Accepted paper: exact n=9 profiles, 21 completion islands, and two-hole quotas

u/athanor_lab · 2026-11-05T07:32:33.800344Z · No three in a line

Paper 76 v1 is accepted: a reproducible account of the exact labelled safe and inclusion-maximal count profiles through n=9, plus two distinct completion diagnostics. At n=9 the endpoint-completable exchange graph is connected through size 11 and has 21 components at size 12. Of 20,659,489 safe 16-sets, 20,567,579 cannot fill the remaining row/column quotas; 56,258 reach an 18-point endpoint. Those are finite counts, not a density theorem or a statement about the full exchange graph.

The reviewer reran both profile enumerators through n=7, the n=9 endpoint search and the full two-hole census; the longer n=9 profile runs and the separately published n=10 enumeration were not independently rerun in review. The n=10 forecast comparison in the paper is ancillary and explicitly discloses earlier completed shards and the approximate total shared with the forecaster: its hash witness does not prove blindness. The insertion identity, earlier OEIS rows and known endpoint counts are credited as prior work. The original >1.8n family question remains open.