Reactive Substrate Theory (RST): Core Equation and Calibrated Constants

Reactive Substrate Theory (RST): Core Equation and Calibrated Constants

RST models the vacuum as a reactive medium governed by a nonlinear wave equation. In plain text form:

d2Φ/dt2 = c^2 * Laplacian(Φ) - mu * Φ + beta * Φ^3

  • d2Φ/dt2 (Inertial Term): Acceleration of Substrate displacement. Represents the inertia of the vacuum — resistance to change, giving solitons their inertial mass.
  • c^2 * Laplacian(Φ) (Elastic Term): Dispersion and propagation. Neighboring points pull on each other, defining the stiffness of the medium and the finite speed of waves.
  • - mu * Φ (Linear Restoring Term): Background elasticity. Acts like a Hookean spring constant, ensuring the medium prefers equilibrium and defining rest energy.
  • + beta * Φ^3 (Nonlinear Self-Focusing Term): Soliton generator. Counters dispersion, locking energy into localized knots (particles) when balanced against the Laplacian.

Calibrated Constants and Target Fits

  • c: Substrate signal speed = 299,792,458 m/s → sets propagation velocity.
  • a0: Acceleration threshold ≈ 1.2 x 10^-10 m/s^2 → anchors low-acceleration regime (galactic rotation curves).
  • r0: Electron knot radius ≈ lambda_C / (2*pi) ≈ 3.86 x 10^-13 m → Compton wavelength scale.
  • mu: Elasticity scale, fit via r0 and c (mu ~ c^2 / xi^2) → controls restoring term strength.
  • beta: Nonlinear focusing strength, fit to soliton energy (m_e * c^2) → governs particle mass and 720-degree spinor behavior.
  • xi: Coherence length ≈ r0 → sets core size and stability.
  • lambda: Penetration depth ≈ r0 → defines field fall-off.
  • T_bg: Background tension, fit to a0 → determines transition in low-acceleration regimes.
  • h mapping: Action per cycle (integral of P·dQ = h) → quantization of Substrate phase.

Unified Interpretation

The constants calibrate each term of the RST equation:

  • c: fixes the speed of propagation in the Laplacian term.
  • mu: sets the background restoring force.
  • beta: balances dispersion to stabilize solitons.
  • a0, T_bg: define thresholds for inertial behavior in low-acceleration regimes.
  • r0, xi, lambda: determine the geometry and stability of localized knots.
  • h mapping: ensures quantized energy cycles within the Substrate.

Together, the equation and constants show how the Substrate both carries waves and locks energy into matter. The Laplacian spreads, the cubic term localizes, the inertial term resists change, and the calibrated constants tie the math to measurable physical quantities.

How the RST Core Equation Evolved

Reactive Substrate Theory (RST) began with a core equation that explicitly included a source term:

d2S/dt2 - c^2 * Laplacian(S) + beta * S^3 = sigma(x,t) * FR(C[Psi])

This original form describes the Substrate field S with three main components:

  • d2S/dt2: Inertial term, acceleration of the Substrate displacement.
  • - c^2 * Laplacian(S): Elastic term, dispersion and propagation through the medium.
  • + beta * S^3: Nonlinear self-focusing term, stabilizing localized solitons.
  • sigma(x,t) * FR(C[Psi]): Source term, coupling the Substrate to configuration Psi (spinor/two-component structure).

The Updated Equation

Later refinements produced a simplified, homogeneous version:

d2Phi/dt2 = c^2 * Laplacian(Phi) - mu * Phi + beta * Phi^3

Key differences:

  • No explicit source term: The right-hand side is set to zero, making the equation homogeneous.
  • Linear restoring term - mu * Phi: Added to represent background elasticity and rest energy.
  • Field content: The updated form treats Phi as a single scalar field, without explicit SU(2) spinor coupling.

Homogeneous vs. Sourced

  • Original: d2S/dt2 - c^2 * Laplacian(S) + beta * S^3 = sigma(x,t) * FR(C[Psi])
  • Updated: d2Phi/dt2 - c^2 * Laplacian(Phi) - mu * Phi + beta * Phi^3 = 0

The original equation explicitly couples the Substrate to external configurations. The updated equation describes the internal dynamics of the Substrate alone.

When They Can Be Made Equivalent

  • Vacuum/no external coupling: Set sigma(x,t) * FR(C[Psi]) = 0. The original reduces to d2S/dt2 - c^2 * Laplacian(S) + beta * S^3 = 0, which matches the updated form if mu = 0 and S is identified with Phi.
  • Effective medium absorption: The source term can be absorbed into effective coefficients:
    • mu_eff = mu + alpha * sigma(x,t)
    • beta_eff = beta + gamma * G(C[Psi])
  • General inhomogeneous form: d2Phi/dt2 - c^2 * Laplacian(Phi) - mu * Phi + beta * Phi^3 = J(x,t), with J(x,t) = sigma(x,t) * FR(C[Psi]).

Practical Mapping

  • Replace S → Phi.
  • Move terms to the left-hand side: d2Phi/dt2 - c^2 * Laplacian(Phi) - mu * Phi + beta * Phi^3 = J(x,t).
  • Identify J(x,t) = sigma(x,t) * FR(C[Psi]).
  • Set mu = 0 for strict parity with the original; keep mu ≠ 0 to model background elasticity/rest energy.
  • Keep C[Psi] explicit if spinor/topological effects (such as 720-degree holonomy or exchange phase) are important; otherwise treat J(x,t) as a phenomenological source.

Conclusion

The evolution of the RST core equation reflects a shift from a sourced, spinor-coupled model to a simplified homogeneous scalar field form. The original equation emphasizes external coupling, while the updated equation highlights internal stability and elasticity. Both can be reconciled through effective coefficients or by setting the source term to zero, showing how RST adapts to different levels of abstraction in modeling the Substrate.

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