Reactive Substrate Theory (RST) v1.1 – Index of Symbols
Reactive Substrate Theory (RST) v1.1 – Index of Symbols
This index lists every symbol used in RST v1.1 along with its meaning, physical role, and where it appears in the formal equations. It is designed as a quick technical reference for readers, researchers, and anyone following the mathematical development of the theory.
Field Symbols
S(x,t)
The Substrate field. A nonlinear, tension-bearing medium whose deformation encodes gravity,
inertia, and the stabilizing forces that hold particles together.
Ψ(x,t)
The Resonance field. Represents structured oscillations that can form stable, particle-like
solitons.
ψ(r)
Radial profile of the time-harmonic Resonance field in spherical symmetry:
Ψ(x,t) = ψ(r) e^{-iωt}.
S₀(r)
Static radial profile of the Substrate field for a spherically symmetric soliton.
Derivatives & Operators
∂ₜ
Partial derivative with respect to time.
∂²ₜ
Second time derivative (acceleration of the field).
∇²
Laplacian operator. In 3D spherical symmetry:
∇²f = f'' + (2/r)f'.
f'
Derivative with respect to radius r.
f''
Second derivative with respect to radius.
Core Constants
c
Wave propagation speed in the Substrate field.
v
Wave propagation speed in the Resonance field.
μ
Linear mass-like parameter in the Resonance equation.
λ (Lambda)
Self-repulsion coefficient of the Resonance field. Appears in λ|Ψ|²Ψ.
β (Beta)
Substrate stiffening coefficient. Appears in β S³ and prevents collapse.
κ (Kappa)
Coupling strength from Substrate to Resonance. Appears in κ S Ψ.
α (Alpha)
Coupling strength from Resonance to Substrate. Appears in α σ |Ψ|².
Source & Response Terms
σ(x,t)
Source distribution controlling where Ψ couples into S. Often localized around a soliton.
C[Ψ]
Characteristic functional of the Resonance field. In v1.1, C[Ψ] = |Ψ|².
F_R(C)
Substrate response functional. In v1.1, F_R(C) = α C.
Effective Quantities
η_eff(|Ψ|²)
Effective Substrate response coefficient. Decreases with amplitude due to βS³ stiffening.
b_eff
Effective cubic coefficient in the Resonance equation:
b_eff = λ − κ η_eff.
a
Effective linear coefficient in the time-harmonic Resonance equation:
a = μ − ω².
Frequency & Energy
ω (Omega)
Internal oscillation frequency of a time-harmonic soliton.
E_total
Total energy of the soliton (Resonance + Substrate). Defines the particle’s mass.
E_ψ
Energy contribution from the Resonance field alone.
Spatial Variables
x
Position vector in 3D space.
t
Time coordinate.
r
Radial distance from the soliton center: r = |x|.
Stability & Soliton Parameters
L
Characteristic width (radius) of a soliton.
A
Peak amplitude of the Resonance field in 1D or radial ansatz.
Tensegrity Condition
The requirement for soliton stability:
κ η_eff > λ.
Special Terms
Soliton
A stable, localized solution of the Ψ–S system.
Substrate Tension Gradient
The spatial variation of S that produces gravitational-like effects.
Stiffening
The nonlinear increase in Substrate resistance as S grows.
Reactive Substrate Theory (RST) v1.1 – Unified Expansion Pack
This block contains the full set of RST v1.1 updates: the roadmap, the gravity explanation, the aether comparison, a beginner-friendly introduction, and a visual schematic of the Substrate– Resonance interaction. It is designed as a complete, self-contained reference for readers exploring RST for the first time.
RST v1.1 Roadmap
RST v1.1 builds on the formal field equations introduced in v1.0 and moves toward a fully computational, testable physical model. The roadmap below outlines the next major milestones.
1. Numerical Soliton Construction
- Implement the 3D radial ODE shooting method
- Generate the first stable Ψ–S soliton profiles
- Identify families of solutions (electron-like, photon-like, proton-like)
2. Parameter Calibration
- Determine realistic ranges for (c, v, μ, λ, β, κ, α)
- Match soliton energies to known particle masses
- Explore scaling laws for composite structures
3. Emergent Gravity Derivation
- Show how Substrate tension gradients reproduce gravitational attraction
- Derive weak-field and strong-field limits
- Compare predictions to GR without singularities
4. Quantum Behavior from Resonance Dynamics
- Linearize Ψ around stable solitons
- Recover Schrödinger-like behavior for small perturbations
- Explain measurement as Substrate-mediated resonance selection
5. Multi-Particle Interactions
- Simulate soliton–soliton interactions
- Study interference, scattering, and bound states
- Explore whether gauge-like behavior emerges naturally
How RST Explains Gravity
In General Relativity, gravity is curvature of spacetime. In RST, gravity is the result of tension gradients in the Substrate field S(x,t). The Substrate equation:
∂²ₜ S − c² ∇² S + β S³ = α σ |Ψ|²shows that matter (Ψ) deforms the Substrate. This deformation changes the effective propagation speed of waves traveling through it. Light and matter follow paths of least Substrate tension, which appear to us as “curved spacetime.”
Key Points
- Mass = Substrate deformation
- Gravity = gradient of that deformation
- No singularities because β S³ prevents infinite compression
- Time dilation arises from slower wave propagation in high-tension regions
Thus, RST reproduces gravitational effects without requiring spacetime curvature as a fundamental entity. Gravity becomes a mechanical consequence of the Substrate’s nonlinear response.
