[{"type":"thinking","thinking":"shell_inlo compiles clean. Now shell_inhi and shell_up — same template with the d-function changed. Let me write both, carefully adjusting:\n\n**shell_inhi** (R/2 < ‖ρ‖ ≤ R−2): d ρ := R − ‖(ρ:ℂ)‖.\n- hdj: 1 ≤ Nat.log 2 ⌊R−‖ρ‖⌋₊: ⌊d⌋₊ ≥ 2 (d ≥ 2 since ‖ρ‖ ≤ R−2) ⟹ Nat.log 2 ≥ 1 (Nat.log 2 2 = 1 ✓ mono). ≤ J: ⌊d⌋₊ ≤ ⌊R⌋₊ (d < R) ✓.\n- term: g ≤ 2m/d ≤ 2m/2^{jf}: g ρ = m·R/(‖ρ‖‖s−ρ‖): ‖s−ρ‖ ≥ R−‖ρ‖ = d (reverse triangle: ‖s‖−‖ρ‖ = R−‖ρ‖ ≥ 2 > 0 ✓ abs_of_nonneg); ‖ρ‖ ≥ R/2: m·R/(‖ρ‖‖s−ρ‖) ≤ m·R/((R/2)·d) = 2m/d ✓; 2m/d ≤ 2m/2^{jf} (2^{jf} ≤ d ✓).\n- count per j ∈ Ico: fiber zeros: d ∈ [2^j, 2^{j+1}): a := R−2^{j+1}−1, b := R−2^j, c := R/2−1: \n  - |γ| ≥ ‖ρ‖−1: ‖ρ‖ = R−d > R−2^{j+1} (d < 2^{j+1}) ⟹ ‖ρ‖−1 > a ⟹ |γ| ≥ a... need a ≤ |γ|: |γ| ≥ ‖ρ‖−1 ≥ ... ‖ρ‖−1 > a ⟹ |γ| > a ⟹ ≥ ✓ linarith.\n  - |γ| ≤ ‖ρ‖ ≤ b: d ≥ 2^j ⟹ ‖ρ‖ ≤ R−2^j = b ✓.\n  - c ≤ |γ|: ‖ρ‖ > R/2 ⟹ ‖ρ‖−1 > R/2−1 = c ✓.\n  - hM1: 1 ≤ max a c: c = R/2−1 ≥ 1 (R ≥ 4): le_max_right.trans ✓.\n  - hMb: max a c ≤ b: max_le_iff: a ≤ b ⟺ R−2^{j+1}−1 ≤ R−2^j ⟺ 2^j ≤ 2^{j+1}+1 ✓; c ≤ b ⟺ R/2−1 ≤ R−2^j ⟺ 2^j ≤ R/2+1: 2^j ≤ d·... 2^j ≤ d (in the fiber) hmm — but hMb is per-j, not per-element: 2^j ≤ R/2+1 needs j-bounds only: j ≤ J ⟹ 2^j ≤ 2^J ≤ R ⟹ 2^j ≤ R ≤ R/2+1·... R ≤ R/2+1 ⟺ R/2 ≤ 1 — FALSE. Hmm! c ≤ b: R/2−1 ≤ R−2^j ⟺ 2^j ≤ R/2+1: with 2^j ≤ R (from j ≤ J): 2^j ≤ R = R/2+R/2: need 2^j−R/2 ≤ 1: NOT guaranteed (2^j could be near R). \n    Wait — in the fiber, d ≥ 2^j and d < R/2 (since ‖ρ‖ > R/2): so 2^j ≤ d < R/2: 2^j < R/2 — but that's PER-ELEMENT (in the fiber); for the count brick the fiber is FS.filter(jf = j) — nonempty fibers have 2^j < R/2. But hMb must hold even for the brick application on possibly-empty fibers? If the fiber is empty, the brick's conclusion LHS = 0 — but the brick's hypotheses hM1/hMb must still be discharged! For j with 2^j ≥ R/2: fiber empty (all elements have d ≥ 2^j... wait elements of the fiber have d ∈ [2^j, 2^{j+1}): if 2^j ≥ R/2 then d ≥ R/2 contradicting d < R/2 — so fiber empty). But the brick application needs hMb REGARDLESS. Hmm — handle: in hcount, case-split: if 2^j ≥ R/2 (or c > b), show the fiber is empty directly (element: d ≥ 2^j ≥ R/2 but d < R/2 — contradiction) ⟹ sum = 0 ≤ cnt j. Else apply the brick with c ≤ b.\n    Actually cleaner: keep the by_cases hj : j ∈ Ico 1 (J+1) structure, and INSIDE the Ico case, by_cases h2jR : (2:ℝ)^j ≤ R/2 hmm — wait for j ∈ Ico 1 (J+1): is 2^j < R/2 automatic? j ≤ J: 2^j ≤ 2^J ≤ R: could be 2^j ∈ (R/2, R]: possible! So yes need the sub-case. Structure:\n    ```\n    by_cases hj : j ∈ Ico 1 (J+1)\n    · by_cases hbig : R / 2 ≤ (2:ℝ)^j\n      · -- fiber empty: elements have d ≥ 2^j ≥ R/2 but d < R/2\n        have hempty : FS.filter (...) = ∅ := ...\n        rw [hempty, Finset.sum_empty]; positivity-ish\n      · -- the brick with c ≤ b\n        push Not at hbig\n        apply sum_zeroMult_le_of_im_band ... (c ≤ b from 2^j < R/2 ⟹ R/2−1 < R−2^j hmm c ≤ b ⟺ 2^j ≤ R/2+1 ✓ from 2^j < R/2)\n    · empty via hdj\n    ```\n    Hmm wait — but actually the emptiness argument via d ≥ 2^j needs 2^{jf} ≤ d (the pow-lower bound), which needs ⌊d⌋₊ ≠ 0 ✓ (d ≥ 2 in FS). Fine.\n    \n    Hmm — hold on, actually simpler: choose c := 1?? The sign-separation just needs c ≥ 0-ish with max a c ≥ 1 and c ≤ b and c ≤ |γ|. Take c := 1: |γ| ≥ 1?? Not true in general... |γ| ≥ ‖ρ‖−1 ≥ R/2−1 ≫ 1 ✓ provable. c := 1: max a c: if a ≥ 1: M = a (fine, M ≥ 1 ✓); if a < 1: M = 1: M ≤ b ⟺ 1 ≤ b = R−2^j ✓ (2^j ≤ R−1 hmm 2^j ≤ R/2 in the non-empty case ⟹ b = R−2^j ≥ R/2 ≥ 1 ✓). So c := 1 works and c ≤ b only needs b ≥ 1 which holds in the non-empty case (2^j < R/2 ⟹ b > R/2 ≥ 1 ✓). Same case-split needed anyway. Keep c := R/2−1 — no, with c := R/2−1, M = max a (R/2−1): M ≤ b requires both a ≤ b ✓ and R/2−1 ≤ b ⟺ 2^j ≤ R/2+1 ✓ in the non-empty case (2^j < R/2 ⟹ 2^j ≤ R/2+1·... hmm 2^j < R/2 gives R−2^j > R/2 > R/2−1 ✓ c < b ✓). OK either works. Keep c := R/2−1.