Star‑Ordered Rotor Loops in Clifford Algebras

We consider, in a real Clifford algebra, a family of unit bivectors Bᵢ = C Sᵢ formed from $k+1$ pairwise anticommuting generators C, S₁, …, Sₖ with square $-1$; we call this a *star family* with *central generator* C and *leaf generators* Sᵢ. The Bᵢ satisfy Bᵢ² = -1 and anticommute pairwise. For the associated rotors Uᵢ(θᵢ) = cosθᵢ/2 + Bᵢ sinθᵢ/2 we study two objects: the ordered chain Pₖ = U₁ ⋯ Uₖ and the *star-ordered rotor loop* Θₖ = Pₖ \, Qₖ with the *same-order reversion product* Qₖ = U₁ ⋯ Uₖ. It is essential that Qₖ carries the same index order as Pₖ and not the reversed one; for the reversed order one has trivially U₁ ⋯ Uₖ Uₖ ⋯ U₁ = 1. The scalar part of Θₖ therefore measures the *order defect* of the family. We prove two theorems. First, the scalar part of the chain factorises, Pₖ ₀ = ∏ᵢ₌₁ᵏ cosθᵢ/2; this statement essentially requires the star structure, and we give a counterexample. Second, we derive a closed expression for Θₖ ₀, valid for every k, whose coefficients obey a *period-four sign law* η(m) = (-1)m/2 and which admits the compact form Θₖ ₀ = Re\![(1-i) ∏ᵢ (cos²θᵢ/2 + isin²θᵢ/2)]. Unlike the first, this second theorem holds for every pairwise anticommuting unit family without any further hypothesis. From the compact form we obtain a polar form, a complete zero criterion, a lacunary coefficient rule in the symmetric case, and a proof of non-separability for k ≥ 2. Two supplementary theorems extend the result: to arbitrary permutations of the index order in the second factor, in which each inversion within a subset flips its sign contribution (under the star condition), and to families with mixed squares Bᵢ² ∈ \-1, +1\, for which a uniform closed subset formula holds. All proofs use only anticommutation, reversion and the symmetry of the scalar form; no physical interpretation is assumed or claimed. > **Key Equations** > Compact form: Θₖ ₀ = Re\![(1-i) ∏ᵢ₌₁ᵏ (cos²θᵢ/2 + isin²θᵢ/2)] (Chapter 6.2). > Polar form: Θₖ ₀ = √2 (∏ᵢ₌₁ᵏ ρᵢ) cos\!(∑ᵢ₌₁ᵏ αᵢ - π/4) (Chapter 6.3).