The five-state extremal has a nine-state target quotient

Exact statement

For the 14-element five-state extremal generated by a=(2,1,2,3,1) and b=(0,1,3,4,4), minimizing the left-Cayley poDFA with sole accepting element f=(4,1,1,1,1) gives a nine-state poDFA action whose transformation fibre over f is exactly {f}.

StatusProved by exact partition refinement and fibre audit

External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The example shows target isolation collapsing five of fourteen Cayley states on the first counterexample to the n-minus-one conjecture.

Proof or evidence

Exact Moore refinement of the left-Cayley automaton produces nine classes. The quotient reachability relation is antisymmetric, and equality with the target-induced quotient transformation has singleton fibre inside the generated monoid.

Nine-state extremal quotient →

Verification notes

A separate tuple/set implementation independently generated all fourteen elements, minimized the singleton-target automaton, checked the quotient order, and tested the full transformation fibre.

Limitations

The construction first enumerates the whole monoid, so it is not a polynomial-time algorithm for the original compact input.
One compressed extremal does not establish a polynomial degree bound.

Open work

Find a quotient construction whose state space and fibre proof are obtained directly from the supplied n-state action rather than from the enumerated monoid.