[{"type":"thinking","thinking":"Now E-028: verify the traced constants numerically and recompute N₀/margins. Script: adapt E-023. Components:\n- V-R: compute R(s) at sample points σ ∈ {−2.5, 0.5, 2}, t ∈ {10, 50, 100, 500, 1000} and check |R(s)| ≤ 6.0 log(t+2) + 40 (expect ~5.9 max from E-021's max 4.75 at A=0.9 — huge slack). R(s) computed via zeta_logder (xi route, mpmath dps 30) minus Σ_{|γ−t|<1} 1/(s−ρ) using zeros from E-027 cache (to 3080 — covers t=1000 fine).\n- V5: certified Z₂ ≤ 0.5699 (compute partial sum Σ_{n≤10^4} Λ(n)/n² + tail (log N + 1)/N — need Λ(n): sympy or mpmath... compute via factorization of n ≤ 10⁴ — trivial sieve).\n- Margin rerun: A = 6.0, B₀ = 40, I_I length 2(√n+1): N₀ new, contiguity, convexity grid, margin at refs, N₁.\n- Reuse M₀ = 1.3352 from E-023 JSON (don't recompute the short segment).\n\nOutput E-028-traced-margin.csv/.json. Runtime: zeta_logder 15 points at dps 30 — each maybe a second (mp.diff + zeta) — fine, foreground.\n\nWrite the script.","signature":""},{"type":"text","text":"Now E-028 — verification of the traced R-bound and margin rerun with the new constants."},{"type":"tool_use","id":"Write_142","name":"Write","input":{"content":"\"\"\"E-028 — task 4.5a support: D-056 traced R-bound verification + margin rerun.\n\nComponents:\n  V-R. Verify the traced reference-point R-bound (D-056 Theorem R)\n       |R(s)| <= 6.0 log(|t|+2) + 40,  sigma in [-5/2, 3], |t| >= 10\n       at sample points sigma in {-2.5, 0.5, 2}, t in {10,50,100,500,1000}\n       (R(s) = zeta'/zeta(s) - sum_{|gamma-t|<1} 1/(s-rho), zeta'/zeta\n       via the E-021 xi-route at dps 30; zeros from the E-027 cache).\n  V5.  Certified Z2 = -zeta'/zeta(2) <= 0.5699: partial sum to 1e4 plus\n       tail bound (log N + 1)/N.\n  S3'. Margin rerun with the TRACED constants A = 6.0, B0 = 40 (was:\n       measured 0.9/1.0, first-estimate 2.0/2.0) and the I_I length fix\n       2(sqrt(n)+1). Recompute N0, contiguity to 4e5, convexity grid to\n       N1, margin references. M0 = 1.3352 imported from E-023 (S1).\n\nOutputs: research/numerics/data/E-028-traced-margin.csv (+ .json)\n\"\"\"\nimport csv as csvmod\nimport json\nimport sys\nimport time\nfrom pathlib import Path\n\nimport mpmath as mp\nimport numpy as np\n\nREPO = Path(__file__).resolve().parents[3]\nDATA = REPO / \"research\" / \"numerics\" / \"data\"\nE021 = DATA / \"E-021-sinf-sf-curve.csv\"\nE023J = DATA / \"E-023-assembly.json\"\nZCACHE = DATA / \"E-027-zeros-cache.npy\"\nCSV = DATA / \"E-028-traced-margin.csv\"\nJSON = DATA / \"E-028-traced-margin.json\"\n\nGAMMA = 0.577215664901532860606512090082402431\nC1L = (GAMMA - 1 - np.log(2 * np.pi)) / 2 - GAMMA / 2\nA_TR, B0_TR = 6.0, 40.0\nT0_PLATT = 3.0e12\n\n\ndef zeta_logder(s):\n    sm = mp.mpc(s.real, s.imag)\n\n    def xi_mp(u):\n        return 0.5 * u * (u - 1) * mp.pi ** (-u / 2) * mp.gamma(u / 2) \\\n            * mp.zeta(u)\n\n    xi_log = mp.diff(xi_mp, sm) / xi_mp(sm)\n    return complex(xi_log - 1 / sm - 1 / (sm - 1)\n                   + mp.mpf(0.5) * mp.log(mp.pi)\n                   - mp.mpf(0.5) * mp.digamma(sm / 2))\n\n\ndef check_R_bound(zeros):\n    mp.mp.dps = 30\n    out = []\n    for sigma in (-2.5, 0.5, 2.0):\n        for t in (10.0, 50.0, 100.0, 500.0, 1000.0):\n            s = complex(sigma, t)\n            val = zeta_logder(s)\n            near = [g for g in zeros if abs(g - t) < 1.0]\n            r = val - sum(1.0 / (s - complex(0.5, g)) for g in near)\n            bound = 6.0 * np.log(t + 2) + 40.0\n            out.append({\"sigma\": sigma, \"t\": t, \"abs_R\": abs(r),\n                        \"bound\": bound, \"holds\": bool(abs(r) <= bound)})\n    return out\n\n\ndef certified_Z2(N=10 ** 4):\n    \"\"\"-zeta'/zeta(2) = sum Lambda(n)/n^2; partial sum + tail.\"\"\"\n    lam = np.zeros(N + 1)\n    sieve = np.zeros(N + 1)\n    for p in range(2, N + 1):\n        if sieve[p] == 0:\n            for m in range(p, N + 1, p):\n                sieve[m] = p\n    for n in range(2, N + 1):\n        p = sieve[n]\n        if p > 0 and abs(p ** round(np.log(n) / np.log(p)) - n) < 0.5:\n            lam[n] = np.log(p)\n    part = float(np.sum(lam[2:] / np.arange(2, N + 1) ** 2))\n    tail = (np.log(N) + 1) / N\n    return part, tail, part + tail\n\n\ndef F_count(T):\n    return (T / (2 * np.pi)) * np.log(T / (2 * np.pi)) - T / (2 * np.pi) \\\n        + 7 / 8\n\n\ndef S_bound(T):\n    return 0.137 * np.log(T) + 0.443 * np.log(np.log(T)) + 1.588\n\n\ndef C0(t):\n    return (F_count(t + 1) - F_count(t)) + S_bound(t + 1) + S_bound(t)\n\n\ndef H_bound(T):\n    return 3 * C0(T) ** 2 + A_TR * np.log(T + 2) + B0_TR\n\n\ndef B_bound(n, m0):\n    n = float(n)\n    T = np.sqrt(n) + 1.0\n    twoN = 2 * (F_count(T) + S_bound(T))\n    I3_long = 4 * T * ((2 / 3) * C0(T) + A_TR * np.log(T + 2) + B0_TR)\n    I3_short = 40.0 * m0\n    I24 = 2 * (np.e ** 4 + 1) * (7 * H_bound(T) + 3.3)\n    r = np.sqrt(n)\n    # I_I length fix: 2T = 2(sqrt(n)+1)\n    logI1 = np.log(2 * (r + 1) * 0.5699) + r / 2 - (2 * r - 2) * np.log(2)\n    I1 = np.exp(logI1) if logI1 > -700 else 0.0\n    return twoN + I3_long + I3_short + I24 + I1 + 1.0\n\n\ndef margin(n, m0):\n    n = np.asarray(n, dtype=np.float64)\n    return 0.5 * n * np.log(n) + C1L * n - 4.0 - B_bound(n, m0) + 1.0\n\n\ndef main() -> int:\n    if JSON.exists() and \"--force\" not in sys.argv:\n        print(f\"cached: {JSON} (use --force to recompute)\")\n        return 0\n    t0 = time.time()\n\n    m0 = json.loads(E023J.read_text())[\"key_values\"][\"M0_use\"]\n    print(f\"M0 imported from E-023: {m0}\")\n\n    print(\"V-R: traced R-bound verification ...\", flush=True)\n    zeros = np.load(ZCACHE)\n    rpts = check_R_bound(zeros)\n    vr = all(p[\"holds\"] for p in rpts)\n    worst = max(rpts, key=lambda p: p[\"abs_R\"] / p[\"bound\"])\n    print(f\"  all {len(rpts)} points hold: {vr}; worst ratio \"\n          f\"{worst['abs_R']/worst['bound']:.3f} at \"\n          f\"sigma={worst['sigma']}, t={worst['t']}\", flush=True)\n\n    print(\"V5: certified Z2 ...