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 December 2025 Abstract This paper describes the odd–even part of the classical Collatz map from a 2-adic perspective. First, it is shown that the odd integers satisfying ν2(3n+ 1) = rform, within the set of odd integers, a single residue class modulo 2r+1 and occur with density 1/2r. 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, and to group all odd integers yielding the same Ainto 4-adic families FA. It is proved that the only possible even closure nodes satisfy A≡2,10 (mod 12) and that F2={1,5,21,85, . . .} is the only family admitting a self-loop in the reduced graph. Global attraction to F2is stated as a conjecture and supported by computational verification up to even closure nodes A≤106. 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 odd trajectory under the classical Collatz map, produced by the author. Each row contains the successive even values appearing in the even tail of the trajectory, namely 3n+ 1,3n+ 1 2,3n+ 1 22, . . . , 3n+ 1 2r−1, up to the last even number before the orbit returns to an odd integer. The pattern of gaps visible in the table reflects a hierarchy of decreasing densities governed by the power of two dividing 3n+ 1. Figure 1: Odd integers nmapped through the even tail up 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 a single residue class modulo 2r+1, and their density among the odd integers is δr=1 2r. 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 = 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 a single class modulo 2r+1 in Zis 1/2r+1 and the density of the odd integers is 1/2, the relative density among the odds is 1/2r+1 1/2=1 2r. Corollary 2.2 (Spacing between successive occurrences).Let drbe the number of odd integers strictly between two consecutive odd integers msatisfying ν2(3m+ 1) = r. Then dr= 2r−1. Proof. The solutions form an arithmetic progression with step 2r+1 in Z. Between two odd integers separated by 2r+1 there are exactly 2r−1odd integers. 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 by families The previous section explains why the lower rows of Figure 1are increasingly sparse. We now use the same 2-adic datum to group odd integers into families determined by a common even closure node, i.e. the last even number in the odd–even step of Collatz. Let nbe an odd integer and let A=3n+ 1 2ν2(3n+1)−1 be the last even number before the orbit returns to an odd one. If 3n+ 1 = 2rq(qodd), then A= 2q, so A/2 = qis exactly the next odd value. In particular, every such Asatisfies A≡2 (mod 4), and in fact A≡2or 10 (mod 12). 2 All odd integers mproducing the same Asatisfy 3m+1=A·2s(s≥0), because dividing 3m+ 1 by 2s−1yields Aas the last even number. For mto be an integer we need A2s≡1 (mod 3). Since 2≡ −1 (mod 3), there are two cases: •If A≡1 (mod 3) (i.e. A≡10 (mod 12)), then 2s≡1 (mod 3), so sis even. Writing s= 2k, 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 A2s≡1 (mod 3) forces sodd. Writing s= 2k+ 1, m=A·22k+1 −1 3=2A·4k−1 3, k = 0,1,2, . . . Definition 3.1 (Family associated with a last even number).Let Abe a last even number in the odd trajectory, with A≡2,10 (mod 12). Define the family associated with Aby FA=         A·4k−1 3:k≥0,if A≡10 (mod 12), 2A·4k−1 3:k≥0,if A≡2 (mod 12). Proposition 3.2 (Disjoint partition into families).Every odd integer nbelongs to a unique family FA. Moreover, if A=Bthen FA∩ FB=∅. Proof. For a given odd n, the factorization 3n+ 1 = 2rqwith qodd is unique; hence the derived value A= (3n+1)/2r−1= 2qis unique. Therefore ncannot satisfy 3n+1=A2sand 3n+1=B2t with A=B, which proves disjointness and uniqueness of membership. Proposition 3.3 (Uniqueness of the 4→2→1cycle in the family graph).Consider the directed graph whose nodes are families FA, and where we draw an edge FA→ FBif the next odd obtained from a representative of FAbelongs to FB. Then the only family admitting a self-loop is F2. Proof. If A= 2, then the next odd is A/2=1, and 1∈ F2, so F2→ F2. If A > 2, the next odd is A/2 = qwith qodd, and qbelongs to a family determined by its own closure node, which is not Ain general. Hence FAcannot map to itself. Proposition 3.4 (Modular classes of even closure nodes).If Ais an even closure node, i.e. A=3n+ 1 2ν2(3n+1)−1, then necessarily A≡2or 10 (mod 12). Proof. For odd n, one has 3n+ 1 ≡4or 10 (mod 12). Dividing by powers of 2until the last even value yields a number congruent to 2or 10 (mod 12). Conjecture 3.5 (Global attraction to the family F2).In the reduced structure by families FA defined from even closure nodes, the family F2={1,5,21,85, . . .} acts as a global attractor. That is, every orbit of the reduced graph starting from a family FA with A>2is conjectured to reach F2in finitely many steps. 