Pair and Triangle Statistics in the Monster's 2A Conjugacy Class

We treat the Monster simple group's 2A conjugacy class as a finite homogeneous ensemble and measure it computationally with mmgroup 1.0.8. Pair statistics: 16,613 uniformly random axes classified against a fixed axis reproduce Norton's exact rank-9 subdegrees (Pearson 2.75 over the five cells with expected count at least five; 831 independent product-order cross-checks agree with the axis-based classification). Triangle statistics: a pure-2A Klein four subgroup is constructed by cubing a class-6A product, and two independent samples of 16,646 random axes are classified by their ordered relations to the resulting three-axis triangle. Each 9x9x9 table is compared with the maximum-entropy distribution having the same three pairwise marginals, fitted by iterative proportional fitting. Natural logarithms throughout: D3 = 0.027729 bits and G^2 = 639.9 in run 1; D3 = 0.026998 bits and G^2 = 623.0 in the preregistered independent replication; pooled D3 = 0.027361 bits per environment. Ten-thousand-replicate parametric bootstraps, with every simulated table refitted to its own pairwise marginals, produce no statistic exceeding the observed value in either run (largest null values 57.0 and 57.1). These are plus-one Monte Carlo estimates with rule-of-three upper confidence limits, not bounds on the true tail probability. A 2x2x2 calculation in the {5A,6A} sector gives Theta = 0.3126 (run 1) and 0.3280 (run 2). Both runs suppress (5A,5A,5A) and enrich permutations of (5A,4B,5A). The triangle-conditioned distribution therefore contains replicated higher-order structure not representable by any model with only one-coordinate and pairwise potentials. No priority is claimed over the underlying taxonomy, which is due to Norton (1985) and Whybrow (2018). What is plausibly new, pending literature review, is the uniform-ensemble measure and the information projection against the maximum-entropy pairwise family. All archives are hash-sealed; MANIFEST.md lists every artifact with its SHA-256 and a verifiable path.