[{"type":"thinking","thinking":"g_float is perfect. But S(k) for k ≥ 1 has errors ~1.3e-9 — way bigger than the table version (3.6e-12). The k≥1 quadrature must be mishandling something. Note S(0) is fine (1.3e-12). For k≥1 the window [lo,hi] = [(k−1)h,(k+1)h] with kinks — error 1.3e-9 suggests kink handling or... wait: for k=1, h=0.005: window [0.005, 0.015], no kinks (log 2 = 0.69). So no kink issue at k=1,2. Error 1.29e-9 — from GL quadrature of a SMOOTH function?? 12-pt GL on a smooth function should be spectral. UNLESS the weight (h − |xs − kh|) has its kink at xs = kh INSIDE the sub-interval [lo, hi] and I didn't split there!! The triangle weight peaks at t = kh — a kink in the INTEGRAND at t = kh. GL over [lo, hi] straddling the kink → slow convergence. Fix: split at t = kh as well: pts include kh. Sub-intervals [lo, kh], [kh, hi] plus kinks. That's the bug.\n\nFix: pts = sorted set {lo, kh, hi} ∪ kinks.","signature":""},{"type":"text","text":"Found it: the triangle weight has its own kink at t = kh, and I didn't split the quadrature there — GL straddles a non-smooth point. Splitting at kh too."},{"type":"tool_use","id":"Edit_55","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\numerics\\scripts\\E-009-suzuki-galerkin.py","new_string":"    lo, hi = (k - 1) * h, (k + 1) * h\n    n_lo = int(np.ceil(np.exp(lo)))\n    n_hi = int(np.floor(np.exp(hi)))\n    pts = ([lo, k * h]\n           + [float(np.log(n)) for n in range(max(n_lo, 2), n_hi + 1)\n              if lo < np.log(n) < hi] + [hi])\n    pts = sorted(set(pts))","old_string":"    lo, hi = (k - 1) * h, (k + 1) * h\n    n_lo = int(np.ceil(np.exp(lo)))\n    n_hi = int(np.floor(np.exp(hi)))\n    pts = [lo] + [float(np.log(n)) for n in range(max(n_lo, 2), n_hi + 1)\n                  if lo < np.log(n) < hi] + [hi]","replace_all":false},"extra_content":null}]