Sorting the Spray: Cumulative Mass Diagnostics and Eigenvector Concentration on Twisted Ulam Grids

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 kEAE and kBV remain single- to few-digit, and a spray class B2 where kBV stays bounded while kEAE grows monotonically across three successive Ulam-grid doublings (four N-grids), with WBA ε = 0. 70 as the sole confirmed B2 cell. The B2 sequence admits an empirical power-law fit kEAE (N) ≈ 0. 5072·N⁰. 7715 on N ∈ 2048, 4096, 8192, 16384, with kBV remaining in 3. 10, 3. 35 across the same grids. A complementary off-diagonal probe at N = 1024 shows that WBA ε = 0. 70 and Lorentzian ε = 1. 02 have nearly identical column entropies and tail fractions (0. 9111 vs. 0. 9011), a one-percentage-point difference that is not statistically meaningful. Paper 49 therefore localises the B2 spray, on the executed grids, to eigenvector concentration rather than gross off-diagonal transition structure. Paper 50 pursues this eigenvector-level mechanism using only previously computed Ulam eigenvectors on the boundary grid. For each eigenvector we sort cell masses and study cumulative mass functions MN (k), discrete percentiles k₅0 (N), k₇5 (N), k₉0 (N), and the top-1% mass fraction m₁% (N) on the four B2 grids for WBA ε = 0. 70 and a B1 Lorentzian comparator. On the executed grids, the diagnostics either reveal widening concentration widths and eigenvector dilution for WBA relative to Lorentzian (a positive eigenvector-flattening classification) or remain within a common envelope, in which case the B2 spray is invisible to sorted cumulative-mass profiles at these resolutions. All statements are strictly finite-grid and eigenvector-level; no claim is made about the continuum invariant density or any asymptotic scaling law as N → ∞.