Functional Renormalization Group Flow and Continuum Limit 2 of a 26-Valent Chebyshev Network

Overview This record contains the preprint/manuscript titled "Functional Renormalization Group Flow and Continuum Limit of a 26-Valent Chebyshev Network". This work serves as a natural dynamical and quantum extension of our companion study (Vacuum rigidity and infrared isotropic wave dynamics in 26-valent Chebyshev networks), which established the static geometry and exact algebraic curvature invariant (I_∞ = 47/512) of a 26-valent Chebyshev (^∞) network. Abstract We investigate the continuum limit and renormalization group flow of a discrete $3+1$ dimensional spacetime network whose fundamental degree of freedom parametrizes local curvature deviations (ε) from a rigid 26-valent Chebyshev lattice. Starting from a lattice Hamiltonian with a nonlinear Z₂-symmetric potential, we integrate out the conjugate momenta to obtain the Euclidean lattice action. Using the Functional Renormalization Group (FRG) via the Wetterich equation in $D=4$ Euclidean dimensions, we analyze the scaling behavior and physical consequences of the model. Key Findings & Highlights Continuum Limit and Triviality: The renormalization group flow demonstrates that the continuum limit is governed by a Gaussian fixed point, corresponding to a free scalar theory with logarithmic corrections (the familiar triviality of φ⁴ theory in $D=4$). Emergent Kinematics: The infrared expansion of the Chebyshev Laplacian yields an isotropic spatial derivative term, establishing a physical propagation velocity vphys = 3c set by microscopic lattice parameters. Protection of the Curvature Invariant: In the symmetric phase, quantum fluctuations do not shift the vacuum from the isotropic configuration (ε = 0). Consequently, the background geometric curvature invariant I_∞ = 47/512 is strictly protected from radiative shifts. Physical Interpretation: The model is rigorously framed as a discrete scalar field theory with a geometric interpretation rather than a complete fundamental theory of quantum gravity, providing a solid effective field theory framework while clearly outlining its boundaries.