Synchronized macro-events and two-letter path blocks are different compilers

Exact statement

The reachable six-state construction of FPRD-SH-X04 presents each pair $(i,2i)$ as one synchronized macro-event. The direct word expression $R_5=(a_0a_0+a_1a_2+a_2a_4+a_3a_1+a_4a_3)^*$ presents the same code as two original-alphabet events along a path. Its length- $2m$ slice is the classical independent language $C^m$ , where $C=\{00,12,24,31,43\}$ . Reachability of the former does not determine the natural-prefix state count of the latter.

StatusSelf-contained comparison; the graph language is explicit in the primary automata-capacity literature

External reviewThe graph-language construction is peer reviewed; the FPRD prefix comparison has no external review.

Context

The two constructions encode the same classical C5 square code but expose it at different semantic layers.

Proof or evidence

The synchronized compiler uses five composite signatures in one step. The direct expression uses two original letters per block; concatenation gives the square-root-five graph language used by Meiburg.

Macro-event versus path compiler →

Verification notes

All ten codeword pairs and all 488,281 alphabet words of lengths zero through eight pass the independent language check; exactly 781 of those words are accepted.

Limitations

Neither construction improves Shannon capacity.
Both list the five codewords explicitly and therefore do not establish succinct asymptotic generation.

Open work

Fix a resource measure that does not count a hard-coded block expression as an automata-specific capacity improvement.