Topological Vacuum Protection and Parameter-Free Finite QFT from B = −1 in Array Cosmology

Abstract

The rendering algebra R12 ∼= M12(C) of Array Cosmology satisfies the closure relation B ≡ e1e2 · · · e12 = −1. We prove that this single algebraic fact forces antiperiodic boundary conditions (APBC) on the internal cyclic structure, with four
consequences of increasing scope.

(I) Vacuum Protection. All zero-modes are topologically forbidden, generating a mass gap mgap = sin(π/12), and the vacuum polarisation vanishes identically: Π(q) = 0 for q ̸= 0 (mod 6).

(II) Complete n-Point Protection. All odd-n-point lattice loop factors vanish unconditionally. All even-n-point factors vanish at generic external momenta. Non-vanishing contributions occur only at self-energy momenta q ∈ {0, 6}, removed by wave-function normalisation. The 3-point vertex correction vanishes for all 122 = 144 momentum assignments (numerically verified to < 10−13).

(III) Convergent Perturbation Series. UV finiteness (Nmax = 2241 < ∞), IR protection (mgap > 0), and APBC loop suppression combine to yield a geometrically convergent perturbation series with ratio α · Nint = 12/137 < 1—in contrast to the asymptotic (divergent) series of continuum QFT.

(IV) Parameter-Free Finite QFT. We define a QFT on R12 with zero free parameters (all 19 SM parameters derived), zero divergences (all loops are finite sums), and no renormalisation. The “running” of coupling constants is identified as a residual artefact of the continuum approximation in which the APBC protection is lost