Exact state–access frontier for the modulus-three trace

Exact statement

The minimum controller size for $K_3$ is three, while the minimum size among presentations with legal-entry diameter at most four is six. Its exact nondominated state/access pairs are therefore $(3,6)$ and $(6,4)$ .

StatusProved; the five-state exclusion is computer-assisted and internally audited

External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

This separates state minimization from access minimization for one fixed Dyck trace.

Hypotheses and scope

The envelope is the one-bracket Dyck language.
Controller cost is legal-entry balanced diameter.
Unused-state padding is allowed in the finite-state lower-bound encoding.

Proof or evidence

Exact trace comparison rules out controllers below three states. A three-state controller has cost six, a six-state controller has cost four, and an exact closed-relation encoding is unsatisfiable through five states at cost four.

Modulus-three frontier →

Verification notes

The six-state witness was independently reconstructed. The lower bound uses the trusted Z3 4.16 kernel and reviewed encoding, with independent CEGIS cross-checks; it is not a proof-assistant certificate.

Limitations

The theorem concerns one fixed trace and one fixed Dyck envelope.
It does not imply a similar state frontier for every modulus.

Open work

Produce a portable unsatisfiability certificate if the finite-state encoding is revisited.

Notes

The unsatisfiability result is computer-assisted and trusts Z3 4.16 plus the reviewed finite encoding. It is not a formally checked proof certificate or a claimed general tradeoff theorem.