[{"type":"thinking","thinking":"MUCH worse — eigenvalues now −9.5 at a=3 and diverging with refinement (−3.2 → −9.5 → −37)! And vᵀQv = 7.3e-5 vs expected 2.44e-4. So the new S(k) is WRONG — worse than the table version. Bug in my new quadrature. Let me check S(k) against the mpmath reference from before: exact S(0) = −9.8352771288e-08, S(1) = −2.3766064586e-07 etc.\n\nPossible bugs:\n1. k=0 case: I integrate (h−xs)·g over [0,h] then double ✔. pts for k=0: [0, h] (no kinks since h < log 2 ✔).\n2. k≥1: weight (h − |xs − kh|) ✔ triangle.\n3. Kink computation: n_lo = ceil(exp(lo)), n_hi = floor(exp(hi)); for n in range(max(n_lo,2), n_hi+1) if lo < log n < hi. OK.\n4. g_float: DIG_COEFF sign: −(|t|/2)(ψ(1/4) − logπ) = +2.686t. DIG_COEFF = float(−(digamma(1/4) − log π)/2) = −(−5.372)/2 = +2.686 ✔.\n5. lerch_comb: −0.25*(PHI1 − e^{−t/2}Φ(e^{−2t})) ✔.\n\nWait — the mpmath reference g_mp earlier gave S exact values which the OLD code matched to 3.6e-12. Let me test the NEW g_float against g_mp at a few points — maybe a bug in g_float (e.g., lerch fast series). Test at t = 0.1 (slow path), 1.0, 3.7, 10 (fast path).","signature":""},{"type":"text","text":"Worse — the new quadrature path has a bug. Testing new `g_float` against the mpmath reference directly."},{"type":"tool_use","id":"Bash_54","name":"Bash","input":{"command":"cd D:/Python/Quantum && ./.venv/Scripts/python.exe -c \"\nimport numpy as np, mpmath as mp\nmp.dps = 25\nsrc = open('research/numerics/scripts/E-009-suzuki-galerkin.py').read().split('def main')[0]\nns = {'__file__': 'research/numerics/scripts/E-009-suzuki-galerkin.py'}\nexec(compile(src, 'e9', 'exec'), ns)\nLAM_LOG = ns['LAM_LOG']\ndef g_mp(t):\n    t = mp.mpf(t); a = abs(t)\n    s = -4*(mp.exp(a/2)+mp.exp(-a/2)-2)\n    n = 2\n    while mp.log(n) <= a:\n        s += (LAM_LOG[n]/mp.sqrt(n))*(a - mp.log(n)); n += 1\n    s -= (a/2)*(mp.digamma(mp.mpf(1)/4) - mp.log(mp.pi))\n    s -= mp.mpf(1)/4*(mp.lerchphi(1,2,mp.mpf(1)/4) - mp.exp(-a/2)*mp.lerchphi(mp.exp(-2*a),2,mp.mpf(1)/4))\n    return s\nfor t in [0.001, 0.1, 0.34, 0.36, 1.0, 3.7, 7.0, 10.0]:\n    mine = ns['g_float'](np.array([t]))[0]\n    ex = g_mp(t)\n    print(t, 'mine=%.12e exact=%.12e err=%.2e' % (mine, ex, abs(mine-float(ex))))\n# S check\nh=0.005\nfor k in [0,1,2,500]:\n    f = lambda s: (h-abs(s))*g_mp(k*h+s)\n    exact = mp.quad(f, [-h, 0, h])\n    mine = ns['S'](k, h)\n    print('S', k, 'mine=%.10e exact=%.10e err=%.2e' % (mine, exact, abs(exact-mp.mpf(mine))))\n\"","description":"Test new g_float and S against mpmath reference","timeout":300000},"extra_content":null}]