Complementing permutations compile to reachable twisted sections

Exact statement

Let $G=J_\psi$ be the conflict graph of a one-support-per-letter policy, use optional-once component expressions, and let $\pi:G\to\overline G$ be a complementing permutation. The synchronized product with a $\pi$ -relabelled copy has exactly $1+|V(G)|$ reachable natural-prefix states, and its last-signature section $\{(v,\pi(v)):v\in V(G)\}$ is independent in $G^{\boxtimes2}$ .

StatusSelf-contained all-length proof; exhaustive complementing-permutation audit passes

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

Context

This separates the expression model from the earlier support-tensor dictionary: the code is now the image of reachable product states under a last-signature projection.

Proof or evidence

The nonedge-to-edge half of complementing makes all composite supports overlap, leaving one initial and one state per macro-letter. The edge-to-nonedge half makes the projected permutation graph independent in the strong square.

Twisted-section compiler →

Verification notes

All 265 complementing-permutation certificates among the 1,099 labelled graphs through five vertices pass 2,544 independence and 2,544 synchronized-blocking pair checks.

Limitations

The underlying self-complementary graph code is classical.
C7 is not self-complementary, so this compiler does not produce its known ten-word square code or 367-word fifth-power code.
Novelty of the natural-prefix compiler and exact resource statement is plausible but unconfirmed.

Open work

Use self-complementary Paley graphs as further square calibrations; use the separate path compiler FPRD-SH-X06 for C7.