Alexander S. Petty

Repetend Rigidity at the Golden Scale

A condition at the golden scale can force a repeating tail to have period one. We establish this for a denominator statistic that counts cyclic shifts of a reference tail, allowing arbitrary coprime integer bases. Let b,q >= 2 be coprime, put h = ord_q(b), and let the positive in...

Carry Boundaries and Bernoulli Spectra in Long Division

A finite character-sum treatment of long-division digit-cylinder observables, proving a carry-boundary reduction, a primitive odd Bernoulli/sawtooth factorization, a visible-boundary theorem,and an exact prime-power second-moment identity with a composite proper-conductor obstruc...

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...