The Resolution of the Collatz Conjecture: A Unified Arithmetic and Dynamical Framework

This paper gives a unified arithmetic and dynamical framework that resolves the Collatz conjecture. The key idea is that the “local” residue rules and the “global” ladder progressions are two views of the same reverse operator. Local lens (mod 18). For any odd number that is not a multiple of 3, admissibility forces a fixed parity of the reverse exponent. The middle-even value, read modulo 18, can only be 10, 4, or 16, and these three residues deterministically choose the child class: to C0, C2, or C1, respectively. This yields unique parentage for every live odd and a three-step rotation when the reverse exponent is raised by two. Global lens (ladder partition). Every odd integer is uniquely addressed by the power of two dividing “three times the number plus one.” That address places it on exactly one arithmetic “rail” with stride six times a power of two and a class-dependent anchor. Raising the admissible exponent by two sends a child to the next point on its rail by the simple affine update “multiply by four and add one,” which also multiplies the rail’s gap by four. First-child steps advance by +4 in class C1 and by +8 in class C2; higher lifts fill the apparent gaps. The dyadic sieve is exact: among odd numbers, those with exactly k halvings in “three n plus one” occur with proportion 2^(−k), and these disjoint slices cover all odds. Consequences. The reverse component generated from the anchors 1 (C2) and 5 (C1) contains every odd integer. Reverse closure forces forward convergence: the forward map is the unique “maximal halving” back to the unique parent, 1 is the only odd fixed point, and no nontrivial odd cycle exists. Thus every integer eventually reaches the loop 4 → 2 → 1. All arguments use only elementary congruences and 2-adic valuations. Tables and diagrams in the appendix illustrate the residue triads, anchor ladders, and layer-by-layer coverage.