Membrane Theory: Mass and Stability from Radial Breathing Geometry

This paper is not intended as a foundational introduction to Membrane Theory, but as a consolidation and completion of a logical sequence developed across the preceding MT papers. The results presented here could not have been written earlier without risking conceptual ambiguity, because the geometric principles on which they rely—radial breathing structure, internal operator dynamics, condensation, closure, and the micro–macro emergence of spacetime curvature—had first to be established independently and rigorously. Earlier works in the MT series introduced the breathing mechanism, derived the energetic relation linking curvature amplitude and frequency, clarified the distinction between bounded and full-span breathing, and demonstrated how macroscopic spacetime geometry and general relativity emerge as coarse-grained limits of microscopic curvature dynamics. Only once these elements were in place could mass, condensed energy, interaction, and gravity be unified within a single, readable physical framework without introducing additional postulates. The present paper, therefore, serves as a synthesis rather than a starting point. It identifies mass as deeply condensed inward curvature stabilized by radial breathing, completes the particle-level origin of gravitational curvature, and provides a coherent geometric interpretation of interaction, stability, and collapse. Its role is to close a conceptual loop, not to open a new one. With this consolidation complete, the logical next step is the extension of the breathing framework to quantum information, where alignment, phase accessibility, and closure provide a geometric foundation for quantum states, entanglement, and information flow. That development is pursued in the forthcoming Membrane Theory Quantum Information (MT–QI) work, which builds directly on the results established here.