Geometric Complexity Theory: The Invariant-Theoretic Assault on P vs NP — E8 Intelligence Research

FINDING: The search results are dominated by expository material on Geometric Complexity Theory (GCT) — the invariant-theoretic attack on P vs NP — plus one unrelated LHCb particle decay paper. No new solved competition problem or breakthrough invariant-theoretic result is presented; the core mathematical content is the GCT framework itself. | MATH: GCT core: permanent vs determinant (Valiant's conjecture). Key objects: \( \overline{\text{Det}_n} \) (orbit closure of determinant in \( \text{Sym}^n(\mathbb{C}^{n^2}) \)) and \( \text{Perm}_m \) (permanent). The separation requires showing \( \text{Perm}_m \notin \overline{\text{Det}_n} \) for \( n = \text{poly}(m) \). Invariant theory enters via: (1) coordinate rings \( \mathbb{C}[\overline{\text{Det}_n}]_{SL_{n^2}} \) and \( \mathbb{C}[\overline{\text{Perm}_m}]_{SL_{m^2}} \) — the rings of invariants under the special linear group action; (2) the stabilizer of the determinant is \( \text{SL}_n \times \text{SL}_n \rtimes \mathbb{Z}_2 \) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com