Intrinsic Crystallographic Organization and Projected Crystal Habit in Coherence Geometry

This paper develops a coherence-geometric representation of crystalline form in which intrinsic crystallographic organization, interfacial habit, growth realization, and observable morphology are treated as distinct but coupled geometric layers. Crystallographic organization is encoded through the Gram geometry of phase-gradient fields, while habit selection, environmental weighting, and observable projection are introduced separately.Two-dimensional fourfold and sixfold constructions recover square and hexagonal habits while rotating covariantly with their internal crystal frames rather than with the Cartesian grid. In three dimensions, a single cubic Gram geometry supports distinct {100}, {110}, and {111} realizations, yielding cube, rhombic-dodecahedral, and octahedral habits. This establishes that crystallographic class does not determine a unique macroscopic habit.A tetrahedral realization further separates relational symmetry from polarity: a balanced rank-three Gram organization supports a signed non-centrosymmetric interfacial realization with exact vertex-to-facet and polarity ratios of 3. A fixed-crystal environmental sweep then shows that changing only a prescribed environmental axis redistributes growth among the same admissible cubic directions, while a body-diagonal null orientation preserves cubic growth. Together the results support the layered architecture \(Q_G \rightarrow \Gamma_G \rightarrow M_G(E) \rightarrow R \rightarrow \Sigma\), separating intrinsic crystallographic organization, habit selection, environment-dependent growth realization, projected crystalline amplitude, and observable morphology. The work is intended as a structural factorization rather than a replacement for Wulff thermodynamics, surface kinetics, transport theory, or phase-field modeling. Dynamic transport, interface–environment feedback, morphological instability, and dendritic realization are left as subsequent extensions.CGI Internal ID: CGI-RSR-000044