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Theorem · final-equivalence normal form

FPRD-T135

Unary final equivalence has a right-packed normal form

Exact statement

For every fixed history length, nonempty finite working alphabet, scaffold degree one, and scaffold distance one, endpoint detours, literal guarded absorptions, causal-garbage changes, and temporal carrier slides generate exactly final behavioral equivalence. Every final fibre has one literal right-packed normal form determined by its final figure and history length.

StatusProved, independently audited, and exhaustively tested on the declared finite domain
External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

At degree one and distance one every row descriptor is MISSING, SELF, the empty word, or the single port 0. Consecutive essential residual carriers are joined by one forced relay pattern, making their chronological positions controllable by a local slide.

Definitions

  • A temporal carrier slide exchanges (ε,0)(\varepsilon,0) with (0,ε)(0,\varepsilon) on adjacent rows whose candidate carrier labels agree.
  • Right-packed means that the essential carriers of the final reduced chain occupy the latest possible consecutive rows, with a canonical missing, recursive, or base terminal block below them.

Hypotheses and scope

  • Fixed length (n), nonempty finite working alphabet, degree (d=1), and distance (k=1).
  • Absorption applies only to a literal old physical self-loop; behavioral recursion alone is not enough.

Proof or evidence

The unique-relay lemma forces each supported carrier gap to have the form epsilon followed by zeros. Preparing one unsupported relay label allows an adjacent temporal slide, strictly decreasing total carrier gap. Processing from the final root downward packs every essential carrier, after which detours, absorption, and garbage moves choose the unique terminal block. The checker enumerated 299,592 histories in 606 final fibres, checked 52,964 absorptions, 68,016 detours, 57,400 slides, and 1,175,352 garbage canonicalizations, with no normal-form collision.

Unary final presentation →

Verification notes

The proof was rederived from the unique-relay and carrier-gap arguments, including the three distinct terminal cases. The complete declared finite checker was rerun successfully on 2026-08-27.

Limitations

  • The theorem is restricted to degree one and distance one.
  • A temporal slide preserves the final figure but may change intermediate figures and causal support.
  • The finite enumeration is corroboration, not the proof of the all-length theorem.

Open work

Investigate which part of right-packing survives at degree two or distance two without assuming a chain of unique relays.

Notes

The proof uses the unique unary relay, carrier-gap descent, and separate canonical bottom cases for missing, the distinguished base, and terminal SELF recursion. It does not extend to degree two, distance two, or an unrestricted final calculus. The 299,592-history audit is finite corroboration only. No novelty claim is made.