FPRD-T157: Stationary transport spectra

1. Bounded stationary persistent presentations

A bounded stationary persistent presentation of a language consists of:

• a finite control set of size $q$;
• a finite working alphabet of size $g$;
• immutable nodes of fixed out-degree $d$;
• one distinguished newest node;
• a fixed observation and descriptor radius $k$;
• one finite transition table.

On each input symbol, the table reads the control and the radius-$k$ unfolded view of the newest node. It changes control and appends one new labelled node. Each new port is missing, points to the new node itself, or points to a node named by an old-root path of length at most $k$. Old nodes never change. Acceptance is determined by finite control.

This is exactly the finite Loff–Moreira–Reis scaffolding-automaton model, viewed as a presentation of the canonical residual dynamics from FPRD-T156. Every reachable presentation configuration determines one full future residual, and appending an input letter commutes with the residual shift.

2. The stationary transport spectrum

For a language $L$, define

$$\operatorname{Tr}(L)= \{(d,k,g,q)\in\mathbb N^4: L\text{ has such a presentation with these resource ceilings}\}.$$

Unused ports, longer permitted paths, unused labels, and unused control states can always be added. Therefore $\operatorname{Tr}(L)$ is upward closed in the componentwise order:

$$(d,k,g,q)\in\operatorname{Tr}(L),\quad (d',k',g',q')\ge(d,k,g,q) \Longrightarrow (d',k',g',q')\in\operatorname{Tr}(L).$$

By Dickson’s lemma, every subset of $\mathbb N^4$ has only finitely many componentwise-minimal elements. Hence, when $\operatorname{Tr}(L)$ is nonempty, there is a unique finite antichain

$$\operatorname{MinTr}(L)$$

such that

$$\operatorname{Tr}(L)=\uparrow\operatorname{MinTr}(L).$$

This finite Pareto basis is an implementation-minimized invariant of $L$ inside the stationary persistent model.

3. Exact model boundary

The spectrum is nonempty exactly when a finite scaffolding automaton recognizes $L$. This is a direct equivalence of definitions, not a new recognition theorem.

By FPRD-T110, every feasible tuple supplies an ordinary PEG for $\operatorname{rev}(L)$. Only this sufficient scaffold-to-PEG direction is used here.

4. Boolean calculus

Complement

Because the machines are deterministic and total, complement only flips the accepting control states. Therefore

$$\operatorname{Tr}(\overline L)=\operatorname{Tr}(L).$$

Pairing

Suppose

$$(d_1,k_1,g_1,q_1)\in\operatorname{Tr}(L_1), \qquad (d_2,k_2,g_2,q_2)\in\operatorname{Tr}(L_2).$$

Create one physical node for each simultaneous pair of component nodes. Its label is the pair of component labels. Its first $d_1$ ports encode the first component and its last $d_2$ ports encode the second. Translate every path by using the corresponding port block. Missing and SELF descriptors translate unchanged. The control is a pair, and the observation radius is the maximum of the two component radii.

Thus any Boolean function $b:\{0,1\}^2\to\{0,1\}$ has

$$(d_1+d_2,\max(k_1,k_2),g_1g_2,q_1q_2) \in\operatorname{Tr}(b(L_1,L_2)).$$

In particular this applies to union, intersection, symmetric difference, and set difference. Iterating gives the corresponding finite-product construction.

5. Why the spectrum helps

The residual tower gives a canonical semantic machine for every language. The transport spectrum asks whether that machine has any bounded stationary persistent realization and, when it does, records all resource tradeoffs after minimizing over implementations.

A positive FPRD-C02 proof can now aim to place each forgotten-action language inside some explicit upward cone. A negative proof must show its spectrum is empty. A bound against one coordinate system or one resource tuple is not enough.

The finite-basis statement is an application of classical Dickson’s lemma, and the presentation equivalence unfolds the scaffolding-automaton definition. No literature-priority claim is made for those ingredients.

6. Reference

Bruno Loff, Nelma Moreira, and Rogerio Reis, The computational power of parsing expression grammars, arXiv:1902.08272.