Empirical Structure of the Gilbreath Decay Constants

Chase, Hunter, and Tao introduced a stationary continuous Gilbreath model in which the top row consists of independent standard exponential variables and c_i := E a(i,j) denotes the expected entry at depth i. They proved sum_{i ≤ n} c_i ≥ log(n+e), computed c_0, ..., c_3 exactly, and could not prove that (c_i) is bounded. This note reports a Monte Carlo study to depth 8192, anchored by new exact rational values of c_4, c_5, and c_6. The principal empirical law is

c_i ≈ C · λ^{s_2(i)} / i,

where s_2(i) is the binary digit sum and the effective λ drifts slowly through approximately 1.14–1.20 in the sampled windows. The conjectural 1/i behavior thus becomes visible after conditioning on digit-sum classes, while pooled data decay more slowly and display a pronounced dyadic sawtooth; at extreme digit sums the modulation saturates below its geometric extrapolation. Complementary finite-depth experiments indicate a polynomial-versus-exponential growth transition for continuous uniform data, a full-row relaxation law of order G^{0.63}–G^{0.66}, and a spike survival distance asymptotic to its amplitude. These findings quantify the decay mechanism of the Gilbreath array that lies beyond the elementary parity wave.