Empty Regions of Digit Polygons: Exact Geometry and Arithmetic Information

A periodic digit expansion draws a polygonal walk inside a fixed square. Its empty regions retain information that total area alone cannot reveal: how the gaps are divided, where they lie, and how their geometry changes with the arithmetic. Empty Regions of Digit Polygons: Exact Geometry and Arithmetic Information develops an analytical account of these connections. For the complete residue chains modulo b² + b − 1 and b² − b − 1, the paper determines every empty component in every admissible base. Interval-covering arguments exclude interior holes and yield exact harmonic area laws. Moments of the boundary gaps recover generalized harmonic numbers of every order. Adjacent inverse bases traverse the same residue cycles in opposite directions, yet change the empty area by an explicit harmonic increment. Individual orbits admit a separate constructive classification. A local four-digit inequality proves that every orbit in either family has nonpositive winding and decomposes its filled region into at most twice its period many rational triangles. Exact crossing formulas determine the complementary components and their areas. Fibonacci and Lucas identities restrict the possible reduced denominators, producing finitely many affine polygon templates for each fixed period. Every period-four orbit is a square. For the first quadratic family, normalized winding multiplicities converge to a continuous profile with an explicit formula in radicals and logarithms throughout the square. Matching-order bounds establish the optimal uniform convergence rate, proportional to the reciprocal of the base. The profile’s central value involves the golden ratio. The paper also identifies the limits of geometric recovery. Distinct denominators can have identical empty sets, while empty locations provide quantitative certificates of uneven residue distribution. A neighboring quadratic family exhibits positive winding, showing why the hypotheses matter. Compact component-count formulas across unrestricted periods, the best convergence constant, and a first-order correction profile remain open. Analytical proofs are accompanied by exact verification, explanatory figures, editable source, calculation code, and reproducible data.