Closed Forms for Powers and Longer Rotor Words

Derive closed forms for scalar projections of powers of the same-order rotor loop and for the remaining longer words, including words of the forms P_k Q_k^n and P_k Q_k P_k Q_k^{-1}, for general k.

Background

The paper studies the same-order rotor loop Θ_k and observes that powers are completely understood for k = 2 through a Chebyshev-polynomial relation. The same reduction is observed numerically for k = 3, but beginning at k = 4 the scalar part of Θ_k2 is no longer determined solely by the scalar part of Θ_k. Consequently, a general treatment would require information about the principal angles of the rotor, and the authors leave both the power problem and the related longer-word problems unresolved.

References

From k = 4, \langle \Theta_k2 \rangle_0 is no longer determined by \langle \Theta_k \rangle_0; a closed form for powers would have to go through the principal angles of the rotor and is open, as are the remaining longer words.

— Star-Ordered Rotor Loops in Clifford Algebras  (2610.21593 - Jenzer, 15 Oct 2026) in Section 8.4, Open Fronts, OF 3: Powers and Longer Words