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 the odd integers satisfying ν2(3n+ 1) = rform an arithmetic progression that, when restricted to the odd integers, appears with density 1/2r−1. This explains the pattern of increasing gaps observed in the graphical representation of the trajectories (Figure 1). The same 2-adic information is then used to introduce the even closure node A=3n+ 1 2ν2(3n+1)−1, which represents the last even number before returning to an odd value. All odd numbers that produce the same Aare grouped into 4-adic families FA. It is proved that the only possible even closure nodes satisfy A≡2,10 (mod 12), and that under the reduced dynamics T(A) = A/2, every family distinct from F2leads immediately to an odd number belonging to F2={1,5,21,85, . . .}. Hence, in the reduced family structure, the cycle 4→2→1is unique and acts as a global attractor. This framework establishes an explicit correspondence with the Structure Theorem for (d, g, h)-maps by Kontorovich and Sinai [0], in which the decreasing 2-adic density plays the same analytical role as the negative drift in their probabilistic formulation. 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 values appearing in the even part of the trajectory, that is, the terms of the form 3n+ 1,3n+ 1 2,3n+ 1 22, . . . , 3n+ 1 2r−1, until reaching the last even number before returning to an odd one. The pattern of visible gaps in the table reflects a structure of decreasing densities governed by the power of two dividing 3n+ 1. 1 Figure 1: Trajectory of odd numbers nup to the last even value in the classical Collatz map. Each row represents the even numbers obtained successively from 3n+1,(3n+1)/2,(3n+1)/22, etc. 2 2-adic Density Lemma Lemma 2.1 (Decreasing 2-adic density).Let mbe an odd integer and r=ν2(3m+ 1) the exponent of the highest power of 2dividing 3m+ 1. Then the odd integers msatisfying ν2(3m+ 1)=rform an arithmetic progression modulo 2r+1, and their density within the set of odd numbers is δr=1 2r−1. Proof of the density. The congruence 3m+ 1 ≡0 (mod 2r)has a unique solution m≡a (mod 2r)because gcd(3,2r)=1. This describes the integers msuch that ν2(3m+ 1) ≥r. To isolate those satisfying exactly ν2(3m+ 1) = r, we consider the congruence modulo 2r+1. Writing m=a+t2rwith t= 0,1and using 3a+ 1 = 2rcwith codd, we have 3m+ 1 = 2r(c+ 3t). If t= 0, then c+3tis odd and hence ν2(3m+1) = r. If t= 1, then c+3 is even and consequently ν2(3m+ 1) ≥r+ 1. Therefore, the mwith ν2(3m+ 1) = rform exactly one class modulo 2r+1. Since within the integers the density of one class modulo 2r+1 is 1/2r+1, and the density of odd integers is 1/2, the relative density within the odds is 1/2r+1 1/2=1 2r−1. Corollary 2.2 (Spacing between successive occurrences).Let drdenote the distance between two consecutive odd numbers satisfying ν2(3m+ 1) = r. Then dr= 2r−1−1. Proof. From the previous Lemma, these odd numbers form an arithmetic progression with step 2r. Since among the integers only half are odd, the effective distance between consecutive terms within the odd numbers is 2r−1. Hence, the number of gaps between two consecutive valid values is dr= 2r−1−1. Observation 2.1. The lemma shows that each lower row of the table represents a progressively rarer subset of odd numbers. For finite levels r, the density is positive, but in the limit r→ ∞ it vanishes. This behaviour reveals the hierarchical and 2-adic nature of the Collatz dynamics: at each level of divisibility by 2, the density of odds reaching that level is halved, which in the graphical representation appears as increasing spacing and a progressively emptier table. 2 3 4-adic Reduction and Compressed Representation of Odd Columns The previous section showed that the odds whose 3n+ 1 has 2-adic valuation rappear with density 1/2r−1within the set of odd integers. This explains why the lower rows of Figure 1 are increasingly sparse. We now use the same 2-adic data to group the columns: if two odd numbers produce, after all possible divisions by 2, the same last even A, then they belong to the same 4-adic family. Thus, the vertical density information is translated into a horizontal family organisation, on which convergence toward family (2) can be studied. Let nbe an odd number and let A=3n+ 1 2ν2(3n+1)−1 be the last even number of its odd Collatz trajectory, that is, the even value just before returning to an odd one. Note that writing 3n+ 1 = 2r·q(qodd), gives A= 2q, so A/2 = qis precisely the following odd number. In particular, every visible last even number satisfies A≡2 (mod 4), and more specifically, A≡2or 10 (mod 12). From this Awe can describe all odd numbers producing that same last even number. They satisfy 3m+1=A·2s, s ≥0, since dividing 3m+ 1 by 2s−1yields 3m+ 1 2s−1=A, which is the definition of a last even number. For mto be integer we need A·2s≡1 (mod 3). As 2≡ −1 (mod 3), two cases occur: •If A≡1 (mod 3) (that is, A≡10 (mod 12)), then 2s≡1 (mod 3), so smust be even. Writing s= 2kwe obtain m=A·22k−1 3=A·4k−1 3, k = 0,1,2, . . . •If A≡2 (mod 3) (that is, A≡2 (mod 12)), then A·2s≡1 (mod 3) requires sto be odd. Writing s= 2k+ 1 gives m=A·22k+1 −1 3=2A·4k−1 3, k = 0,1,2, . . . This leads to the following definition. Definition 3.1 (Family associated with a last even number).Let Abe a visible last even number of the odd Collatz trajectory, with A≡2,10 (mod 12). The family associated with Ais defined 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. 