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

Abstract

Let $\ddot{I}„(n)$ denote the number of positive divisors of $n$. Starting from $n_0=x$, consider the orbit $n_{j+1}=n_j-\ddot{I}„(n_j)$, and let $a(x)$ be its hitting time of zero. Although the average order of $\ddot{I}„$ 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 ${\sum }_{j<a(x)}\ddot{I}„(n_j)=x$ 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 $a(x)\asymp x/\log x$.

Major revision, 14 pages. Corrected the endpoint and displacement identity; added unconditional bounds and a parity constraint; clarified the limitations of the phase-rigidity approach; and reformulated the structural argument as an explicit two-hypothesis conditional framework. Code reproducing the computations through 10^7 is included