← Session 3 — Project kickoff: RH blueprint, staged workflow + archive design
thinking (2247 chars)
Two things to address:
1. E-007b failed with exit 1 — check output.
2. E-008 check failures:
- λ_1 error 1.35e-4 > 1e-4 tolerance. λ_1 truncated = 0.022961 vs known 0.0230957 — relative error 0.58%. This is the tail contribution: missing zeros contribute ~1.3e-4 to λ_1. My tolerance was too tight — the tail bound for n=1: tail_const·1 = 2·2·(8.36)/(2π·9878) = 4·8.36/62075 = 5.4e-4. So expected tail ~ up to 5.4e-4; observed error 1.35e-4 is within the bound ✔. Fix: tolerance should be the tail bound (5.4e-4), not 1e-4.
- tail at n=500: 14.1% > 5% threshold — my "tail_small" check was too optimistic given the crude 2x-safety bound. The bound is conservative (the integral Σ1/γ² estimate vs actual); 14% at n=500 means honest usable range is n ≲ 300. Relax: check tail/lambda < 5% at n=300 instead, and report the usable range.
- The asymptotic ratio λ_n/((n/2)log n) ≈ 0.63 at large n — hmm, known leading term: λ_n ~ (1/2)n log n + ...? Actually known: λ_n = n(A log n + B) + o(n)-ish with A = 1/2? Let me sanity check with λ_1000 = 2191: (1000/2)·6.9 = 3453; ratio 0.63. Known asymptotic: λ_n ~ (n/2) log n as n→∞ per... Actually I recall λ_n = n·(½ log n + ½(γ_E − log 2π...))... The ratio 0.63 at n=2000 should approach 1 if ½ n log n is right leading order. With truncation: missing tail at n=2000 is up to 44% (bound) — the measured 0.63 is depressed by truncation (tail contributes positively!). The true ratio at n=2000 could be up to 0.63/(1−0.45)-ish... So the trend measurement is inconclusive beyond n~300. That's fine — the honest reporting: ratio ≈ 0.63 and slowly rising where measurable; tail bound caps usable n.
Hmm wait, actually let me reconsider the leading asymptotic: λ_n = Σ_ρ(1−(1−1/ρ)^n). Known: λ_n ~ (1/2) n log n? Bombieri–Lagarias: λ_n = (1/2)n log n + O(n)... I believe that's right (the archimedean contribution gives n log n /2... the prime sums give lower order under RH). OK keep ratio reference but note truncation bias.
Fix checks: λ_1 tolerance = tail bound at n=1 (report ratio); usable range: report max n with tail/lambda < 5% (compute from data: solve tail_n/λ_n < 0.05).
3. E-007b exit 1 — need the output. Probably an error (maybe timeout? exit 1 not 143). Read the output file.
E-007b failed — checking its output first, then fixing E-008's tolerance design.
tool: Read ok
input
{
"file_path": "C:\\Users\\lucky\\AppData\\Local\\Temp\\nguyen\\ff74d795-3344-493a-8c08-dc1bd8e633e1\\tasks\\br2amanfv.output"
}
result 2026-08-17 22:58:04Z → 2026-08-24 13:05:13Z
{
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"file": {
"filePath": "D:\\Python\\Quantum\\NGUYEN.md",
"content": " warnings; LiCircleBound + LiSqueeze root-imported; full build GREEN;\n audit-session034-axioms.txt: 5 decls, 3 standard axioms only).\n `C0hat_add_two_le` (\u0108\u2080(R+2) \u2264 5\u00b7log R, R \u2265 16002);\n `not_badPoint_on_gapCircle` (every gap-circle point good);\n `gentire_circle_le` (the circle bound transfers to Gentire);\n **`gentire_const`** (the Schwarz squeeze: max-modulus via Zeta23's\n AnalyticOn.norm_le_of_norm_le_on_sphere + the ball-form Schwarz lemma\n on the \u03b3-gap circle sequence; growth RB \u2264 C\u2080\u2032 + 64\u00b7C\u2081\u00b7Q\u00b3 with\n Q := R^{1/4}, R = Q\u2074 (Real.log_le_rpow_div + pow_le_pow_left\u2080 +\n rpow_inv_natCast_pow), radius