Discrete Curvature Geometry of Prime Gaps

Description This project studies prime numbers through the geometry of their gap sequence. Instead of analyzing primes or gap sizes directly, it analyzes discrete curvature, defined as the second difference of prime gaps. The project shows that curvature is an exact operator for discrete systems, that curvature determines the gap sequence up to a linear trend, and that the geometric trajectory of curvature and slope contains structure that is destroyed by randomized controls even when gap distributions are preserved. The work develops exact discrete calculus identities, curvature energy measures, phase portrait geometry, null model tests, and a preregistered rule based prediction framework for the next prime.