An A-B-C Coordinate Encoding for Exact Boundary-Cycle Extraction on Hexagonal Lattices

Source abstract

We study an integer-coordinate encoding of boundary incidences for finite subsets of the regular hexagonal tiling. Each selected cell contributes six ordered records with endpoints in two role-distinguished classes, point and tail. Identical records from neighboring cells cancel, and the surviving records form a disjoint union of cycles, including the exterior boundaries of disconnected components and the boundaries of holes. We derive the actual even- and odd-row record formulas from the A--B--C transforms used in Hexagones 2.0, prove exact shared-edge identities and the global role-qualified incidence structure, and thereby obtain an unconditional correctness theorem for the abstract coordinate construction. We compare the encoding with an independent exact cube-coordinate half-edge hashing implementation, proving an incidence isomorphism and checking 504 finite regions. We give an expected-linear-time indexed algorithm and distinguish it from the historical fixed-capacity C traversal, whose sentinel and closure conventions do not inherit the theorem automatically. The claim is a compact coordinate realization of a known planar boundary principle, not historical priority or a new three-axis lattice.