Two Solutions to Century-Scale Problems in Fundamental Physics

For over a century, two problems have stood as immovable pillars at the foundation of theoretical physics, resisting every attempt at resolution. The first is the origin of the Born rule, the mysterious squared-amplitude law that governs every quantum measurement yet has never been explained—only postulated. Since Max Born introduced it in 1926, physicists have accepted that probability in quantum mechanics is given by the square of the wavefunction without understanding why. The second is the convergence of Regge calculus to general relativity, a mathematical proof sought since Tullio Regge first formulated discrete gravity on triangulated manifolds in 1961. For sixty-five years, the question of whether discrete spacetime solutions actually approach continuum Einstein solutions in the limit of vanishing edge length remained unanswered, supported by numerical evidence and formal arguments but lacking rigorous proof.

This work presents complete solutions to both problems. The Born rule is derived from deterministic wave dynamics without assuming probability at the fundamental level. A real-valued field satisfying a second-order wave equation oscillates far faster than any detector can resolve. The measurement process integrates over many oscillation cycles, and the time-averaged field intensity at each location determines the likelihood of a detection event. The squared amplitude emerges naturally from the time averaging of an oscillating real field, and probability itself is revealed as an effective description of deterministic threshold crossings with inaccessible fine-grained phase information. No Hilbert space, no state vectors, no measurement axioms are required.

The convergence of Regge calculus is proved using the Lax-Richtmyer framework for numerical analysis, adapted here to the simplicial setting of discrete gravity. The proof establishes that the Regge action approximates the Einstein-Hilbert action with error proportional to the square of the edge length, that the linearized Regge evolution is stable under the Courant-Friedrichs-Lewy condition naturally satisfied in any discretization where the speed of light is fundamental, and that consistency plus stability implies convergence to the continuum Einstein equations with an explicit bound on the metric error. The result provides the mathematical backbone for any physical theory, including quantum gravity, in which spacetime is fundamentally discrete.

Both solutions originate from the Emergence model, a unified framework in which all fundamental physics emerges from wave dynamics on a pre-geometric canvas. These results demonstrate that two of the deepest problems in theoretical physics, each standing for generations as unexplained postulates or unproven conjectures, yield to a single unified approach.