Cyclic pair synchronization has a five-over-four support packing

Exact statement

Give letter $a_i$ the adjacent-pair support $\{i,i+1\}$ on a five-participant cycle. Then $J_\psi=C_5$ , $\Omega(\psi)=2$ , and the tensor-square signatures $\{(i,2i\bmod5):i\in\mathbb Z_5\}$ are pairwise compatible, so $\Omega(\psi^{\otimes2})=5>4=\Omega(\psi)^2$ and $\Omega_\infty(\psi)=\sqrt5$ .

StatusExplicit five-signature certificate; exact Lovász upper bound

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

Context

This is the smallest support policy whose Cartesian tensor packs more disjoint supports than the product of optimal one-layer packs.

Proof or evidence

Adjacent-pair supports intersect exactly along C5. The five displayed words form an independent set in the strong square, and the multiplicative Lovász bound gives the matching upper bound five.

Pentagon policy split →

Verification notes

All cycle-overlap edges for C5, C7, and C9 and all ten pairs in the displayed C5 square code pass exact checks.

Limitations

The capacity theorem and code are classical; novelty is only the automata interpretation.
This claim is policy-level; the separate reachable expression realization is FPRD-SH-X04.
For the seven-cycle policy, the current preprint lower certificate is an explicit recursive code in the two-hundredth tensor power.

Open work

Use FPRD-SH-X04 for the expression-level realization; next compile the typed recursive C7 gadget rather than enumerating its output.