The Quasi-Koszul Endomorphism Complex and Non-Semisimple Enhancement of Modular Spectral Langlands

This paper introduces the quasi-Koszul Endomorphism Complex (qKEC), a finite complex of \mathbf F_\ell[\pi_0(Z_e)]-modules that reduces bad-prime concentration problems in modular spectral Langlands to explicit linear algebra. It proves a comparison theorem identifying the local bad-stratum contribution to Ext^\ast(S_q,S_q) with the \pi_0(Z_e)-hypercohomology of qKEC via Propp’s Block–Getzler sheaf realization and descent along covering groups of the reductive centralizer. The paper gives a complete explicit transparent model computation for G_2 at \ell=3, develops a balanced-tensor transparency criterion and a phantom opacity theorem, proves opacity for F_4(a_3) at \ell=3, and assembles these local results into an Enhanced Tier 5 framework in which the correct bad-prime spectral category is an enlarged quantum Langlands category QL_\ell(\check G) whose bad sector is generated by defect objects (neglectons). The global assembly is formulated abstractly, while the remaining concrete transparent/opaque classification is reduced to finitely many explicit checks by component-group family.