Stationary transport spectra have finite Pareto bases

Exact statement

For every language L, Tr(L) is upward closed in the componentwise order and therefore, by Dickson's lemma, is generated by a unique finite antichain of Pareto-minimal resource tuples whenever nonempty. Complement preserves Tr exactly. If tuples (d_i,k_i,g_i,q_i) present L_i, then every binary Boolean combination has a presentation bounded by (d_1+d_2, max(k_1,k_2), g_1g_2, q_1q_2). Nonempty spectrum is equivalent to recognition by a finite scaffolding automaton and implies a PEG for the reverse language through FPRD-T110.

StatusSelf-contained proof with finite-basis and paired-machine audits

External reviewThe finite-basis step is classical Dickson's lemma and the model is due to Loff, Moreira, and Reis; no external review of this FPRD spectrum formulation is recorded.

Context

A single resource bound is presentation-dependent. The spectrum and its finite minimal basis retain all tradeoffs among degree, locality, labels, and finite control within one invariant.

Hypotheses and scope

Presentations use the finite deterministic Loff--Moreira--Reis scaffold transition form.
Resource coordinates are upper ceilings and are ordered componentwise.
The Boolean product construction processes the same input synchronously in both component presentations.

Proof or evidence

Padding proves upward closure and Dickson's lemma gives the finite basis. Complement flips accepting states. Pairing two virtual nodes inside each physical node partitions the port set and translates component paths. The audit reconstructs all 65,536 upward closures on a four-by-four grid and checks 73,152 paired persistent runs with 369,792 projected rows.

Finite Pareto basis theorem →

Verification notes

The construction was checked for MISSING and SELF descriptors, component path translation, unequal component degrees, empty input, all sixteen Boolean output functions, and reversal-sensitive PEG transfer.

Limitations

The theorem does not compute the Pareto basis of an arbitrary language or decide whether the spectrum is empty.
The pairing bound is sufficient and need not be Pareto-optimal.
The finite-basis claim is a direct use of classical Dickson's lemma; no literature-priority claim is made.
Only the locally formalized scaffold-to-PEG direction is invoked, so the PEG conclusion concerns the reverse language.

Open work

Develop lower bounds that exclude all Pareto tuples simultaneously, or compile projected frontier representations into an explicit feasible tuple.