The Coarse-Grained Primeon State: A Unified Pre-Theory for Gravity and Quantum Mechanics

We present a unified pre-geometric framework in which both classical gravity and quantum mechanical phenomenology emerge as complementary macroscopic limits of a single underlying amplitude–phase state dynamics. The framework is based on a complex state function ℘ = Aeiθ defined over a discrete prime-indexed structure (p-space) and coarsegrained to an effective continuum description. We demonstrate that the same coarse-grained state ℘(x) = A(x)eiθ(x) yields (i) General Relativity when phase structure averages out and amplitude encodes emergent spacetime geometry, and (ii) quantum mechanics when phase gradients remain coherent and generate conserved probability currents. Both Newton’s gravitational constant G and Planck’s constant ~ emerge as regime-specific macroscopic conversion scales rather than fundamental parameters. This dual-limit structure suggests that the apparent incompatibility between GR and QM is an artifact of treating both as fundamental.