Unary quotient without left invariance

Exact statement

For $a^*\amalg a^*$ , merging the three noninitial location states gives the two-state prefix automaton as an ordinary existential quotient, but no such quotient partition is left-invariant.

StatusExact proof and exhaustive partition check

External reviewNo documented external or specialist review of these FPRD results is recorded.

Context

Removing the distinct-label obstruction restores an ordinary quotient but not the stronger predecessor compatibility used in the 2019 and 2021 quotient architecture.

Hypotheses and scope

Both shuffle operands are the same unary starred letter.

Proof or evidence

All noninitial locations must form one block. A one-coordinate location has the initial block as an a-predecessor, while the two-coordinate location does not.

Unary control and predecessor defect →

Verification notes

Direct enumeration of the unique labelled map confirms the ordinary quotient and rejects left invariance; the independent checker is published with FPRD-SH-T06.

Limitations

This is the smallest instance; FPRD-SH-T06 supplies the unary and full starred-word classification.

Open work

Retain this as the smallest exceptional case in the all-length proof FPRD-SH-T06.