[{"type":"thinking","thinking":"Only the hRhi unused warning. hRhi (R ≤ 4k+3) is genuinely unused in THIS theorem (the series bound doesn't need the upper circle radius — it's needed for the LATER Fexpr assembly where Ĉ₀(4k+9/2) ≤ Ĉ₀(R+2)). Hmm — but the natural API for the next stage passes the full gap-circle data. Options: keep hRhi for API uniformity (with underscore prefix to silence), or drop it. The final consumer (5e-v-c) will apply this with R from exists_gapCircle_radius which supplies both bounds — dropping hRhi here means the caller just doesn't pass it. But keeping the signature uniform with not_isNontrivialZero_on_gapCircle (which also only needs hRlo)... For cleanliness, drop hRhi from seriesFun_norm_le_gapCircle. Actually wait — will 5e-v-c need Ĉ₀(4k+9/2) ≤ Ĉ₀(R+2)? That uses hRhi + C0hat_mono at the CALL site, not here. Drop it.","signature":""},{"type":"text","text":"Clean except the unused `hRhi` — dropping it from the signature (callers supply it where needed)."},{"type":"tool_use","id":"Edit_120","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean","old_string":"theorem 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 →","new_string":"theorem seriesFun_norm_le_gapCircle {k : ℕ} (hk : 4000 ≤ k)\n    {R : ℝ} (hRlo : 4 * (k : ℝ) + 5 / 2 ≤ R)\n    (hgap : ∀ ρ : ℂ, IsNontrivialZero ρ →"},"extra_content":null}]