The Goldbach Conjecture as an Informational Economy Principle: A Heuristic Framework from Computational Physics

TL;DR

This paper introduces an informational economy principle inspired by physics to analyze the Goldbach Conjecture, suggesting that prime number differences tend to minimize sums, linking the conjecture to physical limits of computation.

Contribution

It proposes a novel heuristic framework connecting the Goldbach Conjecture to principles of information physics and computational limits, offering a new interpretative perspective.

Findings

Prime differences tend to have minimal sums with high probability

Goldbach Conjecture can be viewed as an informational consistency condition

Violations would imply an informational anomaly

Abstract

This paper presents a heuristic framework for analyzing the Goldbach Conjecture (GC) from the perspective of the physics of information. Through empirical analysis, we propose an Informational Economy Principle (IEP), which posits that differences between prime numbers tend to be resolved by pairs with a minimal sum with overwhelmingly high probability. We argue that this tendency is analogous to least-action principles in physics and is consistent with the fundamental physical limits of computation. Within this framework, the GC can be interpreted as an informational consistency condition for the set of prime numbers. The falsehood of the conjecture would imply a violation of this observed economy, representing an extreme informational anomaly. This approach suggests that the difficulty in proving the GC may not lie solely in mathematical abstraction, but in the decoupling between said abstraction and the physical constraints inherent to information.