Reactive Substrate Theory (RST) v1.0 – Formal Specification
Reactive Substrate Theory (RST) v1.0 – Formal Specification
A Unified Resonance–Substrate Field Model
Reactive Substrate Theory (RST) models reality as a nonlinear, tension-bearing Substrate field coupled to structured Resonance fields. Stable, localized configurations of these fields correspond to “particles” (matter).
1. Fields and Domains
- Spacetime domain: x ∈ ℝ³, t ∈ ℝ.
- Substrate field: S(x,t) ∈ ℝ.
- Resonance field: Ψ(x,t) ∈ ℂ (or ℝ in the simplest scalar version).
2. Core 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
- c: Substrate wave speed (tension propagation).
- v: Resonance wave speed.
- μ: Linear “mass-like” parameter for Ψ.
- λ > 0: Resonance self-repulsion (defocusing nonlinearity).
- β > 0: Substrate stiffening (saturating nonlinearity).
- κ > 0: Substrate → Resonance coupling strength.
- α > 0: Resonance → Substrate source strength.
- σ(x,t): Source distribution (localizes coupling, e.g. around a lump).
Interpretation: |Ψ|² deforms S; the deformed S feeds back on Ψ. Nonlinearities (β S³, λ |Ψ|² Ψ) and this feedback loop allow stable, localized structures.
3. Effective Nonlinearity and Tensegrity Condition
For static, localized configurations, the Substrate response can be approximated as:
S₀(x) ≈ η_eff(|Ψ|²) |Ψ(x)|²
This induces an effective cubic term in the Ψ equation:
b_eff = λ - κ η_eff(|Ψ|²)
Localization condition (tensegrity balance):
b_eff < 0 ⇒ κ η_eff > λ
Substrate focusing (κ η_eff) must overpower bare self-repulsion (λ) for a stable lump to exist. The β S³ term makes η_eff decrease with amplitude, preventing collapse and favoring finite-size structures.
4. Static, Spherically Symmetric Lump (3D Particle Model)
Assume a single, spherically symmetric, time-harmonic lump:
Ψ(x,t) = ψ(r) e^{-i ω t}
S(x,t) = S₀(r)
r = |x|
The 3D Laplacian in spherical symmetry:
∇²f = f''(r) + (2/r) f'(r)
4.1 Static Resonance Equation (3D)
- v² (ψ'' + (2/r) ψ') + (μ - ω²) ψ + λ ψ³ - κ S₀(r) ψ = 0
a = μ - ω²
b_eff = λ - κ η_eff
- v² (ψ'' + (2/r) ψ') + a ψ + b_eff ψ³ = 0
- Bound-state condition: a = μ - ω² > 0.
- Net focusing: b_eff < 0 ⇒ κ η_eff > λ.
4.2 Static Substrate Equation (3D)
- c² (S₀'' + (2/r) S₀') + β S₀³ = α σ(r) ψ²
For moderate amplitudes, S₀(r) ≈ η_eff(ψ²) ψ²(r), with η_eff decreasing as ψ² grows due to β S₀³.
4.3 Boundary Conditions
At r = 0:
ψ'(0) = 0
S₀'(0) = 0
As r → ∞:
ψ(r) → 0
S₀(r) → 0 (or a constant background)
5. 1D Soliton Check (Concept Validation)
In 1D, with ψ(x) = A sech(x/L), the effective equation
- v² ψ'' + a ψ + b_eff ψ³ = 0
is solved exactly if:
a = v² / L²
b_eff = -2 v² / (A² L²)
which implies:
μ - ω² = v² / L² > 0
κ η > λ
A² = 2 v² / [ (κ η - λ) L² ]
This confirms the tensegrity condition and shows that amplitude and size are linked, not arbitrary.
6. Energy of the Lump (Resonance Part)
Static resonance energy density:
E_ψ = ∫ [ (v²/2) (ψ')² + (a/2) ψ² + (b_eff/4) ψ⁴ ] dx
For the 1D sech solution:
E_ψ = 4 v⁴ / [ 3 (κ η - λ) L³ ]
This behaves like a mass: finite, positive, and increasing as the lump narrows. Adding Substrate energy will select a preferred radius.
3D Numerical Shooting Method Setup (RST v1.0)
To construct explicit 3D RST particles, we solve the coupled radial ODEs for ψ(r) and S₀(r) using a shooting method.
1. Radial ODE System
Unknowns: ψ(r), S₀(r). Parameters: c, v, μ, λ, β, κ, α, ω.
Resonance:
- v² (ψ'' + (2/r) ψ') + (μ - ω²) ψ + λ ψ³ - κ S₀(r) ψ = 0
Substrate:
- c² (S₀'' + (2/r) S₀') + β S₀³ = α σ(r) ψ²
Choose σ(r) = 1 in the core region (or simply σ(r) = 1 for a first pass).
2. Initial Conditions at r = 0
Regularity at the origin implies:
ψ(0) = A (shooting parameter)
ψ'(0) = 0
S₀(0) = S_c (central Substrate value, can be fixed or also shot)
S₀'(0) = 0
Typically, you fix S_c = 0 or a small value and treat A (and possibly ω) as shooting parameters.
3. Shooting Strategy
- Choose parameters: c, v, μ, λ, β, κ, α, and a trial ω.
- Pick initial guesses: A (and optionally S_c).
- Integrate outward: Numerically integrate the ODE system from r = 0 to r = R_max using the initial conditions.
- Check asymptotics:
- Does ψ(r) → 0 as r → R_max?
- Does S₀(r) → 0 (or a constant) as r → R_max?
- Adjust A (and/or ω): Use a root-finding method (e.g. bisection, secant) on A (and/or ω) to enforce ψ(R_max) ≈ 0 and S₀(R_max) ≈ 0.
- Converged solution: When boundary conditions at large r are satisfied within tolerance, you have a candidate RST particle profile.
4. Energy Calculation in 3D
Once ψ(r) and S₀(r) are known, compute the total energy:
E_total = 4π ∫₀^∞ [
(v²/2) (ψ')²
+ (a/2) ψ²
+ (b_eff/4) ψ⁴
+ (c²/2) (S₀')²
+ (β/4) S₀⁴
] r² dr
This E_total is the mass-like energy of the RST particle. Varying parameters and looking for minima in E_total will reveal preferred sizes and stability properties.
5. Status and Outlook
- RST v1.0: A fully specified nonlinear field theory with explicit equations and a clear stability mechanism (tensegrity balance).
- 3D soliton viability: The 3D equations fall into a known class that supports spherically symmetric solitons under the derived conditions.
- Next concrete step: Implement the radial ODE system and shooting method numerically to obtain the first explicit RST particle profiles.
