Formal Proof: The Radion-Zeta Stabilization Arithmetical Rigidity in the 26D Bosonic Manifold

This paper provides a formal axiomatic proof establishing that the non-observation of a fifth-force deviation at sub-millimeter scales is the definitive signature of a stabilized 26D Bosonic Manifold. We demonstrate that the Radion field ϕ is arithmetically locked at the self-dual radius R = √2lp through the ramification of rational primes within the cyclotomic domain K = Q(ζ12). By equating the energy density of prime collapse with the Riemann Zeta residue ζ(−1), we prove that the gravitational coupling constant α must yield an arithmetical zero, rendering the vacuum silent and the manifold rigid.