A 2-adic Density Lemma in the Odd Trajectory of the Collatz Map

A 2-adic Density Lemma in the Odd Trajectory of the Collatz Map Miguel Cerdá Bennassar November 2025 Abstract This paper describes the odd–even part of the classical Collatz function from a 2-adic perspective. First, it is shown that odd integers for which 3n+ 1 has exact 2-adic valuation ν2(3n+ 1) = rform an arithmetic progression which, when restricted to the odd integers, occurs with density 1/2r−1. This accounts for the pattern of increasing gaps observed in the graphical representation of the trajectories (Figure 1). Then the same 2-adic datum is used to introduce the even closure node A=3n+ 1 2ν2(3n+1)−1, namely the last even number before the orbit returns to an odd one. All odd integers that produce the same Aare grouped into 4-adic families FA. It is proved that the only possible even closure nodes are those of the form A≡2,10 (mod 12) and that, under the reduced dynamics T(A) = A/2, every family different from F2reaches immediately an odd number in F2={1,5,21,85, . . .}. Thus, in the reduced structure by families the cycle 4→2→1is unique and acts as a global attractor. Keywords: Collatz conjecture, 2-adic dynamics, 4-adic families, modular density, even closure node, 4–2–1cycle. 1 Introduction Figure 1shows a visual representation of the trajectory of odd numbers under the classical Collatz map, produced by the author. Each row contains the successive even values that appear in the even part of the trajectory, namely 3n+ 1,3n+ 1 2,3n+ 1 22, . . . , 3n+ 1 2r−1, up to the last even number before returning to an odd integer. The pattern of gaps in the table reflects a structure of decreasing densities governed by the power of two dividing 3n+ 1. Figure 1: Trajectory of odd integers nup to the last even term in the classical Collatz map. Each row shows the successive even values 3n+ 1,(3n+ 1)/2,(3n+ 1)/22, etc. 1 2 A 2-adic density lemma Lemma 2.1 (Decreasing 2-adic density).Let mbe an odd integer and let r=ν2(3m+ 1) be the largest exponent of 2dividing 3m+ 1. Then the odd integers msatisfying ν2(3m+ 1) = rform an arithmetic progression of modulus 2r+1, and their density among the odd integers is δr=1 2r−1. Density justification. The congruence 3m+ 1 ≡0 (mod 2r)has a unique solution m≡a (mod 2r)because gcd(3,2r)=1. This describes those integers mfor which ν2(3m+ 1) ≥r. To isolate those with exactly ν2(3m+ 1) = rwe consider the congruence modulo 2r+1. Write m=a+t2rwith t= 0,1and use 3a+ 1 = 2rcwith codd. Then 3m+ 1 = 3(a+t2r) + 1 = 3a+ 1 + 3t2r= 2r(c+ 3t). If t= 0, then c+ 3t=cis odd, hence ν2(3m+ 1) = r. If t= 1, then c+ 3 is even, hence ν2(3m+ 1) ≥r+ 1. Therefore the integers with ν2(3m+ 1) = rform exactly one residue class modulo 2r+1. Since the density of one class modulo 2r+1 in Zis 1/2r+1, and the density of the odd integers is 1/2, the relative density among the odd integers is 1/2r+1 1/2=1 2r−1. Corollary 2.2 (Spacing between successive occurrences).Let drbe the distance between two consecutive odd integers msatisfying ν2(3m+ 1) = r. Then dr= 2r−1−1. Proof. From the previous lemma, the integers satisfying the condition form an arithmetic progression with step 2r+1 in Z, i.e. with step 2rwhen we restrict to odd indices. Since only every second integer is odd, the effective spacing among odd numbers is 2r−1, and therefore the number of empty slots between two such odd numbers is 2r−1−1. Observation 2.3. The lemma shows that each lower row of the table represents a sparser and sparser set of odd integers. For finite levels rthe density is positive, but in the limit r→ ∞ it vanishes. This reveals the hierarchical and 2-adic nature of the Collatz dynamics: at each level of divisibility by 2, the density of odd integers reaching it is halved, which in the graphical representation appears as increasing spacing and an increasingly empty table. 