The Subtractive Divisor Orbit: Unconditional Bounds, Parity Constraints, and a Conditional Framework

Abstract

Let $\tau(n)$ denote the number of positive divisors of $n$. Starting from $n_0=x$, consider the orbit $n_{j+1}=n_j-\tau(n_j)$, and let $a(x)$ be its hitting time of zero. Although the average order of $\tau$ suggests $a(x)\asymp x/\log x$, the orbit samples the divisor function endogenously, and no unconditional estimate of this order is known to us. We prove the exact identity $\sum_{j<a(x)}\tau(n_j)=x$ and the unconditional bounds $$ \frac{x}{(\log(2x))^3}\ll a(x)\le \frac{3x}{8}+O\!\left(\frac{\sqrt{x}}{\log x}\right). $$ We also show that the orbit changes parity exactly at square states. On dyadic orbit segments, we establish a local-to-global criterion, a large-value truncation, and a quantitative implication from small relative variance to a step-mass-saturating dynamic near-ladder. Finally, under two explicit hypotheses -- a regularity-or-ladder dichotomy and an anti-ladder estimate -- we obtain $a(x)\asymp x/\log x$.