RST Equation Progression: Original → Generalized → Homogeneous

Equation Progression Diagram

Original   ➝   Generalized   ➝   Homogeneous

(∂t² S − c² ∇² S + β S³ = σ(x,t) ⋅ FR(C[Ψ]))
↓
(d²Φ/dt² − c² ∇² Φ − μ Φ + β Φ³ = J(x,t), with J(x,t) = σ(x,t) ⋅ FR(C[Ψ]))
↓
(d²Φ/dt² − c² ∇² Φ − μ Φ + β Φ³ = 0)

RST Equation Progression: Original → Generalized → Homogeneous

Original Sourced Equation Generalized Complete Equation Homogeneous Special Case
(∂t² S − c² ∇² S + β S³) = σ(x,t) ⋅ FR(C[Ψ]) d²Φ/dt² − c² ∇² Φ − μ Φ + β Φ³ = J(x,t)
with J(x,t) = σ(x,t) ⋅ FR(C[Ψ])		 d²Φ/dt² − c² ∇² Φ − μ Φ + β Φ³ = 0
Field: S (substrate field). Field: Φ (scalar field, replacing S). Field: Φ (scalar field, self-contained).
∂t² S: Inertial term (time evolution of substrate). d²Φ/dt²: Inertial term (vacuum inertia, resistance to acceleration). d²Φ/dt²: Same inertial term, no external source.
− c² ∇² S: Elastic term (wave propagation at finite speed). − c² ∇² Φ: Elastic/dispersion term (finite-speed propagation). − c² ∇² Φ: Same elastic term, applies internally.
+ β S³: Nonlinear self-interaction (stabilizes knots/solitons). + β Φ³: Nonlinear self-focusing (soliton formation, particle stability). + β Φ³: Same nonlinear term, balances dispersion internally.
σ(x,t) ⋅ FR(C[Ψ]): Source term, coupling to external distributions and spinor configurations. J(x,t): General source term, defined identically as σ(x,t) ⋅ FR(C[Ψ]). No source term: RHS = 0, purely intrinsic dynamics.
No linear restoring term: Stability from nonlinear self-interaction and external source. − μ Φ: New linear restoring term (background elasticity, equilibrium). − μ Φ: Restoring term remains, ensures equilibrium without external input.
Equivalence: Matches generalized form if μ = 0 and S ↔ Φ. General case: Includes both intrinsic terms and external source J(x,t). Special case: J(x,t) = 0, reduces to self-contained homogeneous equation.

✅ Takeaway

The progression shows how RST equations evolve: the Original form couples directly to external spinor sources, the Generalized form adds a linear restoring term and unifies intrinsic and external dynamics, and the Homogeneous case removes the source entirely, leaving a self-contained scalar field equation. Together, they illustrate how RST bridges external coupling and internal stability.

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