A Sharp Six-Neighbor Expressivity Threshold for Third-Order Geometric Invariants

Many geometric learners compress a local point cloud into pooled low-order invariants before regression. When does that compression first destroy information? We determine the exact answer for the complete reflection-even third-order correlation in three dimensions. Every single-type cloud with at most five distinct neighbors is reconstructed, up to permutation and O(3), whereas six neighbors admit noncongruent clouds with identical complete third-order information. Equivalently, rank-three positive-semidefinite Gram matrices of order at most five are reconstructible from their unlabelled three-vertex principal-submatrix decks, and order six is the first failure. The five-neighbor result is computer assisted but exact: all B10 = 115,975 edge-equality patterns are reduced to five maximal ambiguity templates, resolved by parallel human-readable geometric/algebraic arguments and thirteen rational ideal-membership certificates. Every exact six-neighbor collision has repeated radii and therefore lies in a measure-zero discriminant; the known four-parameter family is locally a codimension-eleven collision branch. The threshold has direct learning consequences. Below it, every continuous invariant target on a compact domain factors continuously through the representation; at six neighbors, a smooth four-vertex target imposes an analytic Bayes error floor. On 11,000 exact collision pairs, third-order models converge to that floor regardless of data or capacity. Fourth-order cards and a sufficiently deep edge-incidence MPNN remove the error, while depth-0/1 controls with matched parameter counts do not. Perturbation experiments show that the repair persists near the collision manifold. The result separates a representation-level obstruction from basis truncation, optimization, and model size.