Additive Boundaries and Conductor Energies of Periodic Remainder Orbits

A finite character-sum treatment of periodic long division. Signed additive boundaries of digit regions and multiplicative remainder orbits decompose into nonnegative exact-conductor energies. The results include the complete base-15 conductor spectrum, prime-level support for full unit orbits at odd prime powers, an index-two off-resonance formula depending on the prime modulo 8, and a shifted Jacobi-sum expansion for higher-index prime orbits. Originally published May 2026; revised September 11, 2026.