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Formalizability in V#L of Straubing’s combinatorial proof approach

Ascertain whether Straubing’s combinatorial proof technique for the Cayley–Hamilton Theorem can be formalized within the bounded arithmetic theory V#L so as to yield a formal proof of the Pfaffian Cayley–Hamilton Theorem presented in the paper.

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Background

The paper presents a Pfaffian version of the Cayley–Hamilton Theorem and notes multiple possible proof routes. One route would adapt Straubing’s combinatorial argument originally used for the determinant.

While the authors provide a proof via cofactor expansion (PCE) that can be formalized in V#L, they explicitly state uncertainty about whether Straubing’s combinatorial method can itself be formalized in V#L.

References

We can prove this theorem in several ways. One way is to use the combinatorial argument which is used to prove Cayley-Hamilton Theorem for the determinant due to Straubing [8]. However, we do not know whether such proof can be formalized in V#L.

— Formalizing Pfaffian in bounded arithmetic (2404.01728 - Kuroda, 2 Apr 2024) in Section 5 (Cayley-Hamilton Theorem for Pfaffian)