An invariant for networks generated by Collatz-type algorithms

An invariant for networks generated by Collatz-type algorithms Mikhail Bokiy 14.10.2025 Abstract. For an extended class of Collatz-type algorithms, a “network divisibility” invariant was found using a constructive-topological approach. Its main property is that the divisibility of the entire network is decomposed into a product of the divisibilities of its constituent parts raised to powers equal to the natural densities of the network's constituent parts. The invariant is directly related to the convergence of algorithms. Contents 1. Introduction 2. Network divisibility 3. Conclusion 1. Introduction This article continues the approach to the Collatz conjecture proposed previously [1, 2]. In the first article, the problem was conceptualized as a task of the structure of the network generated by Collatz-type algorithms. The second article proposes using geometric mean properties of network structures to study the structure of the corresponding networks. This article refines the definition and reveals the essence of the “network divisibility” invariant discovered in this approach. For the sake of consistency between the articles, we will adhere to the definitions and designations already used earlier. The definition of the Collatz algorithm for positive integers is standard: 3n+1 if n is odd and n/2 if n is even. For the original algorithm, we use the mnemonic notation “3n+1/2” (a reminder of the order of operations) or shorter “3n+1”. The class of Collatz-type algorithms includes algorithms of the general form “an+b/c” (a similar mnemonic notation), where a and c are positive relatively prime numbers, and b is an integer. The Table with keys and pointers (see Fig. 1) refers to a table of expansions of natural numbers in powers of 2 with “keys” (the column of all odd numbers n) and “pointers” (even numbers of the form 3n+1 in rows). Similarly, tables of expansion of all numbers in powers of any base c = 3, 4, 5… are constructed. The Table with keys and pointers helps to topologically represent and explore the network defined by the algorithm. In the case 3n+1, the network topology is a rooted tree. The root is the key to which the algorithm converges. A cycle is a set of several keys, a cyclic variant of the root. In the general case of algorithms of the form an+b/c, the network may consist of several “rooted trees” and a “rootless subnet.” The reason we insist on our own terminology (network, keys, pointers) is that it is more adequate for the object under study. 1 Fig. 1 2. Network divisibility Article [2] introduced an important property — “the divisibility Δ averaged over the entire network, calculated as the geometric mean of the divisors 2p over all even pointers of the network.” Let us define Δ more thoroughly. A well-known analogue is the average divisibility by 2p over the entire series of even natural numbers 2, 4, 6, 8…, which is simply calculated through the sum of the series 2(1+1/2+1/4+1/8+…) = 22 = 4. We prefer to calculate it using the geometric mean over a series of divisors 21, 22, 21, 23…, which gives the same result in the limit. For the simplest algorithm 1n+1/2, if the pointers of its Table are arranged in ascending order, we obtain the same series of all even numbers and the same limit divisibility Δ = 4. It is logical to extend this definition to all algorithms of the form an+b/2. A specific algorithm has its own pointer placement scheme and its own order of divisors. But, as numerical verification on different algorithms has shown, in all cases, the divisibility of the entire network tends to 4 in the limit. A typical change in Δ with an increase in the number of pointers taken into account is shown in Fig. 2. Fig. 2 Such a definition of network divisibility Δ can naturally be extended to the entire class of Collatz-type algorithms of the form an+b/c. This gives rise to a general 2 formula for network divisibility Δ(с) = cc/(c−1), which is easily derived by summing the corresponding series, similar to the case c = 2. A selective numerical check for algorithms with base c = 3, 4, 5 confirms the tendency to converge to the correct value. A formal proof of this statement is routine and of no interest. Definition of network divisibility for Collatz-type algorithms: network divisibility Δ(с) = cc/(c−1) is the limit value of the geometric mean over a series of divisors сp extracted from the pointer scheme of any algorithm of the form an+b/c after sorting the pointers in ascending order. The dependence of Δ only on the parameter c allows us to call network divisibility an invariant for a subset of algorithms with the same base. But this characteristic has another, even more significant property. Let's take an arbitrary Collatz-type algorithm and write down the geometric mean over a partial series of divisors ei from its pointer scheme: Dk = (e1e2e3… ek)1/k, where k is the number of terms in the series. Let us imagine that the network corresponding to this algorithm consists of two parts, for example, a rooted tree and a rootless subnet. Any key and any pointer belongs either to the rooted tree or to the rootless subnet. We group the pointers according to their belonging, count them separately, and start manipulating their divisors fi, gi: Dk = (f1f2f3… fm × g1g2g3… gn)1/k, where m+n = k, Dk = ((f1f2f3… fm)m/m × (g1g2g3… gn)n/n)1/k, Dk = (f1f2f3… fm)m/mk × (g1g2g3… gn)n/nk, Dk = ((f1f2f3… fm)1/m)m/k × ((g1g2g3… gn)1/n)n/k, Dk = (Dm)m/k × (Dn)n/k, where Dm and Dn are the geometric means of the parts. Passing to the limit k→∞, we obtain the main formula of the invariant: Δ(с) = (Δroot)ωroot × (Δunroot)ωunroot = cc/(c−1), where ωroot + ωunroot = 1 is the sum of the limiting shares of all network pointers (which is equivalent to the limiting shares of keys) belonging to the rooted tree and the rootless subnet, respectively. 2 equations with 4 unknowns. Thus, the divisibility of the entire network Δ(c) is decomposed into the product of the divisibilities of the rooted tree Δroot and the rootless subnet Δunroot, raised to powers equal to the natural densities of the network's constituent parts. This conclusion easily generalizes to networks of any composition. This is an invariant for networks generated by Collatz-type algorithms. First considerations in connection with the found invariant. Under the assumption of the coexistence of a rooted tree and an rootless subnet, the inequality Δunroot < Δ(с) < Δroot is obviously true. In addition, the inequality c < Δunroot < a+ is always true, where the lower bound is related to minimal divisibility, and the upper bound is caused by the nature of the rootless subnet [2]. Finding an upper bound for Δroot is difficult. The well-known result of Tao [3] corresponds to the limit ωroot→1, Δroot→4 when c = 2. Based on the invariant, a first prediction can be made. A complete rootless network (its condition ωunroot = 1, Δunroot = Δ(с)) cannot exist when a < Δ(с), and corresponding algorithms of the form an+b/c must have at least one root. This can be verified on previously unexplored algorithms with base c = 3 or more. 3 3. Conclusion The discovery of the invariant confirms the productivity of the constructivetopological approach to the 3n+1 problem and similar algorithms. It also provides additional evidence in favor of the network conceptualization of the task. It turns out that not only numerical sequences are connected in network structures, but also topologically separate rooted trees and rootless subnets are not autonomous, since they share a common limited resource of network divisibility. References [1] Bokiy, M. (2024). A new inherent approach to solving the Collatz 3n+1 problem and its analogues. https://www.academia.edu/124810891 (https://doi.org/10.5281/zenodo.14632983) [2] Bokiy, M. (2025). A distinct proof and extension of the Collatz conjecture. https://www.academia.edu/144160827 (https://doi.org/10.5281/zenodo.17210722) [3] Tao, T. (2019). Almost all orbits of the Collatz map attain almost bounded values. https://arxiv.org/abs/1909.03562 Email address: [email protected] 4