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 integral concentration problems in modular spectral Langlands to explicit linear algebra. It proves a local comparison theorem identifying the bad-stratum contribution to Ext^\ast(S_q,S_q) with \pi_0(Z_e)-hypercohomology of the qKEC via Propp’s Block–Getzler sheaf realization and descent along covering groups of the reductive centralizer. It gives a complete explicit transparent model computation for G_2 at \ell=3, establishes a balanced-tensor transparency criterion, proves a phantom opacity theorem (reducing opacity to a single representation-theoretic verification under an integral lattice hypothesis), and proves unconditional opacity for F_4(a_3) at \ell=3 using an S_4–V_4 coprimality reduction. These results are assembled into an Enhanced Tier 5 framework: the correct bad-prime spectral category is a quantum/non-semisimple enlargement QL_\ell(\check G) whose bad sector is generated by defect objects (“neglectons”) attached to opaque strata.