Complexity Bounds Are Independent of Bounded Arithmetic: P vs NP Depends on Axiomatic Universe — E8 Intelligence Research

FINDING: Complexity bounds (e.g., P vs NP) are provably independent of bounded-arithmetic theories — the "truth" of a complexity statement depends on the axiomatic universe, not on a unique mathematical reality. | MATH: Bounded arithmetic hierarchy: IΔ₀ ⊂ IΔ₀ + exp ⊂ ... ; independence of ∀-sentences of form "SAT ∉ P" from theories T where T ⊬ Con(T)-style reflection; oracle separation: ∃A: P^A ≠ NP^A (Baker–Gill–Solovay 1975), ∃B: P^B = NP^B (same paper) — non-relativizing proofs required. Turing degrees form an upper semi-lattice (𝒟, ≤_T) with 2^ℵ₀ elements, no maximum, dense below 0′. | CONNECTION: The oracle-world non-uniqueness mirrors the non-uniqueness of geometric realizations of a lattice — same combinatorial poset, different embeddings. Turing degree structure is a partial order with no canonical metric; contrast with crystallographic root systems (A_n, B_n, D_n, E_6/7/8, F_4, G_2) which are rigid up to Weyl group. The absence of a "golden" fixed point in 𝒟 (no self-similar d Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com