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Theorem · strong-trace normal form

FPRD-T132

Convergent strong-trace presentation

Exact statement

For every fixed finite scaffold degree d, distance, and working alphabet, least physical-endpoint detours and literal guarded SELF unrolls form a terminating confluent graph-local presentation of complete strong-trace equivalence. Two histories have the same behavioral figure after every prefix exactly when they normalize to the same SELF-eager history, and a length-n history takes at most d*n whole-row moves.

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

Context

Static bisimulation collapse identifies the right semantic object but may ignore chronology. The strong calculus instead rewrites rows while preserving the minimal figure after every prefix.

Definitions

  • An endpoint detour replaces a bounded port word by the least word naming the same physical endpoint.
  • A literal guarded unroll replaces pointers back to an old self-looping representative by SELF at the fresh node and chooses least names for strict successors.
  • A SELF-eager history uses SELF at every available recursive coordinate and least endpoint names elsewhere.

Hypotheses and scope

  • Fixed finite degree d≥1d\ge1, fixed finite observation distance, and finite working alphabet.
  • The moves are graph-local with physical endpoint-equality certificates; they are not required to be bounded contiguous substring rewrites.

Proof or evidence

Both moves preserve all prefix figures and common suffixes. Descriptor lexicographic order proves termination. A one-state extension dichotomy and reduced-prefix lemma force every irreducible row to be the unique least row for its next figure, giving completeness and confluence. At most d detours or one whole-row unroll per row gives the d*n bound. The replayed checker found no split strong-trace class or multiple normal form across its complete one- and two-label domains.

Convergent strong-trace presentation →

Verification notes

The proof was reconstructed from the source audit, including the realizability step for canonical rows and the distinction between object and path coherence. The causal normalization checker was rerun successfully.

Limitations

  • Graph-locality does not imply that a scaffolding automaton can discover or execute the normalizer in finite control.
  • Confluence gives one object normal form; it does not identify every pair of derivations by a finite family of two-dimensional cells.
  • No novelty or external-review claim is made.

Open work

Investigate path-level coherence for the strong calculus without implying that object confluence already supplies a finite two-cell presentation.

Notes

The proof uses the one-state extension dichotomy, existence of the least history for each realizable trace, and reducedness of every SELF-eager live prefix. Graph-local means a bounded neighborhood of the physical scaffold with endpoint-equality certificates; it does not assert a bounded contiguous substring rewrite or an automaton-executable normalizer. Confluence gives one object normal form, not uniqueness of derivations modulo a finite two-cell family. No novelty claim is made.