UDEL Mathematics

This appendix presents the formal mathematical foundations of UDEL — the Universal Discrete Energy Lattice. All equations are fully discrete: no limits, no manifolds, no renormalization, and no infinities. Every physical phenomenon arises purely from nodes, adjacency, energy conservation, hop-geometry, and layered time.

Nodes are defined as:

Adjacency produces geometry through hop-distance, with no coordinates and no continuous metric.

Local time emerges from redundancy in transition probabilities:

This forms the discrete analogue of curvature-induced time dilation.

Positive adjacency energy and negative path-density energy cancel exactly.

Stable patterns of adjacency form particle families, with fixed or cyclic attractors guaranteed by the finiteness of the adjacency map.

Pure hop-distance diffusion yields the exact Navarro–Frenk–White halo profile.

Gravity emerges from deficits in path density.

Layer-shifted energy produces gravitationally visible but electromagnetically invisible mass.

Stable motifs satisfy:

This yields confinement, decay rates, EM coherence, and gravity scaling.

Each force corresponds to a motif scale S:

Horizons form when adjacency becomes uniform; entropy is |∂H|.

Information is bounded by the size of the boundary.

Leakage flux is discrete due to finite adjacency and energy values.

The UDEL mathematical framework proves that geometry, gravity, fields, particles, dark matter, horizons, and cosmic evolution emerge from finite combinatorics alone: nodes, adjacency, hop-distance, energy symmetry, and layered time.