A Geometric Extension of Regulated Accumulation: Spectral Geometry and Einstein Gravity as Low-Spectrum Degeneration

Presenting a geometric and spectral extension of the theory of scale regulated accumulation developed in four preceding papers. The central claim is that spacetime geometry, as encoded by Einstein’s field equations, emerges as a low spectral, additive limit of a more general accumulation–regulation principle. By formulating the regulator dynamics in spectral space using the Laplace–Beltrami operator, demonstrating in three explicit steps how the spectral equation governing regulated accumulation degenerates into Poisson gravity and subsequently into the Einstein field equations. In this framework, geometry is not fundamental but arises as an infrared fixed point of a scale dependent spectral filter acting on accumulated physical quantities. The approach preserves experimental discipline, introduces no new ontologies, and clarifies the geometric meaning of scale, additivity, and emergence.