A Balance-Based Approach to the Collatz Hypothesis

Abstract

This note introduces a novel approach to the Collatz hypothesis, based on the principle of global balance in closed discrete systems. We employ a ternary system of change (+1, 0, –1), viewing any "increase" as internal redistribution rather than the emergence of value from nothing. This framework allows for a fresh perspective on the Collatz problem and reveals universal laws for a broad class of discrete processes.

Introduction

The classical Collatz hypothesis is formulated as a simple iterative rule over natural numbers. Despite its apparent simplicity, it remains unsolved and continues to intrigue mathematicians. Here, we present an alternative approach—using the concept of internal exchange and ternary logic of change.

Axioms and Core Principles

1. System Closure:
   The set of natural numbers is treated as a closed system, with no external creation or annihilation of elementary units (resources, mass, or information).
2. Ternary Change Logic:
   Each element can, at every step, experience one of three changes: increase (+1), decrease (–1), or stability (0).
3. Global Invariant:
   At every step, the sum of all changes equals zero:

   ∑i Δni = 0,   where Δni ∈ {–1, 0, +1}

Reformulation of the Collatz Rule

• For even elements: n → n / 2.
• For odd elements: n → 3n + 1, where the "+1" is an internal redistribution, not an external addition.
• Any gain for one element is offset by a corresponding loss elsewhere, preserving global balance.

Theoretical Result

In any closed discrete system with fully balanced internal exchanges, unbounded increasing trajectories are impossible; all redistribution processes inevitably lead to stable cycles or fixed points.

Applied to the Collatz hypothesis: all sequences must eventually reach a cycle or stabilization, as unbalanced growth cannot occur.

Discussion and Universality

This approach applies not only to Collatz but to any discrete dynamical system governed by the principle of internal exchange. The ternary system enables construction of flow diagrams, identification of cycles, and definition of universal invariants.

Conclusion

The Collatz problem is not a puzzle about individual numbers, but a special case of a global balance law: in closed discrete systems, any local change is an aspect of internal redistribution, and all dynamics manifest as movements along stable cycles.

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Copyright (c) 2025 YEVGENIY S.

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