# Proof: The Goldbach conjecture holds for every even integer greater than 2.

- **Generated:** 2026-03-28
- **Verdict:** UNDETERMINED
- **Audit trail:** [proof_audit.md](proof_audit.md) | [proof.py](proof.py)

## Key Findings

- **0 counterexamples** found among all 499,999 even integers in [4, 1,000,000], verified by two independent computational methods (A1, A2).
- Both methods — trial division and prime sieve — agree completely on the result.
- The literature records computational verification up to 4 x 10^18 (Oliveira e Silva, 2013), with recent extensions by the Gridbach project (2025).
- **No proof or disproof of the conjecture exists** as of March 2026. Finite computation cannot resolve a universal claim over infinitely many integers. Verdict: **UNDETERMINED**.

## Claim Interpretation

**Natural language:** "The Goldbach conjecture holds for every even integer greater than 2."

**Formal interpretation:** The Goldbach conjecture asserts that every even integer greater than 2 can be expressed as the sum of two prime numbers. This is a universal claim over an infinite domain. The formal check counts the number of counterexamples in [4, 10^6] and compares to 0 (operator: `==`, threshold: 0).

**Operator note:** This is an unresolved conjecture. Computational verification up to a finite bound provides evidence but cannot prove the claim for all even integers. The verdict is always UNDETERMINED regardless of computational outcome.

**Claim type:** Open problem.

## Evidence Summary

| ID | Fact | Verified |
|----|------|----------|
| A1 | Primary check: counterexamples in [4, 10^6] via trial division | Computed: 0 counterexamples found (conjecture holds for all 499,999 even integers tested) |
| A2 | Cross-check: counterexamples in [4, 10^6] via prime-pair sieve | Computed: 0 counterexamples found (independent confirmation) |

*Source: proof.py JSON summary*

## Proof Logic

The proof applies two independent computational methods to every even integer in [4, 1,000,000]:

**Method 1 — Trial Division (A1):** For each even n, iterate candidate primes p from 2 to n/2. For each p, test whether both p and (n - p) are prime using `sympy.isprime()`. If any such pair exists, n satisfies the conjecture. This method tests primality on demand for each candidate.

**Method 2 — Prime Sieve (A2):** Pre-generate all primes up to 10^6 using `sympy.primerange()` (Sieve of Eratosthenes). Store them in a sorted list (for ordered iteration) and a set (for O(1) membership lookup). For each even n, iterate the sorted prime list up to n/2 and check whether (n - p) is in the prime set.

The two methods are structurally independent: Method 1 uses per-number primality testing; Method 2 uses a pre-computed sieve with set membership. A bug in one algorithm would not propagate to the other.

Both methods found **0 counterexamples** across all 499,999 even integers, and their results agree exactly.

However, there are infinitely many even integers greater than 2, and no finite computation can verify them all. The conjecture remains unproven.

## Counter-Evidence Search

1. **Has any counterexample ever been found?** Searched for "Goldbach conjecture counterexample found disproved." Reviewed Wikipedia, Physics Forums, Medium, and LessWrong. No counterexample has ever been reported in the literature or computational records.

2. **Has the conjecture been formally proved or disproved?** Searched for "Goldbach conjecture proof solved 2024 2025 2026." Found several claimed proofs in non-peer-reviewed venues (SSRN, SCIRP, preprints.org, ScienceOpen) but none accepted by mainstream mathematics. The conjecture remains open.

3. **What is the current verification bound?** Oliveira e Silva verified the conjecture up to 4 x 10^18 (2013). The Gridbach project (2025) extended this further. Our 10^6 bound is far below the literature record but uses two independent methods for internal consistency.

*Source: proof.py JSON summary*

## Conclusion

**Verdict: UNDETERMINED**

The Goldbach conjecture has been computationally verified for all 499,999 even integers in [4, 1,000,000] by two independent methods (A1, A2), with 0 counterexamples found. The literature confirms verification up to 4 x 10^18 with no counterexample ever discovered.

Nevertheless, this is a **universal conjecture over infinitely many integers**, and no finite computation can constitute a proof. No accepted mathematical proof exists as of March 2026. To resolve this claim, a formal mathematical proof (or a counterexample) would be required.

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Generated by [proof-engine](https://github.com/yaniv-golan/proof-engine) v0.10.0 on 2026-03-28.