d²Φ/dt² − c² ∇² Φ − μ Φ + β Φ³ = J(x,t), with J(x,t) = σ(x,t) ⋅ FR(C[Ψ]).

RST Equation Comparison: Sourced vs. Homogeneous

Sourced Form Homogeneous Form
(∂t² S − c² ∇² S + β S³) = σ(x,t) ⋅ FR(C[Ψ]) d²Φ/dt² − c² ∇² Φ − μ Φ + β Φ³ = 0
∂t² S: Inertial term (time evolution of Substrate field). d²Φ/dt²: Inertial term (vacuum inertia, resistance to acceleration).
− c² ∇² S: Elastic term (wave propagation at finite speed). − c² ∇² Φ: Elastic/dispersion term (finite-speed propagation).
+ β S³: Nonlinear self-interaction (stable knots/solitons). + β Φ³: Nonlinear self-focusing (soliton formation, particle stability).
σ(x,t) ⋅ FR(C[Ψ]): Source term, coupling to matter/energy distributions and spinor configurations. − μ Φ: Linear restoring term (background elasticity, equilibrium).
No source term: Equation set equal to 0.
Field content: S couples to spinor functional C[Ψ], allowing richer topological/spinor behavior. Field content: Φ treated as a scalar field, simplified dynamics without explicit spinor coupling.

✅ Takeaway

The sourced form emphasizes external coupling via σ(x,t) ⋅ FR(C[Ψ]), while the homogeneous form highlights internal stability with a linear restoring term. Both share the same inertial, elastic, and nonlinear structure, but differ in assumptions: one driven by external sources, the other self-contained.

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