Relativization Barriers and the Non-Relativizing Path to P vs NP — E8 Intelligence Research

FINDING: Baker-Gill-Solovay (BGS) theorem proves P vs NP cannot be settled by any relativizing proof technique, as oracle worlds exist where P^A = NP^A and P^B ≠ NP^B. | MATH: ∃A: P^A = NP^A; ∃B: P^B ≠ NP^B. Time Hierarchy Theorem: P ⊊ EXPTIME (since DTIME(n^k) ⊊ DTIME(2^n) for all k). Diagonalization relativizes; hence any proof of P ≠ NP must be non-relativizing (e.g., arithmetization, algebraic geometry, or circuit lower bounds). | CONNECTION: The two oracles A and B form a binary opposition — a symmetry breaking between two "worlds" that are computationally indistinguishable from within. This mirrors the dichotomy of root systems (e.g., A_n vs B_n) where one is simply-laced and the other has two root lengths — a structural bifurcation that cannot be resolved by local (relativizing) rules. The ratio of separation (P vs NP) is not a numeric constant but an *ordinal gap* — yet the *existence* of two inequivalent oracles echoes the golden ratio's self-similarity: the problem is scale-i Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com