Strong unary synchronization restores the left quotient

Exact statement

For $m,n\ge1$ , under strong synchronization on $\Gamma=\{a\}$ , the accessible location automaton and the natural prefix automaton of $(a^m)^*\mathbin{\amalg^{\Gamma}_{s}}(a^n)^*$ are isomorphic cycles of length $\operatorname{lcm}(m,n)$ , with their initial state attached. In particular the prefix automaton is a left-invariant quotient of the accessible location automaton.

StatusSelf-contained orbit proof with parameterized executable audit

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

Context

Strong synchronization removes the one-sided advances that created the predecessor defect in pure shuffle.

Hypotheses and scope

Both components are unary starred words.
The synchronization alphabet is exactly $\{a\}$ .
Only the accessible part of the syntactic location automaton is compared.

Proof or evidence

Every noninitial accessible location lies on the diagonal orbit $(i,j)\mapsto(i+1\bmod m,j+1\bmod n)$ , whose length is $\operatorname{lcm}(m,n)$ . The last-event prefix recursion produces the same orbit, so the identity correspondence is an automaton isomorphism and hence a left quotient.

Mandatory-synchronization theorem →

Verification notes

Exact construction confirms the isomorphism for all 576 pairs with 1≤m,n≤24; an independent event-sequence oracle also agrees with both automata throughout 4,536 bounded word-length comparisons shared with the arbitrary and weak audits.

Limitations

The theorem is stated for the accessible automaton; unreachable syntactic product locations are not silently discarded inside a quotient.
No external review or priority claim is recorded.

Open work

Determine which non-unary strong synchronized families retain the same accessible-orbit quotient.