Reachability and uniqueness of the affine Collatz inversion
Abstract
We study the positive odd integers under the three affine operations A(x) = (2x-1)/3, C(x) = (4x-1)/3, V(x) = 4x+1, where A is legal on inputs 2 (mod 3), C on inputs 1 (mod 3), and V is always legal. These operations are inverse pieces of the accelerated Collatz map T(x) = (3x+1)/2^(v_2(3x+1)) in the precise sense that T (A(x)) = x, T (C(x)) = x, and T (V (x)) = T (x). The first point is uniqueness. The images of A, C, V are separated modulo 8. This gives a single-valued parent map P. Thus every point has at most one ancestry chain, and a collision of two affine closures is an ancestry relation. The counting argument is based on an exact change of coordinates. We can show that an AC-pull has binary mass 2^{-a(w) + 2c(w)), while a V-run of length t has mass 4^{-t}. The map V supplies the required ternary sampling: one V-cycle modulo 3^k visits every residue class exactly once.After truncating the AC-depth at K(n) = ⌊ 1/2 log_3(n)⌋, the discarded mass is o(1), and the retained paths require only O(K(n)) extra binary headroom. Hence the affine closure of 1 occupies density one in the odd integers. Finally, every fixed root has positive affine-closure density after removing at most one finite cycle. Density one then forces a collision between the tree originating at 1 and the tree originating at x, and the parent map turns this collision into x is in the tree originating at 1. Since x is arbitrary, one obtains that the tree originating at 1 is the set of all positive odd integers.
Related
- Basis Rigidity of the AES S-box and Generic Rigidity of Inversion under Affine Transformations
- A Complete Solution to the Conjecture on the Inverse Collatz–Sinogowitz Irregularity Problem
- A variant of Collatz's conjecture over binary polynomials
- Spectral Koopman-Hopf Formula for Reachability with Adversary
- Parity-Word Complexity and Exponent Growth in Non-Bounded Collatz Orbits
BibTeX
@misc{author2026reachability,
title = {{Reachability and uniqueness of the affine Collatz inversion}},
author = {Anonymous Author and Marie Graff},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05748420v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Reachability and uniqueness of the affine Collatz inversion
AU - Anonymous Author
AU - Marie Graff
PY - 2026
DA - 2026/09/14
PB - HAL
UR - https://hal.science/hal-05748420v1
AB - We study the positive odd integers under the three affine operations A(x) = (2x-1)/3, C(x) = (4x-1)/3, V(x) = 4x+1, where A is legal on inputs 2 (mod 3), C on inputs 1 (mod 3), and V is always legal. These operations are inverse pieces of the accelerated Collatz map T(x) = (3x+1)/2^(v_2(3x+1)) in the precise sense that T (A(x)) = x, T (C(x)) = x, and T (V (x)) = T (x). The first point is uniqueness. The images of A, C, V are separated modulo 8. This gives a single-valued parent map P. Thus every point has at most one ancestry chain, and a collision of two affine closures is an ancestry relation. The counting argument is based on an exact change of coordinates. We can show that an AC-pull has binary mass 2^{-a(w) + 2c(w)), while a V-run of length t has mass 4^{-t}. The map V supplies the required ternary sampling: one V-cycle modulo 3^k visits every residue class exactly once.After truncating the AC-depth at K(n) = ⌊ 1/2 log_3(n)⌋, the discarded mass is o(1), and the retained paths require only O(K(n)) extra binary headroom. Hence the affine closure of 1 occupies density one in the odd integers. Finally, every fixed root has positive affine-closure density after removing at most one finite cycle. Density one then forces a collision between the tree originating at 1 and the tree originating at x, and the parent map turns this collision into x is in the tree originating at 1. Since x is arbitrary, one obtains that the tree originating at 1 is the set of all positive odd integers.
ER -