Word derivatives have sharp universal exposure

Exact statement

On the full language carrier $\mathcal P(\Sigma^*)$ , the left derivative $\delta_u(K)=u^{-1}K$ has exposure at most $|u|$ , and this universal bound is sharp. The completed derivative orbit of any language has the same finite images as its literal derivative orbit and is reconstructed exactly from them; for regular languages the tower stabilizes to the usual finite residual automaton.

StatusProved and internally audited

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

Context

A derivative by a word u shifts a finite language table by exactly |u| input symbols, making it the basic calibration of observational exposure.

Definitions

$\delta_u(K)=\{v:uv\in K\}$ .
The level-N observation of a language is its membership table on words of length at most N.

Hypotheses and scope

The sharpness statement is on the full carrier of all languages over the fixed finite alphabet.
Orbit completion is taken in the inverse-limit topology generated by finite membership tables.

Proof or evidence

A table through length N+|u| determines the derivative table through N. The pair K={u} and H=empty agrees below the required depth but is separated after deriving by u, proving universal sharpness. Finite discrete images of an orbit equal those of its closure, yielding the exact completed-orbit reconstruction.

Derivative calibration →

Verification notes

The proof, sharpness witness, orbit-closure argument, regular-language stabilization, and generalized-Dyck exposure-one calibration were rechecked against the M22 audit.

Limitations

Sharpness on the full carrier does not imply sharpness on every selected invariant closure.
The theorem concerns left derivatives and the chosen bounded-word observation tower.

Open work

Determine exact exposure on selected invariant language closures rather than relying on the full-carrier universal bound.

Notes

The all-language derivative coalgebra is classical. M22 uses it as the exact recognition calibration for the exposure-filtered finite-shadow architecture, not as a novelty claim for language derivatives.