A Spectral Theory of Radial Prime Gaps: Towards a Geometric Renormalization of Prime Distribution

This paper introduces a geometric reformulation of prime gap theory based on a radial embedding of prime numbers into a two-dimensional Lebesgue space. Instead of treating primes as isolated arithmetic objects, we represent each prime 𝑝by a disk of area 𝑝, yielding a natural radial coordinate r(p) = √(p/π). The fundamental objects of the theory are not the disks themselves, but the annular regions between consecutive prime disks. These annuli induce a sequence of radial gaps ∆rn = rn+1-rn, which may be interpreted as a geometrically renormalized version of the classical prime gaps gn = pn+1-pn.I develop a measure-theoretic and spectral framework based on these radial gaps, introduce a spectral zeta function, entropy and energy invariants, and derive asymptotic laws supported by numerical experiments. The resulting formalism suggests a natural bridge between prime gap statistics, spectral geometry, and zeta-type constructions.