Canonical-root factor search for Goldbach representations
Abstract
A failed primality test can provide new candidates for a Goldbach search: factor the complement and test its distinct prime factors in turn. We study full breadth-first search under this rule, started at the least prime p0(N) >= N/2. A freshly regenerated exhaustive scan of all 4,999,999 even inputs 4 <= N <= 10^7 finds no canonical-root failure. With the successful endpoint included, the mean number of tested vertices is 7.0527, its maximum is 151, and the maximum first-success depth is 10. Changing only the initial prime produces genuine failures: a complete upper-half-root census through 2 * 10^4 has 301 composite-complement failures at six inputs, none from p0. Exact same-input certificates distinguish eventual failure from shallow delay and show that neither p0 nor p1 universally minimizes success depth. Elementary closure arguments identify the unresolved reachability question. The computations do not prove universal canonical success or the binary Goldbach conjectureAI assistance. Generative AI systems were used during the exploratory research process to propose, test, and critique mathematical arguments, assist with computational experiments, and support manuscript preparation. All claims included in this note were checked against the stated computations or proofs before inclusion. The author remains responsible for the content.