3 The Collatz dynamics can then be represented in a compressed way on the last even numbers: to each A(even) we apply the next Collatz step, which is A7→ A/2, and that odd number A/2 belongs to some other family FB. This gives rise to the directed graph in Figure 2, where some of the lowest families have been drawn, for instance 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 3.1 (Uniqueness of the cycle 4,2,1in the reduced family 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 number in FAthe step 3x+ 1 and exhausting the divisions by 2down to the last even, the odd obtained belongs to FB. Then, in this graph Gthe 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 Gcomposed of distinct families. Proof. If A= 2, the next step is 27→ 1, and 1belongs precisely 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 the family FA, rather to another family determined by that odd number. Therefore, from FAone always leaves toward a different family. This prevents the existence of cycles made of several different families. As a direct consequence, the only cyclic node of the reduced structure also acts as an attracting node for all other families. This can be stated as follows. Proposition 3.2 (Modular classes of last even numbers and convergence to F2).Let Abe the last even number of an odd Collatz trajectory, that is, 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 a single step, an odd number belonging to the family F2={1,5,21,85, . . .}. Proof. If nis odd, 3n+ 1 is even and, modulo 12, only the residues 3n+ 1 ≡4or 10 (mod 12) may appear, because the odd residues modulo 12 are 1,3,5,7,9,11 and 3n+1 alternates between 4,10,4,10,4,10 respectively. In both cases the visible 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 2 is 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,...} which characterises the family F2. Therefore, every family FAwith A>2ends, after a finite number of iterations of the reduced rule, in F2—in fact, after exactly one step. 4 Corollary 3.1 (Unique attractor of the reduced family structure).In the reduced family structure FAdefined from the visible 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 that converge toward the family F2, the only one that self-connects. 2 10 14 22 FA . . . F2: (1,5,21,85, . . .) F10 : (3,13,53,213, . . .) F14 : (9,37,149, . . .) F22 : (7,29,117, . . .) Familias restantes (A≡2,10(mod 12)) Figure 2: Estructura reducida por familias 4-ádicas. Todas las familias FAcon A≡2,10 (mod 12) convergen en la familia F2, única con autoenlace. 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 family (2) reveal a system whose modular structure is completely determined, where each individual trajectory finds its reflection in a unique route of convergence. This correspondence between arithmetic level and functional structure summarises 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, whereas the 4-adic reduction organises the trajectories into horizontal families defined by their even closure node. Both perspectives converge into a compact representation in which all families reduce to the attractor F2, corresponding to the cycle 4→2→1. This approach reveals the 2-adic and modular nature of the Collatz map and offers an analytic way to reinterpret its global behaviour from a finite and deterministic structure. 5 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)) , where d, g ∈Nand h:Z→Zis a function 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 multiplication and translation, are reduced repeatedly by divisibility. The main goal of the structure theorem is to describe the orbits in terms of congruences modulo dg and of the exponent sequences (k1, k2, . . . , km)that specify the divisions by dat each step. A.2. The Collatz function as a particular case The classical Collatz function is obtained with the parameters d= 2,g= 3 and h(x)=1, which gives T(n) = 3n+ 1 2ν2(3n+1) . This is exactly the function used as a basis in 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 the drift and the 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 has µ<0, which implies average contraction. In the present structural setting, the local expansion of an odd–even segment is expressed as G= 3j2−(n−j), where jis the number of odd terms and n−jthe number of divisions by 2. The condition log G<0also characterises contraction of the trajectory. Both expressions quantify the same balance between expansion by 3and reduction by powers of 2. A.4. Modular structure and classification of families The structure theorem of Kontorovich–Sinai classifies the orbits into congruence classes modulo dg and parametrises them by the sequences of exponents νd. Similarly, the present theory organises the trajectories into 4-adic families FAdefined by their even closure node, and into modular zones according to digital roots. The difference lies in the viewpoint: while Kontorovich and Sinai describe a space of probability measures, here we construct a discrete, hierarchical and fully deterministic structure. 6 A.5. Unified dynamical interpretation In both models, global contraction appears in an equivalent way: negative drift implies statistical attraction toward a finite domain, and decreasing 2-adic density explains the hierarchical convergence of all families toward the attractor F2. The reduced map T(A) = A/2plays, in the deterministic framework, the same role that the logarithmic average does in the stochastic model: an iterative dissipation mechanism that concentrates the orbits. A.6. Conclusion The (d, g, h)model provides a general and probabilistic formulation of Collatz-type transformations. The present approach may be interpreted 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 finds an explicit translation in modular and 4-adic terms, where the dynamics become visible as a finite system of connections culminating in 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). Iteration of the number-theoretic function f(x) = x/2(x even), 3x+ 1 (x odd). Advances in Mathematics, 25(1), 42–45. •Lagarias, J. C. (1985). 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