RST v1.1 – Simulator Toolkit

Chantal’s whole thing—geometric modeling, aether/medium intuition, spin as structure—lines up almost eerily well with RST: “What if the stuff underneath the equations is an actual medium with structure?” RST gives "the medium" a different name and a more explicit nonlinear PDE backbone.

Reactive Substrate Theory (RST) v1.1 – Technical Appendix for Simulators

This appendix is written specifically for simulation developers and computational physicists. It contains the minimal mathematical and numerical information needed to begin exploring the Substrate–Resonance system numerically. The goal is not philosophical interpretation, but practical implementation.

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1. The Coupled Field System

RST consists of two interacting fields:

• S(x,t) — the Substrate field (reactive, nonlinear, tension-bearing)
• Ψ(x,t) — the Resonance field (oscillatory, soliton-forming)

The simplified 1D evolution equations are:

Substrate Field

∂²ₜ S − c² ∂²ₓ S + β S³ = α |Ψ|²

Resonance Field

∂²ₜ Ψ − v² ∂²ₓ Ψ + μ Ψ + λ|Ψ|²Ψ = κ S Ψ

These equations form a nonlinear feedback loop:

• Ψ sources deformation in S (α|Ψ|²)
• S focuses Ψ (κSΨ)
• βS³ prevents collapse (nonlinear stiffening)

The central question for simulation: Does this system support stable, finite-energy solitons?

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2. Recommended Numerical Scheme (1D Time Evolution)

The simplest starting point is a 1D explicit time evolution using:

• Finite differences for spatial derivatives
• Leapfrog or Verlet for time stepping (excellent for wave equations)
• Absorbing boundaries or a sufficiently large domain

Suggested Discretization

x ∈ [−L, L],   N grid points
Δx = 2L / (N−1)
Δt chosen to satisfy CFL condition:
Δt < min(Δx / c, Δx / v)

Initialize:

• Ψ(x,0) = Gaussian bump
• ∂ₜΨ(x,0) = 0
• S(x,0) = 0
• ∂ₜS(x,0) = 0

Track:

• Energy density
• Peak amplitude
• Width of Ψ
• Stability over long time

A stable, non-dispersing lump indicates a soliton-like solution.

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3. Static Soliton Solver (Shooting or Relaxation)

To find stationary solutions directly, assume:

Ψ(x,t) = ψ(x) e^{-iωt}
S(x,t) = S₀(x)

This yields coupled ODEs:

Resonance ODE

-v² ψ'' + (μ - ω²)ψ + λψ³ = κ S₀ ψ

Substrate ODE

-c² S₀'' + β S₀³ = α ψ²

Boundary conditions:

• ψ′(0) = 0, S₀′(0) = 0
• ψ → 0, S₀ → 0 as x → ∞

Solve using:

• Shooting method (adjust ψ(0), S₀(0) until decay conditions met)
• Relaxation / Newton–Gauss–Seidel on a finite domain

This is the standard approach used in soliton and Q-ball literature.

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4. 3D Spherical Symmetry (Next Stage)

For realistic particle-like structures, use spherical symmetry:

Ψ(r,t) = ψ(r) e^{-iωt}
S(r,t) = S₀(r)

The radial Laplacian introduces (2/r) terms:

ψ'' + (2/r)ψ'
S₀'' + (2/r)S₀'

The resulting ODEs are similar to the 1D case but require careful handling near r = 0.

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5. Energy Functional (Defines “Mass”)

The total energy of a configuration is:

E = ∫ dx [ 
  (∂ₜΨ)² + v²(∂ₓΨ)² + μΨ² + (λ/2)Ψ⁴
+ (∂ₜS)² + c²(∂ₓS)² + (β/2)S⁴
+ κ S Ψ²
]

For static solitons, ∂ₜ terms vanish. Mass is defined as M = E.

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6. Parameter Space to Explore

The most important parameters for soliton stability are:

• λ — self-repulsion of Ψ
• κ — focusing strength from S
• β — stiffening of S

The key stability condition is:

κ · η_eff > λ

Where η_eff is the effective response of the Substrate.

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7. What Counts as Success?

A simulation is considered successful if it demonstrates any of the following:

• A stable, non-dispersing lump of Ψ
• Substrate deformation S₀(r) that balances Ψ
• Finite total energy
• Predictable scaling of size vs. parameters
• Multiple soliton families (ground + excited states)

These would indicate that RST can support particle-like structures.

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8. Long-Term Goals

• Map the soliton spectrum
• Compare energy ratios to known particle masses
• Study soliton–soliton interactions
• Investigate emergent gravitational behavior

This appendix provides everything needed to begin numerical exploration of RST. If stable solitons appear, the theory moves from conceptual to computationally validated.