3 4-adic reduction and compressed representation of odd columns The previous section showed that odd integers for which 3n+1 has 2-adic valuation rappear with density 1/2r−1among the odds. This explains why the lower rows of Figure 1are increasingly sparse. We now use the same 2-adic datum to group columns: if two odd integers, after exhausting all divisions by 2, produce the same last even number A, then they belong to the same 4-adic family. In this way, the vertical density information is translated into a horizontal organization by families. Let nbe an odd integer and let A=3n+ 1 2ν2(3n+1)−1 2 be the last even number in its odd Collatz trajectory, i.e. the even value appearing just before the next odd value. If 3n+ 1 = 2rq(qodd), then A= 2q, so that A/2=qis exactly the next odd value. In particular, every visible last even number is of the form A≡2 (mod 4), and in fact A≡2or 10 (mod 12). From this Awe can describe all odd integers that produce exactly that same last even number. The general condition they must satisfy is 3m+1=A·2s(s≥0), because dividing 3m+1 by 2s−1we obtain A, which is exactly the definition of last even number. For mto be an integer we need A2s≡1 (mod 3). Since 2≡ −1 (mod 3), we obtain two cases: •If A≡1 (mod 3) (i.e. A≡10 (mod 12)), then 2s≡1 (mod 3), hence smust be even. Writing s= 2kwe get m=A·22k−1 3=A·4k−1 3, k = 0,1,2, . . . •If A≡2 (mod 3) (i.e. A≡2 (mod 12)), then A·2s≡1 (mod 3) forces sto be odd. Writing s= 2k+ 1 we get m=A·22k+1 −1 3=2A·4k−1 3, k = 0,1,2, . . . This gives the following. Definition 3.1 (Family associated with a last even number).Let Abe a visible last even number in the odd Collatz trajectory, with A≡2,10 (mod 12). We define the family associated with A as FA=         A·4k−1 3:k≥0,if A≡10 (mod 12), 2A·4k−1 3:k≥0,if A≡2 (mod 12). All elements of FAgenerate the same even tail and therefore the same column in Figure 1. The Collatz dynamics can then be represented in compressed form on the last even numbers: to each even Awe apply the next Collatz step, namely A7→ A/2, and this odd number A/2 belongs to some other family FB. This yields the directed graph of Figure 2, where some low families have been drawn, e.g. F10 ={3,13,53,213, . . . },F14 ={9,37,149,...},F22 ={7,29,117,...}. All of them point to the family F2={1,5,21,85, . . . }. Proposition 3.2 (Disjoint partition into families).Let nbe an odd integer and let A=3n+ 1 2ν2(3n+1)−1 be its even closure node. Then nbelongs to a unique family FA, and two distinct families FA, FBwith A=Bhave no common elements. 3 Proof. By definition, the family FAconsists exactly of the odd integers msuch that 3m+1=A·2s for some s≥0. For a given odd m, the value of Aobtained by exhausting all divisions by 2in 3m+ 1 is unique, because the factorization 3m+ 1 = 2rqwith qodd is unique. Thus mcannot satisfy 3m+1=A·2sand 3m+ 1 = B·2t with A=B. Hence the families FAform a disjoint partition of the odd integers. Proposition 3.3 (Uniqueness of the cycle 4,2,1in the reduced structure).Consider the directed graph Gwhose nodes are the families FAdefined above from the last even number A=3n+ 1 2ν2(3n+1)−1, and where we draw an arrow FA−→ FB if, after applying to an odd element of FAthe step 3x+ 1 and dividing by 2until the last even is reached, the obtained odd belongs to FB. Then the only family that admits a self-loop FA−→ FA is the family F2={1,5,21,85, . . .}, corresponding to the classical cycle 4→2→1. In particular, there are no directed cycles in Gformed by distinct families. Proof. If A= 2, the next step is 27→ 1, and 1belongs to the same family F2, so we obtain the self-loop F2→ F2and the cycle 4→2→1. If A > 2, since Ais the last even, we have A= 2qwith qodd, and the next step is A7→ q. But qdoes not belong to FA, it belongs to another family determined by that odd. Therefore from FAwe always exit towards a different family. This prevents the existence of cycles with several families. Proposition 3.4 (Modular classes of last even numbers and convergence to F2).Let Abe the last even number of an odd Collatz trajectory, i.e. A=3n+ 1 2ν2(3n+1)−1. Then necessarily A≡2or 10 (mod 12). Moreover, in both cases the reduced map T(A) = A/2produces, in one step, an odd integer that belongs to the family F2={1,5,21,85, . . .}. Proof. If nis odd, 3n+ 1 is even and, modulo 12, only residues 3n+ 1 ≡4or 10 (mod 12) can appear, because the odd residues modulo 12 are 1,3,5,7,9,11 and 3n+ 1 takes the values 4,10,4,10,4,10 respectively. In both cases the last even number is of the form A=3n+ 1 2r−1≡2or 10 (mod 12). If A≡2 (mod 12), the next odd after dividing by 2is A/2≡1 (mod 6), and if A≡10 (mod 12), then A/2≡5 (mod 6). In both cases the resulting odd belongs to the sequence {1,5,21,85,...