The External‑Dimension Theory (EDT) develops a unified framework in which time, gravity, torsion, and quantum structure all emerge from a single ultra‑light scalar field defined on a fundamentally three‑dimensional spatial manifold. Instead of treating spacetime as a primitive four‑dimensional Lorentzian geometry, EDT reconstructs spacetime as an induced Riemann–Cartan manifold whose metric, connection, and torsion are derived functionals of the scalar field and its derivatives. The scalar field evolves monotonically along an external evolution parameter, generating physical time as an emergent quantity. Spatial gradients of the same field induce gravitational acceleration, while cross‑derivatives generate a totally antisymmetric torsion sector that supports stationary rotation without introducing new degrees of freedom. This mechanism bypasses classical no‑go theorems and yields Kerr‑like frame dragging from a single-field ontology. The induced geometry satisfies locality, covariance, and the Newtonian and first post‑Newtonian limits. Static, spherically symmetric solutions are nonsingular, with curvature invariants remaining finite and horizons replaced by smooth redshift surfaces. The rotational sector admits stable, axisymmetric, horizonless compact objects with correct asymptotic behavior. A topological SU(2) holonomy generated by contorsion produces a geometric origin for spin‑1/2, reproducing the double‑valued transformation law of fermions. Cosmologically, the homogeneous scalar field drives a dynamical dark‑energy‑like component, a time‑varying effective Newton constant, and scale‑dependent growth of structure. Linear perturbations yield three gravitational‑wave polarizations—two tensor modes and a scalar breathing mode—with a positive‑definite polarization projector ensuring ghost‑free propagation. The theory predicts distinctive signatures across cosmology, gravitational waves, and strong‑field astrophysics, including modified photon‑ring structure, quasi‑normal‑mode splitting, and Yukawa‑suppressed fifth forces. The manuscript provides the full mathematical development of the theory, including: • derivation of the induced metric and inverse metric, • construction of the Riemann–Cartan connection and contorsion, • static and rotating solutions, • cosmological evolution and perturbation theory, • quasi‑normal modes and gravitational radiation, • SU(2) holonomy and the emergence of spin, • Coleman–Weinberg radiative stability, • and observational forecasts across multiple regimes. EDT offers a coherent, mathematically controlled, and observationally testable alternative to geometric formulations of gravity, unifying temporal flow, gravitational interaction, and quantum structure within a single scalar field.