Cosmic Pace: a theory for the fD law in Einstein's gravity

In the fD model the density of dark energy follows fD, the rate at which linear structure grows. With CMB, DESI DR2 BAO and DES-Dovekie supernovae it fits better than ΛCDM by Δχ² = 13.4 with as many parameters. Its equation of state crosses w = −1 at zc = 0.453. The form was selected among 55, and the law raises the Hubble tension. We construct a classical carrier for the law, Cosmic Pace, in which dark energy is a property of the Universe as a whole. A single global variable follows the growth equation. Its rate sets the dark-energy density on slices of constant mean curvature. These slices enter the action as a constraint, as in the minimally modified gravity VCDM. At infinite inertia of the global variable the homogeneous background is the fD law plus an optional constant ε ρDE,0. Without the constant and without radiation, the regular growing branch crosses where the matter fraction has fallen to 0.563, for any present matter density. The crossing follows from the field equations in a patch model and a planar field model. This holds for nonzero dust and radiation on closed slices that admit no metric of positive scalar curvature, such as the three-torus. At ε = 0 and for irrotational matter at rest, the computed linear perturbations in three dimensions add one global mode and no local one. They show no ghost in the ultraviolet and no gradient instability. Tensor propagation and the weak-field limit agree with general relativity at the orders computed. Inside black holes the cosmic slices pile up at r = 3M/2 in a homogeneous cylinder model. On a fixed background the local quadratic action for k ≠ 0 agrees with reconstructed VCDM, so no planned measurement separates the two. The law is postulated; its origin, nonlinear evolution with collisionless matter and a well-posedness theorem remain open. This record contains the paper (version 1). The reproduction package with the code and the results behind its numbers, tables and figures is a separate record (DOI 10.5281/zenodo.24068456). The companion paper on the origin of the law is DOI 10.5281/zenodo.24068454. Preregistrations and result notes are in the time-stamped records, parts 1 to 5. All records are linked under related works. This work was done mainly in collaboration with the AI system Claude (Anthropic). Some computations were written and run with a second AI coding assistant and reviewed by Claude. Other AI models commented on drafts and suggested checks. The author is responsible for the content.