Behzad Nikzad

math.NT — Number Theory

On Bounds for Reciprocal Sums and Residues in Collatz Orbits, and Their Consequences

Contributed by Behzad Nikzad, Karl Frinkle, Fatemeh Bagherzadeh

We prove uniform bounds for reciprocal sums and residue products along Collatz paths. Let $\mathcal F$ be a finite nonrepeating shortcut path whose input states are greater than $1$, let $\mathcal R_{\mathcal F}$ be the sum of the reciprocals of its odd inputs, and let $\rho_{\mathcal F}$ be its finite-path residue. Then $\mathcal R_{\mathcal F}<1, \qquad \rho_{\mathcal F}<\frac{4}{3}.$ For the first-arrival trajectory of a terminating positive integer $n$, these inequalities give $\mathcal R(n)<1, \qquad \rho(n)=\frac{2^{E(n)}}{3^{O(n)}n}<\frac{4}{3},$ where $E(n)$ and $O(n)$ are the division and odd-operation counts. More generally, when each odd value greater than $1$ is counted once, the corresponding reciprocal-sum and residue bounds hold for every forward orbit, without a termination assumption. The proof combines verified finite height-delay records and residue bounds, canonical-block reciprocal estimates, and an exact finite count organized by gap brackets. Gap brackets, closely related to parity vectors, record the divisions by two between successive odd inputs and organize the finite counting argument; together with a related coordinate representation, they also yield exact formulas for reconstructing finite trajectories. After taking base-$2$ logarithms, the bound $\rho_{\mathcal F}<4/3$ leaves an uncertainty of less than $1/2$ in the relation between the even- and odd-operation counts. Since these counts are integers, either count is determined exactly by the other count and the path endpoints. We obtain nearest-integer restrictions on the changes in $E$ and $O$ between adjacent integers, apart from explicit small exceptions, and prove that adjacent terminating integers of equal height have identical operation counts. Thus every equal-height streak lies in a single operation-count class. We further introduce the charge $\lambda=E-2O$ and associated height and charge slice functions to study the persistence and eventual break of equal-height streaks. We distinguish class records, which are the least starting integers attaining specified values, from delay records, which attain new extrema relative to all smaller starting integers. The resulting record theory shows that $\lambda$-delay records form a subsequence of the odd-count delay records, which form a subsequence of the common division-count and height-delay list. On the class side, the nonpositive charge-class records lie among the odd-count class records, which lie among the division-count class records, which in turn lie among the height-class records. The corresponding hierarchies also hold for first-arrival paths to a common endpoint $z$ when $z=1$ or when $z>2$ is nonperiodic under the shortcut map.