The Endpoint Adjugate Identity: A Universal PIM Relation for Brauer Tree Algebras and the Torus-Crossing Phenomenon for GL(2)

We prove the Adjugate Identity Theorem (AIT) for cyclic-defect Brauer tree blocks in the line-tree, endpoint-exceptional regime. For the Cartan matrix C(e,m), we show that the endpoint row of adj(C(e,m)) is independent of exceptional multiplicity m, with explicit coefficients (-1)ᵉ⁻¹⁻ʲ(j+1).   This yields the Endpoint Adjugate Identity (EAI): a normalized linear relation on projective indecomposable module (PIM) dimensions whose right-hand side is the endpoint simple-module dimension. We also provide specialization to GL(2,F_q), computational verification across tested parameter ranges, and theorem-level extension to the leaf-exceptional setting.   Contextual sections discuss cohomological and local-Langlands alignments as downstream interpretations; these are explicitly separated from the logical proof dependencies of the core theorem.