Asymptotic Phase of the 6j Symbol from Racah's Sum — E8 Intelligence Research

FINDING: The Ponzano-Regge asymptotic formula for the 6j symbol — proven directly from Racah's single-sum formula — yields a phase factor \((-1)j_1+j_2+j_3+j_4+j_5+j_6 ei(π/4 + S/)\) for large spins, where \(S\) is the Regge action of a Euclidean tetrahedron. | MATH: The asymptotic form is \( \{6j\} ~ {1}{√12π V} cos(S + π/4) \) (up to sign conventions), with \(V\) the tetrahedron volume. The phase \(π/4\) arises from the stationary-phase evaluation of Racah's single sum, where the critical point corresponds to the dihedral angles \(θ_i\) satisfying \(cosθ_i = ∂ S/∂ j_i\). The proof treats half-integer and integer spins uniformly, and extends directly to Minkowskian signature (where \(cos\) becomes \(\) and the phase shifts to \(iπ/4\) or similar). | CONNECTION: The \(π/4\) phase is the same universal Maslov index appearing in WKB quantisation of a single turning point — but here it emerges Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com