Taming the Wild Stratum: Class-2 Groups of Bounded Rank-Width

Finite p-groups of nilpotency class 2 already account for the leading term in the enumeration of all finite p-groups, while their classification contains tensor isomorphism and graph interpretation. We isolate a tractable part of this class by bounding the rank-width of the commutation tensor. Over R = Z/p^d, the width of a cut is the number of generators required by six crossing modules. For fixed p, d, r, central extensions with distinguished kernel over R and a site-tree layout of width at most r form a uniformly tree-automatic class; chain layouts give a uniformly word-automatic class. Finite local certificates make the advice language exact.

Two phenomena absent over fields drive the proof: sibling interactions need not be bilinear in chosen interface representatives, and distributed central targets create obligations flowing into a subtree. Finite pairing tables and bounded claim modules handle these effects. From certified advice, every fixed first-order query is decidable in linear time, and a sentence in time O(log |G|); the uniform theories are decidable. The class is closed under direct products and contains several familiar structured families. Yet, for fixed parameters, it represents only 2^{O(m)} isomorphism types of order p^m, and is therefore asymptotically negligible among class-2 p-groups.