The Discrete Emerging Universe

Pattern Field Theory - Discrete Spacetime, Emergent Geometry, Spectral Graph Theory, Lattice Dynamics, Pregeometry

The Discrete Emerging Universe develops and computationally tests a discrete physical architecture in which planar closure, equilibrium, and continuation into depth are derived rather than assumed.

The paper lies at the intersection of foundational physics, discrete geometry, spectral graph theory, lattice dynamics, and emergent-spacetime research.

PHYSICS AND MATHEMATICS CONNECTIONS

- discrete spacetime
- emergent spacetime
- pregeometry
- discrete geometry
- spectral graph theory
- graph Laplacians
- lattice dynamics
- equilibrium and stability
- statistical mechanics
- recursive geometry
- discrete transport
- network physics
- Weyl and Coxeter structures
- E8 symmetry
- quantum foundations

COMPUTATIONAL RESULTS

- A radius-9 Allen Orbital Lattice disk contains 271 sites and 756 undirected edges.
- The complete disk closes geometrically with zero measured defect.
- A homogeneous stationary state exhibits numerical stationarity under the implemented conditions.
- Perturbation tests distinguish dynamical stationarity from geometric completeness.
- Removing lattice structure breaks geometric closure even where a constant field remains dynamically neutral.
- All first planar continuation candidates retain geometric debt.
- Reciprocal depth continuation preserves the completed sixfold relation.
- Recursive continuation retains connectivity and a finite spectral gap through the tested layers.
- The construction distinguishes local closure from continuation into depth.

The central structural result is:

Six closes. Seven continues.

The paper also connects the discrete construction to E8, Weyl-Coxeter symmetry, duplex structure, spectral decomposition, and finite graph stability.

The finite simulations are verification windows for local physical rules rather than claims about the finite size of the universe.