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March 7, 2026Open Access

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

MEMatthew EltgrothUniversity of California, San Francisco

Key Points

The aim is to address bad-prime integral concentration problems in modular spectral Langlands through the quasi-Koszul Endomorphism Complex.
Introduced the quasi-Koszul Endomorphism Complex as a finite complex of modules.
Proved a local comparison theorem between bad-stratum contributions and hypercohomology.
Conducted explicit linear algebra computations for G_2 at =3.
Established a balanced-tensor transparency criterion and a phantom opacity theorem.
Developed an enhanced framework for bad-prime spectral categories.
Identified connections between bad-stratum contributions and hypercohomology using Propp’s realization.
Successfully computed explicit models for G_2 at =3.
Proved unconditional opacity for F_4(a_3) at =3 using coprimality reductions.
Established a framework for managing defect objects and opaque strata.

Abstract

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.

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