Six Closes. Seven Continues. The Governing Rule of the Particle That Can Create a Universe

Pattern Field Theory - Graph Theory, Algebraic Topology, Holonomy, Gauge Structure, Discrete Geometry, Quantum Foundations

Six Closes. Seven Continues. develops a graph-theoretic classification of the distinction between sixfold lateral closure and interlayer continuation in the Allen Orbital Lattice.

The central problem is directly connected to graph topology and gauge structure:

When a seventh interlayer relation is added to a sixfold closed neighbourhood, what graph structures can support physically meaningful residual phase?

PHYSICS AND MATHEMATICS CONNECTIONS

- graph theory
- algebraic topology
- cycle space
- graph homology
- holonomy
- gauge invariance
- lattice gauge structure
- discrete geometry
- network topology
- topological phases
- quantum foundations
- phase transport
- interlayer coupling
- discrete state-space topology

PRINCIPAL RESULTS

- The seventh interlayer relation is structurally distinct from the six lateral incidence relations.
- A single pendant interlayer edge introduces no cycle.
- A pendant edge therefore cannot support gauge-invariant residual holonomy.
- Interlayer-sensitive holonomy becomes topologically possible only when interlayer relations participate in graph cycles.
- Cycle-bearing topology permits residual phase but does not itself determine that phase.
- An explicit Hamiltonian Redistribution map is required to assign the physical edge connection and determine its gauge-invariant cycle holonomy.
- Successor supports are classified into graph classes under the declared interface assumptions.
- Cycle rank is calculated for the retained graph classes.
- Degree-three lateral candidates reduce under dihedral symmetry.

The paper therefore connects Pattern Field Theory directly to known mathematical structures in graph theory, cycle topology, discrete gauge systems, holonomy, and algebraic topology.

It provides the graph-theoretic dependency required for later work involving quantum phase, amplitudes, spin, Chern structure, and Bell-type constructions.