Arbitrary unary synchronization has an atomic left boundary

Exact statement

For $m,n\ge1$ , under arbitrary synchronization on $\Gamma=\{a\}$ , the natural prefix automaton of $(a^m)^*\mathbin{\amalg^{\Gamma}_{a}}(a^n)^*$ is an ordinary existential quotient of its location automaton exactly when $\min(m,n)=1$ , and it is a left-invariant quotient exactly when $m=n=1$ .

StatusSelf-contained necessity and constructive sufficiency proofs with exhaustive boundary audit

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

Context

Arbitrary synchronized moves coexist with the solo moves responsible for the pure-shuffle obstruction, but the added diagonal edge changes the exact quotient boundary.

Hypotheses and scope

Both components are unary starred words.
The synchronization alphabet is exactly $\{a\}$ .
The quotient uses existential block transitions.

Proof or evidence

When m,n≥2, the three initial successors must merge, but an edge among them creates a target self-loop that does not exist. If one exponent is 1, an explicit row-or-column projection gives an ordinary quotient. Predecessor profiles rule out left invariance unless both exponents are 1, where the single noninitial block works.

Arbitrary-synchronization classification →

Verification notes

Exact parameter tests cover all 576 pairs with 1≤m,n≤24; exhaustive partition search checks 10,534 partitions across six smallest boundary instances, and an independent constructor reproduces the forbidden-self-loop control.

Limitations

The result uses the FPRD last-event prefix definition FPRD-SH-D02.
No external review or priority claim is recorded.

Open work

Test whether the forced-initial-block obstruction extends to non-unary arbitrary synchronization.