A Multi-Scale Generative Grammar for Crystal Formation Lattice, Morphology, and the Coupling Between Them

Formal models of physical formation are usually models of states: a configuration is assigned an energy, and the theory selects among configurations. Processes that are irreversible and history-dependent resist this treatment, since their outcomes are distinguished by how they were produced and by nothing about the product itself. Crystal formation contains both kinds at once. Its lattice is selected by energy and is identical in every specimen; its branched morphology is selected by the order of accretion and is unique to one history. This paper models both in a single formalism, generative grammar, and locates the difference between them inside that formalism. Three observations are offered. First, a dynamical process can be written as a generative grammar in which the derivation carries the physics, which makes procedural content a formal object and brings the associated theorems within reach. Second, the two scales of crystal formation appear to occupy different components of a grammar: the lattice scale is a statement about the language, order-independent and energetically selected, while the morphological scale is a statement about the derivation, path-dependent and admitting no variational characterisation; weights on configurations alone then look insufficient to reproduce the statistics of growth. Third, the two scales couple at a single point of the formalism, the applicability condition of one production: the lattice alphabet restricts where growth may occur. Small computations reported here agree with the published simulation literature that this restriction fixes the arm count and orientation of the cluster, and that literature reports it shifting the fractal dimension from the continuum value toward a rigorous bound. Each of the three models is given explicitly, with its production rules, a worked example, and a figure. The lattice-scale formalism is due to others; what is attempted here is a placement of the parts.