Axiomatic Proof of the Goldbach Conjecture: Based on the ECS Framework (Extremum-Conservation-Symmetry)

Author: Zhang Suhang (Luoyang)

Abstract: The Goldbach Conjecture asserts that every even integer not less than 4 can be expressed as the sum of two primes. Within the ECS (Extremum-Conservation-Symmetry) axiomatic framework, incorporating the Maximum Information Efficiency (MIE) principle, this paper presents a conditional proof of this conjecture. We construct a state space of odd splits for each even number, demonstrate that this space is fixed by a symmetry group and possesses a global conserved quantity, and then apply the MIE axiom to show that the extremal efficiency state must be a split into two primes. This proof does not rely on traditional tools of analytic number theory, transforming an existence problem into an extremal existence problem under a fixed symmetry group.

Keywords: Goldbach Conjecture; ECS framework; Maximum Information Efficiency (MIE); extremum principle; symmetry

1 Introduction

The Goldbach Conjecture (1742) is one of the oldest unsolved problems in number theory: every even integer not less than 4 can be expressed as the sum of two primes. After three centuries of research, the closest results include Chen Jingrun's "1+2" (1973) and conditional proofs based on the Generalized Riemann Hypothesis. However, a complete proof of the conjecture has remained elusive.

This paper adopts a distinctly different path. We do not rely on the circle method, sieve methods, or L-function estimates. Instead, we re-examine the problem within the ECS axiomatic framework. The ECS framework consists of three axioms—Extremum (E), Conservation (C), and Symmetry (S)—and its core, the Maximum Information Efficiency (MIE) axiom, has been used in previous work to unify the principle of least action, Murray's Law, Euler's formula, and the Collatz conjecture [1,2,3].

This paper will prove that, under the MIE axiom, the Goldbach Conjecture becomes a necessary corollary. This conclusion relies on no unverified number-theoretic hypotheses, only on the acceptance of the axioms and logical deduction.

2 Review of the ECS Framework

Axiom 1 (Extremum, E): Any long-term stable system must maximize the information processing efficiency per unit energy consumption.

Axiom 2 (Conservation, C): A system satisfying the extremum condition possesses a global invariant that remains unchanged under system transformations.

Axiom 3 (Symmetry, S): The existence of a global conserved quantity is equivalent to the existence of a symmetry group in the system's state space that fixes the system's topological structure.

Previous applications of the ECS framework include:

· The principle of least action as a special case of MIE in conservative mechanics [1];

· Noether's theorem deriving conservation laws and symmetries [2];

· Murray's Law, Euler's formula, and Fermat's principle as cross-disciplinary unified examples [3].

3 ECS Modeling of the Goldbach Conjecture

3.1 Construction of the State Space

For any even number 2n (n \ge 2), define its state space of odd splits:

\mathcal{S}_n = \left\{ (a,b) \in \mathbb{N}^2 \;\middle|\; a + b = 2n,\; 1 \le a \le b,\; a,b \text{ are odd} \right\}.

This space is finite, and each state corresponds to a way of splitting the even number into two odd numbers. If there exists a state (p,q) \in \mathcal{S}_n where both p and q are prime, then the Goldbach Conjecture holds for that even number.

3.2 Symmetry (S)

The state space \mathcal{S}_n possesses the following symmetry transformations:

· (i) Exchange symmetry: (a,b) \mapsto (b,a);

· (ii) Modulo p symmetry: For any prime p, rearrangement of residue classes preserves the sum;

· (iii) Parity symmetry: States are restricted to odd pairs, automatically preserved by the constant sum.

These transformations constitute the symmetry group \mathcal{G}_{\text{Goldbach}}, which fixes the fundamental structure of \mathcal{S}_n.

3.3 Conservation (C)

From the symmetry group \mathcal{G}_{\text{Goldbach}}, a global invariant can be constructed:

\mathcal{C}_{\text{Goldbach}} = \prod_{p \le \sqrt{2n}} \left(1 - \frac{1}{p-1}\right) \times \rho(n),

where \rho(n) is a local density factor. This quantity remains invariant under all transformations of \mathcal{G}_{\text{Goldbach}} and corresponds to the singular series in the Hardy-Littlewood conjecture. The existence of a conserved quantity indicates that the statistical structure of the state space is uniquely determined by the symmetry group, independent of the specific state chosen.

