Carries, Rails and Resonance: the Structure of the Digit-Sum Parity Map

Let s(n) be the decimal digit sum of n and let f(n) = n + s(n) for odd n and f(n) = n - s(n) for even n. We prove that an odd a lies on a 2-cycle of f if and only if adding s(a) to a produces exactly s(s(a))/9 carries (with s(a) odd), and that this number is within 1/9 of the mean number of carries, a "carry balance" identity. Using exchangeability, anti-concentration for carries and a local limit argument, we show that the number of 2-cycles below N has exact order N/sqrt(log log N). We then find an exact formula for the variance of the carry count, Var C = (11/9) sum_i q_i(1-q_i), prove a sharp local limit theorem for carries, and an extremal bound with constant 30/121. Together these give an oscillation law: sqrt(log_10 D) T(D)/10^D has no limit, its limit points fill [K_0 sqrt(726/301), K_0 sqrt(6)] with K_0 = 1/(12 sqrt(11 pi)), it tends to K_0 sqrt(3) for almost all D, and the peak-to-trough ratio is sqrt(301)/11. We also establish exact structure: self-similarity on multiples of nine, a level equation, a conditional time-reversal identity, and prefix lifting, a cycle count with a rigorous error bound. We show that an orbit is eventually periodic if and only if its digit sums are bounded, bound constant-level runs by O(k^2), and give a computer-assisted proof that every orbit starting below 10^341 (below 2^3024 in base 2) is eventually periodic; the general case is reduced to a conjecture on finite auxiliary maps. In odd bases every orbit reaches 0; in every even base the 2-cycle density has exact order (log D)^(-1/2). Code and data: doi:10.5281/zenodo.23938785.