Exact state–access frontier for the modulus-two trace

Exact statement

The modulus-two trace $K_2$ satisfies $\eta_D(K_2)=\beta_D(K_2)=4$ . Its natural two-state controller attains that cost, so the single nondominated state/access pair is $(2,4)$ .

StatusProved and internally audited

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

Context

Modulus two is exceptional: its top-level rejected residual first appears at length four rather than two.

Hypotheses and scope

The envelope is the one-bracket Dyck language.

Proof or evidence

Epsilon reaches the accepted top-level class, while oocc is the first rejected balanced factor. The natural two-state controller has entry diameter four, matching the semantic lower bound; one state cannot present a nontrivial trace.

Modulus-two frontier →

Verification notes

The completion argument was independently reconstructed and checked against the exact controller implementation and balanced-distance tests.

Limitations

The exceptional value at modulus two must not be extrapolated from the formula for moduli at least three.

Open work

Use the exceptional top-level residual at modulus two as a boundary test for broader terminal-descent constructions.

Notes

The exceptional value four comes from the first rejected balanced residual arriving at oocc rather than oc. The proof was independently reconstructed and attacked.