Weak unary synchronization always restores an ordinary quotient

Exact statement

For every $m,n\ge1$ , the backward-receipt prefix automaton of $(a^m)^*\mathbin{\amalg^w_{\{a\}}}(a^n)^*$ is an existential state quotient of its accessible weak location automaton.

StatusAll-length proof with closed-form quotient and exact parameter audit

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

Context

Weak synchronization distinguishes the side used in the current block, giving the prefix automaton enough receipt state to accept a quotient for every unary cycle pair.

Proof or evidence

The quotient maps paired neutral states identically, shifts paired left-receipt states by one in the right coordinate, shifts paired right-receipt states by one in the left coordinate, and merges the two boundary cycles into the resulting seams.

Weak ordinary-quotient theorem →

Verification notes

Exact equality of quotient and target states, edges, initial set, and final set passes for all 1,024 pairs with 1≤m,n≤32; an independent event-sequence oracle confirms the automata's word-level semantics on the audited bounded range.

Limitations

The quotient is ordinary, not left-invariant.
The theorem is restricted to unary cyclic starred components and the FPRD receipt definition.

Open work

Search for a non-unary family in which the shifted receipt projection remains exact.