} that characterizes the family F2. Hence every family FAwith A>2reaches, after a finite number of iterations of the reduced rule, the family F2. 4 Corollary 3.5 (Global attractor of the reduced structure).In the reduced structure of families FAdefined from the last even number of the Collatz map, the cycle 4→2→1is the only closed loop and acts as a global attractor. All families FAwith A>2form a countably infinite set of nodes converging to the family F2, the only one with a self-loop. 2 10 14 22 FA . . . F2: (1,5,21,85, . . .) F10 : (3,13,53,213, . . .) F14 : (9,37,149, . . .) F22 : (7,29,117, . . .) Remaining families (A≡2,10(mod 12)) Figure 2: Reduced structure by 4-adic families. All families FAwith A≡2,10 (mod 12) converge to the family F2, the only one with a self-loop. Comment (Final note).The reduced structure by 4-adic families provides a compact view of the odd–even Collatz dynamics. Its hierarchical organisation and the central role of the family (2) reveal a system whose modular structure is completely determined: every individual trajectory is mirrored in a unique route of convergence. This correspondence between arithmetic level and functional structure synthesises the internal logic of the iterative process. Conclusion This study does not address the full Collatz conjecture, but it does provide an alternative structural framework to describe its odd–even dynamics. From the 2-adic density lemma we obtain a vertical hierarchy of divisibility levels, while the 4-adic reduction organises trajectories into horizontal families defined by their even closure node. Both perspectives converge into a compact representation where all families reduce to the attractor F2, corresponding to the cycle 4→2→1. This approach reveals the 2-adic and modular nature of Collatz and offers an analytic viewpoint to reinterpret its global behaviour through a finite and deterministic structure. Appendix A. Connection with the structure theorem for (d, g, h)- maps A.1. The general framework of Kontorovich–Sinai In 2006, Kontorovich and Sinai introduced a general framework for Collatz-type transformations, called (d, g, h)-maps, defined by T(x) = gx +h(gx) dνd(gx+h(gx)) , 5 where d, g ∈Nand h:Z→Zis periodic modulo d. The exponent νd(y)denotes the highest power of ddividing y. This formalism encompasses a wide class of affine transformations which, after a multiplication and a translation, are repeatedly reduced by divisibility. The main goal of the structure theorem is to describe orbits in terms of congruences modulo dg and of the sequences (k1, k2, . . . , km)specifying the powers of ddivided out at each step. A.2. The classical Collatz map as a particular case The classical Collatz map is obtained with parameters d= 2,g= 3 and h(x)=1, giving T(n) = 3n+ 1 2ν2(3n+1) . This is precisely the map used as the base of the present work, where the parameter ν2(3n+ 1) determines the length of the even segment and, consequently, the even closure node A= (3n+ 1)/2ν2(3n+1)−1. A.3. Correspondence between drift and local expansion factor Kontorovich and Sinai introduce a mean parameter called the drift, µ= log g−d d−1log d, which measures the average tendency of the orbit to expand or contract. For (d, g) = (2,3) one gets µ<0, implying average contraction. In the present structural framework, the local expansion of an odd–even segment is G= 3j2−(n−j), where jis the number of odd steps and n−jthe number of divisions by 2. The condition log G<0also characterises the contraction of the trajectory. Both expressions quantify the same balance between expansion by 3and reduction by powers of 2. A.4. Modular structure and family classification The structure theorem of Kontorovich–Sinai classifies orbits into congruence classes modulo dg and parametrises them by the sequences of exponents νd. Likewise, the present theory organises trajectories into 4-adic families FAdefined by their even closure node, and into modular zones according to residue conditions. The difference lies in the viewpoint: whereas Kontorovich and Sinai work in a