[{"type":"thinking","thinking":"Interesting! Partial after j=100: 43758453032.4 — but in the earlier diag3 run (dps 300, e011 η), partial after j=100 was 4.37584774e+10 — WAIT: 43758477400 vs 43758453032 — CLOSE but differ at the 6th digit! And e011's final: −2.14; new: −24371.\n\nThe partial at j=100 differs: 4.37584774e10 (e011, dps-300) vs 4.3758453032e10 (new, dps-500). Difference 2.4e4. And the final answers differ by 24373. So the difference comes from terms j ≤ 100, visible already at j=100. Hmm — but both η sets were validated against each other to tiny diffs... at dps 100! The e011-η in diag3 was computed at dps 150 (Stieltjes strings 80 digits! recursion at dps 150). η values ~1e-43 at j=99 → e011-η at j~99 accurate to ~1e-102 abs (relative 1e-59). New-η at j=99: from zeta formula, abs error 1e-243, stored 193 digits, η_99 ~ 1.9e-48: relative 1e-145.\n\nTerm at j=100: C(200,100)·η_99 ~ 9e58·1.9e-48 = 1.7e11. Difference between terms: 9e58·(η_99 diff). Partial diff at j=100: 2.4e4 → η-diffs Σ C δ ~ 2.4e4 → δ_99 ~ 2.4e4/9e58 = 2.7e-55?? But comparison showed |eta_true − eta_new| at j=59: 9.69e-70 (η ~ 2.4e-29)... at j=99 (not compared here) e011-η vs new-η: e011-η abs accuracy 1e-102 (dps-150 recursion), new-η 1e-243: diff ~1e-102 → term diff 9e58·1e-102 = 9e-44. NOT 2.4e4.\n\nHmm!! So where does 2.4e4 come from?? Maybe eta_new in THIS run differs from eta_new in the previous comparison run? Both computed identically... The comparison run showed diffs vs eta_true (dps-100 recursion, itself inaccurate ~1e-25 at j=20!). For j ≤ 23, eta_new came from Stieltjes at dps 293 (193+100). eta_true in comparison: dps-100 recursion — inaccurate. Diff at j=20: 1.32e-25 — that was eta_TRUE's error. So eta_new j≤23 is accurate to ~1e-200ish. And diag3's e011-η at j≤23: dps-150 recursion → accurate to ~1e-100ish.\n\nTerm j=20: C(200,20)·η_19 ~ 1.6e27·2.9e-10 = 4.6e17. η_19 accuracy: new 1e-200, e011 1e-100 → term diffs: 1.6e27·1e-100 = 1.6e-73 (e011), negligible. Both fine at j=20.\n\nSo Σ_{j≤100} C·(η diffs) should be ~0 for BOTH vs true. Yet partials differ by 2.4e4 and both claim accurate η. IMPOSSIBLE unless the SUMMATION differs. Summation: same recurrence, dps 300 (diag3) vs dps 500 (this run) — both plenty.\n\nWAIT. Look at partials again: diag3 j=100: 4.37584774e+10 (10 sig figs shown: 43758477400); this run: 43758453032.4. Hmm — 43758477400 vs 43758453032: differ by 24368. Final answers: −2.14 vs −24371.47. Difference: 24373.6 ≈ 24368 + 5.6. So the difference is ALREADY THERE at j=100 and stays.\n\nAnd what about the j=23/24/25 partials: after j=23: −6.64e18; after j=24: +1.61e19; after j=25: −3.73e19. These are HUGE partial sums. At j=50: 3.15e23 ✓ matches diag3's 3.150415562e23. So the two runs agree at j=50 to displayed digits, diverge at j=100.\n\nBetween j=50 and j=100: terms ~1e23 down to 1e11. The η values η_50..99: new-η abs error 1e-243; e011-η (diag3) abs error... recursion dps 150: at j=99 η ~ 1e-48, recursion stable-ish... measured in diag3 vs dps-1700-τ η: diffs 1e-102. So e011-η accurate ≥1e-102 at j=99. Term diffs ≤ C(200,100)·1e-102 = 9e58·1e-102 = 9e-44. So partials at j=100 should agree to ~1e-40. THEY DON'T (2.4e4). CONTRADICTION — unless eta_new ≠ eta_true in THIS run at j 50-100 by MORE than 1e-102.\n\nLet me directly compare: print η_new[99], η from Stieltjes-dps-150 (diag3 style), and from a HIGH-precision Stieltjes reference (dps 500 recursion, 200-digit stieltjes strings) at j = 50..100. Find the truth.","signature":""},{"type":"text","text":"Partials agree at j=50 but diverge by 2.4e4 at j=100 — the discrepancy lives in η₅₀…₉₉. Let me get a rock-solid reference: Stieltjes division at dps 600."},{"type":"tool_use","id":"Bash_47","name":"Bash","input":{"command":"cd /d/Python/Quantum && ./.venv/Scripts/python.exe - <<'EOF'\nsrc = open(\"research/numerics/scripts/E-021-sinf-sf-curve.py\").read()\nsrc = src.replace('if __name__ == \"__main__\":', 'if False:')\nns = {\"__file__\": \"research/numerics/scripts/E-021-sinf-sf-curve.py\"}\nexec(compile(src, \"e21\", \"exec\"), ns)\nzeta_str, fd = ns[\"get_zeta_strings\"](300)\neta_str = ns[\"compute_eta\"](zeta_str, fd)\nimport mpmath as mp, flint\n# high-precision Stieltjes reference\nflint.ctx.prec = 4000\nmp.mp.dps = 1000\ngam = [mp.mpf(flint.acb.stieltjes(k).real.str(900, radius=False)) for k in range(102)]\nN_=[mp.mpf(0)]*102; D_=[mp.mpf(0)]*102; N_[0]=D_[0]=1\nfor m in range(0,101): D_[m+1]=(-1)**m*gam[m]/mp.factorial(m)\nfor m in range(1,101): N_[m+1]=-(-1)**m*gam[m]/mp.factorial(m-1)\nE=[mp.mpf(0)]*102\nfor k in range(0,102):\n    s=N_[k]\n    for i in range(1,k+1): s-=D_[i]*E[k-i]\n    E[k]=s\neta_ref=[E[k+1] for k in range(101)]\nmp.mp.dps = 800\nprint(f\"{'j':>3} {'eta_ref (dps1000)':>25} {'eta_new':>25} {'|diff|':>12}\")\nfor j in range(50, 101, 10):\n    en = mp.mpf(eta_str[j])\n    d = abs(eta_ref[j] - en)\n    print(f\"{j:>3} {mp.nstr(eta_ref[j],15):>25} {mp.nstr(en,15):>25} {mp.nstr(d,3):>12}\")\n# S_f(200) with reference eta\nmp.mp.dps = 600\ns = mp.mpf(0); c = mp.mpf(1)\nfor j in range(1, 201):\n    c = c*(200-j+1)/j\n    s += c*eta_ref[j-1]\nprint(\"S_f(200) [stieltjes ref] =\", mp.nstr(s, 15))\ns2 = mp.mpf(0); c = mp.mpf(1)\neta_new = {j: mp.mpf(v) for j, v in eta_str.items()}\nfor j in range(1, 201):\n    c = c*(200-j+1)/j\n    s2 += c*eta_new[j-1]\nprint(\"S_f(200) [new eta]       =\", mp.nstr(s2, 15))\nEOF","description":"Reference eta at dps 1000, compare and resum","timeout":600000},"extra_content":null}]