3 Comment (Computational evidence).Conjecture 3.5 has been verified computationally for all even closure nodes A≤106with A≡2,10 (mod 12). In this range, no alternative directed cycles were detected in the reduced structure, and all orbits reached A= 2 (equivalently, F2). Details are given in Appendix C. 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. The family F2is the only one admitting a selfloop. Computational verification up to A≤106supports the conjecture that all families reach F2. Conclusion This study does not address a complete proof of the Collatz conjecture, but provides a structural description of its odd–even dynamics based on two complementary ingredients. First, the 2-adic density lemma identifies a hierarchy of levels determined by ν2(3n+ 1) and explains the pattern of increasing gaps observed in the tabular representation of odd trajectories. Second, the even closure node groups odd integers into 4-adic families FAand yields a reduced family graph in which F2is the unique family admitting a self-loop, corresponding to the classical cycle 4→2→1. Global attraction to F2is formulated as a conjecture. In support of it, Appendix Creports computational verification up to A≤106, where no alternative directed cycles were detected and all orbits reached A= 2. A Connection with the structure theorem for (d, g, h)-maps A.1 The Kontorovich–Sinai framework Kontorovich and Sinai introduce 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 periodic modulo d. The exponent νd(y)denotes the highest power of ddividing y. 4 A.2 Collatz as a particular case The classical Collatz map corresponds to d= 2,g= 3 and h(x)=1: T(n) = 3n+ 1 2ν2(3n+1) . This is the map underlying the present work, where ν2(3n+ 1) determines the even-tail length and the even closure node A= (3n+ 1)/2ν2(3n+1)−1. A.3 Drift and local expansion Kontorovich and Sinai define the drift µ= log g−d d−1log d. For (d, g) = (2,3) one gets µ < 0, indicating average contraction. In the present framework, an odd–even segment has local factor G= 3j2−(n−j), and log G<0similarly characterises contraction at the segment level. B Transition from classes 4n+3 to 4n+1 in the simplified dynamics B.1 Motivation For the simplified Collatz map T(n) = (n/2, n even, (3n+ 1)/2, n odd, every odd class n≡3 (mod 4) eventually reaches the class n≡1 (mod 4). B.2 Sketch of the modular mechanism Let n= 4k+ 3. Then T(4k+ 3) = 3(4k+3)+1 2= 6k+ 5, and 6k+ 5 ≡1 (mod 4) iff kis even. If kis odd, writing k= 2t+ 1 yields a new parameter t<k, and iterating reduces the parameter until a 4n+1 is reached. Hence permanence in 4n+3 is finite. Corollary B.1 (Finite permanence of the class 4n+ 3).Under T(n) = (3n+ 1)/2, every trajectory starting at an odd integer n≡3 (mod 4) enters the class n≡1 (mod 4) in finitely many steps. C Computational verification up to even closure nodes A≤106 This appendix reports the verification, for all even closure nodes A≤106with A≡2or 10 (mod 12), that the reduced dynamics does not exhibit alternative directed cycles in that range and that every orbit reaches the fixed node A= 2. 5 C.1 Reduced map on even closure nodes Given an even closure node A≡2,10 (mod 12), the next visible odd is q=A/2. Define the reduced successor of Aas the new even closure node obtained after applying 3q+ 1 and dividing by powers of 2until the last even is reached: T(A) = 3(A/2) + 1 2ν2(3(A/2)+1)−1=3A+ 2 2ν2(3A+2)−2, A ≡2,10 (mod 12).(1) The node A= 2 satisfies T(2) = 2. C.2 Verification algorithm We scanned the set A106={A∈2N:A≤106, A ≡2or 10 (mod 12)}. For each A∈ A106we iterated the map (1), recording visited states until one of the following occurred: 1. the orbit reaches A= 2; 2. a state A= 2 repeats (which would detect an alternative directed cycle). In the explored range, event (1) always occurred and event (2) never did. C.3 Results summary The set A106has cardinality |A106|= 166 667. In the explored range: •Convergence: for every A∈ A106there exists t≥0such that Tt(A)=2. •No alternative cycles detected: no directed cycle different from the self-loop at A= 2 was found. •Transient lengths: letting τ(A)be the smallest tsuch that Tt(A) = 2, we obtained max A∈A106 τ(A) = 166 (attained at A= 820 022),E[τ] = 43.63. Percentiles: P50 = 41,P75 = 58,P90 = 72,P95 = 81,P99 = 101. C.4 Interpretation This verification is not a proof of global convergence for all A, but provides quantified and reproducible evidence that, up to A≤106, the only cyclic behaviour of the reduced dynamics is the self-loop at A= 2. 6 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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Unpublished manuscript, Mallorca. 7