Categorical Modular Local Langlands Correspondence for Finite Reductive Groups: Framework, Proofs, and Extensions

This monograph gives a self-contained account of the categorical modular local Langlands correspondence for finite reductive groups over finite fields. It organizes the proof through a plug-in modular architecture: a universal seven-step categorical engine is established once, while the prime- and type-dependent input is isolated in an identification module based on concentration and blockwise Morita control. The monograph proves the correspondence unconditionally for all split reductive groups at good primes, and for key non-split quasi-split families at good primes, including finite unitary groups via Ennola duality and the Steinberg types ^{2}D_n, ^{2}E_6, and ^{3}D_4. It also records the bad-prime Tier 5 assembly conditionally on Smith–Treumann formality, while pointing to a companion qKEC paper that removes this conditionality via an enhanced quantum Langlands category. In addition, it proves a general Davydov–Yetter rigidity theorem for finite group algebras in characteristic dividing the group order. This monograph is intended as a synthesis-and-assembly work with explicit provenance routing to companion preprints for the identification module and related support-theoretic inputs.