Alexander S. Petty

Why the Golden Ratio Selects the Prime Three

For denominators n = 3m with m = 2a5c and nonnegative integers a and c, the proper fractions split into three residue classes. Counting terminating fractions together with those whose repetend matches 1/n gives alignment (2m - 1)/(3m - 1). This exceeds 1/φ, where φ = (1 + √5)/2,...

The Cubic Law for Digit-Collision Energy

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 a...