Mathematical Foundations of the Bergmann Chaos Field

The Bergmann chaos field δ_chaos is introduced as a stochastic scalar field arising from the integration of the bounce term in Bergmann Bounce Relativity (BBR) over cycle volumes in Bergmann Multi-Cycle (BMC) cosmology. We provide a rigorous mathematical foundation: existence as a positive Radon measure via Gaussian Multiplicative Chaos, complete and detailed proof of weak convergence of regularized approximations, renormalization group flow, and applications to quantum gravity. The field regularizes singularities in spacetime metrics and enables information leaks across cycles. A full proof of weak convergence is given using the theory of Rhodes & Vargas (2014). Numerical Monte-Carlo simulations confirm convergence and scaling behavior. Applications to quantum gravity demonstrate regularization of black-hole singularities and emergence of effective loop-like structures. This establishes δ_chaos as a mathematically well-defined object suitable for integration into stochastic quantum field theory and quantum gravity.