A Structural Reduction of the Collatz Conjecture to One-Bit Orbit Mixing

TL;DR

This paper reduces the Collatz conjecture to a one-bit orbit-mixing problem with explicit formulas and a key theorem showing bias is at the orbit level, simplifying the conjecture to binary orbit behavior.

Contribution

It introduces a fixed-modulus, one-bit orbit-mixing framework and proves a Map Balance Theorem that refines the understanding of residual bias in the Collatz problem.

Findings

Exact low-depth decomposition formulas for the Collatz map at depths 3, 4, 5.

A Map Balance Theorem showing bias difference of exactly 1 among certain residue classes.

Reduction of the Collatz conjecture to binary orbit visits modulo 32.

Abstract

We reduce the Collatz conjecture to a fixed-modulus, one-bit orbit-mixing problem. Working with the compressed odd-to-odd Collatz map, we prove exact low-depth decomposition formulas at depths K = 3, 4, 5, reducing block-discrepancy terms to explicit run statistics. We then prove a Map Balance Theorem: among the 2^(K-3), 1 burst residues modulo 2^K that initiate gaps, the counts mapping to gap starts congruent to 3 versus congruent to 7 (mod 8) differ by exactly 1 for every K >= 5. Thus all residual bias is orbit-level, not map-level. For the dominant n congruent to 1 (mod 8) class, the gap outcome depends on a single binary variable, bit 4 of the orbit value at burst-ending times, reducing the conjecture to whether every orbit visits two residue classes modulo 32 with sufficient balance along a sparse subsequence.