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

- **Generated:** 2026-03-28
- **Reader summary:** [proof.md](proof.md)
- **Proof script:** [proof.py](proof.py)

## Claim Specification

| Field | Value |
|-------|-------|
| Subject | Goldbach conjecture |
| Property | counterexamples in range [4, 10^6] |
| Operator | == |
| Threshold | 0 |
| Claim type | open_problem |
| Operator note | The Goldbach conjecture asserts that every even integer > 2 is the sum of two primes. This is an unresolved conjecture — no proof or disproof exists as of 2026. Computational verification up to a finite bound provides evidence but cannot prove the claim for all even integers. The literature records verification up to 4 x 10^18 (Oliveira e Silva, 2013). Our script verifies up to 10^6 with two independent methods. Verdict is always UNDETERMINED regardless of computational outcome. |

*Source: proof.py JSON summary*

## Fact Registry

| ID | Label | Type | Key |
|----|-------|------|-----|
| A1 | Primary check: counterexamples in [4, 10^6] via trial division | A | primary_check |
| A2 | Cross-check: counterexamples in [4, 10^6] via prime-pair sieve | A | cross_check |

*Source: proof.py JSON summary*

## Full Evidence Table

### Type A (Computed) Facts

| ID | Fact | Method | Result |
|----|------|--------|--------|
| A1 | Primary check: counterexamples in [4, 10^6] via trial division | goldbach_trial_division() — isprime() per candidate | 0 counterexamples in [4, 1000000] |
| A2 | Cross-check: counterexamples in [4, 10^6] via prime-pair sieve | goldbach_prime_sieve() — primerange sieve + set lookup | 0 counterexamples in [4, 1000000] |

*Source: proof.py JSON summary*

## Computation Traces

```
even integers checked: (VERIFICATION_BOUND - 4) // 2 + 1 = (1000000 - 4) // 2 + 1 = 499999
trial division vs prime sieve counterexample count: 0 == 0 = True
no counterexamples in [4, 1000000]: 0 == 0 = True
```

*Source: proof.py inline output (execution trace)*

## Cross-Check Agreement

| Description | Value A | Value B | Agree | Independence |
|-------------|---------|---------|-------|--------------|
| Trial division vs prime sieve methods | 0 | 0 | True | Mathematically independent: Method 1 uses per-number isprime() calls; Method 2 pre-generates a prime set via sieve and uses set membership. Different algorithms, different data structures. |

*Source: proof.py JSON summary*

## Adversarial Checks (Rule 5)

### Check 1: Has any counterexample to Goldbach's conjecture ever been found?
- **Verification performed:** Searched: 'Goldbach conjecture counterexample found disproved'. Reviewed Wikipedia, Physics Forums, Medium articles, and LessWrong. No counterexample has ever been reported.
- **Finding:** No counterexample exists in the literature or computational records.
- **Breaks proof:** No

### Check 2: Has the Goldbach conjecture been formally proved or disproved?
- **Verification performed:** Searched: '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 the mainstream mathematical community. The conjecture remains open as of March 2026.
- **Finding:** No universally accepted proof or disproof exists. The conjecture is listed as an open problem by all major references.
- **Breaks proof:** No

### Check 3: What is the current computational verification bound?
- **Verification performed:** Searched: 'Goldbach conjecture verified computational bound 2026'. Oliveira e Silva verified up to 4 x 10^18 (2013). The Gridbach project (2025) extended this further. A March 2025 preprint reports empirical verification beyond 4 quintillion.
- **Finding:** Verified computationally up to at least 4 x 10^18. Our 10^6 bound is far below the literature record but uses two independent methods.
- **Breaks proof:** No

*Source: proof.py JSON summary*

## Hardening Checklist

- **Rule 1:** N/A — pure computation, no empirical facts.
- **Rule 2:** N/A — pure computation, no empirical facts.
- **Rule 3:** N/A — no time-dependent logic in this proof.
- **Rule 4:** CLAIM_FORMAL with operator_note explicitly documents that this is an open problem, the operator choice (== 0 counterexamples), and that the verdict is always UNDETERMINED.
- **Rule 5:** Three adversarial checks performed: searched for counterexamples, proof claims, and current verification bounds. None break the proof.
- **Rule 6:** N/A — pure computation, no empirical facts. Cross-check uses a mathematically independent algorithm (trial division vs. prime sieve).
- **Rule 7:** Computations use `compare()` and `explain_calc()` from `scripts/computations.py`. No hard-coded constants or inline formulas.
- **validate_proof.py result:** PASS (13/13 checks passed, 0 issues, 0 warnings)

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