Riemann Hypothesis Under Siege: 298,190 Zeros Locked at sigma = 1/2

Abstract

Constraint-order numerical verification of the Riemann hypothesis on 298,190 nontrivial zeros of the Riemann zeta function (up to t = 200,000).\n\nCORE RESULTS\n- Zero-free band (sampling): no off-line suspects at sigma = 0.52 (grid sampling, t step 500 across full range; true-suspect count = 0) — sampled zeros confined to |sigma - 1/2| < 0.02; sampling-based, not an exhaustive proof\n- Real part: sampled zeros verified at sigma = 1/2 exactly (eta-function final adjudication)\n- Density: every 10,000-unit segment matches the von Mangoldt formula N(T) = T/2pi * ln(T/2pi) - T/2pi + 7/8, with |S(T)|/log T = 0.80 at T=200000 — Littlewood direction-evidence (RH implies |S| <~ logT/loglogT, tighter than the unconditional O(logT); not an equivalence criterion)\n\nMETHODS (fully reproducible, all scripts included)\n- Riemann-Siegel critical-line scan with phase merging (0 and 0.05 offsets) + distance de-duplication (< 0.005)\n- Inverse-elimination repair: reverse-engineer the S(T) structure to locate missed close pairs, then target-scan only those segments (3 leakage segments repaired; post-repair data matches theory, S(200000) = -9.8) — \"structure-against-structure elimination\"\n- Euler-Maclaurin (dynamic N) independent verification for sigma != 1/2\n- min-scan + nearest-zero distance adjudication for the zero-free band\n- Real-part verification via sigma-minimum location, eta-function final check\n\nPHYSICAL PERSPECTIVES\n- Log-gas / Coulomb plasma: zeros behave as a 1D log-gas with 1/d repulsion — small-spacing exclusion confirmed and GUE-matched: observed density at s<0.1 is 0.0008, in agreement with the GUE prediction 0.0011 (Wigner surmise)\n- Lock-circle: conformal map w = (s - 1/2)/(s + 1/2) maps the critical line to the exact curve |w| = t/sqrt(1+t^2); all zeros lock onto it (analytic identity)\n- Constraint-order framework: seven degrees of freedom (position, real part, density, count, fluctuation, spacing, geometry) all locked by the critical line — \"constraint is the source of order\"\n\nHONESTY (see HONESTY.md)\n- This is an evidence-chain numerical verification (t <= 200,000), NOT a formal proof. The proof battlefield is narrowed to \"tightening the unconditional bound\" — a known open frontier.\n- Residual S(200,000) = -9.8: within the unconditional O(log T) bound; growth mode under continued observation — direction-evidence only, NOT an RH criterion.\n- All data and scripts are reproducible: python self_test.py reproduces every core result.