Research — the binary Goldbach program

An ongoing research program aimed at the binary Goldbach conjecture: every even $N > 2$ is a sum of two primes, $N = p + q$ .

The program

The goal of the program is an unconditional proof of the binary Goldbach conjecture: for every even integer $N > 2$ there exist prime numbers $p$ and $q$ such that $N = p + q$ . All work presented in this section is organized around that single question.

The program operates in what we call an asset-forensics mode. Every claim entering the project is recorded with an explicit status label; negative results are preserved with the same care as positive ones — a refuted approach remains on record together with the reason it fails; and machine verification — formal proofs in Lean 4 and certified interval arithmetic in numerical work — is used wherever possible. The methodology page describes this discipline in detail.

Current status

The conjecture remains open. Open problem Nothing on this site should be read as a claimed proof, or as a claim that a proof is close. The central obstruction identified so far is the fixed- $N$ parity barrier: classical sieve methods do not distinguish primes from products of two primes once the target $N$ is fixed. The parity-barrier page explains why existing methods stall at this point and maps the obstructions in detail.

Pages in this section

The problem and its status — the statement of the conjecture, its history, and the current frontier in the literature.
Results — the partial results of the program, each recorded with an explicit status label.
The parity barrier — why existing methods stall, with a map of the obstructions.
Formalization in Lean 4 — the machine-checked components of the program.
Numerical experiments — empirical observations, clearly separated from proof.
Methodology — how claims are recorded, challenged, and retired.

Suggested reading order. Specialists may wish to begin with the problem statement and the parity-barrier page, then turn to the results and their machine-checked counterparts. General readers may prefer to start with the problem page and the numerical experiments, which give a concrete picture of the conjecture, before the more technical material.