Semantic lower bounds for presentation cost

Exact statement

For a controlled-Dyck language, entry-context residual access $\eta_D$ , top-level balanced residual access $\gamma_D$ , and first balanced acceptance flip $\tau_D$ satisfy $\beta_D\ge\eta_D\ge\gamma_D\ge\tau_D$ . The value $\eta_D$ is exactly computable by finite completion-summary saturation. Together with any minimized presentation it yields a finite minimax sandwich; endpoint equality certifies optimality. The defect $\sigma_D=\beta_D-\eta_D$ is nonnegative and limit-computable from above, and zero is semidecidable.

StatusProved and internally audited

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

Context

The hierarchy distinguishes a quick change of acceptance from delayed top-level residual behavior and from delayed behavior visible only inside nested legal contexts.

Definitions

$\tau_D$ is the shortest balanced word changing the acceptance value at the empty boundary.
$\gamma_D$ is access to balanced residual classes at the top level.
$\eta_D$ takes the worst balanced-factor class arrival over every legal entry context.
$\sigma_D=\beta_D-\eta_D$ measures any remaining gap between separately observable entry behavior and one finite controller realizing it all.

Hypotheses and scope

The Dyck envelope is fixed and L has a finite controller presentation.

Proof or evidence

Every semantic class arrival forces a balanced controller distance. A finite paired-controller saturation summarizes which endpoint pairs are separated by a legal completion of each unfinished stack. Punctured-threshold and first-component families prove that both inequalities below beta can be strict.

Semantic access hierarchy →

Verification notes

Production tests, exact trace comparisons, completion-summary saturation, 103,168 literal membership cells, seven mutation separators, and independent implementations were reconciled. The review explicitly retains the open beta_D=eta_D boundary.

Limitations

No positive sewing defect is known, and universal equality is not claimed.
A positive sandwich gap for one supplied presentation does not prove that the presentation is nonoptimal.
No PEG, CFG, unrestricted-VPA, or novelty consequence is claimed.

Open work

Find either a positive sewing-defect example or a proof that every controlled-Dyck trace has zero defect.

Notes

The bad-presentation family demonstrates necessity of minimization, not a language obstruction. Equality beta_D=eta_D is not claimed in general. No novelty or PEG/CFL consequence is claimed.