Repetend Rigidity at the Golden Scale
Year: 2026
Type: preprint
Abstract: 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 integer m divide a power of b. Among the rows k/(qm) with 1 <= k <= qm-1, give cyclic shifts of the 1/(qm) tail weight one and terminating rows weight lambda in [0,1]. This produces the score S_(b,q,lambda)(m) = [hm+lambda(m-1)]/(qm-1). Write tau = (sqrt(5)-1)/2. For rational lambda and integers r >= 1, its real interpolant satisfies S-tilde_(b,q,lambda)(tau^(-r)) = tau exactly when h = 1 and (q,lambda,r) is in {(2,0,2),(3,1,2)}. The exclusion combines classical Fibonacci coordinates with the absence of unit orders in [q/2,2q/3) for q >= 5. Thus an equality at an irrational scale constrains the actual modular period, and the one-digit condition follows from the classification. For the one-digit profile, a unique admissible real weight exists exactly when q is in {2,3} and r >= 2; every later-power weight is irrational.
Author: Petty, Alexander S.
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7208816447
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