Hypercube Geometry Reveals Complexity Collapse in Atomic Concept Learning

A recent submission on arXiv (2608.02930) explores the geometry of hypercubes and hyperplanes in the context of higher-arity atomic concept learning. The authors highlight that the r-dimensional hypercube of ground atoms lacks structural uniformity, with logical complexity being arranged by hyperplanes. Notably, every hyperplane, except for the full diagonal, reduces to a finite number of elementary-equivalence classes, with a bound that remains constant regardless of term depth. In contrast, the full diagonal presents an exceptional case, with its class count increasing indefinitely. This asymmetry mirrors the reduction-theoretic nature of the concepts. Utilizing a higher-dimensional framework from previous research, the findings are reinterpreted through canonical simple concepts and minimal orderings, leading to a classification of hyperplane behavior in higher dimensions and illustrating that complexity is localized rather than global.