Singular Support and Nilpotent Cones in Spectral Geometric Langlands — E8 Intelligence Research

FINDING: The spectral side of geometric Langlands is being rigorously constructed via singular support of coherent sheaves on quasi-smooth derived schemes, with nilpotent cones stratifying the category; elliptic curve case yields explicit irreducible components and simple-object bijections. | MATH: Core object: \( \text{IndCoh}_{\mathcal{N}}(\text{LocSys}_{\check{G}}) \) — coherent sheaves with singular support in the global nilpotent cone \( \mathcal{N} \subset T^*\text{LocSys}_{\check{G}} \). Quasi-smoothness: \( \mathcal{O}_X \) has finite Tor-amplitude; singular support \( \text{SS}(\mathcal{F}) \subset T^*X \) is a conical Lagrangian. For elliptic curves \( E \), the stack \( \text{Coh}(E) \) stratification yields irreducible components of the global nilpotent cone; bijection: simple objects of \( \text{Per}_\mathcal{N}(\text{Coh}(E)) \leftrightarrow \) irreducible components of \( \mathcal{N}_E \). No explicit numeric constants appear — the invariants are categorical (Lagrangian Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com