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

Edward Y. Chang

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.

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

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.

Paper Structure

This paper contains 35 sections, 20 theorems, 30 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The compressed Collatz map
  4. Burst indicators and block frequencies
  5. Block-TV framework
  6. Exact low-depth decompositions
  7. Depth $K = 3$: one-parameter control
  8. Depth $K = 4$: two new parameters
  9. Depth $K = 5$: four new parameters
  10. Expanding-set reduction to run statistics
  11. Universal run-statistic bounds
  12. The Map Balance Theorem
  13. Gap-length characterisation
  14. The balance theorem
  15. The single-bit bottleneck
  16. ...and 20 more sections

Key Result

Proposition 3.1

Every binary sequence with density $\rho$ and block-alternation rate $q = m/T$ satisfies: Consequently, $\varepsilon_3 = 2|q - \rho(1{-}\rho)| + o(1)$: the depth-$3$ block-TV is a single scalar.

Figures (2)

  • Figure 1: Map Balance Theorem verification. Left: the counts $C_3(K)$ and $C_7(K)$ grow exponentially and track each other almost exactly. Right: their difference alternates $+1, -1, +1, \ldots$ for all $K = 5, \ldots, 19$, confirming $|C_3 - C_7| = 1$.
  • Figure 2: Running bit-$4$ balance along the orbit of $n_0 = 837\,799$ ($43$ burst--gap blocks). The fraction of burst-ending values with $n \equiv 9 \pmod{32}$ (among those with $n \equiv 1 \pmod{8}$) fluctuates around $1/2$. The shaded bands show the $\pm 1\sigma$ and $\pm 2\sigma$ envelopes for a fair-coin (Bernoulli$(\tfrac{1}{2})$) process of the same length; the orbit's trajectory stays within the $2\sigma$ band, consistent with unbiased sampling but far from converged after only $43$ blocks.

Theorems & Definitions (41)

  • Definition 2.1: Run statistics
  • Proposition 3.1: $K = 3$ pair identities
  • proof
  • Proposition 3.2: $K = 4$ exact decomposition
  • proof
  • Proposition 3.3: $K = 5$ support reduction
  • Proposition 3.4: Finite-state growth rate
  • Proposition 3.5: $K = 4$ expanding-set identity
  • Proposition 3.6: $K = 5$ expanding-set identity
  • Proposition 3.7: Deterministic bounds
  • ...and 31 more