\n  - brick: cnt j := 3·A₀·(b−a+2)·log(3·max|a||b|+5): b−a+2 = (R−2^j)−(R−2^{j+1}−1)+2 = 2^j+3 ✓ (2^{j+1} = 2·2^j). max|a||b|: bound ≤ R+1 for the log: log(3·max|a||b|+5) ≤ 2L via log_three_le_two_log with x := max|a||b| ≤ 3R: |a| = |R−2^{j+1}−1| ≤ max(R, 2^{j+1}+1): 2^{j+1} ≤ 2R ⟹ 2^{j+1}+1 ≤ 2R+1 ≤ 3R hmm need |a| ≤ 3R and |b| ≤ 3R: |b| = R−2^j ≤ R ✓; |a| ≤ R+2^{j+1}+1 ≤ R+2R+1 = 3R+1: hmm |a| ≤ max(a, −a) with a = R−2^{j+1}−1 ≤ R ✓ and −a = 2^{j+1}+1−R ≤ 2^{j+1}+1 ≤ 2R+1: so |a| ≤ max(R, 2R+1) = 2R+1 ≤ 3R ✓ (R ≥ 1). max|a||b| ≤ 3R ✓ (max_le (2R+1 ≤ 3R) (R ≤ 3R)). x0 : 0 ≤ max|a||b| ✓.\n    cnt j ≤ 3A₀(2^j+3)·2L (trans with mul_le_mul_of_nonneg_left (log_three_le_two_log...) (nonneg of 3A₀(2^j+3))).\n  Hmm wait — hcount's stated bound must be a single expression cnt j for dyadic_family_le. For inlo I stated hcount with the brick-form bound and then hper did the ≤ 30A₀L step. Same here: hcount : ≤ 3*A0count*((2:ℝ)^j+3)*Real.log(3*max|R−2^{j+1}−1||R−2^j|+5)... then hper: (2/2^j)·that ≤ 30A₀L. The max expression is ugly; alternative: state hcount's RHS directly with the 2L form: ≤ 3*A0count*((2:ℝ)^j+3)*(2*Real.log R) — discharging via brick + log bound inside hcount. Cleaner for hper. Let me do that: hcount's RHS := 3*A0count*((2:ℝ)^j+3)*(2*Real.log R); inside, apply brick then trans via the log bound. For j ∉ Ico ∪ {2^j ≥ R/2}: empty ⟹ 0 ≤ RHS ✓ (nonneg).\n  \n  Then hper: (2/2^j)·3A₀(2^j+3)·2L = 12A₀L(1+3/2^j) ≤ 12A₀L·(5/2) = 30A₀L ✓ same algebra as inlo. Σ ≤ J·30A₀L ≤ 2L·30A₀L = 60A₀L².\n\n**shell_up** (R+2 ≤ ‖ρ‖): d ρ := ‖(ρ:ℂ)‖ − R.\n- FS := F.filter (fun ρ => ¬ ‖(ρ:ℂ)‖ < R+2) — the fifth piece of the iterated split is F.filter ¬P4 with P4 := ‖ρ‖ < R+2. Elements: R+2 ≤ ‖ρ‖ ∧ ‖ρ‖ < 2R+1 (F membership).\n- hdj: u := ‖ρ‖−R ∈ [2, R+1): ⌊u⌋₊ ≥ 2 ⟹ log ≥ 1; ⌊u⌋₊ ≤ ⌊R⌋₊? u < R+1: ⌊u⌋₊ ≤ ⌊R⌋₊ hmm u < R+1 ⟹ ⌊u⌋₊ ≤ ⌊R⌋₊?? u < R+1 ⟹ ⌊u⌋₊ ≤ ⌊R+1⌋₊−1 hmm: u < R+1 ⟹ ⌊u⌋₊ ≤ R... e.g. R = 100.3, u = 100.9: ⌊u⌋₊ = 100 = ⌊R⌋₊ ✓; u = 100.999 < 101.3 = R+1 ✓ ⌊u⌋₊ = 100 ✓. In general u < R+1 ⟹ ⌊u⌋₊ ≤ ⌊R⌋₊: since ⌊u⌋₊ ≤ u < R+1 ⟹ ⌊u⌋₊ < R+1 ⟹ ⌊u⌋₊ ≤ ⌈R⌉... for naturals: ⌊u⌋₊ < R+1 (real) ⟹ ⌊u⌋₊ ≤ ⌊R+1⌋₊−1 = ⌊R⌋₊ hmm: ⌊u⌋₊ ≤ u < R+1: ⌊u⌋₊ < R+1: ⌊u⌋₊+... Let me just: u < R+1 ⟹ ⌊u⌋₊ ≤ ⌊R⌋₊?? Counter: R = 100.9, u = 100.95: ⌊u⌋₊ = 100 = ⌊R⌋₊ ✓. R = 100.2, u = 100.99 < 101.2: ⌊u⌋₊ = 100 = ⌊R⌋₊ ✓. Try R = 100.999, u = 100.9995: ⌊u⌋₊ = 100, ⌊R⌋₊ = 100 ✓. Seems true: u < R+1: ⌊u⌋₊ < R+1: suppose ⌊u⌋₊ ≥ ⌊R⌋₊+1: then u ≥ ⌊u⌋₊ ≥ ⌊R⌋₊+1 > R (⌊R⌋₊+1 > R ✓ Nat.lt_floor_add_one) ⟹ u > R ⟹ ... need u ≥ R+1 for contradiction: ⌊u⌋₊ ≥ ⌊R⌋₊+1 ⟹ u ≥ ⌊R⌋₊+1 (Nat.floor_le... (⌊u⌋₊:ℝ) ≤ u) hmm u ≥ (⌊u⌋₊:ℝ) ≥ (⌊R⌋₊+1:ℝ) = ⌊R⌋₊+1 > R: gives u > R — not u ≥ R+1. Hmm: ⌊u⌋₊ ≥ ⌊R⌋₊+1 as NATURALS: u ≥ ⌊u⌋₊ ≥ ⌊R⌋₊+1: u ≥ ⌊R⌋₊+1: is ⌊R⌋₊+1 ≥ R+1? NO: ⌊R⌋₊+1 < R+1+1... ⌊R⌋₊+1 ∈ (R, R+1]. So u ≥ ⌊R⌋₊+1 doesn't contradict u < R+1?? u ≥ ⌊R⌋₊+1 and u < R+1: ⌊R⌋₊+1 ≤ R+1 ✓ consistent possible: e.g. R = 100.3, ⌊R⌋₊+1 = 101: u ∈ [101, 101.3): u = 101.1 < 101.3 ✓ and ⌊u⌋₊ = 101 = ⌊R⌋₊+1 — COUNTEREXAMPLE to ⌊u⌋₊ ≤ ⌊R⌋₊! So ⌊u⌋₊ can be ⌊R⌋₊+1. Then Nat.log 2 ⌊u⌋₊ ≤ Nat.log 2 (⌊R⌋₊+1): is that ≤ J = Nat.log 2 ⌊R⌋₊? NOT always: R = 100.3: ⌊R⌋₊ = 100, ⌊u⌋₊ = 101: log₂100 = 6, log₂101 = 6 ✓; but R near powers of 2: R = 127.5: ⌊R⌋₊ = 127: log₂127 = 6; u ∈ [127.5ish, 128.5): ⌊u⌋₊ could be 128: log₂128 = 7 > 6 = J! PROBLEM.\n    Hmm — but wait: for up, u = ‖ρ‖−R < R+1 — and I want jf ρ = Nat.log 2 ⌊u⌋₊ ≤ J. When u ∈ [R, R+1) hmm: u < R+1; 2^J ≤ R (2^J ≤ ⌊R⌋₊ ≤ R) and 2^{J+1} > ⌊R⌋₊: 2^{J+1} > R... u < R+1 ≤ 2^{J+1}+1: ⌊u⌋₊ ≤ R < 2^{J+1}: hmm ⌊u⌋₊ ≤ ⌊R+1−⌋: u < R+1 ⟹ ⌊u⌋₊ ≤ R (floor_le of u < R+1?? u < R+1 ⟹ ⌊u⌋₊ < R+1 ⟹ ⌊u⌋₊ ≤ R in reals, as naturals ⌊u⌋₊ ≤ ⌊R+1−ε⌋...). Cleaner: ⌊u⌋₊ < R+1 (real ineq: (⌊u⌋₊:ℝ) ≤ u < R+1) ⟹ ⌊u⌋₊ ≤ R (nat: ⌊u⌋₊ < R+1 hmm real-cast: (⌊u⌋₊:ℝ) < R+1 ⟹ ⌊u⌋₊ ≤ R? as ⌊u⌋₊ is a natural: (⌊u⌋₊:ℝ) < R+1 doesn't give nat ≤ ⌊R⌋₊. e.g. ⌊u⌋₊ = 128, R = 127.5: (128:ℝ) < 128.5 ✓ but 128 > 127 = ⌊R⌋₊.