Finite shadows reconstruct the Dyck residual dynamic

Exact statement

The depth- $N$ quotient keeps every stack of height at most $N$ and places all deeper stacks and failure in one tail class. It has $1+\sum_{h=0}^{N}k^h$ states, and its inverse limit reconstructs exactly the live finite stacks and failure. A word of length $r$ induces a natural graded map from level $N+r$ to level $N$ ; a same-level action by a closer is impossible in general.

StatusProved and internally audited

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

Context

A finite observation depth distinguishes every shallow stack literally and treats every state requiring a longer witness as one unresolved tail.

Definitions

$X_{k,N}=\{s\in[k]^*:|s|\le N\}\sqcup\{\mathrm{tail}_N\}$ .
The quotient size is $g_k(N)=1+\sum_{h=0}^{N}k^h$ ; for $k=1$ this is $N+2$ , and for $k\ge2$ it is $\Theta(k^N)$ .

Hypotheses and scope

The bonding maps retain a shallow stack and collapse anything deeper to the lower-level tail.
An action by a word of length r is compared across the shifted levels N+r and N.

Proof or evidence

A coherent inverse-limit thread either stabilizes at one finite stack or remains in every tail, which is exactly failure. The level shift is necessary because a closer can expose one symbol of information: a height-N+1 stack and failure coincide at level N but a matching closer distinguishes them there.

Finite shadows and reconstruction →

Verification notes

The reconstruction, state count, Boolean/full-suffix quotient equivalence, pair-separation spectrum, and same-level obstruction were checked in the M21 proof and closure audit.

Limitations

One finite shadow is an approximation, not the exact residual system.
The tail class combines different deep stacks with failure at that observation depth.

Open work

Publish a diagram of the bonding maps and one explicit failure of same-level closer action.

Notes

The degree shift is essential: matching pop can expose one previously hidden stack frame, so closer actions generally do not descend as endomorphisms of a single bounded quotient.