Crystallization of Free Energy into the C₂–C₃ Matter Phase

This paper connects three previously developed elements of the present research program—the three-step relational closure C₃, the C₂–C₃ electromagnetic-field model, and the energy–matter phase-transition model—by interpreting matter formation as a structural transition from a free-energy phase. In the earlier framework, one complete C₃ cycle consists of three relational updates, and the action associated with one complete cycle was defined as h₀. A dimensionless phase-coupling ratio α = rω/c was also introduced, where r is a phase radius, with α = 1 interpreted as a candidate boundary between wave-like and matter-like states. Subsequent C₂–C₃ studies examined how relational inconsistency injected into a localized closure can be converted into an emission mode. The new proposal developed here is that electromagnetic energy, once emitted, need not propagate by successively exciting pre-existing empty cells. Instead, it may propagate as a self-maintaining group of free energy units. When the local energy density increases, relational elements are proposed to form tetrahedral C₃ closures and then crystallize into a space-filling rhombohedral structure. A basic matter cell is represented by 2C₃ + C₂, thereby providing a concrete geometrical candidate for the previously phenomenological energy–matter phase transition. We further propose that the relative rotation of the two C₃ closures may distinguish neutral and charged lepton-like structures, and that the relative frequency 2ω arising from counter-rotation may provide a geometrical interpretation of the factor of two in the Zitterbewegung frequency. These latter correspondences remain hypotheses. The derivation of the charge magnitude e, neutrino structure, pair-production constraints, and the wave–particle duality of freely propagating energy are left for subsequent work.