Funded shared-family study — 2026-10-14

The shared study derived exact local period-seven sweep rules, built a reduced word interpreter, and identified three concrete edge-return families. R1 has an informal universal derivation; R2 and R3 remain finite-tested candidates. The variable edge word still has no closed grammar or invariant. No whole-machine proof was started and no kill is claimed.

The full derivations, parameter restrictions, finite checks, and remaining obstruction are in the canonical family study. Its formulas use the reflected line-6 orientation; they are not written in this machine's orientation.

Reflecting the tape and renaming A→E, B→B, C→C, D→A, E→D, F→F maps the line-6 table to this one. This machine's blank-A start corresponds to canonical blank D, not canonical blank A. Canonical blank D reaches A on the first zero of 0101 at position −3 after five defined transitions. That separate start was retained in every replay.

The reduced interpreter matched all 141 saved slow-edge configurations plus the independent C endpoint at 100,000,000 steps. After reflection, 46 of those edge snapshots have state A; the exact prefix rules explain 24 of the 45 consecutive A-edge returns. The final return checkpoint has no next recorded return and is excluded from that denominator. Removing the maximal periodic suffix leaves a 141-bit irregular prefix at the final checkpoint.

The three general edge-return families were checked as constructed canonical configurations; R1 also has an informal universal derivation. They apply under reflection with the corresponding state renaming, but the study does not claim that the three particular edge words occur in this machine's own blank-A run. The whole-run initial obligation remains separate.

Recommendation: keep whole-machine formalization parked until a closed transformation of the growing edge word is established. Additional fixed-prefix examples alone will not settle it. Mechanism confidence remains medium, and no pseudo-random arithmetic recurrence or confirmed cryptid classification was established.

AI assistance: these observations, scripts and checks were produced with the help of GPT-6 Astra (OpenAI).