The cofinite regularity spectrum of F-irregular graphs

For a fixed graph F, the F-degree of a vertex is the number of copies of F containing it; copies are not required to be induced. We prove that, for every connected non-star graph F with at least three vertices and every sufficiently large integer r, there is a connected r-regular graph whose vertex F-degrees are pairwise distinct. The hosts have diameter two and r + o(r) vertices. Stars are the only obstruction. The proof combines a construction with prescribed degree and a localization estimate for rooted homomorphism differences. A spectral decomposition separates the leading rooted statistics, while a scale-preserving reparametrization avoids cancellation without changing the prescribed degree. Four fixed adjacency matrices supply the seed inputs, which are verified by exact arithmetic. This manuscript is a preprint. The accompanying supplementary material contains the finite adjacency-matrix inputs and exact-arithmetic verification resources.