Terminal-descent trace has zero sewing defect

Exact statement

Let (K_3) contain epsilon and the nonempty one-bracket Dyck words whose terminal run of closing brackets has length $2\bmod 3$ . Its semantic entry access and trace-minimax entry cost agree: $\eta_D(K_3)=\beta_D(K_3)=4$ . Hence its sewing defect is zero.

StatusProved; finite-state lower bound is computer-assisted and internally audited

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

Context

The example tests whether contextual behavior must be more expensive to sew into one finite controller. Here the semantic lower bound is attained, so the proposed positive defect disappears.

Definitions

$K_3=\{\epsilon\}\cup\{w\in D\setminus\{\epsilon\}:t(w)\equiv2\pmod3\}$ .
$t(w)$ is the length of the final consecutive run of closing brackets.

Hypotheses and scope

D is the one-bracket Dyck language.

Proof or evidence

The three-state controller resets on each opener and counts the terminal closer run modulo three. A semantic nested-context witness forces cost at least four; an explicit six-state controller attains four. Exact trace saturation proves equivalence.

Terminal-descent tradeoff →

Verification notes

The semantic witness and six-state presentation were independently reconstructed and checked by exact balanced-product equivalence and exact entry distances.

Limitations

The example has zero defect; it is not evidence for a positive defect.
The exact state/access frontier is recorded separately in FPRD-T55.
No novelty claim is made.

Open work

Determine whether any controlled-Dyck trace has positive sewing defect.

Notes

The example collapses a candidate positive sewing defect. It is an exact fixed-envelope theorem and carries no PEG, CFG, unrestricted-VPA, C001, or novelty claim.