Left Green height is not target word distance

Exact statement

Among all 32,896 unordered transformation-generator pairs on four states, 4,473 generate L-trivial monoids. Their maximum left-Cayley height is 6, while the maximum shortest right-Cayley distance of any target across the same search is 3; ideal height is therefore a strict overestimate of target certificate length.

StatusExact finite computation; semigroup control rather than a minimal-DFA classification

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

Context

Exponential Green height in unrestricted transformation semigroups was a tempting route to long certificates, but the relevant quantity is target reachability inside an L-trivial action.

Hypotheses and scope

All unordered generator pairs, including equal generators, on four labelled states are included.
L-triviality is tested exactly on each finite generated monoid; no accessibility or DFA-minimality filter is imposed.

Proof or evidence

Exact closure and Cayley-graph computations give height histogram 1:122, 2:1218, 3:2112, 4:636, 5:360, 6:24, plus the trivial height-zero monoid. Rank-conditioned target distances never exceed three.

Four-state Green-height control →

Verification notes

The executable output records every aggregate count and an explicit ten-element height-six example.

Limitations

This finite control does not exclude a useful asymptotic relation between height and distance under additional hypotheses.
The extremal pair is not claimed to present a minimal four-state DFA.

Open work

Use extension-distinguishable target profiles or actual target geodesics rather than principal-ideal height as the obstruction measure.