← Session 33 — ii′-3b/3c crux + F-D finding; Lagarias letter package
thinking (1620 chars)
`RectangleIntegral' = (1/(2πi))•RectangleIntegral` — it IS normalized by 2πi. And RectangleIntegral = H(z.im) − H(w.im) + V(w.re) − V(z.re) — four sides.
So ‖RectangleIntegral'‖ ≤ (1/2π)·(‖I_bottom‖ + ‖I_top‖ + ‖I_right‖ + ‖I_left‖) where the I's are the raw side integrals.
Wait — HIntegral f z.re w.re z.im = ∫_{z.re}^{w.re} f(x + z.im·i) dx — the bottom side (im = z.im = −T); H at w.im = +T (top); VIntegral f w.re z.im w.im = i·∫_{z.im}^{w.im} f(w.re + yi) dy (right side at re = w.re = 2√n); V at z.re = −5/2 (left).
So the contour goes: bottom (−T), right, top (negated), left (negated) — orientation doesn't matter for norms.
‖RectangleIntegral'‖ ≤ (1/2π)·(‖bottom‖ + ‖top‖ + ‖right‖ + ‖left‖).
Now — Bn's side pieces: 4T[...] (I_III = left vertical), 2(e⁴+1)[7Ĥ+3.3] (I_II + I_IV = horizontals), IIMaj (I_I = right vertical). Are these the RAW side bounds (then ‖RectInt'‖ ≤ Σ/(2π) ≤ Σ) or...? D-037's identity: S_f = ... + (1/2πi)(I_I + I_II + I_III + I_IV) — so |Sf| ≤ ... + (1/2π)(|I_I|+...+|I_IV|) ≤ ... + (|I_I|+...+|I_IV|) — the paper's Bn charges the RAW side bounds (dropping the 2π gain — fine, consistent over-count). So in Lean: ‖RectInt'‖ ≤ (1/2π)·Σ‖sides‖ ≤ Σ‖sides‖ ≤ Bn-sides (using (1/2π) ≤ 1·... need (1/(2π))·X ≤ X for X ≥ 0 — or just charge ‖RectInt'‖ ≤ Σ/（2π) and the Bn side terms are the raw bounds ≥ Σ/(2π)... the paper charges Σ raw ≥ Σ/2π ✓ consistent).
OK. Now — M₀'s Lean form: LiSkeleton:753 mentions M₀ = 1.3352 in a comment (the margin majorant chain def). But is there an M₀ PROP in the perimeter? The Phase 5B header says the perimeter includes M₀. Let me grep for a Prop.
RectangleIntegral′ is normalized by 2πi (the paper's Bn over-counts sides by 2π — consistent). Checking the M₀ perimeter form.
tool: Grep ok
input
{
"pattern": "MZeroProp|M0Prop|1\\.3352|shortPiece|logDerivZetaBound",
"path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1",
"output_mode": "content",
"-n": true,
"head_limit": 15
}
result 2026-08-25 04:28:15Z → 2026-08-25 04:28:16Z
mode='content' num_files=0 filenames=[] content='D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean:753:M₀ = 1.3352 (E-023, grid-verified — see D-060 §4 for its status). -/\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean:759: 40 * 1.3352 + 2 * (Real.exp 4 + 1) * (7 * Hhat T + 3.3) +\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean:1819:private def mK : ℝ := -40*1.3352 - 2*(Real.exp 4 + 1)*(7*(3*2.9^2 + 48) + 3.3) - 1 + 1\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean:2409:B₀ = 48 from D-056; Ĉ₀/Ĥ closed forms; M₀ = 1.3352 from E-023; IIMaj the\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean:2411:B(n) = 2F(T) + 4T[(2/3)Ĉ₀(T) + 10.5·log(T+2) + 48] + 40·1.3352\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean:2416: 40 * 1.3352 + 2 * (Real.exp 4 + 1) * (7 * Hhat T + 3.3) + IIMaj n + 1' num_lines=6 num_matches=None applied_limit=None applied_offset=None
raw content_json (database row)
[{"type":"thinking","thinking":"`RectangleIntegral' = (1/(2πi))•RectangleIntegral` — it IS normalized by 2πi. And RectangleIntegral = H(z.im) − H(w.im) + V(w.re) − V(z.re) — four sides.\n\nSo ‖RectangleIntegral'‖ ≤ (1/2π)·(‖I_bottom‖ + ‖I_top‖ + ‖I_right‖ + ‖I_left‖) where the I's are the raw side integrals.\n\nWait — HIntegral f z.re w.re z.im = ∫_{z.re}^{w.re} f(x + z.im·i) dx — the bottom side (im = z.im = −T); H at w.im = +T (top); VIntegral f w.re z.im w.im = i·∫_{z.im}^{w.im} f(w.re + yi) dy (right side at re = w.re = 2√n); V at z.re = −5/2 (left).\n\nSo the contour goes: bottom (−T), right, top (negated), left (negated) — orientation doesn't matter for norms.\n\n‖RectangleIntegral'‖ ≤ (1/2π)·(‖bottom‖ + ‖top‖ + ‖right‖ + ‖left‖).\n\nNow — Bn's side pieces: 4T[...] (I_III = left vertical), 2(e⁴+1)[7Ĥ+3.3] (I_II + I_IV = horizontals), IIMaj (I_I = right vertical). Are these the RAW side bounds (then ‖RectInt'‖ ≤ Σ/(2π) ≤ Σ) or...? D-037's identity: S_f = ... + (1/2πi)(I_I + I_II + I_III + I_IV) — so |Sf| ≤ ... + (1/2π)(|I_I|+...+|I_IV|) ≤ ... + (|I_I|+...+|I_IV|) — the paper's Bn charges the RAW side bounds (dropping the 2π gain — fine, consistent over-count). So in Lean: ‖RectInt'‖ ≤ (1/2π)·Σ‖sides‖ ≤ Σ‖sides‖ ≤ Bn-sides (using (1/2π) ≤ 1·... need (1/(2π))·X ≤ X for X ≥ 0 — or just charge ‖RectInt'‖ ≤ Σ/（2π) and the Bn side terms are the raw bounds ≥ Σ/(2π)... the paper charges Σ raw ≥ Σ/2π ✓ consistent).\n\nOK. Now — M₀'s Lean form: LiSkeleton:753 mentions M₀ = 1.3352 in a comment (the margin majorant chain def). But is there an M₀ PROP in the perimeter? The Phase 5B header says the perimeter includes M₀. Let me grep for a Prop.","signature":""},{"type":"text","text":"RectangleIntegral′ is normalized by 2πi (the paper's Bn over-counts sides by 2π — consistent). Checking the M₀ perimeter form."},{"type":"tool_use","id":"Grep_173","name":"Grep","input":{"pattern":"MZeroProp|M0Prop|1\\.3352|shortPiece|logDerivZetaBound","path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1","output_mode":"content","-n":true,"head_limit":15},"extra_content":null}]