3.4 Extremum and the MIE Axiom (E)

Define the information efficiency of a state (a,b) \in \mathcal{S}_n as:

\eta(a,b) = \frac{I(a,b)}{E(a,b)},

where:

· The information measure I(a,b) = \log_2 a + \log_2 b (coding length);

· The energy consumption E(a,b) is the algebraic complexity of factorizing the two numbers: primes are indecomposable with minimal energy consumption; composite numbers can be further factorized, incurring higher energy consumption.

The MIE axiom asserts that the system must attain the global maximum of \eta(a,b).

Lemma 1: \eta(a,b) attains its global maximum if and only if both a and b are prime.

Proof: If a is composite, then a = uv with u,v \ge 2. Its factorization complexity satisfies E(a) > E(u) + E(v), implying I(a)/E(a) < \max\{\log_2 u/E(u), \log_2 v/E(v)\}. Therefore, the information efficiency of a composite number is strictly lower than that of the combination obtained by replacing it with its factors. Through a finite number of factorization steps, any composite number is eventually replaced by a series of prime numbers, and this process strictly increases information efficiency. Hence, the global maximum must occur when both a and b are prime. ∎

3.5 Existence Corollary

Theorem 1 (ECS-Goldbach): In the state space \mathcal{S}_n fixed by the symmetry group \mathcal{G}_{\text{Goldbach}}, an MIE extremal state necessarily exists, and this state is a split into two primes. That is, for every even number 2n (n \ge 2), there exists at least one pair of primes (p,q) such that p+q=2n.

Proof: The state space \mathcal{S}_n is finite. By the MIE axiom, the system must attain the global maximum of the information efficiency \eta(a,b). By Lemma 1, this maximum is attained if and only if both a and b are prime. Therefore, the extremal state belongs to \mathcal{S}_n and corresponds to a split into two primes. If such an extremal state did not exist, the system could not achieve the efficiency maximum required by the MIE axiom, a contradiction. Hence, a split into two primes must exist. ∎

Corollary: The Goldbach Conjecture holds within the ECS framework.

4 Comparison with Mainstream Approaches

Aspect Circle/Sieve Methods (e.g., Hardy-Littlewood, Chen) ECS Axiomatic Method (This Paper)
Core Tools Exponential sum estimates, error term control Symmetry, conserved quantity, MIE extremum principle
Strength of Conclusion Asymptotic formulas, "1+2", conditional on Riemann Hypothesis Holds for all even numbers (under MIE axiom)
Explains "Why"? No Yes (efficiency optimality)
Assumptions Generalized Riemann Hypothesis (for some results) MIE axiom

The ECS framework does not aim to replace existing methods but rather provides a novel axiomatic perspective, revealing the underlying structural necessity of the conjecture.

5 Conclusion

This paper has provided a conditional proof of the Goldbach Conjecture within the ECS axiomatic framework. We constructed the state space of odd splits for each even number, identified its symmetry group and conserved quantity, and applied the MIE extremum axiom to deduce the necessary existence of a split into two primes.

The distinctive features of this proof are:

1. Independence from traditional analytic number theory tools: No need for the circle method, sieve methods, or L-function estimates;
2. Logical closure: Given acceptance of the axioms, the conclusion is necessary;
3. Explanatory power: Reveals the essential reason "why" prime pairs exist—they are the unique split states with maximal information efficiency.

Just as with the treatment of the Collatz Conjecture in previous work [1,2,3], the Goldbach Conjecture becomes not a technical difficulty within this framework, but a necessary corollary of the axiomatic system.

References

[1] Zhang, S. The Axiom of Maximum Information Efficiency (I): From Least Action to MIE.
[2] Zhang, S. The Axiom of Maximum Information Efficiency (II): Derivation of Conservation Laws and Symmetries.
[3] Zhang, S. Cross-disciplinary Unification of the Maximum Information Efficiency Axiom: Murray's Law, Polyhedron Law, and Fermat's Principle.
[4] Chen, J. R. On the representation of a larger even integer as the sum of a prime and the product of at most two primes. Sci. Sin., 1973.
[5] Hardy, G. H., Littlewood, J. E. Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes. Acta Math., 1923.

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Appendix: Supplement to the Proof of Information Efficiency Optimality

For a state (a,b) \in \mathcal{S}_n, define the information per unit energy as \eta(a) = \log_2 a / E(a). Then \eta(a,b) = (E(a)\eta(a) + E(b)\eta(b))/(E(a)+E(b)), i.e., a weighted average. Since primes have the maximum \eta value, the weighted average of any combination containing a composite number is strictly less than that of a pure prime combination. Hence, the global maximum uniquely corresponds to a split into two primes.