Cosmological Dualities: Hubble Expansion and Black Hole Singularities as Geometric Limits of a Relational Projection (R → 0)

Standard cosmology treats the expansion of the universe (Dark Energy) and gravitational collapse (Black Holes) as opposing dynamical phenomena. We propose they are symmetric geometric manifestations of a single underlying mechanism: the projection of information relative to a fundamental unity.

Building on a relational geometric framework, we define a Relational Projection Tensor that generates the observable universe vector. We derive:

1. The Hubble Law: The recession velocity is a geometric projection rate V(R) = ω·R, where the fundamental frequency ω ≡ H₀ ≈ 2.27 × 10⁻¹⁸ rad/s emerges naturally, eliminating the need for Dark Energy.

2. The Black Hole Limit: The singularity is reinterpreted not as infinite mass, but as the geometric limit R → 0, where relational difference vanishes and projection frequency diverges (ω → ∞).

This model suggests the universe is not expanding into a void, but undergoing an internal geometric reorganization towards its origin.