Absolute minimization of the UF–FRP separation

Determine whether a finite monoid of size less than 36 admits a unit-separated, h-substitutable pair with exact unique relative prime factorization but without the finite relative presentation property (FRP).

Background

The paper constructs a 36-element finite observer quotient in which every live non-unit relative class has a unique exact prime factorization, while infinitely many valid prime-return rules occur; consequently, unique factorization does not imply FRP. The authors verify that every proper quotient of this particular 36-element monoid has FRP, establishing quotient-minimality only within that construction.

The unresolved issue is whether another finite monoid of smaller cardinality can realize the same UF-plus-not-FRP separation. The existing computation therefore does not provide an absolute lower bound of 36.