Honesty audit of the round-1 reduction to 12 graphs. no-K6 is elementary (a unit K6 is a regular unit 5-simplex, needs R^5). The K_{1,3,3} prune is LOAD-BEARING: re-running the nauty enumeration, of the 74 graphs passing the other filters (independence(H)<=5 and complement maximal-triangle-free) only 12 survive the K_{1,3,3}-free filter -- the prune removes 62. So the reduction to 12 depends on K_{1,3,3} being non-realisable in R^4 (BPSSV Lemma 11: the three colour classes lie on mutually orthogonal circles spanning >=4 dimensions, forcing 4 points onto a circle that then meets a unit circle in 3 points -- a contradiction; re-derived and found correct). Independently, our machinery shows K_{1,3,3}'s complex variety is positive-dimensional, i.e. the obstruction is real-geometric, matching BPSSV.