The de Bruijn-Newman flow on the Davenport-Heilbronn function: when do the off-line zeros reach the line?

The Davenport-Heilbronn function f has a Riemann-type functional equation and 1,268 certified pairs of zeros off the critical line below height 20,000. Under the de Bruijn-Newman heat flow H_t(z) = int exp(t u^2) Phi_f(u) exp(izu) du every such pair moves to the real axis and collides there. The paper derives the kernel Phi_f explicitly, proves it is even and strictly positive, gives the heat-kernel form H_t(z) = E xi(1/2 + W + iz) with W ~ N(0, 2t), and fixes the normalisations against de Bruijn, Newman, Polymath and Rodgers-Tao (a factor 4 in t). An isolated pair at depth a collides at t* = a^2/2; every real zero accelerates the collision. Predictions (collision-time ratios, their dependence on depth and density, the census maximum, height drift, gap statistics after the flow, and the Epstein zeta function of m^2 + 5n^2 as a second specimen) were registered before any trajectory was integrated; the computation is a zero-dynamics ODE for all pairs, and four registered pairs are certified by direct evaluation of H_t in ball arithmetic (all four completed). The bracket 36/625 < Lambda_f <= (1/2)(sigma* - 1/2)^2 and the certified one-pair lower bound at height 240.4 are due to the zeta-lab project; the certificate here reproduces their bound and raises it slightly, to Lambda_f > 0.0576473. What is added is the per-pair collision law across all census pairs, the Epstein comparison and the gap test. Scored as registered: P1, P2(a-c), P3 and P6 pass; every interior pair collides before t*; the census maximum of Lambda_f is attained at the 240.405 pair; P2(d), P4(b) and P5(c) fail and are reported. The flow moves the gaps near former pairs toward GUE but moves the far gaps away from it. No hypothesis (RH, GRH, simplicity of zeros) is assumed. Inputs: the certified census doi:10.5281/zenodo.23676415. This version (4 November 2026) records that all four route-B certificates are complete and updates the citations. Written jointly with Claude (Anthropic); the human author is responsible for the content.