choice Q \u2265 2 + (128\u00b7C\u2081+2\u00b7C\u2080\u2032)\u00b7\u2016z\u2080\u2016/\u03b5\n \u27f9 dist \u2264 \u03b5 \u27f9 Gentire \u2261 Gentire 0); **THE CAPSTONE\n `partialFraction_identity (hZC : ZeroCountProp)`**: \u2203 B : \u2102, \u2200 s\n good, logDeriv riemannZeta s = B + seriesFun s \u2212 1/s \u2212 1/(s\u22121) +\n (log \u03c0)/2 \u2212 \u03c8(s/2)/2 \u2014 the partial-fraction identity Lean-proved\n modulo ONLY the declared perimeter (first perimeter use in the 5e\n chain: the \u03b3-gap circles need ZeroCountProp). Banked: le_or_lt\n absent (le_or_gt + the dist/2 contradiction closes the squeeze);\n set-Q + rw [\u2190 hQ] after push_cast for the rpow_inv_natCast_pow",
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raw content_json (database row)
[{"type":"thinking","thinking":"Two things to address:\n\n1. E-007b failed with exit 1 — check output.\n2. E-008 check failures:\n - λ_1 error 1.35e-4 > 1e-4 tolerance. λ_1 truncated = 0.022961 vs known 0.0230957 — relative error 0.58%. This is the tail contribution: missing zeros contribute ~1.3e-4 to λ_1. My tolerance was too tight — the tail bound for n=1: tail_const·1 = 2·2·(8.36)/(2π·9878) = 4·8.36/62075 = 5.4e-4. So expected tail ~ up to 5.4e-4; observed error 1.35e-4 is within the bound ✔. Fix: tolerance should be the tail bound (5.4e-4), not 1e-4.\n - tail at n=500: 14.1% > 5% threshold — my \"tail_small\" check was too optimistic given the crude 2x-safety bound. The bound is conservative (the integral Σ1/γ² estimate vs actual); 14% at n=500 means honest usable range is n ≲ 300. Relax: check tail/lambda < 5% at n=300 instead, and report the usable range.\n - The asymptotic ratio λ_n/((n/2)log n) ≈ 0.63 at large n — hmm, known leading term: λ_n ~ (1/2)n log n + ...? Actually known: λ_n = n(A log n + B) + o(n)-ish with A = 1/2? Let me sanity check with λ_1000 = 2191: (1000/2)·6.9 = 3453; ratio 0.63. Known asymptotic: λ_n ~ (n/2) log n as n→∞ per... Actually I recall λ_n = n·(½ log n + ½(γ_E − log 2π...))... The ratio 0.63 at n=2000 should approach 1 if ½ n log n is right leading order. With truncation: missing tail at n=2000 is up to 44% (bound) — the measured 0.63 is depressed by truncation (tail contributes positively!). The true ratio at n=2000 could be up to 0.63/(1−0.45)-ish... So the trend measurement is inconclusive beyond n~300. That's fine — the honest reporting: ratio ≈ 0.63 and slowly rising where measurable; tail bound caps usable n.\n\n Hmm wait, actually let me reconsider the leading asymptotic: λ_n = Σ_ρ(1−(1−1/ρ)^n). Known: λ_n ~ (1/2) n log n? Bombieri–Lagarias: λ_n = (1/2)n log n + O(n)... I believe that's right (the archimedean contribution gives n log n /2... the prime sums give lower order under RH). OK keep ratio reference but note truncation bias.\n\n Fix checks: λ_1 tolerance = tail bound at n=1 (report ratio); usable range: report max n with tail/lambda < 5% (compute from data: solve tail_n/λ_n < 0.05).\n\n3. E-007b exit 1 — need the output. Probably an error (maybe timeout? exit 1 not 143). Read the output file.","signature":""},{"type":"text","text":"E-007b failed — checking its output first, then fixing E-008's tolerance design."},{"type":"tool_use","id":"Read_282","name":"Read","input":{"file_path":"C:\\Users\\lucky\\AppData\\Local\\Temp\\nguyen\\ff74d795-3344-493a-8c08-dc1bd8e633e1\\tasks\\br2amanfv.output"},"extra_content":null}]