This monograph establishes a rigorous, coordinate-free unification of quantum gauge fields and gravity by reconstructing the space-time continuum natively within the axiomatic language of higher topos theory. We replace the classical continuous geometric background withthe smooth, cohesive ∞-topos H = Sh∞(ArithSite), modeling the underlying arena as a global adélic variety evaluated over the restricted product ring AQ = R ×∏︁ u2032 p Qp. Under this formulation, the active physical force carriers are re-indexed as cohesive ∞-groupoid curvature stacks Ψ∞ ∈ H(X, BGA)conn, where the active Standard Model gauge fields are locked to a 12-bit message payload derived from the Extended Binary Golay Code G24 and its rootless embeddings within the Leech horizon. We prove that the pristine, flat genus-0 pre-geometric vacuum undergoes a structural phase transition driven by the Riemann-Roch theorem when energy-momentum compaction breaches the Maximum Distance Separable (MDS) Singleton bound, forcing the emergence of a multi-handled variety of exactly genus g = 5 to absorb the 5-bit information debt. The 12 active gauge bosons manifest as the mandatory harmonic connection sections required to smooth out parallel transport across these handles, with the strong color octet anchored tonon-commutative S4 layers and the electroweak triplet regulated by dihedral D3 switchingsymmetries. Furthermore, we demonstrate that the infinite loop divergences of quantum gravity collapse into a finite, self-regulating arithmetic boundary governed by even-weight modular Eisenstein series (E2 through E12) and pinned down by Kummer irregular prime walls at 5 and 691. By evaluating the scale-dependent gravitational beta function against the logarithmic derivative of the Riemann zeta function, we deliver a non-perturbative proof of an Asymptotic Safety fixed point (g∗ ≡ 0) locking gravity to trivial freedom precisely at the non-trivial Riemann zeros along the waveguide axis R⌉(s) = 1/2. Symmetries, matter fields, and universal constants (ℏ, c, G, ∆mass) are thus unmasked as the intrinsic comparison isomorphisms of a beautifully balanced cosmic structure, regularized unconditionally by the volume and measure-theoretic stability of the adélic continuum.