Abstract We establish that the R12 rendering algebra of Array Cosmology carries a natural adelic structure dictated by the prime factorisation 12 = 2 2 × 3. Eleven A-layer results are proved (using only standard number theory): ten theorems and one corollary. (1) The structure constant ω = e 2πi/12 = ζ12 defines the cyclotomic field Q(ζ12) whose Galois group Gal(Q(ζ12)/Q) ∼= (Z/12Z)∗ ∼= Z2 × Z2 has four elements corresponding to the four Volma types of R12. (2) The four Dirichlet characters mod 12 define four L-functions; the special value L(1, χ1) = π/(2√3) (where χ1 is the lift of the primitive character χ−3 to mod 12) provides a number-theoretic invariant of the electroweak sector, while theWeinberg angle sin2 θW = ln 2/3 is independently derived from the Landauer principle (Paper V). (3) The tensor factorisation M3(C) ⊗ M4(C) = M12(C) is the algebraic realisation of Rankin-Selberg functoriality GL(3) × GL(4) → GL(12). (4) The ramification structure of Q(ζ12) determines the gauge hierarchy: p = 3 ramified → SU(3)C, p = 2 ramified → SU(2)L × U(1)Y, p ≥ 5 unramified → energy desert. (5) The Z3 centre obstruction (ω3 residual phase) is the Artin conductor at p = 3, providing the number-theoretic origin of colour confinement.