← Session 33 — ii′-3b/3c crux + F-D finding; Lagarias letter package
thinking (1339 chars)
Good. Now I know:
- `ZeroCountProp`: unit-window [t, t+1] mult-weighted ≤ C0hat(t+1), t ≥ 2.
- `ZeroCountProp2`: two-unit |γ−t| ≤ 1 ≤ 0.6L+0.886LL+2.7, t ≥ 10.
- `NoLowZeros`: |γ| ≥ 14.
- `RHUpTo T₀`: zeros with |γ| ≤ T₀ have re = 1/2.
- `exists_zero_avoiding_T (hZC)`: T ∈ [a, a+1], clearance 1/(3·C0hat(a+1)) against zeros with im ∈ [a, a+1].
For the bridge ‖Σ zero-terms‖ ≤ 2·(count ≤ FAlt T): I need a GLOBAL count N(T) ≤ FAlt T. Does that exist in Lean? The perimeter has only the unit-window form. The global count could be built by summing unit windows: zeros with |γ| ≤ T, im > 0: cover [0, T] by unit windows [k, k+1]: ≤ T·C0hat(T+1)... that's ≈ T·(0.433 log T) vs FAlt T ≈ (1/2π)T log T ≈ 0.159 T log T — the unit-window summation gives 0.433·T·log T > FAlt! So the global N(T) ≤ FAlt is NOT derivable from the unit-window perimeter — it's a SEPARATE perimeter fact (Trudgian's N(T) directly). D-062 L4 says the perimeter includes "Trudgian 2014 explicit N(T), S(T) bounds... the 2F(T) zeros term via FAlt" — so FAlt's majorant property (N(T) ≤ FAlt T) should be a declared perimeter hypothesis. But I don't see it in the current perimeter list! Let me check: is there a `ZeroCountGlobalProp` or similar? Grep FAlt usage/hypotheses.
Also check how lambda_pos_finite_height uses FAlt (marginTilde F n with F := FAlt presumably).
Checking how the global N(T) ≤ FAlt count enters the perimeter (the bridge needs it).
tool: Grep ok
input
{
"pattern": "FAlt|FBoundProp|hF\\b|countMajorant|Ncount.*FAlt",
"path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean",
"output_mode": "content",
"-n": true,
"head_limit": 40
}
result 2026-08-25 04:22:52Z → 2026-08-26 04:53:50Z
mode='content' num_files=0 filenames=[] content='D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean:41: · rw [_root_.riemannZeta_conj, hρ.1, map_zero]\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean-42- · rw [Complex.conj_re]; exact hρ.2.1\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean-43- · rw [Complex.conj_re]; exact hρ.2.2\n--\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiTightR.lean:2241:lemma iteratedDeriv_riemannZeta_conj (n : ℕ) {z : ℂ} (hz : z ≠ 1) :\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiTightR.lean-2242- iteratedDeriv n riemannZeta ((starRingEnd ℂ) z)\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiTightR.lean-2243- = (starRingEnd ℂ) (iteratedDeriv n riemannZeta z) := by\n--\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiTightR.lean:2246: exact _root_.riemannZeta_conj z\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiTightR.lean-2247- | succ m ihm =>\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiTightR.lean-2248- have hzid := ihm hz\n--\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiTightR.lean:2268:lemma analyticOrderAt_riemannZeta_conj {z : ℂ} (hz : z ≠ 1) :\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiTightR.lean-2269- analyticOrderAt riemannZeta ((starRingEnd ℂ) z) = analyticOrderAt riemannZeta z := by\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiTightR.lean-2270- have hzconj : ((starRingEnd ℂ) z) ≠ 1 := by\n--\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiTightR.lean:2284: rw [iteratedDeriv_riemannZeta_conj k hz, hz1 k hk, map_zero],\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiTightR.lean-2285- by\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiTightR.lean:2286: rw [iteratedDeriv_riemannZeta_conj n hz]\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiTightR.lean-2287- intro hbad' num_lines=20 num_matches=None applied_limit=20 applied_offset=None
raw content_json (database row)
[{"type":"thinking","thinking":"Good. Now I know:\n- `ZeroCountProp`: unit-window [t, t+1] mult-weighted ≤ C0hat(t+1), t ≥ 2.\n- `ZeroCountProp2`: two-unit |γ−t| ≤ 1 ≤ 0.6L+0.886LL+2.7, t ≥ 10.\n- `NoLowZeros`: |γ| ≥ 14.\n- `RHUpTo T₀`: zeros with |γ| ≤ T₀ have re = 1/2.\n- `exists_zero_avoiding_T (hZC)`: T ∈ [a, a+1], clearance 1/(3·C0hat(a+1)) against zeros with im ∈ [a, a+1].\n\nFor the bridge ‖Σ zero-terms‖ ≤ 2·(count ≤ FAlt T): I need a GLOBAL count N(T) ≤ FAlt T. Does that exist in Lean? The perimeter has only the unit-window form. The global count could be built by summing unit windows: zeros with |γ| ≤ T, im > 0: cover [0, T] by unit windows [k, k+1]: ≤ T·C0hat(T+1)... that's ≈ T·(0.433 log T) vs FAlt T ≈ (1/2π)T log T ≈ 0.159 T log T — the unit-window summation gives 0.433·T·log T > FAlt! So the global N(T) ≤ FAlt is NOT derivable from the unit-window perimeter — it's a SEPARATE perimeter fact (Trudgian's N(T) directly). D-062 L4 says the perimeter includes \"Trudgian 2014 explicit N(T), S(T) bounds... the 2F(T) zeros term via FAlt\" — so FAlt's majorant property (N(T) ≤ FAlt T) should be a declared perimeter hypothesis. But I don't see it in the current perimeter list! Let me check: is there a `ZeroCountGlobalProp` or similar? Grep FAlt usage/hypotheses.\n\nAlso check how lambda_pos_finite_height uses FAlt (marginTilde F n with F := FAlt presumably).","signature":""},{"type":"text","text":"Checking how the global N(T) ≤ FAlt count enters the perimeter (the bridge needs it)."},{"type":"tool_use","id":"Grep_168","name":"Grep","input":{"pattern":"FAlt|FBoundProp|hF\\b|countMajorant|Ncount.*FAlt","path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean","output_mode":"content","-n":true,"head_limit":40},"extra_content":null}]