We formulate a conditional three-stage cosmological programme that distinguishes statistical localization, real geometric reduction and physical evolution. A six-dimensional real metric with a positive rank-two vertical block yields a four-dimensional Lorentzian quotient only under explicit regularity, projectability and inertia assumptions. Its causal projection and finite heavy-mode elimination are made explicit. A common derivative-coupled action governs the neutral scalar sector, with action-derived continuity, conditional mass crossing and a healthy positive-vacuum limit; its relation to the charged SL6C sector is exhibited. Three-state statistical coexistence is realized by a normalized positive finite evolution, while weighted phase conversion requires compensating energy exchange to preserve the Friedmann constraint. Finite propagation errors and background constraint residuals quantify separate consistency requirements. A gauge-aware Maxwell calculation identifies the metric characteristic cone; ordinary positive-mass propagation gives a delay decreasing with energy and does not establish a dimensional transition. A retained Gaussian transverse-stress benchmark supplies conditional gravitational-wave spectra and redshifted scales. The results identify precisely which geometric, statistical, dynamical and observational constructions are still required for a complete emergence theorem. The accompanying Lean source contains the original proof bodies.