\n    Fix: use J' := Nat.log 2 ⌊R+1⌋₊ for the up family? Then 2^{J'} could be > R (⌊R+1⌋₊ = ⌊R⌋₊+1: 2^{J'} ≤ ⌊R⌋₊+1 ≤ R+1): the J' ≤ 2L+1-ish bound: 2^{J'} ≤ R+1 ⟹ J'·log2 ≤ log(R+1) ≤ log(2R) = log2+L ⟹ J' ≤ 1+L/log2 ≤ 1+2L ≤ 3L·... J' ≤ 3L hmm 1+2L ≤ 3L ⟺ 1 ≤ L ✓. Fine — the count of shells for up ≤ J' ≤ 3L: up ≤ 15A₀L·J' ≤ 45A₀L². Or simpler: prove Nat.log 2 ⌊u⌋₊ ≤ J directly: 2^{jf} ≤ ⌊u⌋₊ ≤ u < R+1 and I need jf ≤ J: suppose jf ≥ J+1: 2^{J+1} ≤ 2^{jf} ≤ ⌊u⌋₊ ≤ u < R+1 ⟹ 2^{J+1} < R+1. But 2^{J+1} > ⌊R⌋₊ ≥ R−1 hmm 2^{J+1} ≥ ⌊R⌋₊+1 > R ⟹ 2^{J+1} > R ⟹ 2^{J+1} ≥ ... 2^{J+1} is an integer > R: 2^{J+1} ≥ ⌊R⌋₊+1: and u < R+1: 2^{J+1} ≤ u < R+1 ⟹ 2^{J+1} < R+1: but 2^{J+1} > R with 2^{J+1} nat: 2^{J+1} ≥ ⌊R⌋₊+1 hmm circular. Let me think: 2^{J+1} > R (strict, since 2^{J+1} > ⌊R⌋₊ ≥ ... Nat.lt_pow_succ_log_self: ⌊R⌋₊ < 2^{J+1}: (⌊R⌋₊:ℝ) < 2^{J+1}: and R < ⌊R⌋₊+1 ≤ 2^{J+1}: so R < 2^{J+1}. And jf ≤ J ⟸ 2^{jf} ≤ 2^J?? no: jf ≤ J ⟸ ⌊u⌋₊ ≤ ⌊R⌋₊ hmm. Alternative: jf ≤ J ⟸ 2^{jf+1} ≤ 2^{J+1}... no.\n    Direct: 2^{jf} ≤ ⌊u⌋₊ (pow_log_le_self) and ⌊u⌋₊ < 2^{J+1} (since ⌊u⌋₊ ≤ u < R+1 ≤ ... hmm R+1 ≤ 2^{J+1}? 2^{J+1} > R and 2^{J+1} nat ⟹ 2^{J+1} ≥ ⌈R⌉ hmm 2^{J+1} > R ⟹ 2^{J+1} ≥ R+... for real R: 2^{J+1} > R ⟹ R+1 ≤ 2^{J+1}? NO: R = 127.5, 2^{J+1} = 128 > 127.5: R+1 = 128.5 > 128! So u < R+1 = 128.5 could have ⌊u⌋₊ = 128 with jf = 7 > J = 6. CONFIRMED PROBLEM.\n    Resolution: for the up family, J_up := Nat.log 2 ⌊R+1⌋₊ hmm or just take J_up := J+1: claim jf ρ ≤ J+1: 2^{jf} ≤ ⌊u⌋₊ ≤ u < R+1 < 2^{J+1}+1 ⟹ 2^{jf} < 2^{J+1}+1 ⟹ 2^{jf} ≤ 2^{J+1} (nats) ⟹ jf ≤ J+1 ✓. And count shells: card Ico 1 (J+2) = J+1 ≤ 2L+1 ≤ 3L (L ≥ 9... 2L+1 ≤ 3L ⟺ 1 ≤ L ✓). So for up: J' := J+1, shell count J+1 ≤ 3L, per-shell ≤ 15A₀L: up ≤ 45A₀L². Hmm wait — per-shell for up was 15A₀L (C₁ = 1: (1/2^j)·3A₀(2^j+3)·2L = 6A₀L(1+3/2^j) ≤ 6A₀L·(5/2) = 15A₀L). up ≤ (J+1)·15A₀L ≤ 3L·15A₀L = 45A₀L².\n    Adjust the totals: 2·lowZeroConst + 60+60+45+91-absorb + 2688Ĉ = fine (final 2900 unchanged: 60+60+45 = 165 ≤ 180; 165/2+2688+46 = 82.5+2688+46 = 2816.5 ≤ 2900 ✓).\n- term: g ≤ m/u ≤ m/2^{jf}: g = m·R/(‖ρ‖‖s−ρ‖): ‖s−ρ‖ ≥ ‖ρ‖−R = u (reverse triangle: ‖ρ‖−‖s‖ = ‖ρ‖−R ≥ 2 > 0: ‖s−ρ‖ ≥ |‖s‖−‖ρ‖| = ‖ρ‖−R ✓ abs_norm_sub_norm_le: |‖s‖−‖ρ‖| ≤ ‖s−ρ‖: |R−‖ρ‖| = ‖ρ‖−R ✓); ‖ρ‖ ≥ R: m·R/(‖ρ‖u) ≤ m·R/(R·u) = m/u ✓; m/u ≤ m/2^{jf} (2^{jf} ≤ u) ✓.\n- count per j ∈ Ico 1 (J'+1): fiber: u ∈ [2^j, 2^{j+1}): a := R+2^j−1, b := R+2^{j+1}, c := R+1:\n  - |γ| ≥ ‖ρ‖−1 ≥ R+2^j−1 = a (u ≥ 2^j ⟹ ‖ρ‖ ≥ R+2^j) ✓.\n  - |γ| ≤ ‖ρ‖ ≤ b: u < 2^{j+1} ⟹ ‖ρ‖ < R+2^{j+1} ⟹ ≤ b ✓.\n  - c ≤ |γ|: ‖ρ‖−1 ≥ R+1 ⟺ u ≥ 2 ✓.\n  - hM1: max a c = a (a ≥ c ⟺ 2^j ≥ 2 ⟺ j ≥ 1 ✓) ≥ R ≥ 1 ✓... max a c ≥ c = R+1 ≥ 1 ✓ (le_max_right).\n  - hMb: max a c ≤ b: a ≤ b ⟺ 2^j−1 ≤ 2^{j+1} ✓; c ≤ b ⟺ R+1 ≤ R+2^{j+1} ⟺ 1 ≤ 2^{j+1} ✓.\n  - brick cnt: b−a+2 = 2^j+3 ✓; max|a||b| = b = R+2^{j+1} (a ≥ 0 ✓ |a| = a ≤ b ✓): log(3b+5) ≤ 2L: b ≤ 3R: 2^{j+1} ≤ 2R: j ≤ J' = J+1: 2^j ≤ 2^{J+1} = 2·2^J ≤ 2R ⟹ 2^{j+1} = 2·2^j ≤ 4R?!?! Hmm: j ≤ J+1 ⟹ 2^j ≤ 2^{J+1} ≤ 2R... so 2^{j+1} ≤ 4R: b ≤ R+4R = 5R: 3b+5 ≤ 15R+5: log(15R+5) ≤ log(16R) = 4log2+L ≤ 4+L ≤ ... 4 ≤ L? L ≥ 9 ✓: ≤ 2L?? log(16R) = log16+L = 4log2+L ≤ 4·1+L = 4+L ≤ 2L ⟺ 4 ≤ L ✓. Hmm OK but that uses log 16 = 4 log 2. Ugh — wait actually: do I need 2^{j+1} ≤ 2R for the brick per j ∈ Ico 1 (J'+1)? j ranges to J' = J+1: at j = J+1: 2^j = 2^{J+1} ≤ 2R: 2^{j+1} = 2^{J+2} ≤ 4R. So b ≤ 5R, log(3·5R+5) = log(15R+5) ≤ log(16R) (5 ≤ R) = log16+L ≤ 2L ✓ (4 ≤ L). Let me just bound: log(3b+5) ≤ 2L via: 3b+5 ≤ 16R ⟸ 15R+5 ≤ 16R ⟸ 5 ≤ R ✓; log(16R) = log 16 + L; log 16 = log(2^4) = 4 log 2 ≤ 4 ≤ L·... 4 ≤ L ⟺ e^4 ≤ R: e^4 ≤ 2.72^4 = 54.8 ≤ R ✓ — hmm need another exp fact. OR: log 16 ≤ 4 (16 ≤ e^4 ⟸ e^4 ≥ 2.7^4 = 53.14 ✓ norm_num) and 4 ≤ L·(4/9)?? 4 ≤ L ⟸ L ≥ 9 ≥ 4 ✓ — as numbers: log 16 ≤ 4 ≤ L? L ≥ 9 ✓ so log 16 + L ≤ 4 + L ≤ L + L = 2L needs 4 ≤ L ✓ (L ≥ 9). But log 16 ≤ 4 needs the exp fact 16 ≤ e^4: e^4 = (e^1)^4 ≥ 2.7^4 = 53.1441 ≥ 16 ✓ norm_num — fine, ~5 lines.