Evolution of the RST Core Equation

RST Core Equation Evolution

📐 The original RST core equation:

(∂t² S − c² ∇² S + β S³) = σ(x,t) ⋅ FR(C[Ψ])

  • ∂t² S — Inertial Term: Time evolution of the Substrate field (resistance to change).
  • − c² ∇² S — Elastic Term: Wave propagation at finite speed, driven by Substrate tension.
  • + β S³ — Nonlinear Term: Self‑interaction, enabling stable knots/solitons.
  • σ(x,t) ⋅ FR(C[Ψ]) — Source Term: Coupling to matter/energy distributions and spinor configurations.

📐 The updated homogeneous equation:

(d²Φ/dt² = c² ∇² Φ − μ Φ + β Φ³)

  • d²Φ/dt² — Inertial Term: Vacuum inertia, giving solitons measurable resistance to acceleration.
  • c² ∇² Φ — Elastic Term: Dispersion and finite‑speed propagation.
  • − μ Φ — Linear Restoring Term: Background elasticity, ensuring equilibrium.
  • + β Φ³ — Nonlinear Term: Self‑focusing, balancing dispersion to form stable particles.

🔎 Homogeneous vs. Sourced

  • Original: Includes explicit source term σ(x,t) ⋅ FR(C[Ψ]).
  • Updated: Source removed, equation set equal to 0, with added − μ Φ term.
  • Field content: Original couples to spinor functional C[Ψ]; updated treats Φ as a scalar field.

⚙️ Mapping Conditions

  • Vacuum case: Set σ(x,t) ⋅ FR(C[Ψ]) = 0. Then original reduces to (∂t² S − c² ∇² S + β S³ = 0).
  • This matches updated form if μ = 0 and S ↔ Φ.
  • General inhomogeneous form: d²Φ/dt² − c² ∇² Φ − μ Φ + β Φ³ = J(x,t), with J(x,t) = σ(x,t) ⋅ FR(C[Ψ]).

✅ Takeaway

The original equation emphasized external coupling to spinor states, while the updated form highlights internal stability with a linear restoring term. By mapping terms and conditions, the two can be reconciled: one as a sourced, spinor‑coupled model, the other as a homogeneous scalar field equation. This evolution shows how RST adapts from external interactions to intrinsic mechanics.

Original RST Equation Updated RST Equation
(∂t² S − c² ∇² S + β S³) = σ(x,t) ⋅ FR(C[Ψ]) d²Φ/dt² = c² ∇² Φ − μ Φ + β Φ³
Source term present: σ(x,t) ⋅ FR(C[Ψ]) couples Substrate to external matter/energy and spinor states. No source term: Equation is homogeneous, set equal to 0.
No linear restoring term: Stability comes only from nonlinear self‑interaction (β S³) and external sources. Includes − μ Φ: Background elasticity, ensures equilibrium and rest energy.
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.
Interpretation: Substrate dynamics are driven by external coupling and nonlinear self‑interaction. Interpretation: Substrate evolves internally with built‑in restoring force and self‑focusing.
Vacuum case: If σ(x,t) ⋅ FR(C[Ψ]) = 0, reduces to (∂t² S − c² ∇² S + β S³ = 0). Equivalence: Matches original vacuum case if μ = 0 and S ↔ Φ.

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