Structural Recurrence and a Non-Halting Certificate for a Six-State Busy Beaver Turing Machine

This deposit records a self-contained executable non-halting certificate for the six-state, two-symbol Turing machine 1RB---_0LC1LD_0RD1LC_1RE0LB_0RF1RD_1RA0RD. The certificate was developed from the blank-tape trajectory by identifying and proving a recurrent symbolic configuration family. The resulting proof establishes reachability of an initial canonical epoch, exact symbolic successor rules within each epoch, closure of the overflow transition into the next epoch, and exclusion of the machine’s halting transition throughout the certified orbit. Together these results yield an infinite sequence of canonical returns and therefore non-halting. The deposited proof materials include the symbolic constructors, local transition certificates, dependency structure, executable verification code, and final non-halting theorem. The complete verifier passes on the frozen certificate bundle. The final certificate archive was recorded with SHA-256: 804d325fd16e5dcda2ceabf1feac6e00cad2c6c74f4da3e05460297d3466ae55 Following completion of the executable certificate, the non-halting result was independently confirmed in Rocq by mxdys, providing a separate formal verification of the machine’s non-halting behaviour. The discovery process used a structural-recurrence methodology developed within the Soma–Noēsis (SN) research programme. SN was used as a discovery and structural-compression layer to identify recurrent behaviour beneath the rapidly changing raw Turing-machine configurations; the final certificate itself is expressed entirely in conventional Turing-machine transition mathematics and can be checked independently of SN. This deposit is intended to establish a dated research record of the result, certificate, and verification status. The underlying proprietary SN mathematical framework and discovery machinery are not disclosed in this release.