The General Variational Foundation of Rotational‑Order Geometry: Exact Derivation of the Geometric Stress‑Energy Tensor

This paper establishes the rigorous variational foundation for the rotational‑order geometric unification framework. We start from the fundamental action S[g_{μν}, nᵃ] involving the internal S³ orientation field constrained by nᵃnₐ = 1. By performing the variation with respect to the effective metric g^{eff}_{μν} and properly treating the constraint via a Lagrange multiplier, we explicitly derive the modified Einstein field equations: G_{μν}[g^{eff}] = 1/M_{Pl}² (T^{mat}_{μν}+T^{n}_{μν}).We obtain the exact, non‑perturbative geometric stress‑energy tensor T^{n}_{μν}, and project it onto a homogeneous FRW background. Remarkably, we find that the O(η²) corrections to the energy density ρₙ and pressure pₙ share the same sign, + f²/2 η² ṅ⁴. This non‑trivial result closes the logical gap in previous phenomenological applications and provides the essential mathematical machinery for analyzing black hole interiors, galaxy rotation curves, and cosmological dark energy.