The Chronosphere: A Discrete Spacetime Model for the Origin of Mass Hierarchies and Emergent Gravity

A mathematically rigorous model of discrete spacetime, based on a three-dimensional lattice of "chronons" confined inside a spherical boundary — the Chronosphere, is developed in this work. The incommensurability between the continuous spherical boundary and the discrete lattice structure generates a topological solid angle deficit proportional to the number π-3, which is explicitly computed from the geometry of the icosahedron. With the help of regularized lattice sums and the Euler–Maclaurin formula, effective continuum equations with Robin boundary conditions containing the factor π-3 are derived. Solving the fourth-order wave equation by the phase function method yields a mass spectrum with a universal logarithmic correction: m_n^2 = Λ^2 (n + δ + (π-3)/(4π) ln n + O(1/n)), which agrees with empirical hadron spectra. From the microscopic model of an elastic medium with defects, an effective gravitational action with velocity-dependent corrections is derived. The post-Newtonian parameters are calculated and a comparison with observational constraints is carried out, revealing the need for fine-tuning at the 10^-2 level. A phase shift of the gravitational wave signal from a black hole–neutron star merger of order ΔΨ ~ 1 rad, achievable for modern detectors, is predicted. The work lays the mathematical foundation of the Chronosphere hypothesis, connecting the discrete geometry of spacetime with observable masses and gravitational phenomena.