Theory of Heterocrystals: Unit-Cell Rupture, Emergent Gauge Fields, and Quantum Geometry

We develop a theory of the heterocrystalline state, in which inequivalent local unit-cell or amorphous order fields form an extensive architecture that cannot be reduced to one compatible cell orbit or to a macroscopic mixture. The fundamental field is a unit-cell rupture operator, , defined on the direct sum of connection-curvature, symmetry-orbit-rupture, and metric-jump spaces. Its thermodynamic-limit norm density is the structural order parameter. A positive , combined with a selected finite wavevector or persistent interface-network density, distinguishes the heterocrystalline state from crystals, quasicrystals, mesocrystals, amorphous solids, polycrystals, and conventional composites. A Landau–Brazovskii theory yields equilibrium and metastable transitions into modulated heterocrystalline phases. Symmetry-allowed response maps couple to pseudo-gauge potentials, topological wall networks, narrow bands, and quantum metric. We audit ten public material systems, identify five present exemplars and two conditional quantum cases, and predict six architectures with response-specific falsifiers. The framework promotes from an interfacial irregularity to an extensive state variable and supplies scale tests, phase-boundary signatures, and registered structure–spectroscopy experiments capable of rejecting the theory.