The Adjugate Transition Bridge: Forced Regularization of Non-Semisimple Modular Data

This preprint establishes the Adjugate Transition Bridge (ATB): a sectorwise factorization of the non-semisimple modular S-data (the S_b/ S-matrix) for the Drinfeld double D(kG) in characteristic p |G|. The main identity expresses modular data through block Cartan matrices and Brauer character tables, yielding the canonical transition operator Tadj=adj(C)/|D| and an inversion relation of the form C\,T\,φ=φ on the Brauer-character span. A central contribution is rigidity/uniqueness: combining Davydov–Yetter rigidity (in the stated regime), one-dimensionality of the modified trace space, and Brauer linear independence, ATB shows that the regularization/shearing operator compatible with trace normalization is forced (rather than chosen). The preprint also proves intrinsic existence of the regularized period using modified trace technology in factorisable finite tensor categories, and verifies the identities explicitly in the worked example D(kS_3) at p=2,3. A companion preprint constructs the categorical lift (modular coherent Springer object and derived Morita equivalence); the present paper provides the K_0-level forcing and certification mechanism.