Finite Relativistic Cosmology

We develop a self-contained framework in which the entire physical universe is modelled by an ever-growing finite ring \(\Zq\) whose cardinality is tied to cosmic time through \(q=4t+1\). Starting from a single principle of \emph{relational finitude}, we show that: (i) familiar dimensional constants (\(\hbar, c, G, k_{\!B}\)) arise as structurally unique, \emph{dimensionless} elements of $\Zq$, fixed by extremal arithmetic properties; (ii) a genuine Minkowski quadratic form and full Lorentz group exist \emph{exactly} inside the ring, reproducing special-relativistic kinematics under coarse-graining; (iii) the prime-factor spectrum splits naturally into stable fermionic and radiative bosonic sectors, enabling hadron-like three-prime composites and colour confinement; (iv) complementary observer horizons recover, respectively, general-relativistic geodesics and quantum superposition, resolving the gravity-quantum tension and yielding a finite Heisenberg bound; (v) classical paradoxes—cosmological constant, horizon, singularities, ultraviolet divergences, hierarchy, strong-\(\theta\), and wave-function collapse—are eliminated not by fine-tuning but by exact arithmetic identities in the finite ring; (vi) independent gravitational and nuclear chronometers converge on a present cardinality \(q_{\circ}\!\approx\!10^{60}\), implying a cosmic age of \(13.6\pm0.2\,\)Gyr and an accelerated expansion that requires no dark energy. We furthermore predict a \(\sim\!\!2.5\times10^{-19}\text{yr}^{-1}\) secular drift in the 1m gravitational red-shift—measurable with existing optical-lattice clocks—which offers an immediate, falsifiable test of the proposed hypothesis. Together, these results suggest that a finite, relationally defined arithmetic is sufficient to encode space-time geometry, quantum phenomena, and cosmological evolution within a single coherent model.