Semantic and presentation cost for all terminal-descent moduli

Exact statement

For terminal-descent traces, $\eta_D(K_2)=\beta_D(K_2)=4$ , while for every $m\ge3$ , $\eta_D(K_m)=\beta_D(K_m)=2(m-1)$ . A uniform $2m$ -state height-residue/obligation controller attains the semantic lower bound for every modulus.

StatusProved and internally audited

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

Context

This theorem identifies the language-level access cost of the complete modulus family and gives a finite controller that attains it.

Definitions

$K_m=\{\epsilon\}\cup\{w\in D\setminus\{\epsilon\}:t(w)\equiv m-1\pmod m\}$ .

Hypotheses and scope

The modulus m is at least two and D is the one-bracket Dyck language.

Proof or evidence

At a nested entry height, every completion either uses the bare descent, which distinguishes one residue, or contains a later opener, which erases the incoming residue. This yields two contextual classes and the exact lower bound. A height-residue/obligation-bit controller realizes the same class arrivals.

All-modulus theorem →

Verification notes

The two-class completion proof survived an independent attack and was reconciled with exact controller traces, balanced products, and entry-distance computations.

Limitations

The 2m-state construction is an access-optimal upper bound, not a state-minimality theorem.
No positive sewing defect, novelty, or broader language-class consequence is claimed.

Open work

Determine the minimum number of states needed to attain optimal access beyond moduli two, three, and four.

Notes

The proof classifies every completion at a fixed entry height and gives a matching finite controller. It does not decide the minimum number of controller states needed to attain the optimal cost.