Updated RST field equations

Reactive Substrate Theory: Updated Field Equations and Term Breakdown

Reactive Substrate Theory (RST) models reality as a nonlinear, tension-bearing Substrate field coupled to structured Resonance fields. Below is the updated, concrete version of the theory, including the soliton (particle-like lump) analysis and the role of the Substrate’s self-nonlinearity.

1. Core fields

Substrate field S(x,t): A continuous, nonlinear, tension-bearing medium (the “reactive substrate”).
Resonance field Ψ(x,t): Structured excitations (resonances) living in and acting on the Substrate.

2. Updated RST field equations (Model A)


Substrate equation:
  ∂²S/∂t² - c² ∇²S + β S³ = α σ(x,t) |Ψ|²

Resonance equation:
  ∂²Ψ/∂t² - v² ∇²Ψ + μ Ψ + λ |Ψ|² Ψ = κ S Ψ

Interpretation: The Resonance field Ψ sources and deforms the Substrate S via |Ψ|², and the deformed Substrate feeds back on Ψ via the κ S Ψ coupling. Nonlinear terms (β S³ and λ |Ψ|² Ψ) and the feedback loop together allow stable, localized, particle-like structures.

3. Term-by-term breakdown

3.1 Substrate equation


∂²S/∂t² - c² ∇²S + β S³ = α σ(x,t) |Ψ|²

∂²S/∂t²: Inertial term for the Substrate; how S accelerates in time.
- c² ∇²S: Wave/tension term. The constant c sets the propagation speed of disturbances in S (analogous to a wave speed or “substrate light speed”).
+ β S³: Substrate self-nonlinearity. For β > 0, large deformations of S are resisted: the Substrate “stiffens” as |S| grows, preventing runaway collapse.
= α σ(x,t) |Ψ|²: Source term. The Resonance intensity |Ψ|² drives the Substrate.
- α: Strength of the coupling from Resonance to Substrate.
- σ(x,t): Source distribution; localizes where Ψ actually couples into S (e.g., around a particle-like lump).
- |Ψ|²: Local resonance “density” or intensity.

New insight: The β S³ term makes the Substrate response saturating: at small S, the response is approximately linear; at large S, the cubic term limits deformation. This turns the effective coupling between Ψ and S into an amplitude-dependent quantity, stabilizing finite-size structures.

3.2 Resonance equation


∂²Ψ/∂t² - v² ∇²Ψ + μ Ψ + λ |Ψ|² Ψ = κ S Ψ

∂²Ψ/∂t²: Inertial term for the Resonance field.
- v² ∇²Ψ: Wave/spread term. The constant v sets the propagation speed of resonance modes.
+ μ Ψ: Linear “mass-like” term. Sets a characteristic frequency scale for Ψ.
+ λ |Ψ|² Ψ: Self-nonlinearity of the Resonance field.
- For λ > 0, this is defocusing/repulsive: Ψ tends to spread out.
= κ S Ψ: Feedback from the Substrate.
- κ: Strength of the coupling from Substrate to Resonance.
- S Ψ: Local Substrate deformation modifies the effective “mass” and potential felt by Ψ.

New insight: The effective cubic term in the Ψ equation becomes λ_eff = λ - κ η_eff(|Ψ|²), where η_eff encodes how strongly S responds to |Ψ|². For localized lumps to exist, λ_eff must be negative (net focusing), which requires the Substrate feedback (κ η_eff) to overpower the bare self-repulsion λ.

4. Soliton (particle-like lump) in 1D: what we learned

In a 1D, static, single-lump approximation, we take


Ψ(x,t) = ψ(x) e^{-i ω t}
S(x,t) = S₀(x)

and use a soliton ansatz


ψ(x) = A sech(x / L)

Under reasonable approximations (small Substrate deformation, then corrected by β S₀³), this ansatz solves the effective resonance equation if:

Bound-state condition:
```
μ - ω² = v² / L² > 0
```
The resonance frequency ω must lie below √μ for a localized mode.
Focusing vs. defocusing balance:
```
λ_eff = λ - κ η_eff(|ψ|²) < 0  ⇒  κ η_eff > λ
```
The Substrate’s focusing effect must dominate the bare self-repulsion of Ψ.
Amplitude–size relation:
```
A² = 2 v² / [ (κ η - λ) L² ]  (in the small-Substrate limit)
```
The lump’s height and width are linked; it is not arbitrary.
Particle-like energy: for the resonance field alone,
```
E_ψ = 4 v⁴ / [ 3 (κ η - λ) L³ ]
```
giving a mass-like energy scale and showing that narrower lumps cost more energy.

Role of β again: The β S₀³ term makes η → η_eff(|ψ|²) decrease with amplitude, preventing runaway focusing and favoring finite-amplitude, finite-size lumps. This is the mathematical expression of the “tensegrity” idea: the Substrate’s stiffness and the Resonance’s tendency to localize balance to create stable structures.

5. What’s new in this updated RST formulation

From interpretation to equations: RST is now expressed as a concrete nonlinear field theory with explicit PDEs for S and Ψ, not just a conceptual picture.
Explicit coupling structure: The choices
```
C[Ψ] = |Ψ|²,  F_R(C) = α C,  G(S,Ψ) = κ S Ψ
```
define a specific, testable Model A of RST.
Existence of particle-like solitons: In 1D, with a time-harmonic ansatz and a sech profile, the equations admit localized, finite-energy solutions under clear parameter inequalities.
Energetic interpretation: These lumps have a well-defined energy E_ψ that can be treated as a mass-like quantity, linking RST parameters to potential particle masses.
Saturating stability mechanism: The β S³ term in the Substrate equation naturally prevents collapse and selects finite-amplitude, finite-size structures, matching the tensegrity intuition.

6. Next directions

3D generalization: Extend the soliton analysis to full 3D and look for stable, localized “balls” of resonance.
Relativistic form: Cast the equations in a manifestly relativistic form and compare to known field theories.
Particle mapping: Explore whether families of RST solitons can reproduce known particle properties (mass, charge, spin).
Parameter fixing: Relate (c, v, μ, λ, β, κ, α) to measured physical constants.

This block is ready to paste into a Blogger post as an updated, mathematically explicit statement of Reactive Substrate Theory, including the new soliton and stability insights.

Conspiranon