Exact profiles for depth thresholds

Exact statement

For generalized-Dyck words whose stack height reaches at least $r\ge1$ , $q_{k,r}^{\ge}(N)=1+\sum_{h=0}^{N}k^h+\sum_{h=\max(0,2r-N)}^{r-1}k^h$ and $a_{k,r}^{\ge}(N)=\max(2r,N)$ . For words staying below (r), $q_{k,r}^{<}(N)=1+\sum_{h=0}^{\min(N,r-1)}k^h$ and $a_{k,r}^{<}(N)=\max(1,\min(N,r-1))$ .

StatusProved and internally audited

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

Context

These families isolate the cost of remembering whether a nesting threshold has been crossed while retaining the exact typed stack.

Definitions

$D_{k,\ell}^{\ge r}$ contains balanced words whose stack reaches height at least r.
$D_{k,\ell}^{<r}$ is the complementary below-threshold branch inside the Dyck language.

Hypotheses and scope

Bracket rank k and threshold r are positive integers.

Proof or evidence

Post-threshold rows are indexed by visible typed stacks. A pre-threshold stack of height h becomes visible exactly when the horizon reaches 2r-h. Shortest height paths give the access formulas.

Depth-threshold profiles →

Verification notes

All pre/post, typed-stack, cross-height, horizon-zero, threshold-one, and neutral-symbol cases were checked in the proof and independent enumerations.

Limitations

A fixed horizon-zero row count of two does not bound access across the family: the value 2r is unbounded.
The theorem makes no novelty claim.

Open work

Use the threshold families as calibrations when comparing access certificates across equivalent controllers.

Notes

This is an exact finite-controller family theorem. It does not imply that scalar count determines access, and it makes no novelty claim.