FPRD-T158: Finite future observers internalize into stationary transport

1. Setting

Let a finite scaffolding automaton $A$ recognize a language $L\subseteq\Sigma^*$. Write its resource tuple as

$$(d,k,g,q),$$

where $d$ is node degree, $k$ is descriptor and observation distance, $g$ is the working-alphabet size, and $q$ is the number of control states.

For $h\ge 0$, put

$$R_{k,h}= \begin{cases} 0,&h=0\text{ or }k=0,\\ k+(h-1)(k-1),&h\ge 1\text{ and }k\ge 1. \end{cases}$$

FPRD-T153 proves that every answer after a continuation of length at most $h$ is determined by the current control and the unfolded old-root view of radius $R_{k,h}$. The radius is uniformly sharp.

2. Finite future observers

Fix a nonempty finite set $U\subseteq\Sigma^*$ and a Boolean function

$$b:\{0,1\}^{U}\to\{0,1\}.$$

The associated future-observer language is

$$\operatorname{Obs}_{U,b}(L) =\{p\in\Sigma^*:b((1_L(pu))_{u\in U})=1\}.$$

This operation asks, after every prefix, for a fixed finite vector of future answers and then applies an arbitrary Boolean decision rule.

Let

$$D_U=\max_{u\in U}R_{k,|u|+1}.$$

3. The theorem

If

$$(d,k,g,q)\in\operatorname{Tr}(L),$$

then

$$(d,D_U,g,q\,2^{|U|}) \in\operatorname{Tr}(\operatorname{Obs}_{U,b}(L)).$$

The degree and working alphabet do not change. The compiled presentation stores the original control together with the current $|U|$-bit future-answer vector. Its transition reads far enough backward to compute all answers after the current input symbol followed by a word in $U$.

Proof

The compiled memory graph is exactly the graph produced by $A$. Its control after reading $p$ is

$$\left(q_p,(1_L(pu))_{u\in U}\right),$$

where $q_p$ is the control of $A$ after $p$.

Suppose the next symbol is $a$. The original next control, node label, and port descriptors depend only on the radius-$k$ view. For each $u\in U$, the new observer bit is the answer obtained by running $A$ on the fixed word $au$ from the old configuration. This word has length $|u|+1$, so FPRD-T153 says that the bit is a function only of the old control and the radius-$R_{k,|u|+1}$ view. All these data are available inside the radius-$D_U$ view. Because the alphabets, state set, and view set are finite, one finite transition table computes the original node update and the whole new observer vector.

The empty-input observer vector is a fixed initial constant. Induction on $|p|$ proves the displayed control invariant. Accepting exactly when $b$ accepts the stored vector recognizes $\operatorname{Obs}_{U,b}(L)$. There are at most $q2^{|U|}$ compiled control states, while the graph alphabet and degree are unchanged.

4. Consequences

For one fixed suffix $u$, take $U=\{u\}$ and let $b$ select its only coordinate. The right quotient

$$L/u=\{p:pu\in L\}$$

therefore satisfies

$$(d,R_{k,|u|+1},g,2q)\in\operatorname{Tr}(L/u).$$

Arbitrary finite unions, intersections, symmetric differences, thresholds, and other Boolean combinations of fixed right quotients are covered by the same construction. The theorem is stronger than applying FPRD-T157 to separate quotient machines: all observer coordinates share one unchanged persistent graph.

The locality bound is uniformly sharp for this construction. The chain witnesses from FPRD-T153 give, for every $k\ge 1$ and every suffix length $t\ge 0$, two old configurations with identical radius $R_{k,t+1}-1$ views but different answers after one current symbol and a length-$t$ suffix. No smaller universal radius can compute every such update. This is a worst-case statement; individual languages and observer families may admit smaller or entirely different presentations.

5. Research boundary

Finite fixed future tests do not threaten bounded stationary transport. They can be compiled into control at a finite, explicit locality cost. Consequently, a lower-bound invariant for FPRD-C02 must survive all finite right-context observations. It must detect unboundedly moving or coherently branching demand, not a fixed finite menu of future questions.

The proof combines the native scaffolding-automaton model with the readable-cone lemma of FPRD-T153. Closure under quotient-like operations is classical across many language families. The claimed contribution here is the explicit resource bound and shared-graph construction for the stationary transport spectrum. No literature-priority or independent-review claim is made.