The Cubic Law for Digit-Collision Energy

Abstract

For an odd prime base b, the lag-one digit-collision energy is a weighted Dedekind-sum average over the prime-square carrier grid. This paper proves the unconditional cubic law E_b = b^3 + O(b^2 (log b)^2), with the leading cubic split exactly into a diagonal contribution b^3/3 and an off-diagonal contribution 2b^3/3. The proof uses Rademacher reciprocity, a coprime Euler sum, and a first-moment estimate for Dedekind sums. Determining the next coefficient is a separate analytic problem. Revised August 2026.