\n    Alternatively avoid: log(3b+5) with b ≤ 5R: 3b+5 ≤ 15R+5 ≤ 15R+R = 16R (R ≥ 5)... same. OK — OR simpler: note 3b+5 ≤ 3b+5·(b/1)... Let me just do: log(3b+5) ≤ log(3b+5b) = log(8b) (5 ≤ 5b ⟺ 1 ≤ b ✓ b ≥ R ≥ 1) = log8+log b = 3log2+log b ≤ 3 + log b... log b ≤ log(5R) = log5+L: log5 ≤ L (5 ≤ R): total ≤ 3+L+L ≤ L/3·... 3 ≤ L ⟹ ≤ 2L+L?? NO: log8+log b ≤ (L/3) + ... ugh. CLEANEST: 3b+5 ≤ 8b (5 ≤ 5b ✓) ⟹ log ≤ log 8 + log b: log 8 ≤ 3 (8 ≤ e^3 ⟸ e^3 ≥ 2.7^3 = 19.683 ✓) and log b ≤ log(5R) = log 5 + L ≤ L/9·... log 5 ≤ L ⟸ 5 ≤ R ✓: so log(3b+5) ≤ 3 + (L + L·)... = 3 + log5 + L ≤ 3 + L + L = 3+2L ≤ 2L+L/3·... NO — 3+2L vs 2L: exceeds. log(3b+5) ≤ 2L fails this way. Go with the 16R route: log(3b+5) ≤ log(16R) = 4log2 + L: 4log2 ≤ 4·0.7 = 2.8?? log2 ≤ 1 (have) ⟹ 4log2 ≤ 4: total ≤ 4+L ≤ L+L = 2L ⟺ 4 ≤ L ✓ (L ≥ 9, linarith). ✓ no new exp facts: 4·log2 ≤ 4 (log_two_le_one) and 4 ≤ L (from L ≥ 9). And 3b+5 ≤ 16R ⟸ b ≤ 5R (15R+5 ≤ 16R ⟺ 5 ≤ R ✓).\n    So for up: log bound: log(3b+5) ≤ 2L with b = R+2^{j+1} ≤ 5R for j ≤ J+1. And cnt j ≤ 3A₀(2^j+3)·2L. per-shell (1/2^j)·that ≤ 6A₀L(1+3/2^j) ≤ 15A₀L. Σ over Ico 1 (J+2): card J+1 ≤ 2L+1 ≤ 3L: up ≤ 45A₀L².\n\n    Hmm wait — hcount is per j : ℕ (all j). For j ∈ Ico 1 (J'+1) with J' := J+1: apply brick. Outside: empty via hdj (jf ∈ Ico 1 (J'+1) for elements). The j ∈ Ico case also needs 2^{j+1} ≤ 4R for the log: j ≤ J' = J+1 ✓ as computed.\n\n- The d-function's hdj for up: 1 ≤ Nat.log 2 ⌊‖ρ‖−R⌋₊ (⌊u⌋₊ ≥ 2 ⟸ u ≥ 2 ⟸ ‖ρ‖ ≥ R+2) and ≤ J' = J+1: shown above (2^{jf} ≤ ⌊u⌋₊ ≤ u < R+1 hmm — direct nat route: 2^{jf} ≤ ⌊u⌋₊ (nat) and ⌊u⌋₊ ≤ u < R+1 ≤ 2^{J+1}+... let me redo cleanly: claim jf ≤ J+1 ⟸ 2^{jf} ≤ 2^{J+1} (pow injective... Nat.pow_le_pow_iff_right (2 > 1)... or Nat.log_mono: jf = Nat.log 2 ⌊u⌋₊ ≤ Nat.log 2 ⌊R+1−ε⌋...). Route: ⌊u⌋₊ ≤ u < R+1 ⟹ (⌊u⌋₊:ℝ) < R+1: and 2^{J+1} ≥ ⌊R⌋₊+1 > R hmm: 2^{J+1} > ⌊R⌋₊ (nat, lt_pow_succ) ⟹ 2^{J+1} ≥ ⌊R⌋₊+1 (nat) ⟹ (2^{J+1}:ℝ) ≥ ⌊R⌋₊+1 > R (Nat.lt_floor_add_one). So R < 2^{J+1}: (⌊u⌋₊:ℝ) < R+1 ≤ 2^{J+1}... R+1 ≤ 2^{J+1}?? R < 2^{J+1} and 2^{J+1} ∈ ℕ: R+1 ≤ 2^{J+1} ⟺ ⌈R⌉ ≤ 2^{J+1}: R < 2^{J+1} with 2^{J+1} nat ⟹ ⌈R⌉ ≤ 2^{J+1} ✓ (ceil ≤ since 2^{J+1} is a nat > R... hmm: (2^{J+1}:ℝ) > R ⟹ ⌈R⌉₊ ≤ 2^{J+1}: Nat.ceil_le... yes: ⌈R⌉₊ ≤ 2^{J+1} ⟸ R ≤ (2^{J+1}:ℝ) (Nat.le_ceil... reverse: ceil_le: ⌈R⌉₊ ≤ n ⟺ R ≤ n: R < 2^{J+1} ⟹ R ≤ 2^{J+1} ⟹ ⌈R⌉₊ ≤ 2^{J+1} — hmm that's ⌈R⌉₊ not R+1. (⌊u⌋₊:ℝ) < R+1: ⌊u⌋₊ ≤ ⌈R⌉₊: (⌊u⌋₊:ℝ) < R+1 ⟹ ⌊u⌋₊ ≤ ⌈R⌉₊ (nat: ⌊u⌋₊ < R+1 as reals ⟹ ⌊u⌋₊+1 ≤ ... e.g. R = 100.2: R+1 = 101.2: ⌊u⌋₊ < 101.2 ⟹ ⌊u⌋₊ ≤ 101 = ⌈R⌉₊ ✓ general: ⌊u⌋₊ ≤ ⌈R⌉₊ ⟸ (⌊u⌋₊:ℝ) < R+1 — since ⌊u⌋₊ nat and ⌊u⌋₊ < R+1 ⟹ ⌊u⌋₊ ≤ ⌈R⌉₊?? ⌈R⌉₊ = smallest nat ≥ R: any nat n < R+1 has n ≤ ⌈R⌉₊?? n = 101, R = 100.2: ⌈R⌉ = 101 ✓; R = 100.9: ⌈R⌉ = 101: n < 101.9 ⟹ n ≤ 101 ✓; R = 101.0 exactly: ⌈R⌉ = 101: n < 102 ⟹ n ≤ 101 ✓. Yes: n < R+1 ⟹ n ≤ ⌈R⌉₊ (n+... n ≤ R+... since n < R+1 ⟹ n ≤ R (n nat, R... no: n < R+1 ⟹ n ≤ ⌈R⌉₊: proof: n−1 < R ⟹ n−1 ≤ ⌊R⌋₊... I'll take the route: ⌊u⌋₊ ≤ u and 2^{jf} ≤ ⌊u⌋₊ (nats) ⟹ (2^{jf}:ℝ) ≤ u < R+1: and (2^{J+1}:ℝ) > R: both powers of 2 (nat): 2^{jf} < R+1 ≤ ... hmm I want 2^{jf} ≤ 2^{J+1}: 2^{jf} < R+1 and 2^{J+1} > R: 2^{jf}, 2^{J+1} ∈ ℤ (nats): if 2^{jf} > 2^{J+1} then 2^{jf} ≥ 2^{J+2} > 2R ≥ R+1 hmm 2^{J+2} = 2·2^{J+1} > 2R ≥ R+1 (R ≥ 1) — contradicts 2^{jf} < R+1 ✓. So 2^{jf} ≤ 2^{J+1} ⟹ jf ≤ J+1 ✓ (Nat.pow_le_pow_iff_right... `Nat.pow_le_pow_iff_right (hb : 1 < b) : b^m ≤ b^n ↔ m ≤ n` ✓).\n\nOK. Also for inhi's hdj: jf = Nat.log 2 ⌊R−‖ρ‖⌋₊ ≤ J: ⌊d⌋₊ ≤ ⌊R⌋₊ (d < R ✓ since d = R−‖ρ‖ < R) ⟹ mono ✓. Fine (no +1 issue since d < R strictly).\n\nLet me now write shell_inhi and shell_up. Also the cnt RHS: for inhi I decided cnt j := 3*A0count*((2:ℝ)^j+3)*(2*Real.log R) directly. For inlo I kept the brick-form; for consistency and simplicity in hper, maybe use the 2L-form for inhi/up. inlo already works — leave it.\n\nActually for the assembly totals: inlo ≤ 60A₀L², inhi ≤ 60A₀L², up ≤ 45A₀L².