℂ} (hs : ‖s‖ = R) :\n    ‖seriesFun s‖ ≤ 2 * lowZeroConst +\n      2900 * A0count * C0hat (4 * (k : ℝ) + 9 / 2) * (Real.log R) ^ 2 := by\n  classical\n  have hsz : ¬ IsNontrivialZero s :=\n    not_isNontrivialZero_on_gapCircle hk hRlo hgap hs\n  have hR : (16002 : ℝ) ≤ R := by\n    have h1 : (4000 : ℝ) ≤ (k : ℝ) := by exact_mod_cast hk\n    linarith [hRlo]\n  have hlogR : (9 : ℝ) ≤ Real.log R := nine_le_log_of_16002_le hR\n  have hlogRnn : (0 : ℝ) ≤ Real.log R := by linarith [hlogR]\n  have hĈ2 : (2 : ℝ) ≤ C0hat (4 * (k : ℝ) + 9 / 2) := two_le_C0hat (by\n    have hkR : (1 : ℝ) ≤ (k : ℝ) := by exact_mod_cast (by omega : 1 ≤ k)\n    linarith)\n  have hA0 := A0count_pos.le\n  -- the shell bounds and the tail\n  have hsh1 := shell_low hs hsz hR\n  have hsh2 := shell_inlo hs hsz hR\n  have hsh3 := shell_inhi hs hsz hR\n  have hsh4 := shell_crit hk hRlo hgap hs hsz\n  have hsh5 := shell_up hs hsz hR\n  have htail := series_tail_le hs hR\n  have hsplit0 := seriesFun_norm_le_split (s := s) (R := R)\n  -- the partition identity (iterated filter splits)\n  have hsplit1 := Finset.sum_filter_add_sum_filter_not (circleZeros R)\n    (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} => ‖(ρ : ℂ)‖ ≤ 3) (fun ρ => ‖seriesTerm s ρ‖)\n  have hsplit2 := Finset.sum_filter_add_sum_filter_not\n    ((circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} => ¬ ‖(ρ : ℂ)‖ ≤ 3))\n    (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} => ‖(ρ : ℂ)‖ ≤ R / 2) (fun ρ => ‖seriesTerm s ρ‖)\n  have hff2 : ((circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n        ¬ ‖(ρ : ℂ)‖ ≤ 3)).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} => ‖(ρ : ℂ)‖ ≤ R / 2)\n      = (circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n          ¬ ‖(ρ : ℂ)‖ ≤ 3 ∧ ‖(ρ : ℂ)‖ ≤ R / 2) := Finset.filter_filter _ _ _\n  rw [hff2] at hsplit2\n  have hff2' : ((circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n        ¬ ‖(ρ : ℂ)‖ ≤ 3)).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} => ¬ ‖(ρ : ℂ)‖ ≤ R / 2)\n      = (circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n          ¬ ‖(ρ : ℂ)‖ ≤ 3 ∧ ¬ ‖(ρ : ℂ)‖ ≤ R / 2) := Finset.filter_filter _ _ _\n  rw [hff2'] at hsplit2\n  have hsplit3 := Finset.sum_filter_add_sum_filter_not\n    ((circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n      ¬ ‖(ρ : ℂ)‖ ≤ 3 ∧ ¬ ‖(ρ : ℂ)‖ ≤ R / 2))\n    (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} => ‖(ρ : ℂ)‖ ≤ R - 2) (fun ρ => ‖seriesTerm s ρ‖)\n  have hff3 : ((circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n        ¬ ‖(ρ : ℂ)‖ ≤ 3 ∧ ¬ ‖(ρ : ℂ)‖ ≤ R / 2)).filter\n        (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} => ‖(ρ : ℂ)‖ ≤ R - 2)\n      = (circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n          (¬ ‖(ρ : ℂ)‖ ≤ 3 ∧ ¬ ‖(ρ : ℂ)‖ ≤ R / 2) ∧ ‖(ρ : ℂ)‖ ≤ R - 2) :=\n    Finset.filter_filter _ _ _\n  rw [hff3] at hsplit3\n  have hff3' : ((circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n        ¬ ‖(ρ : ℂ)‖ ≤ 3 ∧ ¬ ‖(ρ : ℂ)‖ ≤ R / 2)).filter\n        (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} => ¬ ‖(ρ : ℂ)‖ ≤ R - 2)\n      = (circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n          (¬ ‖(ρ : ℂ)‖ ≤ 3 ∧ ¬ ‖(ρ : ℂ)‖ ≤ R / 2) ∧ ¬ ‖(ρ : ℂ)‖ ≤ R - 2) :=\n    Finset.filter_filter _ _ _\n  rw [hff3'] at hsplit3\n  have hsplit4 := Finset.sum_filter_add_sum_filter_not\n    ((circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n      (¬ ‖(ρ : ℂ)‖ ≤ 3 ∧ ¬ ‖(ρ : ℂ)‖ ≤ R / 2) ∧ ¬ ‖(ρ : ℂ)‖ ≤ R - 2))\n    (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} => ‖(ρ : ℂ)‖ < R + 2) (fun ρ => ‖seriesTerm s ρ‖)\n  have hff4 : ((circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n        (¬ ‖(ρ : ℂ)‖ ≤ 3 ∧ ¬ ‖(ρ : ℂ)‖ ≤ R / 2) ∧ ¬ ‖(ρ : ℂ)‖ ≤ R - 2)).filter\n        (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} => ‖(ρ : ℂ)‖ < R + 2)\n      = (circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n          ((¬ ‖(ρ : ℂ)‖ ≤ 3 ∧ ¬ ‖(ρ : ℂ)‖ ≤ R / 2) ∧ ¬ ‖(ρ : ℂ)‖ ≤ R - 2) ∧ ‖(ρ : ℂ)‖ < R + 2) :=\n    Finset.filter_filter _ _ _\n  rw [hff4] at hsplit4\n  have hff4' : ((circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n        (¬ ‖(ρ : ℂ)‖ ≤ 3 ∧ ¬ ‖(ρ : ℂ)‖ ≤ R / 2) ∧ ¬ ‖(ρ : ℂ)‖ ≤ R - 2)).filter\n        (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} => ¬ ‖(ρ : ℂ)‖ < R + 2)\n      = (circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n          ((¬ ‖(ρ : ℂ)‖ ≤ 3 ∧ ¬ ‖(ρ : ℂ)‖ ≤ R / 2) ∧ ¬ ‖(ρ : ℂ)‖ ≤ R - 2) ∧ ¬ ‖(ρ : ℂ)‖ < R + 2) :=\n    Finset.filter_filter _ _ _\n  rw [hff4'] at hsplit4\n  -- the three congruences dropping the redundant conjuncts\n  have hc3 : (circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n        (¬ ‖(ρ : ℂ)‖ ≤ 3 ∧ ¬ ‖(ρ : ℂ)‖ ≤ R / 2) ∧ ‖(ρ : ℂ)‖ ≤ R - 2)\n      = (circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n          ¬ ‖(ρ : ℂ)‖ ≤ R / 2 ∧ ‖(ρ : ℂ)‖ ≤ R - 2) := by\n    ext ρ\n    simp only [Finset.mem_filter]\n    constructor\n    · rintro ⟨hF, ⟨-, h2⟩, h3⟩\n      exact ⟨⟨hF, h2⟩, h3⟩\n    · rintro ⟨hF, h2, h3⟩\n      refine ⟨⟨hF, ⟨?