A Unified Cosmological Model from the Spatial Substrate S-Field: Fifth Interaction, Topological Dark Sectors, and Layered Chronology

This work presents the first-principles derivation of the dark energy equation of state w(a) = w0 + gamma(a-1) with gamma proportional to xi, showing that a single parameter xi simultaneously determines both w0 and wa. The standard Lambda-CDM model leaves the nature of dark energy unexplained, faces persistent small-scale problems in particle dark matter, and leaves the ontology of time ambiguous. We propose a framework of a fifth fundamental interaction mediated by the spatial substrate scalar field (S-field) - not the laboratory fifth force coupled to baryon number. The model rests on the axiom of spatial pre-existence: the S-force separates out first from the unified field and builds the primordial spatial topology; gravity separates out next and, through metric coupling, disciplines the unmeasurable substrate chronology into measurable physical time, which divides into cosmic time and local natural time. Dark energy arises from the global topological relaxation of the S-field; local topological distortions retained across cycles give dark matter. The two share a single origin. We distinguish two objects usually conflated as the scalar breathing mode. The S-singularity intrinsic oscillation (10^43 Hz) drives space construction and symmetry breaking, and lies far beyond the reach of any planned gravitational-wave detector. The cosmological S-breathing mode, some 63 orders of magnitude lower in frequency, has an effective mass anchored by the dark-energy density to mu_eff ~ 0.03 H_0 (10^-20 Hz); its signature is an S-induced gravitational slip on cosmological scales. Its dispersion relation omega_phys^2 = k^2/a^2 + mu_eff^2 separates two sources: mu_eff^2 does not redshift, while k^2/a^2 does, providing a macroscopic yardstick for cosmic time. The scalar sector contains only one independent dynamical degree of freedom - a decisive criterion. The power spectrum derivation path is given; its numerical implementation is left to future work.