\n\nLet me write shell_inhi now (cloning shell_inlo's structure with d := R−‖ρ‖ and the sub-case split for 2^j ≥ R/2).","signature":""},{"type":"text","text":"shell_inlo compiles. Now shell_inhi — same template, d := R−‖ρ‖, with the extra sub-case for fibers where 2^j ≥ R/2 (empty)."},{"type":"tool_use","id":"Bash_90","name":"Bash","input":{"command":"cd \"D:/Python/Quantum/research/lean/NguyenRH1\" && python - << 'EOF'\np = 'NguyenRH1/LiCircleBound.lean'\nsrc = open(p, encoding='utf-8').read()\nnew_section = '''/-! ## 8b. The inner-high and upper shells -/\n\n/-- **The inner-high shell** (R/2 < ‖ρ‖ ≤ R−2): ≤ 60·A₀·(log R)².\nDyadic in d := R−‖ρ‖ ∈ [2, R/2). -/\nlemma shell_inhi {s : ℂ} {R : ℝ} (hs : ‖s‖ = R) (hsz : ¬ IsNontrivialZero s)\n    (hR : (16002 : ℝ) ≤ R) :\n    ∑ ρ ∈ (circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n        ¬ ‖(ρ : ℂ)‖ ≤ R / 2 ∧ ‖(ρ : ℂ)‖ ≤ R - 2), ‖seriesTerm s ρ‖\n      ≤ 60 * A0count * (Real.log R) ^ 2 := by\n  classical\n  have hRpos : (0 : ℝ) < R := by linarith [hR]\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  set J := Nat.log 2 ⌊R⌋₊ with hJ\n  have hJle : (J : ℝ) ≤ 2 * Real.log R := Nat.log_floor_two_le hR\n  have h2JR : (2 : ℝ) ^ J ≤ R := two_pow_log_floor_le hR\n  set FS := (circleZeros R).filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n    ¬ ‖(ρ : ℂ)‖ ≤ R / 2 ∧ ‖(ρ : ℂ)‖ ≤ R - 2) with hFS\n  have hdj : ∀ ρ ∈ FS,\n      1 ≤ Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊ ∧ Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊ ≤ J := by\n    intro ρ hρ\n    rw [hFS, Finset.mem_filter] at hρ\n    obtain ⟨-, hρR2, hρRm2⟩ := hρ\n    have hρR2' : (R / 2 : ℝ) < ‖(ρ : ℂ)‖ := lt_of_not_ge hρR2\n    have hd2 : (2 : ℝ) ≤ R - ‖(ρ : ℂ)‖ := by linarith [hρRm2]\n    have hfl2 : (2 : ℕ) ≤ ⌊R - ‖(ρ : ℂ)‖⌋₊ := by\n      rw [Nat.le_floor_iff (by linarith [hd2] : (0 : ℝ) ≤ R - ‖(ρ : ℂ)‖)]\n      exact_mod_cast hd2\n    have h1 : (1 : ℕ) ≤ Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊ := by\n      have h2 := Nat.log_mono_right (b := 2) hfl2\n      have h3 : Nat.log 2 2 = 1 := by decide\n      rw [h3] at h2\n      exact h2\n    have h2 : Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊ ≤ J := by\n      apply Nat.log_mono_right (b := 2)\n      apply Nat.floor_mono\n      have hnn : (0 : ℝ) ≤ ‖(ρ : ℂ)‖ := norm_nonneg _\n      linarith [hnn]\n    exact ⟨h1, h2⟩\n  have hterm : ∀ ρ ∈ FS, ‖seriesTerm s ρ‖\n      ≤ 2 * (zeroMult (ρ : ℂ) : ℝ) / (2 : ℝ) ^ (Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊) := by\n    intro ρ hρ\n    rw [hFS, Finset.mem_filter] at hρ\n    obtain ⟨-, hρR2, hρRm2⟩ := hρ\n    have hρR2' : (R / 2 : ℝ) < ‖(ρ : ℂ)‖ := lt_of_not_ge hρR2\n    have hd2 : (2 : ℝ) ≤ R - ‖(ρ : ℂ)‖ := by linarith [hρRm2]\n    have hpowle : (2 : ℝ) ^ (Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊) ≤ R - ‖(ρ : ℂ)‖ := by\n      have hfl : ⌊R - ‖(ρ : ℂ)‖⌋₊ ≠ 0 := by\n        rw [Ne, Nat.floor_eq_zero]\n        push Not\n        linarith [hd2]\n      have h1 := Nat.pow_log_le_self 2 hfl\n      have h2 : ((2 ^ Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊ : ℕ) : ℝ) ≤ (⌊R - ‖(ρ : ℂ)‖⌋₊ : ℝ) := by\n        exact_mod_cast h1\n      have h3 : ((2 ^ Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊ : ℕ) : ℝ)\n          = (2 : ℝ) ^ (Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊) := by\n        push_cast\n        ring\n      rw [h3] at h2\n      have h4 : (⌊R - ‖(ρ : ℂ)‖⌋₊ : ℝ) ≤ R - ‖(ρ : ℂ)‖ :=\n        Nat.floor_le (by linarith [hd2])\n      linarith [h2, h4]\n    have hsr : R - ‖(ρ : ℂ)‖ ≤ ‖s - (ρ : ℂ)‖ := by\n      have h := abs_norm_sub_norm_le s (ρ : ℂ)\n      rw [hs, abs_of_nonneg (by linarith [hd2] : (0 : ℝ) ≤ R - ‖(ρ : ℂ)‖)] at h\n      exact h\n    have hρpos : (0 : ℝ) < ‖(ρ : ℂ)‖ := by linarith [hρR2', hR]\n    have hsrpos : (0 : ℝ) < ‖s - (ρ : ℂ)‖ := by linarith [hsr, hd2]\n    have hid := seriesTerm_norm_eq_div hs hsz ρ\n    rw [hid]\n    have hm : (0 : ℝ) ≤ (zeroMult (ρ : ℂ) : ℝ) := Nat.cast_nonneg _\n    have h1 : R / (‖(ρ : ℂ)‖ * ‖s - (ρ : ℂ)‖)\n        ≤ R / (‖(ρ : ℂ)‖ * (R - ‖(ρ : ℂ)‖)) := by\n      apply div_le_div_of_nonneg_left hRpos.le (mul_pos hρpos (by linarith [hd2]))\n      exact mul_le_mul_of_nonneg_left hsr hρpos.le\n    have h2 : R / (‖(ρ : ℂ)‖ * (R - ‖(ρ : ℂ)‖)) ≤ 2 / (R - ‖(ρ : ℂ)‖) := by\n      rw [div_le_div_iff₀ (mul_pos hρpos (by linarith [hd2] : (0 : ℝ) < ‖(ρ : ℂ)‖ * (R - ‖(ρ : ℂ)‖)))\n        (by linarith [hd2] : (0 : ℝ) < R - ‖(ρ : ℂ)‖)]\n      have h3 : (R - ‖(ρ : ℂ)‖) * (2 : ℝ) = 2 * (R - ‖(ρ : ℂ)‖) := by ring\n      have h4 : R * (R - ‖(ρ : ℂ)‖) ≤ (‖(ρ : ℂ)‖ * (R - ‖(ρ : ℂ)‖)) * 2 := by\n        have h5 : R ≤ 2 * ‖(ρ : ℂ)‖ := by linarith [hρR2']\n        have h6 : (0 : ℝ) ≤ R - ‖(ρ : ℂ)‖ := by linarith [hd2]\n        nlinarith [h5, h6, hρpos]\n      linarith [h3, h4]\n    have h3 : (2 : ℝ) / (R - ‖(ρ : ℂ)‖) ≤ 2 / (2 : ℝ) ^ (Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊) := by\n      apply div_le_div_of_nonneg_left (by norm_num : (0 : ℝ) ≤ 2) (by positivity)\n      exact hpowle\n    calc (zeroMult (ρ : ℂ) : ℝ) * (R / (‖(ρ : ℂ)‖ * ‖s - (ρ : ℂ)‖))\n        ≤ (zeroMult (ρ : ℂ) : ℝ) * (R / (‖(ρ : ℂ)‖ * (R - ‖(ρ : ℂ)‖))) :=\n          mul_le_mul_of_nonneg_left h1 hm\n      _ ≤ (zeroMult (ρ : ℂ) : ℝ) * (2 / (R - ‖(ρ : ℂ)‖)) :=\n          mul_le_mul_of_nonneg_left h2 hm\n      _ ≤ (zeroMult (ρ : ℂ) : ℝ) * (2 / (2 : ℝ) ^ (Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊)) :=\n          mul_le_mul_of_nonneg_left h3 hm\n      _ = 2 * (zeroMult (ρ : ℂ) : ℝ) / (2 : ℝ) ^ (Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊) := by ring\n  have hcount : ∀ j : ℕ,\n      ∑ ρ ∈ FS.filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n          Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊ = j), (zeroMult (ρ : ℂ) : ℝ)\n      ≤ 3 * A0count * ((2 : ℝ) ^ j + 3) * (2 * Real.log R) := by\n    intro j\n    have hRHSnn : (0 : ℝ) ≤ 3 * A0count * ((2 : ℝ) ^ j + 3) * (2 * Real.log R) :=\n      mul_nonneg (mul_nonneg (mul_nonneg (by norm_num : (0 : ℝ) ≤ 3) A0count_pos.le)\n        (by positivity)) (mul_nonneg (by norm_num : (0 : ℝ) ≤ 2) hlogRnn)\n    by_cases hj : j ∈ Finset.Ico 1 (J + 1)\n    · rw [Finset.mem_Ico] at hj\n      by_cases hbig : R / 2 ≤ (2 : ℝ) ^ j\n      · -- the fiber is empty: d ≥ 2^j ≥ R/2 contradicts d < R/2\n        have hempty : FS.filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n            Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊ = j) = ∅ := by\n          apply Finset.eq_empty_iff_forall_notMem.mpr\n          intro ρ hρ\n          rw [Finset.mem_filter] at hρ\n          obtain ⟨hρF, hρj⟩ := hρ\n          rw [hFS, Finset.mem_filter] at hρF\n          obtain ⟨-, hρR2, hρRm2⟩ := hρF\n          have hρR2' : (R / 2 : ℝ) < ‖(ρ : ℂ)‖ := lt_of_not_ge hρR2\n          have hd2 : (2 : ℝ) ≤ R - ‖(ρ : ℂ)‖ := by linarith [hρRm2]\n          have hpowle : (2 : ℝ) ^ j ≤ R - ‖(ρ : ℂ)‖ := by\n            have hfl : ⌊R - ‖(ρ : ℂ)‖⌋₊ ≠ 0 := by\n              rw [Ne, Nat.floor_eq_zero]\n              push Not\n              linarith [hd2]\n            have h1 := Nat.pow_log_le_self 2 hfl\n            rw [hρj] at h1\n            have h2 : ((2 ^ j : ℕ) : ℝ) ≤ (⌊R - ‖(ρ : ℂ)‖⌋₊ : ℝ) := by exact_mod_cast h1\n            have h3 : ((2 ^ j : ℕ) : ℝ) = (2 : ℝ) ^ j := by push_cast; ring\n            rw [h3] at h2\n            have h4 : (⌊R - ‖(ρ : ℂ)‖⌋₊ : ℝ) ≤ R - ‖(ρ : ℂ)‖ :=\n              Nat.floor_le (by linarith [hd2])\n            linarith [h2, h4]\n          linarith [hpowle, hbig, hρR2']\n        rw [hempty, Finset.sum_empty]\n        exact hRHSnn\n      · push Not at hbig\n        have hj2 : (2 : ℝ) ≤ (2 : ℝ) ^ j := by\n          have h1 : (2 : ℝ) ^ 1 ≤ (2 : ℝ) ^ j :=\n            pow_le_pow_right₀ (by norm_num) (by omega : 1 ≤ j)\n          linarith [h1]\n        have hM1 : (1 : ℝ) ≤ max (R - (2 : ℝ) ^ (j + 1) - 1) (R / 2 - 1) := by\n          have h1 : (1 : ℝ) ≤ R / 2 - 1 := by linarith [hR]\n          exact le_trans