The direct pentagon block expression has eleven prefix states and a six-state reversible quotient

Exact statement

For $R_5=(a_0a_0+a_1a_2+a_2a_4+a_3a_1+a_4a_3)^*$ , the Broda--Maia--Moreira--Reis backward prefix construction has the eleven states $\varepsilon$ , $R_5a_i$ , and $R_5a_i a_{2i}$ . Merging $\varepsilon$ with the five completed-block states gives a minimal six-state partial reversible DFA for the same language.

StatusExact symbolic derivation and independent automaton audit

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

Context

This corrects the tempting but invalid transfer of the synchronized macro-event compiler's six-state count to the direct two-letter word expression.

Proof or evidence

The backward recursion closes on one initial, five halfway, and five completed-block prefix states. Quotienting the six block-boundary states gives the standard reversible block DFA.

Eleven-to-six direct block compiler →

Verification notes

The natural automaton and quotient agree on all 488,281 alphabet words of lengths zero through eight; exactly 781 are accepted. Reversibility and distinguishability witnesses for minimality pass exactly.

Limitations

The six-state quotient is the classical block generator recorded by Meiburg, not a new capacity certificate.
The natural prefix automaton itself is not reversible.

Open work

Compare natural-prefix and compact-generator size only after fixing a uniform resource model.