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Definition

FPRD-D37

Observation levels for scaffold histories

Exact statement

For fixed finite scaffold degree, observation distance, and working alphabet, a scaffold construction history is a word over a finite row alphabet. Its chronological physical history, complete strong trace of minimal rooted behavioral figures, final behavioral figure, and acceptance readout are distinct observations. A presentation move must state which of those observations it preserves.

StatusFixed and self-contained; no completeness or novelty claim
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

One row appends a labelled node whose ports are missing, point to the fresh node, or follow bounded port words from the preceding top. The same row history can be observed at several resolutions, and preserving a coarser resolution does not imply preserving a finer one.

Definitions

  • The physical history records every created node and port endpoint.
  • The figure at a prefix is the minimal rooted labelled port graph with the same unfolding behavior.
  • The strong trace is Tr⁡(H)=(F1(H),…,Fn(H))\operatorname{Tr}(H)=(F_1(H),\ldots,F_n(H)), while the final observation is only π(H)=Fn(H)\pi(H)=F_n(H).
  • Acceptance is a finite readout of the final observed neighborhood and is coarser still.

Hypotheses and scope

  • Finite scaffold degree, finite observation distance, and finite working alphabet.
  • Histories compared by strong or final equivalence have equal length.

Proof or evidence

The publication gives the row semantics, minimal-figure construction, strong and final congruences, and an explicit three-row example with equal final figures but different strong traces.

Histories, observations, and two congruences →

Verification notes

The definition was checked against the governed claim ledger, the paper, and its implication-oriented proof audit. No theorem is asserted by this entry.

Limitations

  • Strong equivalence is finer than final equivalence; neither is physical equality.
  • The ambient history space is not identified with the runs of one particular automaton.
  • A result about acceptance alone must not be promoted to a result about figures or histories.

Open work

Use the observation ladder consistently when classifying later scaffold moves and lower bounds.

Notes

Strong equivalence retains the figure after every prefix; final equivalence forgets those intermediate figures. Physical endpoint equality is finer than behavioral equality. The definition makes no completeness or novelty claim and does not identify the ambient history algebra with the generated orbit of one particular automaton.