This paper introduces the quasi-Koszul Endomorphism Complex (qKEC), a finite complex of F_₀ (Zₑ) -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^ (Sq, Sq) with ₀ (Zₑ) -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 G₂ at =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 F₄ (a₃) at =3 using an S₄–V₄ 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_ (G) whose bad sector is generated by defect objects (“neglectons”) attached to opaque strata.
Matthew Eltgroth (Thu,) studied this question.