Proof of the Permuted Loop Formula Without the Star Condition

Prove that the permuted loop formula for \(\langle \Theta_k(\tau) \rangle_0\) remains valid for nonidentity permutations without assuming the star condition, by identifying the required cancellation of odd off-diagonal terms.

Background

Theorem 5.5 establishes the permuted loop formula under the star condition. The paper notes that the identity permutation does not require this condition because odd off-diagonal terms cancel in pairs, whereas that pairing argument does not work termwise for nonidentity permutations. Numerical tests on the triangle family suggest that the formula may nevertheless remain valid without the star condition for nonidentity permutations. A general proof would require a coarser grouping of the odd off-diagonal terms, which the authors have not found.

References

Nevertheless, numerical evidence suggests that the formula also holds without the star condition in that case: tested on the triangle family from Remark 4.4, which violates the star condition, under all six permutations of {1, 2, 3} with random parameter sets; the largest deviation encountered was in the range of double machine precision. A general proof would have to capture the cancellation of the odd off-diagonal terms in coarser groups than pairs; we have not determined such a grouping.

— Star-Ordered Rotor Loops in Clifford Algebras  (2610.21593 - Jenzer, 15 Oct 2026) in Remark 5.7, Section 5.3: Arbitrary Index Orders