The published 367-code extension is the unique 128-state residual minimum

Exact statement

For coordinate order $(0,2,3,4,1)$ , the minimal partial DFA accepting the published 367-word code has residual-layer counts $(1,7,49,61,9,1)$ , hence 128 states and 424 transitions. Across all eight maximum extensions of the fixed 327-word core and all 120 coordinate orders, the published extension is the unique code attaining 128 states; the other minima are 129 or larger.

StatusMyhill--Nerode proof for each order; all 960 extension/order pairs checked exactly

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

Context

A small-state win is possible only in the word-path model; an injective last-signature image of size 367 itself needs at least 367 states.

Proof or evidence

Bottom-up equality of accepted-suffix sets gives the minimal acyclic partial DFA. Exhausting all coordinate orders gives the exact layer counts, while comparison of all eight maximum extensions singles out the code printed by Polak and Schrijver.

Residual-optimal published extension →

Verification notes

The raw trie has 1,059 states; the best residual quotient has 128 states and 424 transitions. Six coordinate orders attain that pair for the published extension, and none of the other seven extensions reaches 128 states.

Limitations

The 128-state automaton is not reversible: same-letter transitions merge at three residual layers.
Optimality is among the eight maximum extensions and coordinate permutations, not among arbitrary encodings or transducers.

Open work

Determine whether Gao's typed product operation admits a residual quotient computed compositionally from child packages.