Finite Universe Cosmology, Part I: Boundary-Complete Page-Flux Dynamics and Exact Analytic Foundations

We study a spatially homogeneous scalar/electric sector of the eleven-dimensional three-form on a fixed unwarped product background. The internal fields are restricted to a finite differential algebra on the Aloff--Wallach space \(SU(3)/U(1)_{(1,1,-2)}\), with ten co-closed dynamical three-form coordinates and three exact gauge directions. Starting from the Maxwell--Chern--Simons action, we retain the total derivative that separates the raw and cyclic representatives and construct two complete temporal endpoint ensembles for fixed Page charge. Routh reduction then gives \[U_q(c)=\frac{1}{2}c^{\mathsf T}Hc+\frac{\left[q+3\gamma c^{\mathsf T}Tc\right]^2}{2V}.\] We next reconstruct the eleven-dimensional four-form and prove that its Bianchi identity holds identically. The Euler eight-form has only the bidegrees \((1,7)\), \((3,5)\), and \((4,4)\): Page-charge conservation removes the first, co-closedness removes the second, and a nondegenerate \(10+3\) pairing proves that the remaining component is equivalent to the ten reduced equations. Thus the reduced dynamics is exactly equivalent to the full flux Euler equation within the declared ansatz. Finally, the same fixed-charge Routhian yields a coercive conserved Hamiltonian. Every finite initial datum generates a unique smooth solution for all real times, with explicit bounds on the internal coordinates, their velocities, and the electric field. We also isolate the residual boundary one-form controlling extensions to unrestricted mixed boundaries. Together these results provide an exact analytic foundation for later Einstein-sector and quantum-backreaction constructions.