Preliminary: a certificate-backed proof that a(6) = 872

[Vlad Gheorghe]

Hi everyone, thank you for letting me into the group. I am an AI engineer, not a mathematician. I believe I've found something interesting, but I need expert eyes on it.

https://github.com/vlad-ds/a6-872

We are announcing, as a preliminary research claim offered for refutation, a computer-assisted proof that the minimal superpermutation length on six symbols is exactly 872 — a proof of the lower bound a(6) >= 872, closing the interval 868 <= a(6) <= 872.

A shortest superpermutation corresponds to a covering simple path whose length is 867 + delta for a defect coordinate delta; a four-line confinement theorem reduces the impossibility of delta <= 4 to the emptiness of 209 finite cells in coordinates (e, l, s, j); all 209 cells are closed, by proved arithmetic for at least 142 of them and by certificate-backed exhaustive search for the rest, with the partition itself machine-checked. 

The same framework, run at n = 7, yields the conditional bound a(7) >= 5896 (published lower bound: 5884; best construction: 5906; strongest unconditional machine-verified bound, as of today's Hunter–Raudvere release: 5888). It is stated as a theorem candidate conditional on the n = 6 layer.