Emergent Mass and Coupling Ratios from Resonant Knot Solitons in R-Theory (Work Model) a minimal route from an R-Lagrangian to a discrete spectrum

This record reports the main methods + results step of the R-Theory work model Emergent Mass and Coupling Ratios from Resonant Knot Solitons. The core proposal is simple: what we call “particles” can be modeled as stable knots in a field—topological solitons (Hopfions)—formed within a coherence order parameter associated with the zero-field (𝕆). In this framework, rest mass is defined operationally as the minimum static energy of a stable knot in a given topological sector Q. The key target is therefore not absolute units, but dimensionless predictions such as energy (mass) ratios E_Q/E_1 and basic, field-defined coupling proxies. To make the claim testable, the record starts from a specified field Lagrangian / energy functional, and then performs direct numerical minimization within fixed topology to obtain stable configurations. It presents preliminary computed results for Q=1..4: (i) optimized energy estimates and ratio tables, (ii) stability diagnostics via linearized mode analysis (low-lying \omega^2 spectrum) to separate real minima from numerical artifacts, and (iii) sensitivity checks against discretization and solver choices. A compact “Run Card” standardizes what to report so independent implementations can compare results cleanly. This is explicitly a work model: the goal is not a final particle-physics fit, but a transparent demonstration—open to replication or refutation—that a resonance/topology-based field model can produce a discrete, stable energy hierarchy with reproducible analysis hooks. Keywords: R-Theory; resonance; zero-field; coherence field; Lagrangian field model; knot solitons; Hopfions; topological charge; emergent mass; energy ratios; numerical stability; reproducibility.