Periodic Carry Boundaries and Orbit Energies in Long Division

A finite character-sum treatment of the periodic side of long division. The paper develops signedboundary flux for periodic remainder orbits, proves a conductor-energy decomposition for oddcharacter moments, establishes a full-orbit prime-power energy law, gives an index-two mod 8orbit-energy formula, and reduces higher-index subgroup orbit boundaries to explicit Jacobi-typecharacter sums.