Collision Cascades, Telescopic Lattice Equilibria, and Coupled Relaxation in the de Bruijn–Newman Calogero–Moser Flow

High-precision (100+ decimal digits) numerical and analytical investigation of the zero dynamics of the de Bruijn–Newman family H_t(z) under backward (t < 0) and forward (t > 0) flows, governed by the rational Calogero–Moser (CM) pole system. Key contributions of this preprint and dataset (Stages A–AH): 1. Backward Collision Spectrum (t < 0): Complete mapping of the first 41 pair collisions (n <= 81), determining exact collision depths t_c(n), survivor zeros, and local compression factors R_n in [0.68, 18.73] (median 5.02).2. Exact Pole Cancellation Identity: Derivation of d(delta^2)/dt = 8 - 2*delta*Delta_other, showing that the internal partner poles (+/- 4/delta) cancel the stability threshold +4 identically, leaving purely the third-body field difference Delta_other (identified via PV Im(xi'/xi) with regularized prime sine sums).3. Closed-Form Frozen-B Dynamics & Witness Accumulation: Exact solution of the autonomous frozen-B ODE, establishing delta*B = 4 as the stable attractor, predicting 30/31 protected backward collision depths within ~10%, and proving via Taylor reduction that tight Lehmer pairs (delta -> 0, e.g., n = 18860) accumulate at 0^- with the two-particle speed t_2p = -delta^2 / 8.4. Falsification of Static and Local Shortcuts: Quantitative falsification of local normal forms, instantaneous t=0 repulsion on wide pairs (Pairs 46, 64, 79), cubic curvature bounds, conditional xi'/xi bounds, and static high-altitude O(1) bounds on Delta_other(0).5. Telescopic Crystal Identity & Cluster Shielding: Proof that an infinite equally spaced zero lattice evaluates identically to the telescopic series delta*B = 4 * sumₖ₌₁^infty 1/(k(k+1)) = 4, while finite lower-boundary truncation explains the universal empirical ceiling delta*B -> 3.67–3.73 < 4 and cluster boundary shielding (verified in 89/89 cases).6. Coupled Dynamic Rescue: Demonstration that transient spikes near tight Lehmer neighbors (delta*B_0 ~ 113) undergo rapid multi-particle relaxation, collapsing Delta_other by 5x within t = 0.25 (Pair 1497) while the gap contracts by only 1.2x. The three remaining analytic barriers to an unconditional proof of Lambda <= 0 are explicitly formulated. Author: Eskelinen, J.M.E.License: Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0).