h1 (le_max_right _ _)\n        have hMb : max (R - (2 : ℝ) ^ (j + 1) - 1) (R / 2 - 1) ≤ R - (2 : ℝ) ^ j := by\n          apply max_le_iff.mpr\n          constructor\n          · have h1 : (2 : ℝ) ^ (j + 1) = 2 * (2 : ℝ) ^ j := by rw [pow_succ']\n            have h2 : (0 : ℝ) < (2 : ℝ) ^ j := by positivity\n            linarith [h1, h2]\n          · linarith [hbig]\n        have hsub : ∀ ρ ∈ FS.filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n              Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊ = j),\n            R - (2 : ℝ) ^ (j + 1) - 1 ≤ |(ρ : ℂ).im| ∧ |(ρ : ℂ).im| ≤ R - (2 : ℝ) ^ j ∧\n              R / 2 - 1 ≤ |(ρ : ℂ).im| := by\n          intro ρ hρ\n          rw [Finset.mem_filter] at hρ\n          obtain ⟨hρF, hρj⟩ := hρ\n          rw [hFS, Finset.mem_filter] at hρF\n          obtain ⟨-, hρR2, hρRm2⟩ := hρF\n          have hρR2' : (R / 2 : ℝ) < ‖(ρ : ℂ)‖ := lt_of_not_ge hρR2\n          have hd2 : (2 : ℝ) ≤ R - ‖(ρ : ℂ)‖ := by linarith [hρRm2]\n          have hpowle : (2 : ℝ) ^ j ≤ R - ‖(ρ : ℂ)‖ := by\n            have hfl : ⌊R - ‖(ρ : ℂ)‖⌋₊ ≠ 0 := by\n              rw [Ne, Nat.floor_eq_zero]\n              push Not\n              linarith [hd2]\n            have h1 := Nat.pow_log_le_self 2 hfl\n            rw [hρj] at h1\n            have h2 : ((2 ^ j : ℕ) : ℝ) ≤ (⌊R - ‖(ρ : ℂ)‖⌋₊ : ℝ) := by exact_mod_cast h1\n            have h3 : ((2 ^ j : ℕ) : ℝ) = (2 : ℝ) ^ j := by push_cast; ring\n            rw [h3] at h2\n            have h4 : (⌊R - ‖(ρ : ℂ)‖⌋₊ : ℝ) ≤ R - ‖(ρ : ℂ)‖ :=\n              Nat.floor_le (by linarith [hd2])\n            linarith [h2, h4]\n          have hpowup : R - ‖(ρ : ℂ)‖ < (2 : ℝ) ^ (j + 1) := by\n            have h1 : ⌊R - ‖(ρ : ℂ)‖⌋₊ < 2 ^ (Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊ + 1) :=\n              Nat.lt_pow_succ_log_self (by norm_num) _\n            rw [hρj] at h1\n            have h2 : ⌊R - ‖(ρ : ℂ)‖⌋₊ + 1 ≤ 2 ^ (j + 1) := h1\n            have h3 : R - ‖(ρ : ℂ)‖ < (⌊R - ‖(ρ : ℂ)‖⌋₊ : ℝ) + 1 := Nat.lt_floor_add_one _\n            have h4 : ((⌊R - ‖(ρ : ℂ)‖⌋₊ + 1 : ℕ) : ℝ) ≤ ((2 ^ (j + 1) : ℕ) : ℝ) := by\n              exact_mod_cast h2\n            have h5 : ((2 ^ (j + 1) : ℕ) : ℝ) = (2 : ℝ) ^ (j + 1) := by push_cast; ring\n            rw [h5] at h4\n            push_cast at h4\n            linarith [h3, h4]\n          have him1 := abs_im_ge_norm_sub_one ρ\n          have him2 : |(ρ : ℂ).im| ≤ ‖(ρ : ℂ)‖ := abs_im_le_norm _\n          exact ⟨by linarith [him1, hpowup], by linarith [him2, hpowle],\n            by linarith [him1, hρR2']⟩\n        have hbrick := sum_zeroMult_le_of_im_band\n          (FS.filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n            Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊ = j)) hM1 hMb hsub\n        have hab : R - (2 : ℝ) ^ j - (R - (2 : ℝ) ^ (j + 1) - 1) + 2 = (2 : ℝ) ^ j + 3 := by\n          have h1 : (2 : ℝ) ^ (j + 1) = 2 * (2 : ℝ) ^ j := by rw [pow_succ']\n          linarith [h1]\n        rw [hab] at hbrick\n        -- the log bound: max|a||b| ≤ 3R\n        have hmaxle : max |R - (2 : ℝ) ^ (j + 1) - 1| |R - (2 : ℝ) ^ j| ≤ 3 * R := by\n          have h1 : |R - (2 : ℝ) ^ j| = R - (2 : ℝ) ^ j := by\n            apply abs_of_nonneg\n            have h2 : (2 : ℝ) ^ j ≤ (2 : ℝ) ^ J :=\n              pow_le_pow_right₀ (by norm_num) (by omega : j ≤ J)\n            linarith [h2, h2JR, hbig]\n          have h2 : |R - (2 : ℝ) ^ (j + 1) - 1| ≤ 3 * R := by\n            have h3 : (2 : ℝ) ^ (j + 1) = 2 * (2 : ℝ) ^ j := by rw [pow_succ']\n            have h4 : (2 : ℝ) ^ j ≤ (2 : ℝ) ^ J :=\n              pow_le_pow_right₀ (by norm_num) (by omega : j ≤ J)\n            have h5 : R - (2 : ℝ) ^ (j + 1) - 1 ≤ R := by\n              have h6 : (0 : ℝ) < (2 : ℝ) ^ (j + 1) := by positivity\n              linarith [h6]\n            have h6 : -((2 : ℝ) ^ (j + 1) - 1) ≤ 2 * R + 1 := by\n              nlinarith [h3, h4, h2JR, hR]\n            rw [abs_le]\n            constructor <;> linarith [h5, h6, hR]\n          have h3 : |R - (2 : ℝ) ^ j| ≤ 3 * R := by\n            rw [h1]\n            have h4 : (0 : ℝ) < (2 : ℝ) ^ j := by positivity\n            linarith [h4, hR]\n          exact max_le h2 h3\n        have hlogle : Real.log (3 * max |R - (2 : ℝ) ^ (j + 1) - 1| |R - (2 : ℝ) ^ j| + 5)\n            ≤ 2 * Real.log R :=\n          log_three_le_two_log hR\n            (le_trans (abs_nonneg _) (le_max_left _ _) |>.trans hmaxle) hmaxle\n        exact hbrick.trans (mul_le_mul_of_nonneg_left hlogle\n          (mul_nonneg (mul_nonneg (by norm_num : (0 : ℝ) ≤ 3) A0count_pos.le)\n            (by positivity)))\n    · have hempty : FS.filter (fun ρ : {ρ : ℂ // IsNontrivialZero ρ} =>\n          Nat.log 2 ⌊R - ‖(ρ : ℂ)‖⌋₊ = j) = ∅ := by\n        apply Finset.eq_empty_iff_forall_notMem.mpr\n        intro ρ hρ\n        rw [Finset.mem_filter] at hρ\n        obtain ⟨hρF, hρj⟩ := hρ\n        have hdjρ := hdj ρ hρF\n        rw [hρj] at hdjρ\n        rw [Finset.mem_Ico] at hj\n        omega\n      rw [hempty, Finset.sum_empty]\n      exact hRHSnn\n  -- the closed-form per-shell bound\n  have hper : ∀ j ∈ Finset.Ico 1 (J + 1),\n      (2 / (2 : ℝ) ^ j) * (3 * A0count * ((2 : ℝ) ^ j + 3) * (2 * Real.log R))\n      ≤ 30 * A0count * Real.log R := by\n    intro j hj\n    rw [Finset.mem_Ico] at hj\n    have h2j0 : (0 : ℝ) < (2 : ℝ) ^ j := by positivity\n    have hj2 : (2 : ℝ) ≤ (2 : ℝ) ^ j := by\n      have h1 : (2 : ℝ) ^ 1 ≤ (2 : ℝ) ^ j :=\n        pow_le_pow_right₀ (by norm_num) (by omega : 1 ≤ j)\n      linarith [h1]\n    have hA0 := A0count_pos.le\n    have h1 : (2 / (2 : ℝ) ^ j) * (3 * A0count * ((2 : ℝ) ^ j + 3) * (2 * Real.log R))\n        = 12 * A0count * Real.log R * (1 + 3 / (2 : ℝ) ^ j) := by\n      field_simp\n      ring\n    rw [h1]\n    have h2 : 3 / (2 : ℝ) ^ j ≤ 3 / 2 := by\n      apply div_le_div_of_nonneg_left (by norm_num : (0 : ℝ) ≤ 3) (by norm_num) hj2\n    have h3 : (0 : ℝ) ≤ 1 + 3 / (2 : ℝ) ^ j := by\n      have h4 : (0 : ℝ) ≤ 3 / (2 : ℝ) ^ j := div_nonneg (by norm_num) h2j0.le\n      linarith [h4]\n    have h4 : (1 : ℝ) + 3 / (2 : ℝ) ^ j ≤ 5 / 2 := by linarith [h2]\n    have h5 : (12 : ℝ) * A0count * Real.log R * (1 + 3 / (2 : ℝ) ^ j)\n        ≤ (12 * A0count * Real.log R) * (5 / 2) :=\n      mul_le_mul_of_nonneg_left h4\n        (mul_nonneg (mul_nonneg (by norm_num : (0 : ℝ) ≤ 12) hA0) hlogRnn)\n    have h6 : (12 : ℝ) * A0count * Real.log R * (5 / 2) = 30 * A0count * Real.log R := by\n      ring\n    rw [h6] at h5\n    exact h5\n  -- assembly via dyadic_family_le\n  have hmain := dyadic_family_le FS (fun ρ => R - ‖(ρ : ℂ)‖) J\n    (fun ρ => ‖seriesTerm s ρ‖) 2\n    (fun j => 3 * A0count * ((2 : ℝ) ^ j + 3) * (2 * Real.log R))\n    (by norm_num) hdj hterm hcount\n  calc ∑ ρ ∈ FS, ‖seriesTerm s ρ‖\n      ≤ ∑ j ∈ Finset.Ico 1 (J + 1), (2 / (2 : ℝ) ^ j) *\n          (3 * A0count * ((2 : ℝ) ^ j + 3) * (2 * Real.log R)) := hmain\n    _ ≤ ∑ j ∈ Finset.Ico 1 (J + 1), 30 * A0count * Real.log R := Finset.sum_le_sum hper\n    _ = (J : ℝ) * (30 * A0count * Real.log R) := by\n        rw [Finset.sum_const, Nat.card_Ico, Nat.add_sub_cancel, nsmul_eq_mul]\n    _ ≤ 60 * A0count * (Real.log R) ^ 2 := by\n        have hA0 := A0count_pos.le\n        have h1 : (J : ℝ) * (30 * A0count * Real.log R)\n            ≤ (2 * Real.log R) * (30 * A0count * Real.log R) :=\n          mul_le_mul_of_nonneg_right hJle\n            (mul_nonneg (mul_nonneg (by norm_num : (0 : ℝ) ≤ 30) hA0) hlogRnn)\n        have h2 : (2 * Real.log R) * (30 * A0count * Real.log R)\n            = 60 * A0count * (Real.log R) ^ 2 := by ring\n        rw [h2] at h1\n        exact h1\n\nend NguyenRH1\n\nend'''\nsrc = src.replace('''end NguyenRH1\n\nend''', new_section)\nopen(p, 'w', encoding='utf-8').write(src)\nEOF\nexport PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean NguyenRH1/LiCircleBound.lean 2>&1 | head -40","description":"Add shell_inhi and compile","timeout":300000},"extra_content":null}]