When Aggregate Boolean Invariants Stop Determining Structure: A Minimal PN-Equivalence Failure and an Infinite Collision Family

A Boolean function can be described by several mathematically meaningful invariants, including Hamming weight, deterministic decision-tree depth, certificate complexity, sensitivity, influence, algebraic degree, and Fourier-level energy. We investigate whether a heterogeneous collection of such invariants is sufficient to determine a Boolean function up to permutation and complementation of its input variables. We define a fingerprint Φ consisting of eight components: Hamming weight, exact decision-tree depth, separate pointwise certificate-complexity profiles for outputs 0 and 1, sensitivity profile, influence profile, algebraic degree, and Fourier-level energy. Exhaustive computation shows that Φ separates all PN-equivalence classes for n ≤ 4, yielding respectively 3, 6, 22, and 402 fingerprint classes, with no cross-PN collisions. The first failure occurs at n = 5. An exhaustive census of all 65,536 five-variable Boolean functions of algebraic degree at most two identifies three cross-PN collision groups. In the largest group, three distinct PN classes have the same fingerprint despite their quadratic interaction graphs having three, four, and five edges, with different degree sequences, connectivity, and triangle structure. We provide an explicit minimal witness and prove that XOR with parity on fresh variables preserves equality of the full fingerprint while retaining PN inequivalence. This construction yields collisions in every dimension n ≥ 5. The results establish a precise boundary between agreement of aggregate Boolean invariants and recoverability of interaction topology.