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Theorem · structural normal form and one-dimensional existence criterion

FPRD-T161

Every scaffolding automaton has a unary two-control structural normal form

Exact statement

Let a scaffolding automaton have tuple (d,k,g,q) with k>=1, put D=sum_(i=1)^k d^i, and m=ceil(log_2(1+gq)). It has an equivalent tuple (D+m,2,1,2). Shortcut ports expose every source endpoint in one hop, and marker ports encode each ordinary source label-control pair as local SELF/missing geometry. The zero marker code is reserved for the distinguished base node. Hence Tr(L) is nonempty exactly when it contains some tuple (D,2,1,2).

StatusSelf-contained constructive proof with combined compiler and structural-code audits
External reviewA targeted literature search found the original scaffold model but no published unary-label, two-control structural normal form; no specialist review or priority claim is recorded.

Context

FPRD-T160 removes distance above two by shortcut closure. This theorem simultaneously moves the working alphabet and full source control into local graph geometry, leaving graph degree as the only unbounded coordinate on a canonical existence slice.

Hypotheses and scope

  • The source is a finite deterministic scaffolding automaton with fixed degree d, distance k>=1, working alphabet Gamma of size g, and q controls.
  • The native scaffold has its distinguished base node, and every later node has one source label and is created together with one reached source control.
  • Old nodes are immutable, marker ports may be SELF or missing, and shortcut descriptors have distance at most two.

Proof or evidence

Reserve marker code zero for the distinguished base and encode each of the gq ordinary label-control pairs by a nonzero m-bit word, where m=ceil(log_2(1+gq)). A radius-two compiled view reads current markers, reaches every source path endpoint through one shortcut, and reads that endpoint's markers. The compiled transition reconstructs the exact source state and radius-k view, installs the T160 shortcuts plus the next pair code, and keeps only the source acceptance bit in two physical controls. The audit checks 3,060 runs, 18,456 transitions, 315,280 shortcut edges, 270,240 marker bits, and all 1,296 pair codes for sizes through eight.

Structural scaffold normal-form theorem →

Verification notes

The checker covers source degrees one and two, source distances one through three, empty input, all binary prefixes through length seven, missing endpoints, SELF markers, exact shortcut closure, unary physical labels, two physical controls, acceptance preservation, the reserved base code, and the sharp marker-width inequality.

Limitations

  • The degree D+m is sufficient and need not be Pareto-optimal.
  • The theorem collapses recognition expressiveness, not descriptional complexity or degree tradeoffs.
  • The four-dimensional transport spectrum remains useful because smaller alphabets, controls, distance, and degree can trade against one another.
  • Structural encoding is a standard pointer-machine idea; no novelty or literature-priority claim is made without specialist review.

Open work

Study the minimum canonical structural degree delta_str(L), including composition laws and lower bounds that survive every finite label and control encoding.