Exact state–access frontier for the modulus-four trace

Exact statement

The modulus-four trace $K_4$ satisfies $\eta_D(K_4)=\beta_D(K_4)=6$ . Its natural state-minimal four-state controller has cost eight; an explicit six-state controller has cost six; and no controller with at most five states attains cost six. The exact frontier is $(4,8)$ and $(6,6)$ .

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

Modulus four repeats the state-versus-access separation from modulus three, but with a different semantic optimum and natural-controller cost.

Hypotheses and scope

The envelope is the one-bracket Dyck language.
Controller cost is legal-entry balanced diameter.

Proof or evidence

The semantic completion argument gives cost six. Exhaustive comparison rules out trace presentations below four states; the natural controller realizes (4,8), and an explicit six-state controller realizes (6,6). Exact CEGIS excludes cost six through five states.

Modulus-four frontier →

Verification notes

A separate one-hot CEGIS encoding reached unsatisfiable at five states, and the six-state witness was independently reconstructed and checked. Both lower-bound routes still trust Z3 rather than a proof-assistant certificate.

Limitations

The result does not establish the minimum optimal-cost state count at modulus five or any general even/odd law.

Open work

Seek a checkable certificate for the five-state exclusion and avoid extrapolating an unproved minimum-state law.

Notes

The generalized CEGIS decision and independent one-hot decision are exact but still trust Z3. This is not a proof-assistant certificate.