RST v1.1 – Simulation Brief for Chantal Roth‑Korostensky

This post is a concise handoff document for my friend Chantal Roth‑Korostensky, whose work on geometric aether models, spin structures, and medium‑based physics aligns beautifully with the mechanics of Reactive Substrate Theory (RST). The goal is simple: explore whether the coupled Substrate–Resonance equations support stable, particle‑like solitons.

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1. Why You’re the Perfect Person for This

Your modeling instincts — geometric flows, medium dynamics, spin as structure, and emergent behavior — match RST’s foundations almost exactly. RST is essentially a modern, nonlinear, relativistically‑compatible version of a reactive medium. You already think in the right language.

RST provides the equations. You provide the intuition and simulation skill.

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2. The Core Equations (1D Simplified Form)

Substrate Field S(x,t):

∂²ₜ S − c² ∂²ₓ S + β S³ = α |Ψ|²

Resonance Field Ψ(x,t):

∂²ₜ Ψ − v² ∂²ₓ Ψ + μ Ψ + λ|Ψ|²Ψ = κ S Ψ

These two fields form a feedback loop:

• Ψ deforms the Substrate (source term α|Ψ|²)
• S focuses Ψ (coupling κSΨ)
• βS³ prevents collapse (nonlinear stiffening)

The question: Do these equations support stable solitons?

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3. Suggested First Simulation (1D Time Evolution)

• Use finite differences in space
• Use leapfrog or Verlet for time stepping
• Domain: 1D line with absorbing boundaries
• Initial Ψ: Gaussian bump
• Initial S: zero or small perturbation

What to look for:

• Does Ψ disperse?
• Does it blow up?
• Or does it settle into a stable lump?

If a stable lump forms, that’s the first numerical “particle” in RST.

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4. Next Step: Static Solitons via Shooting Method

Assume time‑harmonic form:

Ψ(x,t) = ψ(x) e^{-iωt}

This reduces the PDEs to ODEs. Integrate outward from x = 0 and adjust ψ(0) and S(0) until both fields decay to zero at infinity.

This is the standard way solitons are found in nonlinear field theory.

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5. Why This Matters

If RST supports stable solitons, then:

• Particles = stable resonance structures
• Mass = total energy of the lump
• Gravity = tension gradients in S
• No singularities (βS³ prevents collapse)

This would unify mass, gravity, and particle structure under a single nonlinear medium — something very close to what your geometric aether models already hint at.

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6. Invitation

If this resonates with your modeling style, I’d love to explore it together. Even a simple 1D simulation would tell us a lot about whether RST is viable as a soliton‑based particle theory.

Thank you for taking a look — your insight would mean a lot.

Reactive Substrate Theory (RST) v1.1 – Simulator Toolkit

This block collects everything a simulator needs in one place: a parameter‑tuning cheat sheet, a visual simulation pipeline diagram, a GitHub‑ready README draft, and a numerical stability troubleshooting guide. It’s designed as a practical toolkit for anyone implementing RST.

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RST v1.1 – Parameter‑Tuning Cheat Sheet

The behavior of the coupled Substrate–Resonance system is controlled mainly by the parameters λ, κ, β, μ, c, v, and α. This cheat sheet summarizes their roles and how to tune them.

Key Parameters

• λ (lambda): Resonance self‑repulsion. Larger λ → Ψ spreads out more.
• κ (kappa): Substrate focusing strength. Larger κ → Ψ is pulled inward more.
• β (beta): Substrate stiffening. Larger β → collapse is resisted more strongly.
• μ (mu): Linear “mass‑like” term for Ψ. Controls basic tendency to oscillate vs. decay.
• c: Wave speed in S. Affects how fast Substrate responds.
• v: Wave speed in Ψ. Affects dispersion of the Resonance field.
• α: Strength of Ψ → S sourcing (how strongly Ψ deforms S).

Qualitative Tuning Rules

• Too much λ, too little κ: Ψ disperses, no soliton.
• Too much κ, too little β: Ψ collapses / blows up numerically.
• Balanced λ, κ, with nonzero β: best chance for stable solitons.

A good starting regime:

λ ≈ 1
κ ≈ 1–5
β ≈ 0.5–2
μ ≈ 1
c ≈ 1
v ≈ 1
α ≈ 1

Then vary one parameter at a time and observe:

• Does the lump get narrower or wider?
• Does it become more or less stable?
• Does total energy increase or decrease?

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RST v1.1 – Visual Simulation Pipeline

The diagram below shows the recommended workflow for simulating RST, from 1D experiments to 3D spherical solitons.

