From a Single Elastic Master Equation to Schrödinger, Dirac, and Einstein Dynamics Projection Limits of the Methane Metauniverse

This paper presents a compact and purely derivational analysis showing how the Schrödinger, Dirac, and Einstein equations emerge as reduction limits of a single elastic master equation defined on internal degrees of freedom. Within the Methane Metauniverse (MMU) framework, observable time, energy, spin, and gravitation arise through projection, phase orientation, and continuum limits, without introducing independent quantization rules, operator axioms, or spacetime postulates. The work documents explicit reduction steps linking internal elastic dynamics to nonrelativistic quantum mechanics, relativistic spinor structure, and general relativity, including the appearance of the cosmological constant as an elastic prestress. Standard spectroscopic effects such as the Balmer and Rydberg scalings, Zeeman, Stark, and Lamb shifts are identified as higher-order or externally perturbed projections of the same master equation. This manuscript is intended as a technical and mathematical complement to the foundational MMU presentation, clarifying how established physical equations can be understood as effective descriptions of a single underlying geometric structure rather than as independent theoretical foundations.