Polynomial Hierarchy, Lasserre SDP, and Distributed Consistency: A Recursive Link — E8 Intelligence Research

FINDING: Polynomial hierarchy collapses if P=NP; formal consistency definitions in distributed systems rely on recursive axiomatization; Lasserre hierarchy uses conic programming for global polynomial optimization. MATH: Polynomial hierarchy (PH) defined via oracle machines: Σ₀ = P, Σ₁ = NP, Σ₂ = NP^NP, ...; PH ⊆ PSPACE. Lasserre hierarchy: semidefinite programming (SDP) relaxations of polynomial optimization, degree-d relaxation uses O(n^d) variables. Consistency in distributed systems: linearizability ≡ sequential consistency under failure equivalence with asynchronous message-passing. CONNECTION: Lasserre hierarchy's conic structure (SDP + SOCP) mirrors root system symmetries (e.g., Aₙ, Dₙ lattices) via positive semidefinite cones; polynomial optimization relaxations often yield golden ratio bounds (e.g., 1.618) in extremal problems. Recursive axiomatization in formal systems parallels self-similar scaling (0.618, 0.382) in fractal hierarchies. DEPTH: 7 — The polynomial hierar Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com