FPRD-T170: The causal selection chain rule

1. Two stages of hidden history

Let

$$\Gamma\xrightarrow{\pi}\Sigma\xrightarrow{\rho}\Delta$$

be letter-to-letter projections, let $A\subseteq\Gamma^*$, and put

$$B=\pi(A),\qquad C=\rho(B)=(\rho\circ\pi)(A).$$

Suppose $T_1,\ldots,T_m$ are synchronous finite-state selectors covering $B$ through $A$, and $U_1,\ldots,U_n$ are selectors covering $C$ through $B$:

$$B=\bigcup_{i=1}^{m}T_i^{-1}(A), \qquad C=\bigcup_{j=1}^{n}U_j^{-1}(B).$$

Each composite $T_i\circ U_j:\Delta^*\to\Gamma^*$ is again a synchronous causal selector for $\rho\circ\pi$.

2. FPRD-T170: multiplicative composition

If every $T_i$ has at most $s$ states and every $U_j$ has at most $t$ states, then the $mn$ pairwise composites form an accepting selector cover and each has at most $st$ states. Equivalently,

$$(m,s)\in\operatorname{Sel}(A,\pi), \quad (n,t)\in\operatorname{Sel}(B,\rho)$$

implies

$$(mn,st)\in\operatorname{Sel}(A,\rho\circ\pi).$$

The same statement holds horizon by horizon:

$$(m,s)\in\operatorname{Sel}_h(A,\pi), \quad (n,t)\in\operatorname{Sel}_h(B,\rho) \Longrightarrow (mn,st)\in\operatorname{Sel}_h(A,\rho\circ\pi).$$

Proof

The product-state machine for $T_i\circ U_j$ first lets $U_j$ read the current $\Delta$-letter and emit one $\Sigma$-letter. It immediately feeds that letter to $T_i$, which emits one $\Gamma$-letter. Its state is the pair of component states, so its state set has size at most $st$. Moreover,

$$(\rho\circ\pi)(T_i(U_j(w)))=w.$$

If $w\in C$, some $U_j(w)$ belongs to $B$, and then some $T_i(U_j(w))$ belongs to $A$. Conversely, any selected word in $A$ projects to $w$, so there are no false positives. The proof is unchanged when restricted to words of length at most $h$.

3. Obstruction localization

The contrapositive gives a useful diagnostic rule:

$$\operatorname{Sel}(A,\rho\circ\pi)=\varnothing \Longrightarrow \operatorname{Sel}(A,\pi)=\varnothing \ \text{or}\ \operatorname{Sel}(B,\rho)=\varnothing.$$

Thus an unbounded hidden-choice obstruction in a finite projection pipeline must already occur at some individual stage. This direction is exact. The converse need not hold because a later projection can erase distinctions that made an earlier stage difficult.

Give feasible pairs escape value infinity and infeasible pairs the finite escape length from FPRD-T169. The horizonwise chain rule yields

$$e_{A,\rho\circ\pi}(mn,st) \ge \min\bigl(e_{A,\pi}(m,s),e_{B,\rho}(n,t)\bigr).$$

A product budget cannot fail before at least one of its two constituent budgets fails.

4. Sharper hierarchical transport bound

Assume

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

Applying FPRD-T167 first to the $m$ inner selectors and then to the $n$ outer selectors preserves their shared two-stage structure. Under the uniform state bounds above it gives

$$\left( mnd, k, g^{mn}, q^{mn}s^{mn}t^n \right) \in\operatorname{Tr}(C).$$

Treating all $mn$ composites independently would instead give the weaker control bound $q^{mn}s^{mn}t^{mn}$. The improved exponent records that each outer selector state is shared across its $m$ inner branches.

For nonuniform state sizes $s_i$ and $t_j$, the corresponding control bound is

$$q^{mn} \left(\prod_{i=1}^{m}s_i\right)^n \left(\prod_{j=1}^{n}t_j\right).$$

5. Algebraic interpretation and limits

Coordinatewise multiplication

$$(m,s)\odot(n,t)=(mn,st)$$

is associative and has identity $(1,1)$. The causal selection spectrum is therefore a resource-valued, composition-compatible invariant of finite letter-projection chains. It is an upper-bound calculus: a composite hidden presentation may admit better selectors than those obtained by pairwise composition.

• The theorem requires the intermediate language to be exactly $B=\pi(A)$.
• It applies to deterministic synchronous finite-state selectors.
• It gives a sufficient product profile, not the Pareto-minimal profile of the composite projection.
• Empty selection spectrum remains presentation-relative and is not by itself an SCA or PEG lower bound.
• The construction is a finite-state product argument. No literature-priority claim is made without specialist review.