[{"type":"thinking","thinking":"Now I understand the perimeter definitions. Key facts:\n\n- `ZeroCountProp`: ∀ t ≥ 2, mult-weighted count in [t, t+1] ≤ C0hat(t+1).\n- `C0hat t = 0.433·log(t+2) + 0.886·log log(t+2) + 2.9`.\n- `NoLowZeros`: no nontrivial zeros with |γ| ≤ 14.\n- `isNontrivialZero_conj` (from LiCircleBound) for the negative side.\n\nNow, the piece-3 design (journal-035 + session-035 journal \"Status/next\"):\n\n**The far difference series.** `∑' ρ, if ρ ∈ nearZerosSub t then 0 else seriesDiffTerm s s₀ ρ` where `seriesDiffTerm s s₀ ρ = m·(1/(s−ρ) − 1/(s₀−ρ)) = m·(s₀−s)/((s−ρ)(s₀−ρ))`.\n\nNorm bound per term: ‖seriesDiffTerm‖ = m·‖s₀−s‖/(‖s−ρ‖·‖s₀−ρ‖) ≤ 4.5·m/(x·√(1+x²)) where x = |t−γ| (since ‖s₀−s‖ = |2−s.re| ≤ 4.5, ‖s−ρ‖ ≥ |t−γ| = x, ‖s₀−ρ‖ = √((2−re ρ)² + (t−γ)²) ≥ √(1+x²)).\n\nFor ρ far (not near), x = |t−γ| ≥ 1. On interval k ≤ x < k+1: term ≤ 4.5·m/(k√(1+k²)).\n\n**Interval decomposition.** We need to sum over the zero subtype, partitioned by k = ⌊|t−γ|⌋. This is a tsum over a subtype with a finset split... The cleanest Lean approach might be:\n\nFor each k from 1 to K_max = ⌈t⌉ or so, the zeros with k ≤ |t−γ| < k+1 form a finite set, with mult-weighted count bounded via the perimeter:\n- Right side: γ ∈ [t+k, t+k+1): count ≤ C0hat(t+k+1) via ZeroCountProp at t+k (needs t+k ≥ 2 ✓).\n  Then C0hat(t+k+1) ≤ C0hat(2t+1) for k ≤ t, and C0hat(2t+1) ≤ C0hat(t+1) + 0.55 (the L′ conversion).\n- Left side: γ ∈ (t−k−1, t−k]: count ≤ C0hat(t−k+1) ≤ C0hat(t+1).\n  Wait, the perimeter ZeroCountProp is for windows [a, a+1] with a ≥ 2. The left interval (t−k−1, t−k] needs t−k−1 ≥ 2, i.e. k ≤ t−3. For k up to t−3, both sides are covered by ZeroCountProp. For k > t (i.e. γ < 0 or γ > 2t), we need the tail bound 0.5828 via A0count.\n\nHmm, wait — but the heights < 14 are empty by NoLowZeros. The left intervals with t−k < 14 are empty. Hmm, but that doesn't change the bound (count ≤ C0hat still holds, just loose). Actually NoLowZeros is used for... let me re-read:\n\n\"the height-< 14 intervals empty by NoLowZeros\"\n\nHmm. For the left side when t is small (t ∈ [10, 14]), the windows below height 14 would need a ≥ 2 for ZeroCountProp, but heights below 14 are empty. But actually — hmm, when t ∈ [10,14), left intervals go down to negative γ... Since NoLowZeros says |γ| ≥ 14 for ALL zeros, when t < 14 there are NO zeros with γ ∈ [0, t] at all! And zeros with γ < 0 have |t−γ| = t + |γ| ≥ t + 14 ≥ 24, so they fall in the k ≥ 24 interval — the k > t tail.\n\nWait, more carefully: for γ < 0, x = |t−γ| = t + |γ| ≥ t + 14. For γ > 2t, x = γ − t > t. So \"k > t tail\" = zeros with x > t, covering γ < 0 (x ≥ t+14) and γ > 2t (x > t).\n\nFor the k > t tail: the intervals are [k, k+1) with k > t. On the right (γ > 2t): γ ∈ [t+k, t+k+1), count ≤ C0hat(t+k+1) — but this grows like log(t+k), and Σ 4.5/(k√(1+k²))·C0hat(t+k+1) converges but the bound would be C0hat-dependent... hmm, that's not how the 0.5828 was computed.\n\nLet me reconsider: journal-035 says \"the k > t tail ≤ 0.5828 (Σ log(3k)/k² and Σ 1/k² via the same telescopes)\". This suggests using the UNCONDITIONAL A0count (Jensen-type): the mult-weighted count in a unit band at height y is ≤ 3·A₀·3·log(3y+5)-ish (from LiCircleBound's `sum_zeroMult_le_of_im_band`: band [a,b] ⟹ ≤ 3·A₀·(b−a+2)·log(3·max|a||b|+5)).\n\nHmm wait, but that would give a bound involving A₀. Let me check: for k > t ≥ 10, the count in [t+k, t+k+1] via the band lemma: ≤ 3·A₀·3·log(3(t+k+1)+5). And A₀ = A0count is some explicit constant. Then the tail Σ_{k>t} 4.5/(k√(1+k²)) · 9·A₀·log(3(t+k+1)+5)... that would be ~9·A₀·4.5·Σ log(3k)/k² — depends on A₀.\n\nHmm, \"0.5828\" — let me see. Σ_{k≥11} 1/(k√(1+k²)) ≤ Σ 1/k² ≤ 1/10 = 0.1 (via the telescope Σ_{k≥11} 1/(k(k−1)) = 1/10). And Σ_{k≥11} log(3k)/k²-ish... The journal says \"the k > t tail ≤ 0.5828 (Σ log(3k)/k² and Σ 1/k² via the same telescopes)\". \n\nHmm — I think the actual plan: for the tail, use the band lemma with count ≤ 3·A₀·3·log(3(t+k+1)+5) ≤ 9·A₀·log(9k+8)... with t ≤ k: t+k+1 ≤ 2k+1 ≤ 3k, so log(3(t+k+1)+5) ≤ log(9k+8) ≤ log(12k) = log 12 + log k ≤ 2.485 + log k. Hmm.\n\nThen Σ_{k≥11} 4.5/(k√(1+k²)) · 9·A₀·(2.485 + log k)... With A₀ small this could be 0.5828. What's A₀? Let me look at A0count's definition in LiCircleBound.\n\nActually, let me just read the relevant part of LiCircleBound.lean to see the band lemma and A0count. Also `sum_zeroMult_le_of_im_band` — \"band [a,b] + sign-separation c ⟹ 3·A₀·(b−a+2)·log(3·max|a||b|+5)\".\n\nLet me grep for it.","signature":""},{"type":"text","text":"Now let me look at the LiCircleBound band-count lemma and A0count (needed for the k > t tail), plus check how `isNontrivialZero_conj` is stated."},{"type":"tool_use","id":"Grep_9","name":"Grep","input":{"pattern":"theorem sum_zeroMult_le_of_im_band|def A0count|theorem A0count|isNontrivialZero_conj|theorem zero_tail_inv_sq_le|def lowZeroConst","path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean","output_mode":"content","-n":true},"extra_content":null}]