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Collatz Analysis Two Stage Tree and Multiset Calculus
2025-12-29
The Collatz map T(n)=n/2 for even n and T(n)=3n+1 for odd n admits classical affine descriptions via parity vectors. The shortcut map compresses each odd event into the macro step (3n+1)/2, obscuring intermediate algebraic states. We introduce a two-stage branching formalism that decomposes the odd-step operation into two explicit sub-operations: a rewrite step R (expressing n=2x+1) followed by a forced follow-up C (mapping x→3x+2). This decomposition reveals intermediate states invisible in classical parity-vector representations and yields an explicit monomial expansion for the trajectory offset σ_N (w). We prove that complete two-stage words compress under RC→O to recover the standard affine form, establishing a precise equivalence criterion and canonical matching rule (k,D,Σ). The framework naturally connects to 2-adic formulations through residue-class ‘locking’ conditions modulo 2^D(w) .
Additionally, we develop a signed-multiset calculus on generators {g_j} that encodes binary arithmetic via local rewrite rules (Carry, Annihilation, Borrow). We prove this system is terminating and confluent, yielding unique canonical binary normal forms. Within this calculus, we derive an explicit bit-complement formula for 2^D-3^k and reformulate the classical cycle equation in multiset language, enabling digit-by-digit analysis of cycle constraints.
Ссылка для цитирования:
Aliabdali F. 2025. Collatz Analysis Two Stage Tree and Multiset Calculus. PREPRINTS.RU. https://doi.org/10.24108/preprints-3114186
Список литературы
1. L. Collatz, “On the motivation and origin of the (3n+1)-problem,” J. Qufu Normal Univ., Nat. Sci. Ed. 12(3), 9–11 (1986).
2. J. C. Lagarias, “The 3x + 1 problem and its generalizations,” Amer. Math. Monthly 92(1), 3–23 (1985).
3. J. C. Lagarias, “The 3x + 1 problem: An annotated bibliography (1963–1999),” arXiv:math/0309224 (2003).
4. J. C. Lagarias (ed.), The Ultimate Challenge: The 3x + 1 Problem, AMS (2010).
5. R. Terras, “A stopping time problem on the positive integers,” Acta Arith. 30(3), 241–252 (1976).
6. T. Tao, “Almost all orbits of the Collatz map attain almost bounded values,” Forum Math. Pi 10, e12 (2022).
7. R. E. Crandall, “On the ‘3x + 1’ problem,” Math. Comp. 32(144), 1281–1292 (1978).
8. G. J. Wirsching, The Dynamical System Generated by the 3n + 1 Function, Lecture Notes in Math. 1681, Springer (1998).
9. S. Eliahou, “The 3x + 1 problem: New lower bounds on nontrivial cycle lengths,” Discrete Math. 118, 45–56 (1993).
10. H. G. Senge and E. G. Straus, “PV-numbers and sets of multiplicity,” Period. Math. Hungar. 3, 93–100 (1973).
11. C. L. Stewart, “On the representation of an integer in two different bases,” J. Reine Angew. Math. 319, 63–72 (1980).
12. K. R. Matthews, “Generalized 3x + 1 mappings: Markov chains and ergodic theory,” in The Ultimate Challenge: The 3x + 1 Problem, AMS, 79–103 (2010).
13. D. J. Bernstein, “A non-iterative 2-adic statement of the 3x + 1 conjecture,” Proceedings of the American Mathematical Society 121(2) (1994), 405–408.
