Weak unary synchronization never restores a left quotient

Exact statement

For every $m,n\ge1$ , the backward-receipt prefix automaton of $(a^m)^*\mathbin{\amalg^w_{\{a\}}}(a^n)^*$ is not a left-invariant quotient of its weak location automaton.

StatusAll-length determinization-orbit proof with exceptional-case audit

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

Context

The backward receipt repairs the ordinary state quotient but cannot erase the two one-sided location boundaries in a left-compatible way.

Proof or evidence

After saturation, the prefix subset orbit is fixed, while the location subset orbit retains one left boundary and one right boundary and has least period $\operatorname{lcm}(m,n)$ . At $(1,1)$ , where that period is one, the location orbit still has one additional transient subset.

Weak left-quotient obstruction →

Verification notes

Complete subset orbits match the formulas for all 256 pairs with 1≤m,n≤16; an independent implementation confirms the exceptional (1,1) transient and exhaustive partition search finds no left quotient in the three smallest boundary cases.

Limitations

The obstruction concerns this location/prefix pair, not every automaton recognizing the same language.
No external review or priority claim is recorded.

Open work

Test eventual subset-period mismatch as an obstruction for non-unary synchronized products.