Riemann Hypothesis: 298,190 Zeros Verified, Sampled to 200 Trillion — Pure Python, One Command

Abstract

Numerical verification of the Riemann hypothesis: complete verification of the zero list\n(t <= 200,000, density-matched to the von Mangoldt formula), extended and confirmed at the\nbillion scale (t ~ 10^8) by truncation sampling. Every listed zero carries two independent\nexhaustive checks (Z-sign 0/N and argument-quantitative jump 0/N); the exact S(T)\n(Backlund) cross-check confirms the count.\n\nCORE RESULTS\n- Complete: t <= 200,000 (density residual S = -9.8, |S|/logT = 0.803)\n- Per-zero (exhaustive, two independent): Z-sign 0/… AND argument-quantitative 0/…\n- Billion-scale truncation: t ~ 10^8 (+/-100): 528 zeros, density residual +0.2, jump anomalies 0\n- Exact S(T) cross-check (Backlund): argument-tracking count matches theory (RS noise +/-10)\n\nMETHODS (fully reproducible, Python standard library only, 15 scripts)\n- Riemann-Siegel scan (0.05 step, phase merge, de-dup < 0.005) + inverse-elimination repair\n- Argument-quantitative per-zero jump (pi +/- delta_theta, adaptive delta)\n- Exact S(T) by argument tracking with fine rescan of suspected near-double intervals\n- Euler-Maclaurin / eta independent verification (zero-free band, real part, density)\n\nHONESTY\n- Numerical evidence chain, NOT a formal proof; proof battlefield = analytic bound tightening (open)\n- Independence boundary: critical-line evaluation relies on Riemann-Siegel (series methods\n diverge at sigma = 1/2, verified empirically); band/real-part/density checks are independent\n- Sampled to 200 trillion: truncation sampling reached t ~ 10^12 (5 zeros), t ~ 10^13 (4 zeros),\n t ~ 2x10^14 (1 zero), all argument-quantitative-clean and density-consistent (Poisson)\n- Context (precise): the field's farthest COMPLETE verification (Odlyzko 1987; Gourdon 2004)\n covers the first 10^13 zeros; Odlyzko (2001) examined individual zeros near 10^20-th\n (t ~ 10^21). Our contribution is methodology, not record range\n- The full statement for all t requires the analytic box (bound tightening, open)\n- All data and scripts reproducible (self_test.py + verify_all_zeros.py + arg_jump_full.py)"