The Brunner Bridge: A Universal Invariant for Quantum-Relativistic Scaling with Numerical Validation

The Brunner Bridge defines a unique rational, dimensionless invariant , with , that satisfies three irreducible conditions: divergence at UV/IR limits, a Planck-scale pivot, and duality symmetry. Derived variationally from the Einstein-Hilbert and Dirac actions, it provides a regulator for quantum gravity with falsifiable predictions for LIGO O4 echoes, Hawking radiation discretization, and Planck-mass remnants. Numerical simulations validate discrete gravitational wave echo signatures, with complementary predictions for laboratory Bose-Einstein condensate analogues. This work positions the Brunner Bridge as an analytic, testable regulator alongside string theory, loop quantum gravity, and asymptotic safety frameworks.