← Session 29 — ii′-5e-v: circle bound (far-tail + series bounds)
thinking (800 chars)
Now let me read the current LiCircleBound.lean to see what's there and plan chunk 3 — the main seriesFun circle bound `seriesFun_norm_le_gapCircle`.
The task (from STATE.md NEXT ACTION):
1. ‖seriesFun s‖ ≤ ∑' ρ, m·R/(‖ρ‖·‖s−ρ‖) via norm_tsum_le_tsum_norm + per-term norm identity from one_div_sub_add_one_div (needs ζ s ≠ 0 on the circle — discharged via re ≥ 1 nonvanishing, dist(R/2,ℤ) ≥ 1/4, γ-gap + threshold lemma).
2. THRESHOLD LEMMA (k ≥ 4000): 16·Ĉ₀(4k+9/2)·26 ≤ 4k+2-ish — explicit linear/rational arithmetic.
3. ONE tsum_split_finset at F := {‖ρ‖ ≤ 2R} + tail via zero_tail_inv_sq_le.
4. The finite part by Finset.filter: low / inner-low / inner-high / critical / upper bands, each dyadic with Ncount_le_of_interval_gen.
Let me read the current file first to see exactly what's available.
Now let me read the current LiCircleBound.lean to see the existing machinery.
tool: Read
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean"
}
raw content_json (database row)
[{"type":"thinking","thinking":"Now let me read the current LiCircleBound.lean to see what's there and plan chunk 3 — the main seriesFun circle bound `seriesFun_norm_le_gapCircle`.\n\nThe task (from STATE.md NEXT ACTION):\n1. ‖seriesFun s‖ ≤ ∑' ρ, m·R/(‖ρ‖·‖s−ρ‖) via norm_tsum_le_tsum_norm + per-term norm identity from one_div_sub_add_one_div (needs ζ s ≠ 0 on the circle — discharged via re ≥ 1 nonvanishing, dist(R/2,ℤ) ≥ 1/4, γ-gap + threshold lemma).\n2. THRESHOLD LEMMA (k ≥ 4000): 16·Ĉ₀(4k+9/2)·26 ≤ 4k+2-ish — explicit linear/rational arithmetic.\n3. ONE tsum_split_finset at F := {‖ρ‖ ≤ 2R} + tail via zero_tail_inv_sq_le.\n4. The finite part by Finset.filter: low / inner-low / inner-high / critical / upper bands, each dyadic with Ncount_le_of_interval_gen.\n\nLet me read the current file first to see exactly what's available.","signature":""},{"type":"text","text":"Now let me read the current LiCircleBound.lean to see the existing machinery."},{"type":"tool_use","id":"Read_4","name":"Read","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean"},"extra_content":null}]