Paper 22a — Two-Loop Renormalization of a Sedenion Scalar Field and Gravitational Coupling

We compute the two-loop MS-bar beta-function of S₈₄, the real 84-component scalar field theory whose quartic coupling tensor is the adjacency matrix of the sedenion zero-divisor pair graph ZD*(A₄).

The only new algebraic input is Tr(A⁴) = 4032, obtained in one line from the known spectral data of the graph and verified by GAP 4.15.1.

Three structural properties of ZD*(A₄) — triangle-free, constant [A³]_{ab} = 12 on all edges, and edge-transitivity of the automorphism group — simplify the Machacek–Vaughn two-loop formula to give β₂(λ) = -189λ³/(4π⁴), closing Open Problem 20.4.

The two-loop renormalization group equation shows that the tachyonic instability of the irr₇ sector persists at all scales below the Landau pole independently of the renormalization scheme, closing Open Problem 20.1.

The fourteen Level-2 modes remain massless at two-loop order by a triple algebraic protection, confirming that the 3:1 ratio dim(Level-3)/dim(Level-2) is preserved under renormalization-group flow.

Under the IIB-II identification of the sedenion norm with the gravitational coupling, the one-loop result of the preceding paper yields the first falsifiable prediction δG_eff/G ≈ -8.41λ [Conjecture, IIB-II].