== proposed (projection)
   group               states   rho_n >= rho_(n-1)
   APS                   1429                37.6%
   CHA fun1                83                28.9%
   CHA flat fun3-6       1763                28.2%
   CHA other              237                25.3%
   AUC of S, flat states against:
     APS            0.422
     CHA fun1       0.409
     CHA other      0.386

== reading (3), as for m>=2
   group               states   rho_n >= rho_(n-1)
   APS                    990                27.5%
   CHA fun1                36                27.8%
   CHA flat fun3-6        659                31.3%
   CHA other              204                16.7%
   AUC of S, flat states against:
     APS            0.284
     CHA fun1       0.345
     CHA other      0.229

bar to beat (three-point order estimate, m >= 2): 0.2% canonical, 5.8% flat