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Theorem · complete distance normalization for scaffolding automata

FPRD-T160

Every scaffolding automaton has a distance-two normal form

Exact statement

If a scaffolding automaton has tuple (d,k,g,q) with k>=2, then it has an equivalent tuple (sum_(i=1)^k d^i,2,g,q). The compiler gives each physical node one shortcut port for every original port word of length at most k. The compiled radius-two view reconstructs the original radius-k view, distinguishing base from missing at shortcut endpoints; at most two hops install every shortcut on the next node. Hence SA_<=2=SA. Combined with FPRD-T159, SA_<=1=REG is a strict subset of SA_<=2=SA.

StatusSelf-contained constructive proof with machine, symbolic-descriptor, and strictness audits
External reviewA targeted literature search found the original scaffold model but no published distance-two normalization; no specialist review or priority claim is recorded.

Context

FPRD-T159 identifies distance one with regular languages. The shortcut compiler shows that the next level already has the full recognition power of every finite scaffold distance.

Hypotheses and scope

  • The source is a finite deterministic scaffolding automaton with fixed degree d and distance k>=2.
  • Each new original port is missing, SELF, or an old-root path of length at most k.
  • Old nodes are immutable and cannot point forward to the new node.

Proof or evidence

Each compiled node stores direct shortcut ports for all nonempty original port words of length at most k. The compiled radius-two view recovers the original radius-k unfolded view and distinguishes base from missing at shortcut endpoints. Symbolically following any such word from the new node either remains at SELF, becomes missing, or exits once to an old path t and then follows a suffix s; the latter endpoint is named by at most two shortcut hops [t][s]. The audit checks 6,132 compiled runs, 43,032 view reconstructions, 423,148 shortcut edges, and 18,697,728 symbolic endpoint equalities.

Distance-two normal-form theorem →

Verification notes

The checker covers original degrees one and two, distances two through four, empty old-root paths, repeated SELF choices, old-node cycles, missing paths, all direct descriptor rows in two exhaustive finite domains, and the nonregular distance-two selector.

Limitations

  • The degree bound sum d^i is sufficient and need not be Pareto-optimal.
  • The construction preserves language and one node per input symbol but can greatly increase port degree.
  • Distance remains a succinctness resource even though it is not an expressiveness hierarchy above two.
  • Shortcut closure is a standard pointer-machine idea; no novelty or literature-priority claim is made without specialist review.

Open work

Replace distance-based expressiveness questions with degree-versus-distance succinctness bounds and seek lower bounds against the entire distance-two normal form.