Singularity Avoidance in Grand Relativity: Modified TOV Equations and the Planck Floor Mechanism

We extend the Grand Relativity (GRL) framework by deriving the modified Tolman-Oppenheimer-Volkoff (TOV) equations that govern hydrostatic equilibrium in the presence of a dynamical scalar field. We demonstrate that the coordinated scaling of fundamental constants (c ∝ ϕ, G ∝ ϕ2, m ∝ ϕ−2) produces a geometric pressure barrier that prevents gravitational collapse to infinite density. The scalar field gradient generates an effective repulsive force scaling as ϕ−3, which dominates over gravity as ϕ → 0. This mechanism creates a ‘Planck Floor’—a minimum physical scale where thelocally-measured Planck length grows to meet the shrinking matter distribution, halting collapse at finite physical density. We find the core stabilizes at a physical size of L ∼ 10−24 m (approximately 1011 times the fundamental Planck length), corresponding to extreme coordinate radii in the Einstein frame. The Planck Floor scale follows LPlanck ∝ M 1/3, making it significantly larger than the fundamental Planck scale for astrophysical black holes. We show that the event horizon (rs ∼ 3 km) and physical core are fundamentally decoupled through ϕ-dependent scaling, resolving the classical singularity problem through field dynamics rather than quantum effects. The framework suggests black holes contain frozen, finite-density cores rather than singularities.