Polynomial bound for prime-prefix escape

Prove or disprove that the prime-prefix escape radius $eta_{\rm cut}(L,h)$ is bounded by a polynomial in the number $q$ of relative primes and the canonical structural thickness $\tau$.

Background

Under PTLD, the paper identifies the cut-separation parameter with the shortest length of a word in a prime class that escapes the prefix class induced by a nontrivial cut of its canonical representative.

This parameter controls the length of cut-separation witnesses in the explicit characteristic sample for the strong learner. It vanishes for lexically anchored systems, but no general polynomial bound in the canonical parameters is established.

References

Can $\eta_{\rm cut}(L,h)$ be bounded by a polynomial in $q$ and $\tau$? Equivalently, do the finitely many false prefix classes exposed by cuts of canonical prime words always admit polynomial-length escape witnesses?

— Relative Prime Factorization and Finite-State Presentations under Fixed Finite-Monoid Observation  (2609.03643 - Kuriyama, 3 Sep 2026) in Section 13, “Open problems and next steps,” Question: Polynomial prime-prefix escape

For which familiar restricted CFG subclasses can $q$, $\tau$, and $\eta_{\rm cut}(L,h)$ nevertheless be bounded polynomially in the size of the given generating grammar? A positive answer for a natural linear fixed-$h$ subclass would connect the general weak polynomial bounds of the fixed-$h$ reconstruction paper directly to polynomial strong structural learning.

— Relative Prime Factorization and Finite-State Presentations under Fixed Finite-Monoid Observation  (2609.03643 - Kuriyama, 3 Sep 2026) in Section 13, “Open problems and next steps,” Question: Representation-size transfer under restricted presentations