Entropic Spray and Eigenvector Concentration in Ulam Discretisations of the Twisted Bird-Map Boundary Grid: N-Scaling and a Negative Off-Diagonal Probe (Paper 49 in the Non-Holomorphic Fractal Series)

Paper 48 established a discrete two-class taxonomy within numerical Regime B on the twisted Bird-map boundary grid: a coherent class B1 where both k_EAE and k_BV remain single- to few-digit, and a spray class B2 where k_BV stays bounded while k_EAE grows monotonically across three successive Ulam-grid doublings (four N-grids), with WBA ε = 0.70 as the sole confirmed B2 cell [12]. Paper 49 asks: what discrete numerical process produces the B2 spray? We pursue two complementary approaches on the locked grids N ∈ {2048, 4096, 8192, 16384}. First, we fit an empirical power law to the four-grid B2 sequence and obtain k_EAE(N) ≈ 0.5072·N^0.7715 with no visible saturation on the four computed grids, while k_BV remains bounded in [3.10, 3.35] (Proposition 2.1, grid-scoped). Second, a direct comparison of off-diagonal transition-mass structure between WBA and Lorentzian Ulam matrices at N = 1024 using column Shannon entropy and a bandwidth of 51 cells (5% of N) yields nearly identical tail fractions (0.9111 vs. 0.9011) and relative entropies, a difference of approximately one percentage point that is not statistically meaningful. This null result demonstrates that the B2 spray is not driven by a simple, global inflation of off-diagonal mass in the Ulam matrix. It localises the B2 spray on the executed grids to eigenvector concentration rather than gross off-diagonal transition structure.