RSFT Foundational Derivations

This technical manuscript, titled "RSFT Foundational Derivations" and authored by Anthony James Bell on March 31, 2026, serves as a formal mathematical supplement to versions 3. 2 through 3. 5. 1 of Rotational Substrate Field Theory. The work is designed to derive the fundamental laws of physics from a single scalar Lagrangian density defined on a 13-point cuboctahedral nodal lattice at the Planck scale. Core Theoretical Framework The theory proposes that all physical phenomena emerge from a continuous rotational medium known as the substrate. Within this framework, the RSFT Lagrangian is governed by parameters for spin-field stiffness and inertia, the ratio of which determines the speed of light. Particles are modeled using a Soliton Ansatz, where they are viewed as stable, dynamically rotating topological vortex-ring solitons rather than static points. Furthermore, electromagnetic fields are shown to emerge directly from the kinematics of this substrate. Key Derivations and Achievements The document outlines several successful theoretical "checks" to validate the RSFT model. Electromagnetism (U (1) ) is fully derived from the phase rotation symmetry of the Lagrangian, while the Weak Force (SU (2) ) is identified through three conserved Noether currents associated with the vortex rotation. The Strong Force (SU (3) ) is mapped to the specific geometry of the cuboctahedral lattice, where eight triangular faces correspond to the eight Gell-Mann generators. In the gravitational sector, the theory recovers the full nonlinear Einstein field equations from substrate back-reaction and provides an exact nonperturbative Schwarzschild solution. Dark Matter is explained as "impedance-mismatched" vortices that remain electromagnetically invisible due to extreme frequency differences but still couple gravitationally. Additionally, the lattice structure resolves singularities by replacing them with Planck-density cores. Identified Challenges and Open Problems Despite these derivations, the manuscript identifies critical unresolved issues, most notably the Hierarchy Problem, where a discrepancy of 10^57 exists between the substrate parameters required for gravity versus those required for electromagnetic charge. The Cosmological Constant remains an open challenge, as the vacuum energy density requires extreme fine-tuning to match physical observations. While the work qualitatively proves quark confinement via the Wilson loop area law, establishing a quantitative confinement scale at the correct energy remains a primary goal. Authorship and AI Assistance Anthony James Bell is the sole author and creator of the RSFT conceptual framework and its core intellectual architecture. The document includes an AI Assistance Disclosure stating that Claude (Anthropic) assisted with formal mathematical steps, tensor algebra, and LaTeX typesetting during an interactive dialogue in March 2026. Gemini (Google) was previously utilized for mathematical consistency checks in earlier versions of the theory.