measure-theoretic setting, here we build a discrete, completely deterministic hierarchical structure. A.5. Unified dynamical interpretation In both models, global contraction appears in analogous form: negative drift implies statistical attraction to a finite domain, and the decreasing 2-adic density explains the hierarchical convergence of all families toward the attractor F2. The reduced map T(A) = A/2plays, in this deterministic setting, the same role as the logarithmic average in the stochastic model: an iterative dissipation mechanism concentrating the orbits. A.6. Conclusion The (d, g, h)model provides a general and probabilistic formulation of Collatz-type maps. The present approach can be seen as its 2-adic structural counterpart: a discrete realisation in which congruence classes materialise as hierarchical families and finite graphs. In this way, the structure theorem admits an explicit translation into modular and 4-adic terms, where the dynamics becomes visible as a finite system of connections ending in the cycle 4→2→1. 6 Appendix B. Transition from classes 4n+3 to 4n+1 in the simplified dynamics B.1. Motivation An essential empirical property of the simplified Collatz dynamics T(n) = (n/2, n even, (3n+ 1)/2, n odd, is that every odd class n≡3 (mod 4) eventually reaches a number in the class n≡1 (mod 4). This guarantees that trajectories cannot remain forever in the subclass 4n+ 3, which in the structural model means that no 4-adic family can form an autonomous cycle made only of 4n+3 odd integers. B.2. Modular analysis of the transition Let n= 4k+ 3. Applying the simplified Collatz map we get T(4k+ 3) = 3(4k+3)+1 2=12k+ 10 2= 6k+ 5. Observe that 6k+ 5 ≡1 (mod 4) if kis even,6k+ 5 ≡3 (mod 4) if kis odd. Thus the residue modulo 4depends on the parity of k: only those with even kleave the class 4n+ 3 in one step. B.3. Persistence and decrease of the internal parameter If kis odd, the new value 6k+ 5 still belongs to the class 4n+ 3. We can write k= 2t+ 1, hence T(4(2t+ 1) + 3) = T(8t+ 7) = 6(2t+ 1) + 5 = 12t+ 11. The next step is T(12t+ 11) = 3(12t+ 11) + 1 2=36t+ 34 2= 18t+ 17. Reducing modulo 4we obtain 18t+ 17 ≡2t+ 1 (mod 4), so that 18t+ 17 ≡(1 (mod 4), t even, 3 (mod 4), t odd. Thus, if tis even, the sequence enters the class 4n+ 1 in this second step, and if tis odd it remains one step more in the class 4n+ 3, but with a smaller internal parameter (since t<k). At each persistence in the class 4n+ 3 this parameter is roughly halved, therefore the process cannot continue indefinitely: after a finite number of iterations the trajectory necessarily reaches the class 4n+ 1. 7 B.4. Structural consequence This can be interpreted in terms of an internal parameter measuring the distance to the next number of type 4n+1. Each time we apply T(n)to an element of 4n+3, that parameter strictly decreases, so the process cannot stay in that class forever. In the context of 4-adic families FA, this transition implies that any family whose generators are of type 4n+ 3 will, after finitely many steps, reach a family generated by a 4n+ 1, i.e. a family whose even closure node satisfies A≡2,10 (mod 12). Corollary .6 (Finite permanence of the class 4n+ 3).Under the simplified Collatz map T(n) = (3n+ 1)/2, every trajectory starting at an odd integer n≡3 (mod 4) enters the class n≡1 (mod 4) in a finite number of steps. Consequently, the classes 4n+ 3 cannot form autonomous cycles or closed subdynamics. B.5. Interpretation in the family model In the family model FA, the class 4n+ 3 corresponds to the upper members of a generation feeding the even node A. The forced transition 4n+ 3 →4n+ 1 ensures that each family has a single direction of descent, i.e. a unique “exit” leading to the next segment. Combined with the uniqueness of the self-loop in Section 3, this property completes the convergence argument: all descending trajectories end up in the family F2, associated with the cycle 4→2→1. References •Terras, R. A stopping time problem on the positive integers. Acta Arithmetica 30 (1976), 241–252. •Everett, C. J. (1977). 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A 2-adic Density Lemma in the Odd Trajectory of the Collatz Map. Unpublished manuscript, Mallorca. 8