Alexander S. Petty

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...

Why the Golden Ratio Selects the Prime Three

Fix a base b, a prime p not dividing b with b congruent to 1 mod p, and a positive integer m whose prime factors all divide b. The repetend alignment satisfies alpha(pm) = (2m-1)/(pm-1). Among odd primes, only p = 3 admits alignment exceeding 1/phi, with threshold m >= phi^2. Any...