Version of June 3rd, 2026, following technical development within the Absolute Frame Theory programme. We show that the gauge structure $SU(3)_c \times SU(2)_L \times U(1)_Y$ of the Standard Model emerges as a topological property of the embedding interface between a four-dimensional observable manifold $\mathcal M$ and a higher-dimensional substratum $\mathcal A$, as posited by the Absolute Frame Theory (AFT) [P. E. Valenzuela, preprint, under review in \emph{Foundations of Physics} (2026)]. By analyzing the natural fiber bundle structure of the embedding $X:\mathcal{M} \to \mathcal{A}$, we identify the structure group of the normal bundle as $SO(N-4)$, and demonstrate that a parsimony argument based on dimensional counting under the constraint of reproducing the Standard Model gauge structure forces $N=10$ as the minimal admissible dimension. We further show that (i) the natural connection on the normal bundle is an $\mathfrak{so}(N-4)$-valued one-form, whose components are the gauge field candidates; (ii) the Nyquist mode of the channel between $\mathcal M$ and $\mathcal A$ induces a compatible complex structure on the normal fiber whose stabilizer in $SO(6)$ is $U(3) \simeq (SU(3) \times U(1))/\mathbb{Z}_{3}$, realizing the Pati-Salam strong-hypercharge structure as a derived geometric consequence; (iii) the chiral electroweak factor $SU(2)_L$ emerges from the spin double cover of the tangent bundle of $\mathcal A$ through the decomposition $\text{Spin}(10) \supset \text{Spin}(4) \times \text{Spin}(6) \simeq (SU(2)_L \times SU(2)_R) \times SU(4)$, surviving the signature transition from Euclidean $\mathcal A$ to Lorentzian $\mathcal M$ as an internal symmetry; and (iv) the hypercharges of the Standard Model fermions are derived as $Y = 2T^{3}_{R} + (B-L)$, with explicit numerical verification for the eight fermions of a generation. A G\"odelian analysis evaluating dimensions $N<10$ shows that the observable complexity of the Standard Model places a structural lower bound on $N$. These results, combined with the entropic-gravity derivation of the embedding tension in the foundational paper of the AFT framework, close Conjecture VI of that work: no fundamental forces exist within $\mathcal M$; all four interactions are manifestations of the same underlying $\mathcal M$-$\mathcal A$ embedding structure.