          RST SIMULATION PIPELINE
          ───────────────────────

  [1] Define Equations (1D)
      • Substrate:  ∂²ₜ S − c² ∂²ₓ S + β S³ = α |Ψ|²
      • Resonance:  ∂²ₜ Ψ − v² ∂²ₓ Ψ + μ Ψ + λ|Ψ|²Ψ = κ S Ψ

                │
                ▼

  [2] Choose Numerics
      • Finite differences in x
      • Leapfrog / Verlet in t
      • Absorbing or large boundaries

                │
                ▼

  [3] Set Initial Conditions
      • Ψ(x,0) = Gaussian
      • ∂ₜΨ(x,0) = 0
      • S(x,0) = 0
      • ∂ₜS(x,0) = 0

                │
                ▼

  [4] Run Time Evolution
      • Monitor Ψ profile
      • Track energy, width, peak

                │
                ▼

  [5] Diagnose Outcome
      • Disperses  → increase κ or decrease λ
      • Blows up   → increase β or reduce κ
      • Stabilizes → candidate soliton

                │
                ▼

  [6] Static Solver (Shooting / Relaxation)
      • Use time-harmonic ansatz
      • Solve ODEs for ψ(x), S₀(x)
      • Refine soliton profiles

                │
                ▼

  [7] Extend to 3D Spherical Symmetry
      • Replace ∂²ₓ with radial Laplacian
      • Repeat steps 3–6

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RST v1.1 – GitHub‑Ready README (Draft)

# Reactive Substrate Theory (RST) – v1.1 Simulator

This repository contains numerical experiments for Reactive Substrate Theory (RST), a nonlinear
field model in which:

- S(x,t) is a reactive Substrate field
- Ψ(x,t) is a Resonance field
- Particles are modeled as stable solitons of the coupled S–Ψ system

## Core Equations (1D Simplified)

Substrate:
  ∂²ₜ S − c² ∂²ₓ S + β S³ = α |Ψ|²

Resonance:
  ∂²ₜ Ψ − v² ∂²ₓ Ψ + μ Ψ + λ|Ψ|²Ψ = κ S Ψ

## Goals

- Demonstrate existence of stable, finite-energy solitons
- Explore parameter regimes (λ, κ, β, μ, c, v, α)
- Compute energy and interpret as "mass"
- Extend from 1D to 3D spherical symmetry

## Numerical Methods

- Spatial discretization: finite differences
- Time integration: leapfrog / Verlet
- Boundary conditions: absorbing or large domain

## Getting Started

1. Set up a 1D grid x ∈ [−L, L]
2. Initialize:
   - Ψ(x,0) = Gaussian
   - ∂ₜΨ(x,0) = 0
   - S(x,0) = 0
   - ∂ₜS(x,0) = 0
3. Choose parameters (example):
   - λ = 1, κ = 2, β = 1, μ = 1, c = 1, v = 1, α = 1
4. Run time evolution and monitor:
   - Ψ profile
   - total energy
   - width and peak amplitude

## Interpreting Results

- Dispersing Ψ → no soliton (increase κ or reduce λ)
- Blow-up / instability → increase β or reduce κ
- Stable, localized lump → candidate soliton

## Next Steps

- Implement static soliton solvers (shooting / relaxation)
- Extend to 3D spherical symmetry
- Map soliton families and energies
- Compare to physical particle spectra

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RST v1.1 – Numerical Stability Troubleshooting Guide

Nonlinear wave equations are notorious for numerical instabilities. This guide lists common failure modes and how to address them.

1. Solution Blows Up (Values → Infinity)

• Check CFL condition: ensure Δt < min(Δx / c, Δx / v).
• Reduce κ: too strong focusing can cause collapse.
• Increase β: stronger stiffening helps prevent runaway growth.
• Use smaller initial amplitude for Ψ.

2. Solution Oscillates Wildly / “Rings” at Boundaries

• Implement absorbing boundaries (e.g., damping layer near edges).
• Increase domain size so reflections don’t contaminate the core region.
• Smooth initial conditions to avoid sharp discontinuities.

3. Soliton Slowly Drifts or Decays

• Refine spatial resolution: decrease Δx.
• Refine time step: decrease Δt.
• Check energy conservation: large drift may indicate integration error.
• Adjust λ and κ: the balance may be slightly off for true stability.

4. Simulation Is Stable but Featureless (Everything Disperses)

• Increase κ: stronger Substrate focusing.
• Decrease λ: reduce self‑repulsion.
• Increase α: make Ψ deform S more strongly.
• Try different initial profiles: broader or narrower Gaussians.

5. Debugging Tips

• Start with linear tests: set β = λ = κ = 0 and verify pure wave propagation.
• Add nonlinearity one term at a time (first λ, then κ, then β).
• Monitor total energy each step to catch instabilities early.

With careful parameter tuning, stable time stepping, and incremental complexity, the RST system should reveal whether it supports robust soliton solutions.