Polynomial witness boundary for L-trivial actions

Exact statement

It remains open whether every positive L-trivial transformation-membership instance has a polynomial-length word witness in the degree of the supplied action. The canonical target-residual degree d_f gives a witness of length at most d_f−1; whether d_f is always polynomial is the current sufficient-bound question.

StatusUnresolved; the n-minus-one conjecture is refuted, but neither NP membership nor PSPACE hardness is established

External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The canonical observer removes the ambiguity over which quotient to seek: its states are exactly the distinct left-context profiles of the requested target.

Proof or evidence

The target-residual theorem converts a polynomial d_f bound directly into polynomial word certificates. The exact binary family has cubic source size but quadratic d_f, while the Green-height probe rules out one coarse obstruction proxy. Neither a universal polynomial bound nor a superpolynomial obstruction is known.

What remains open →

Verification notes

Every positive and negative result was kept distinct from this unresolved general question.

Limitations

Failure to find a compact action is not evidence that none exists.
A superpolynomial d_f family would refute the canonical-observer route but would not alone prove superpolynomial shortest witnesses.

Open work

Prove a polynomial upper bound on target-residual degree from L-trivial fibre evolution, or construct a structured superpolynomial target-profile or geodesic family.