On the Dependence of Gravitational Coupling on Gauge Dynamics A Vanishing Theorem for Gravity in the Absence of Gauge Interactions

We examine whether gravity can exist as a universal interaction independently of gauge dynamics. Using the Adler–Zee spectral representation of the induced Einstein–Hilbert term, we prove a Vanishing Theorem: when gauge couplings are taken to zero along trajectories that preserve the measured low-energy mass spectrum, the induced Newton constant necessarily vanishes. This establishes that gauge dynamics are not merely contributors to induced gravity, but are required for its existence. We further incorporate an empirical constraint, the Gauge-Charged Dominance Principle, which states that local stress–energy sourcing laboratory and Solar-System gravity is overwhelmingly generated by gauge-charged sectors. A numerical evaluation using lattice inputs for Quantum Chromodynamics confirms that the visible Standard Model induces a negligibly small contribution to Newton’s constant. We conclude that the observed Planck scale cannot arise from visible-sector dynamics alone, implying either a fundamental Einstein–Hilbert term or additional hidden gauge sectors.