_, h2⟩⟩, h3⟩\n      exact not_le_of_gt (by linarith [lt_of_not_ge h2, hR] : (3 : ℝ) < ‖(ρ : ℂ)‖)\n  rw [hc3] at hsplit3\n  have hc4 : (circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n        ((¬ ‖(ρ : ℂ)‖ ≤ 3 ∧ ¬ ‖(ρ : ℂ)‖ ≤ R / 2) ∧ ¬ ‖(ρ : ℂ)‖ ≤ R - 2) ∧ ‖(ρ : ℂ)‖ < R + 2)\n      = (circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n          ¬ ‖(ρ : ℂ)‖ ≤ R - 2 ∧ ‖(ρ : ℂ)‖ < R + 2) := by\n    ext ρ\n    simp only [Finset.mem_filter]\n    constructor\n    · rintro ⟨hF, ⟨-, h3⟩, h4⟩\n      exact ⟨⟨hF, h3⟩, h4⟩\n    · rintro ⟨hF, h3, h4⟩\n      refine ⟨⟨hF, ⟨⟨?_, ?_⟩, h3⟩⟩, h4⟩\n      · exact not_le_of_gt (by linarith [lt_of_not_ge h3, hR] : (3 : ℝ) < ‖(ρ : ℂ)‖)\n      · exact not_le_of_gt (by linarith [lt_of_not_ge h3, hR] : (R / 2 : ℝ) < ‖(ρ : ℂ)‖)\n  rw [hc4] at hsplit4\n  have hc5 : (circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n        ((¬ ‖(ρ : ℂ)‖ ≤ 3 ∧ ¬ ‖(ρ : ℂ)‖ ≤ R / 2) ∧ ¬ ‖(ρ : ℂ)‖ ≤ R - 2) ∧ ¬ ‖(ρ : ℂ)‖ < R + 2)\n      = (circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n          ¬ ‖(ρ : ℂ)‖ < R + 2) := by\n    ext ρ\n    simp only [Finset.mem_filter]\n    constructor\n    · rintro ⟨hF, -, h4⟩\n      exact ⟨hF, h4⟩\n    · rintro ⟨hF, h4⟩\n      refine ⟨⟨hF, ⟨⟨?_, ?_⟩, ?_⟩⟩, h4⟩\n      · exact not_le_of_gt (by linarith [le_of_not_gt h4, hR] : (3 : ℝ) < ‖(ρ : ℂ)‖)\n      · exact not_le_of_gt (by linarith [le_of_not_gt h4, hR] : (R / 2 : ℝ) < ‖(ρ : ℂ)‖)\n      · exact not_le_of_gt (by linarith [le_of_not_gt h4, hR] : (R - 2 : ℝ) < ‖(ρ : ℂ)‖)\n  rw [hc5] at hsplit4\n  -- the final absorption: 165A₀L² + 2688ĈA₀L + 91A₀L ≤ 2900A₀ĈL²\n  have hL2 : Real.log R ≤ (Real.log R) ^ 2 := by nlinarith [hlogR]\n  have hnn1 : (0 : ℝ) ≤ A0count * (Real.log R) ^ 2 := mul_nonneg hA0 (pow_two_nonneg _)\n  have hnn2 : (0 : ℝ) ≤ C0hat (4 * (k : ℝ) + 9 / 2) * A0count :=\n    mul_nonneg (by linarith [hĈ2]) hA0\n  have hfin : 2 * lowZeroConst + ((60 * A0count * (Real.log R) ^ 2 +\n        (60 * A0count * (Real.log R) ^ 2 + (2688 * C0hat (4 * (k : ℝ) + 9 / 2) * A0count *\n          Real.log R + 45 * A0count * (Real.log R) ^ 2))) + 91 * A0count * Real.log R)\n      ≤ 2 * lowZeroConst + 2900 * A0count * C0hat (4 * (k : ℝ) + 9 / 2) * (Real.log R) ^ 2 := by\n    nlinarith [hĈ2, hlogR, hA0, hL2, hnn1, hnn2, lowZeroConst_nonneg]\n  linarith [hsplit0, hsplit1, hsplit2, hsplit3, hsplit4, hsh1, hsh2, hsh3, hsh4, hsh5,\n    htail, hfin]\n\nend NguyenRH1\n\nend"},"extra_content":null}]