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Theorem · structural characterization

FPRD-T130

Generated reachability chains and regressive navigation

Exact statement

Every legally generated rooted scaffold and its minimal behavioral figure are weakly acyclic reachability chains. Conversely every finite behaviorally reduced labelled rooted reachability-chain port graph has a legal scaffold presentation at some finite distance. After adjoining a missing sink, every port action is regressive and the finite port-word navigation monoid is R-trivial.

StatusProved, independently rederived, and finitely stress-tested
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

A new scaffold node may point only to itself or into the previous top's reachable component. This chronological restriction forces a total reachability order from every current top.

Definitions

  • Weakly acyclic means that the only directed cycles are self-loops.
  • A port action ff on the ordered figure is regressive when xf≤xxf\le x for every state xx.
  • A monoid is R\mathcal R-trivial when Green's R\mathcal R-classes are singletons.

Hypotheses and scope

  • Legally generated chronological scaffolds for the forward direction.
  • Finite, rooted, labelled, behaviorally reduced reachability-chain port graphs for the converse.

Proof or evidence

Creation time proves weak acyclicity; induction on appended nodes proves total reachability. Conversely, build states from least to greatest and name every strict successor by a port word from the previous top. Pointwise regressiveness collapses any Green-R pair. The replayed checker reconstructed 21,368 accessible chain graphs and verified R-triviality for 3,585 generated figure monoids.

Generated-chain theorem →

Verification notes

The proof, converse distance bound, and Green-relation argument were rechecked; the complete-domain checker was rerun successfully.

Limitations

  • The navigation alphabet is the internal port alphabet of one finite figure, not the streamed input alphabet.
  • The theorem does not make the recognized language regular or R-trivial.
  • No braid, free-tree-monoid, or novelty claim is made.

Open work

Position the reachability-chain characterization against standard ordered transformation monoids without transferring it to the streamed language.

Notes

The navigation alphabet is the internal port alphabet of one finite historical figure, not the streamed input alphabet, so the claim does not make the recognized language regular or R-trivial. The result does not identify a particular port monoid with a free tree monoid and makes no braid or novelty claim.