\", flush=True)\n    part, tail, z2 = certified_Z2()\n    v5 = bool(z2 <= 0.5699)\n    print(f\"  partial {part:.6f} + tail {tail:.6f} = {z2:.6f} \"\n          f\"<= 0.5699: {v5}\", flush=True)\n\n    print(\"S3': margin rerun with traced constants ...\", flush=True)\n    ngrid = np.arange(20, 400001)\n    marg = margin(ngrid, m0)\n    pos = ngrid[(marg > 0) & (np.gradient(marg, ngrid) > 0)]\n    N0 = int(pos[0]) if len(pos) else -1\n    cont = bool(np.all(marg[ngrid >= N0] > 0)) if N0 > 0 else False\n    margN0 = float(marg[ngrid == N0][0]) if N0 > 0 else float(\"nan\")\n    print(f\"  N0 (traced 6.0/40) = {N0} (contiguous to 4e5: {cont}, \"\n          f\"margin(N0) = {margN0:.1f})\", flush=True)\n\n    logn = np.unique(np.logspace(np.log10(max(N0, 20)), np.log10(9e24),\n                                 4000).astype(np.float64))\n    bmarg = margin(logn, m0)\n    conv = bool(np.all(np.diff(bmarg, 2) > -1e-6 * bmarg[:-2].max()))\n\n    N1 = int(3_000_000_000_000 - 1) ** 2\n    refs = {int(n): float(margin(n, m0)) for n in\n            [N0, 100000, 500000, 10 ** 6, 10 ** 9, 10 ** 12, 10 ** 18,\n             N1]}\n    for k, v in refs.items():\n        print(f\"  margin({k:.3e}) = {v:.4e}\", flush=True)\n    v2 = bool(N0 > 0 and cont and conv and N0 < 500000)\n    print(f\"  convexity grid: {conv}; N0 < 5e5 (MV coverage): \"\n          f\"{N0 < 500000}\", flush=True)\n\n    with open(CSV, \"w\", newline=\"\") as f:\n        w = csvmod.writer(f)\n        w.writerow([\"n\", \"margin_traced_6_40\"])\n        for n in range(20, 400001, 20):\n            w.writerow([n, f\"{float(margin(n, m0)):.6f}\"])\n\n    meta = {\n        \"experiment\": \"E-028\",\n        \"task\": \"4.5a support: D-056 traced R-bound verification + \"\n                \"margin rerun with A=6.0, B0=40\",\n        \"script\": \"research/numerics/scripts/E-028-traced-margin.py\",\n        \"checks\": {\n            \"V_R_traced_bound_holds\": {\"pass\": vr, \"points\": rpts},\n            \"V5_Z2_certified\": {\"pass\": v5, \"partial\": part,\n                                \"tail\": tail, \"bound\": z2},\n            \"V2_margin_positive_convex\": v2,\n        },\n        \"key_values\": {\n            \"M0_from_E023\": m0,\n            \"N0_traced_6_40\": N0,\n            \"margin_N0\": margN0,\n            \"overlap_factor_vs_MV_5e5\": 500000 / N0 if N0 > 0 else None,\n            \"margin_at\": {str(k): v for k, v in refs.items()},\n            \"N1\": N1,\n        },\n        \"elapsed_seconds\": round(time.time() - t0, 1),\n    }\n    JSON.write_text(json.dumps(meta, indent=2))\n    print(f\"  wrote {CSV.name}, {JSON.name}; elapsed \"\n          f\"{meta['elapsed_seconds']}s\")\n    return 0 if (vr and v5 and v2) else 1\n\n\nif __name__ == \"__main__\":\n    sys.exit(main())\n","file_path":"D:\\Python\\Quantum\\research\\numerics\\scripts\\E-028-traced-margin.py"},"extra_content":null}]