A Kupisch Criterion for Higher Auslander Koszul Duals

For a connected nonsemisimple split basic Nakayama algebra B, let c_i and d_i be the lengths of the projective cover and injective envelope of the same simple module. We prove that the Koszul dual E of its Auslander algebra has global dimension max_i(c_i+d_i-2) and dominant dimension min_i(c_i+d_i-2). Thus the higher Auslander condition is exactly constancy of c_i+d_i, with the dimension-one convention stated separately. The proof combines an explicit graded morphism calculation with the credited Auslander-Koszul transfer theorem of Chan, Iyama and Marczinzik, version 2. We deduce the hereditary characterization for linear Nakayama algebras and a complete two-simple cyclic classification. Nonselfinjective positive examples include (2r+1,2r+2). The (3,4) series has both dimensions equal to 5, contrary to a negative example in the earlier preprint; independent exact quadratic-dual resolutions are supplied. This is a complete classification of the Nakayama subclass, not a solution of the general representation-finite question in UnsolvedMath record OWR-12697690-008. The English five-page AI-assisted preprint includes739 exact radical checks and97 independent minimal-resolution checks. It is unrefereed; no absolute-priority or formal-verification claim is made.