Hierarchical Variational Geometry and Effective Nonlocality of Gravity

The standard cosmological ΛCDM model, despite its phenomenological success, relies on dark matter and dark energy — entities that lack any confirmed microscopic foundation. We propose an alternative: the effects attributed to dark matter and dark energy arise as natural consequences of the hierarchical structure of spacetime, rather than as manifestations of new fields or particles. We develop a formalism of dissipative hierarchical geometry, in which spacetime is a fibration over an additional layer (scale) parameter. Its central object is the second variation of the action, from which we rigorously derive: a projection onto the observed layer, a canonical splitting of the dynamics into a reversible Hamiltonian part and an irreversible dissipative part, an emergent effective metric on the observed layer, and a nonlocal interaction kernel between spacetime points. The framework yields the following results without introducing new fields or particles: Nonlocality as a projection artifact. Effective nonlocality is not postulated but follows from integrating out the fast layers of the hierarchy. A spectral cutoff of the kernel automatically keeps the speed of gravitational waves equal to the speed of light on small scales. Dark-matter effects from geometry. The effective metric acquires extra terms that, on galactic scales, reproduce the gravitational potential of a massless dust-like fluid and explain flat rotation curves. Topological phase. Under a specific rank condition on the curvature of an associated Lagrangian submanifold, the system enters a topological phase with the structure of the complex projective plane and a fractional Chern class, in which nonlocal effects dominate. Regularization of singularities. Dissipation channels geometric energy into fast microscopic layers, bounding the growth of curvature and replacing the cosmological singularity with a geometric bounce. Thermodynamics of geometry. A geometric effective temperature and a non-negative entropy production emerge naturally and define an objective arrow of geometric time. Geometric memory. A persistence functional makes the current geometry depend on its own history, predicting a hysteresis effect in galaxy mergers. Multilinear geometry. The ordinary bilinear metric is generalized to a four-linear form contracted with two environment vectors of dissipation and memory, encoding the influence of neighboring layers of the hierarchy. Observational predictions. The theory predicts: a universal profile of galaxy rotation curves, oscillations of the radial acceleration with a Feigenbaum-type scaling, a strong empirical correlation between a decoherence parameter of the cosmic microwave background and gravitational lensing, a frequency-dependent departure of the gravitational-wave speed from that of light on cosmological scales, and specific anomalies in the CMB angular power spectrum at large angles. The construction realizes Occam's razor in its strongest form: we do not multiply entities but generalize the geometric structure itself. Dark matter, dark energy, singularity regularization, and the arrow of time follow from a single variational principle.