The Arithmetic of Digit Polygons — Exact Addition, Spectral Orthogonality, and Carry Identities

This paper develops a rigorous addition theory for digit polygons, building on the companion result that the signed area of the digit polygon of 1/p is exactly a negative semidefinite Dirichlet energy. Given two primes p and q coprime to 10, the two digit blocks are extended to a common period L equal to the least common multiple of their individual periods and added in two distinct layers: carry-free superposition and base-10 normalization. For the carry-free sum, an exact polarization identity is proved expressing the area of the sum in terms of the two individual areas and a symmetric interaction form. The interaction form is then characterized spectrally: it vanishes identically on a pair of period subspaces if and only if the gcd of those periods is at most 2, a sharp criterion proved via explicit Fourier support analysis. For the base-10 normalization step, which is nonlinear due to carry propagation, an exact carry-defect identity is derived expressing the area change as the difference of two Dirichlet energies. Complete lag-correlation formulas are given for the self-area of the carry correction vector and for its interaction with the carry-free sum. An exact one-step recursion is established for iterated sums of reciprocal primes. Several stronger claims present in earlier drafts — a universal sign for the carry defect, a universal asymptotic attractor, and Mertens-type equilibration — are explicitly identified as unproved and listed as open problems. All results in the paper are proved in full.