Spacetime as Source-Local Rollout and the Conditional Emergence of Three-Dimensional Effective Geometry in The Emergent Frame

Abstract

The Emergent Frame (TEF) treats spacetime not as a pre-existing common background but as relational structure continuously realized by closed matter through spacetime rollout. This paper gives a conditional mathematical realization of that view using a source-local ensemble of helical rollout trajectories whose transverse relational organization is represented by a weighted two-dimensional cell complex. In the homogeneous transported-complex ground state, intrinsic untwisting yields a product of spatial rollout depth and the transverse complex, producing cubic volume growth with d_f = 3. Under explicit long-scale graph-analytic conditions, the same construction gives normal diffusion with d_w = 2 and a bulk static Green scale G(x,y) ≍ d(x,y)^−1. The familiar 1/r behavior therefore arises as a forward consequence of the candidate rollout geometry, while hydrogenic 1/n^2 gross scaling remains only a low-energy consistency check. The construction is deliberately conditional: the two-dimensional quadratic-growth organization of the full rollout ensemble is a TEF structural postulate, not a result forced by one isolated helix. The same ontology also suggests a source-local cosmogenesis—a continuous “TEF Big Bang” in which each closed matter source realizes its own rollout universe. That interpretation is not yet a quantitative cosmological model; a shared effective world must ultimately emerge from compatibility among many source-local rollout structures.