RST vs Aether: Why They’re Not the Same
RST is often compared to historical “aether” theories, but the resemblance is superficial. The differences are fundamental.
1. The Aether Was Static; RST Is Dynamic
The classical aether was imagined as a fixed background medium. RST’s Substrate is a fully dynamic nonlinear field with its own wave propagation, stiffness, and feedback.
2. The Aether Was Passive; RST Reacts
The Substrate in RST responds to resonance intensity via:
σ(x,t) · F_R(C[Ψ])This reactivity is what stabilizes particles and generates gravity.
3. The Aether Could Not Explain Mass or Gravity
RST explains both as emergent properties of Substrate deformation and tension gradients.
4. The Aether Was Ruled Out; RST Is Compatible with All Data
RST does not predict a preferred rest frame or violate relativity. It reinterprets GR and QM without contradicting their measurements.
RST for Beginners
Reactive Substrate Theory is built on a simple idea:
Reality is made of a reactive medium (the Substrate) and stable vibrations within it (Resonances).
The Two Fields
- Substrate (S): A continuous field that carries tension.
- Resonance (Ψ): A vibration pattern that can become stable and form a particle.
How Particles Form
A particle is a soliton — a stable lump of Ψ held together by the Substrate’s tension. The Substrate pushes inward, the resonance pushes outward, and the two forces balance.
Why This Matters
- Particles have size, not infinite density
- Gravity emerges naturally
- No singularities, no virtual particles, no multiverse
- Everything is mechanical and continuous
Visual Diagram of the S–Ψ Interaction
Below is a conceptual schematic showing how the Substrate (S) and Resonance (Ψ) interact to form a stable particle-like structure.
┌───────────────────────────────┐
│ Resonance Field Ψ │
│ (localized vibration pattern) │
└───────────────┬───────────────┘
│
▼
Ψ increases Substrate tension
│
▼
┌────────────────────────────────────────────────────┐
│ Substrate Field S │
│ (reactive medium with stiffening β S³ term) │
└───────────────────┬────────────────────────────────┘
│
▼
Substrate pushes back (κ S Ψ focusing)
│
▼
Balance = stable soliton (particle)
This loop — resonance → substrate deformation → substrate focusing → resonance stabilization — is the core mechanism of RST.
Reactive Substrate Theory (RST) v1.1 – Glossary
This glossary provides clear, concise definitions of all major concepts used in Reactive Substrate Theory (RST). It is designed as a quick‑reference guide for readers exploring the theory for the first time or reviewing the formal specification.
A
Amplitude (of Ψ)
The local strength of the Resonance field. Higher amplitude means stronger Substrate deformation.
α (Alpha)
Coupling constant controlling how strongly the Resonance field sources the Substrate.
Aether (Classical)
A historical concept of a static, passive medium. Not related to RST. The Substrate is dynamic,
reactive, and nonlinear.
B
β (Beta)
The Substrate stiffening coefficient. The term βS³ prevents collapse and eliminates singularities.
beff (Effective Cubic Coefficient)
The combined nonlinear effect of Resonance self‑repulsion and Substrate focusing:
b_eff = λ − κ η_eff
C
c
Wave propagation speed in the Substrate field.
Collapse (Forbidden in RST)
Infinite compression of matter or spacetime. Prevented by the βS³ stiffening term.
Coupling
The interaction between S and Ψ. Ψ deforms S; S focuses Ψ.
D
Deformation (of S)
The change in Substrate tension caused by the presence of a Resonance field.
Dispersion
The tendency of Ψ to spread out due to λ|Ψ|²Ψ self‑repulsion.
E
Energy (Total)
The integrated energy of both fields. Defines the mass of a soliton.
ηeff (Effective Substrate Response)
Amplitude‑dependent factor describing how strongly S reacts to |Ψ|².
F
FR(C[Ψ])
The Substrate response functional. In v1.0/v1.1, this is proportional to |Ψ|².
Focusing (Substrate)
The inward pressure exerted by S on Ψ, stabilizing the soliton.
G
Gravity (RST Interpretation)
A gradient in Substrate tension, not curvature of spacetime.
L
λ (Lambda)
Self‑repulsion coefficient of the Resonance field.
Lump (Particle)
A stable soliton formed by the Ψ–S interaction.
M
Mass (RST Definition)
The total energy stored in the soliton’s Substrate deformation and Resonance oscillation.
N
Nonlinearity
A term that grows faster than linearly with field amplitude. Essential for soliton stability.
P
Particle (RST)
A stable, localized soliton of the Ψ–S system.
R
Resonance Field (Ψ)
The structured vibration pattern that forms matter.
Reactive Substrate (S)
The nonlinear medium whose tension dynamics give rise to gravity, mass, and stability.
Roadmap (RST v1.1)
The next steps toward numerical soliton construction and emergent physics.
S
Soliton
A stable, self‑reinforcing wave packet that maintains shape due to nonlinear balance.
σ(x,t)
Source distribution controlling where Ψ couples into S.
Stiffening
The increase in Substrate resistance as S grows. Prevents singularities.
Substrate Field (S)
The reactive medium underlying all physical phenomena in RST.
T
Tensegrity Condition
The requirement for particle stability:
κ η_eff > λ
Time Dilation (RST)
Slower wave propagation in high‑tension regions of the Substrate.
V
v
Wave propagation speed in the Resonance field.
Virtual Particles (Not Used in RST)
RST replaces them with real Substrate tension interactions.
W
Wavefunction (RST Interpretation)
A real resonance structure, not a probability cloud.
Ψ (Psi)
Ψ
The Resonance field responsible for matter formation.
