Global Blockwise Morita Equivalences for the Spectral Langlands Category

This preprint proves unconditional global blockwise Morita equivalences for \(\IndCoh(\LocSys_{\check G}(X))\) in positive characteristic k=\mathbb F_\ell. For each spectral Bernstein block s and each depth stratum d, it establishes a fiberwise Morita equivalence between the heart of the restricted t-structure and the product of module categories over the finite-dimensional endomorphism algebras of the local Springer sheaves S_\rho^k. When the stratum has finitely many k-points, this product collapses to a single finite-dimensional algebra \Lambda_{s,d}. The proof combines three unconditional inputs from earlier companion work — the spectral support package, the fiber algebra theorem for modular singular support, and the Springer projective-generator theorem — with two new ingredients developed here: the spectral Bernstein decomposition and spectral parabolic induction. The paper also proves Hecke intertwining, parabolic compatibility, and global gluing, and recovers both the characteristic-zero Arinkin–Gaitsgory theory and the depth-zero GL_n local Langlands picture as special cases. The results are presented as the Morita-level abelian-categorical foundation for a later braided monoidal upgrade via the Springer Hopf diagonal and Schauenburg reconstruction.