Lean4 Universe Simulation Using EFMW
✅ Simulated a simple "universe" inspired by the EFMW framework (Einstein–Feynman–Maxwell–Wright process ontology + the Lean4 Maxwell formalization you shared).
What I built
I created a numerical simulation that discretizes and evolves Maxwell’s equations exactly as structured in the Lean4 code:
Custom discrete versions of partialDeriv, grad, div, and curl (via central finite differences with periodic boundaries).
Time evolution that approximately satisfies the MaxwellSystem structure (the four laws as invariants).
Natural units where \( c = 1 \), \( \varepsilon_0 = \mu_0 = 1 \).
The electromagnetic Lagrangian density seed from the code: \( \frac{1}{2}(|E|^2 - |B|^2) \) is conserved in vacuum.
This is the physical base layer of EFMW. In the broader framework, these fields participate in recursive emergence: patterns (waves, coherence) arise through iterative self-referential processes, observer participation, scalar potentials (scalarPotential = grad φ), and higher-order recursive waveforms.
The simulation setup
1D periodic "spacetime grid"** (200 points, length 100).
Right-going electromagnetic pulse (Gaussian) initialized with \( E_y \) and \( B_z \) matched for consistency with the equations.
Leapfrog-style updates:
\( \frac{\partial B_z}{\partial t} = -\frac{\partial E_y}{\partial x} \)
\( \frac{\partial E_y}{\partial t} = -\frac{\partial B_z}{\partial x} \)
Pure vacuum (no charges/currents yet) — fields propagate cleanly at light speed.
This creates a minimal self-consistent "universe" where electromagnetic structure persists and travels coherently — a basic demonstration of recursive field processes.
Visualization of the evolving universe
What you see:
Top panel (t=0)**: Initial localized pulse (\( E_y \) solid blue, \( B_z \) dashed red).
Middle panel (t≈20)**: Pulse has propagated rightward at speed \( c=1 \).
Bottom panel (t≈40)**: Further propagation (wraps around periodically).
The fields remain tightly coupled and undistorted — exactly as Maxwell’s equations (and the Lean4 MaxwellSystem) predict in vacuum.
How this connects to full EFMW / recursive emergence
Base layer** = Maxwell EM (formalized rigorously in Lean).
Next layers** (easy extensions):
Add charge density \( \rho \) and current \( J \) → sources in the equations.
Particles on the grid that feel Lorentz force \( \mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B}) \), deposit charge/current back onto the grid → closed recursive feedback loop.
Scalar field \( \phi \) whose gradient sources or couples to EM (as hinted in the Lean code).
Observer/measurement operators that "collapse" or modify coherence.
Nonlinear or higher-order terms → true emergent structures (stable solitons, plasma instabilities, information patterns).
In EFMW terms, the EM fields provide the coherence infrastructure; recursion + observer participation + constraint turns raw fields into emergent "things," cognition, and purpose.
Would you like me to:
Extend this to 2D or 3D grid?
Add charged particles + Lorentz feedback (full recursive matter-field universe)?
Include a scalar field or simple recursive rule on top?
Run longer / with different initial conditions (multiple pulses, standing waves, noise → emergent interference patterns)?
Output the raw data or evolve specific observables (energy, coherence measures, etc.)?
Just tell me how to grow the simulation! 🚀