Peano-arithmetic provability of unbounded periods

Prove in Peano Arithmetic that the periods of the first rows of arbitrary finite Laver algebras are unbounded, namely that \(\lim_{n\to\infty}\tau^n(1)=\infty\).

Background

For a finite Laver algebra SnS_n, τn(a)\tau^n(a) denotes the least positive integer p≤np\leq n such that a⋆p=na\star p=n. The paper notes that unboundedness of τ2n(1)\tau^{2^n}(1) is known, but existing proofs use the large-cardinal axiom I3I3, and related growth results are not provable in Primitive Recursive Arithmetic.

The authors conjecture that the broader unboundedness statement for τn(1)\tau^n(1), with arbitrary algebra sizes nn, is nevertheless provable in Peano Arithmetic. This remains unresolved in the paper.

References

It is provable in Peano Arithmetic that $\lim_{n\to\infty}(\taun(1)) = \infty$.

— Non-dyadic Laver algebras  (2610.06106 - Aguilera et al., 5 Oct 2026) in Section 5, "Periods of Laver algebras"

For every $a \geq 2$, there is $n$ such that $\taun(1) = a$. Indeed, there are infinitely many such $n$.

— Non-dyadic Laver algebras  (2610.06106 - Aguilera et al., 5 Oct 2026) in Section 5, "Periods of Laver algebras"