Exposure-filtered finite observation

Exact statement

A complete finite-observation tower is a compact Hausdorff state space $X$ equipped with separating continuous surjections $q_N:X\to X_N$ to finite discrete quotients and compatible bonding maps. An action $F:X\to X$ has exposure at most $e$ when equality at observation level $N+e$ always forces equality of the images at level $N$ .

StatusPrecisely stated and self-contained

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

Context

Exposure measures how many extra levels of finite input information are needed to predict an action at a chosen output resolution.

Definitions

The quotients satisfy $q_N=\pi_{N+1,N}\circ q_{N+1}$ , and equality in every quotient separates points of $X$ .
Exposure $e$ means $q_{N+e}(x)=q_{N+e}(y)\Rightarrow q_N(Fx)=q_N(Fy)$ for every $N$ .

Hypotheses and scope

The exact state space is compact Hausdorff.
Each finite observation space is finite discrete and each quotient map is continuous and onto.

Proof or evidence

These assumptions make the canonical map from X into the inverse limit continuous and bijective; compactness and Hausdorffness then make it a homeomorphism.

Finite observations and exposure →

Verification notes

The definition and completeness assumptions were reconciled with the M22 specification, proof, and closure audit. On 9 September 2026, a worked binary-digit example was checked in separate same-model passes for exact exposure, continuity, tower dependence, and boundary cases; this is not external expert validation.

Limitations

Continuity alone need not give one finite exposure bound across all levels.
The filtration depends on the selected observation tower.

Open work

Find a useful application of the exposure distinction; the worked binary-digit example now explains zero, finite, and unbounded exposure.

Notes

Exposure is an upper bound and is monotone in e. Least exposure, when it exists, is a derived invariant. The definition records a uniform additive observation budget rather than an arbitrary continuity modulus.