Primitive-cell-resolved Crystallography for Moir\'{e} Bilayers from Imaging

Accurate geometric decoding of moir\'{e} bilayers from imaging is essential for engineering quantum systems. Existing schemes, limited by identity or aligned assumptions requiring diagonal beating-to-moir\'e transformations, do not apply to general non-aligned geometries and become underdetermined when buried layers are unresolved. We establish a primitive-cell-resolved moir\'{e} crystallography framework that treats the beating-to-moir\'{e} relation in full generality and introduces a complete descriptor set $\{\theta_r,\boldsymbol{\varepsilon},(T_{Mt},T_{Mb}),N_B\}$, where the integer moir\'{e}--layer matrices $(T_{Mt},T_{Mb})$ and the beating number $N_B$ determine the commensurate unit cell. A hybrid analytical--numerical workflow reconstructs buried-layer lattices, solves Diophantine constraints to obtain $(T_{Mt},T_{Mb})$ and $N_B$, and extracts $(\theta_r,\varepsilon_b,\theta_u,\varepsilon_u)$ with Poisson effects and tensile/compressive branches treated on equal footing. Reanalyzing twisted bilayer graphene, we identify a $N_B=3$ primitive cell rather than a $N_B=9$ aligned supercell, reducing the atomistic basis threefold and correcting the moir\'{e} Brillouin-zone construction. The framework provides a crystallographically consistent route from imaging to primitive-cell-resolved atomistic and many-body models.