Carry Boundaries and Bernoulli Spectra in Long Division

A finite digit rule on a fixed prefix determines an exact carry table after its bulk and terminal endpoint are removed. Primitive odd character coefficients factor into the classical Bernoulli response of the sawtooth and the signed boundary flux. Ordinary fluxes determine the unit boundary; induced carry transforms use conductor-depth weights at prime powers. The terminal collision boundary satisfies an exact prime-power mean law. At base 15, a signed divisor ledger changes the mean from 60 to 28. Parseval gives the finite energy decomposition, with unit-index correlations expressed through Dedekind sums. Originally published May 2026; revised September 11, 2026.