Backward receipt for weak synchronized prefixes

Exact statement

Define $u\overleftarrow{\bowtie}_{\Gamma,D,E}v=\{z^R:z\in u^R\bowtie_{\Gamma,D,E}v^R\}$ , where the published auxiliary operator on the right records solo synchronized letters since the previous joint event. The reversal-dual parameters record solo letters before the next joint event and give a correct final-event $R_\varepsilon$ -recursion for weak synchronized shuffle.

StatusSelf-contained reversal proof with exact unary graph audit

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

Context

The cited weak location automaton carries history from the previous synchronization; prefix recursion needs the opposite temporal boundary.

Definitions

At $D=E=\varnothing$ , reversal preserves the weak word relation because every block between joint events remains one-sided after reversal.
A joint terminal event resets $D,E$ ; a solo terminal event adds its letter to the corresponding set and is forbidden if the other set already contains it.

Proof or evidence

Reversal converts each final event into the corresponding first event of reversed component words, so the three final-event clauses are exactly the published forward clauses in dual orientation.

Backward-receipt definition and proof →

Verification notes

The induced unary N/L/R receipt graph was generated independently and used by the exact quotient and determinization checks.

Limitations

This is an FPRD definition, not a construction attributed to the source authors.
The public theorem application is currently restricted to unary cyclic starred components.

Open work

Test whether the receipt remains finite and useful for non-unary or nested weak shuffles.