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Metadata 
bibkey: jenzer2026starorderedrotorloops
authors: Nik Jenzer
year: 2026
title: Star-Ordered Rotor Loops in Clifford Algebras
doi: 10.48550/arXiv.2610.21593
url: https://arxiv.org/abs/2610.21593v1
claim: The source's scope remark asks whether the permuted loop formula holds for anticommuting unit bivectors without the star condition.
strata_touched:
  - D5/S3/QuadraticForms/JenzerPermutedRotorLoopAlgebra
  - D5/S3/QuadraticForms/JenzerPermutedRotorLoopWords
  - D5/S3/QuadraticForms/JenzerPermutedRotorLoop
license: citation-only
triage: anchor

Verified locator

DOI: 10.48550/arXiv.2610.21593

Source: https://arxiv.org/abs/2610.21593v1

Source statement

Remark 5.7 in the TeX, rendered Remark 5.5, page 19, “On the Scope, with Numerical Evidence”:

The hypotheses of Theorem 5.5 can be sharpened at two points. First, for τ = id no star condition is needed: that is Theorem 5.2, whose pairing argument cancels the odd off-diagonal terms in pairs. Second, this pairing argument fails termwise for τ ≠ id. 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.

Theorem 5.5, page 18, “Permuted Loop Formula”:

Let B₁, …, Bₖ be an anticommuting unit family of type −1 that satisfies the star condition, and let τ be a permutation of {1, …, k}. Then

The settlement removes the star condition and retains every other source hypothesis. Fin k is the increasing relabelling of the source’s one-based set. The scalar projection is the native empty-blade coefficient, read through the linear exterior equivalence; Clifford multiplication remains native.

Verbatim definitions

The following source sentences retain the TeX mathematical notation. D2.1 and D2.2 occur on page 8; D2.3 and D2.4 on page 9; D3.2 and D3.3 on page 10; D5.1 on page 16; D5.2 on page 18. The ordered chain occurs in section 1.1 on page 3.

Let \(V\) be a real vector space of finite dimension \(n\) with a
non-degenerate quadratic form \(q\), and let
\(\mathcal{C} = \mathrm{Cl}(V, q)\) be the associated Clifford algebra,
generated by \(V\) under the relation \(v^2 = q(v)\) for \(v \in V\).
For the foundations we follow \citep{Lounesto2001} and \citep{Porteous1995}; for
the notation in geometric algebra \citep{DoranLasenby2003} as well as
\citep{HestenesSobczyk1984}.

For an orthonormal basis \(e_1, \ldots, e_n\) of \(V\), the products
\[e_A = e_{a_1} e_{a_2} \cdots e_{a_r}, \qquad A = \{a_1 < a_2 < \cdots < a_r\} \subseteq \{1, \ldots, n\},\]
with \(e_\emptyset = 1\) form a basis of \(\mathcal{C}\) as a real
vector space, \(\dim \mathcal{C} = 2^n\). We call \(e_A\) a \emph{basis
blade} of grade \(r = |A|\).

\begin{definition}[Scalar Projection]\label{definition:D2-1}
 For
\(x = \sum_A x_A e_A \in \mathcal{C}\),
\(\langle x \rangle_0 := x_\emptyset \in \mathbb{R}\) is called the
\emph{scalar part} of \(x\); the map
\(\langle \cdot \rangle_0 : \mathcal{C} \to \mathbb{R}\) is the
\emph{scalar projection} or grade-0 projection. It is linear.

\end{definition}

\begin{definition}[Reversion]\label{definition:D2-2}
 \emph{Reversion} is the unique
anti-automorphism \(\widetilde{\;\cdot\;}\) of \(\mathcal{C}\) with
\(\widetilde v = v\) for \(v \in V\); on a blade of grade \(r\) it acts
as \[\widetilde{e_A} = (-1)^{r(r-1)/2} e_A, \qquad r = |A|,\] and for a
product \(\widetilde{xy} = \widetilde y \, \widetilde x\) holds. The
grade sign \((-1)^{r(r-1)/2}\) is found in \citep[Eq.~92]{Hitzer2012}, \citep[Eq.~4.52]{DoranLasenby2003} and \citep[\S 22.1]{Lounesto2001}.

In particular, \(\widetilde B = -B\) for every bivector \(B\), since
\(r = 2\) gives \((-1)^{1} = -1\).

\end{definition}

\begin{definition}[Rotor]\label{definition:D2-3}
 Let \(B \in \mathcal{C}\) be a
bivector with \(B^2 = -1\) and \(\theta \in \mathbb{R}\). Then
\[U(\theta) = \exp\!\left(\frac{\theta}{2} B\right) = \cos\frac{\theta}{2} + B \sin\frac{\theta}{2}\]
is called the \emph{rotor} generated by \(B\) with parameter \(\theta\).

The exponential series converges and, because \(B^2 = -1\), yields
exactly the given trigonometric form. We have
\(\widetilde{U(\theta)} = \cos\frac{\theta}{2} - B \sin\frac{\theta}{2} = U(-\theta)\)
and
\(U(\theta)\widetilde{U(\theta)} = \cos^2\frac{\theta}{2} - B^2 \sin^2\frac{\theta}{2} = 1\).

\end{definition}

\begin{definition}[Subset Product]\label{definition:D2-4}
 Let
\(B_1, \ldots, B_k \in \mathcal{C}\) be given. For
\(S = \{i_1 < i_2 < \cdots < i_m\} \subseteq \{1, \ldots, k\}\) we set
\[B_S := B_{i_1} B_{i_2} \cdots B_{i_m}, \qquad B_\emptyset := 1.\] The
factors always stand in ascending index order. We write throughout
\[c_i := \cos\frac{\theta_i}{2}, \qquad s_i := \sin\frac{\theta_i}{2}, \qquad w_i := s_i^2 = \sin^2\frac{\theta_i}{2} \in [0,1]\]
and for a subset \(S\)
\[a_S := \prod_{i \in S} s_i \prod_{i \notin S} c_i.\]

\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}

\pagebreak

\end{definition}

\begin{definition}[Anticommuting Unit Family]\label{definition:D3-2}
 A family
\(B_1, \ldots, B_k\) of bivectors of a real Clifford algebra is called
an \emph{anticommuting unit family} of type \(-1\) if
\[B_i^2 = -1 \quad (1 \le i \le k), \qquad B_i B_j = -B_j B_i \quad (i \neq j).\]

\end{definition}

\begin{definition}[Star Condition]\label{definition:D3-3}
 An anticommuting unit family
satisfies the \emph{star condition} if
\[\langle B_S \rangle_0 = 0 \qquad \text{for all } S \neq \emptyset.\]

The choice of names is justified by Lemmata 3.1 and 3.5: every star
family is an anticommuting unit family and satisfies the star condition.
We do not investigate the converse; Chapter 8.4 names it as an Open
Front (OF 1).

\end{definition}

\begin{definition}[Period-Four Sign Law]\label{definition:D5-1}
 We set for
\(m \in \mathbb{N}_0\)
\[\eta(m) := (-1)^{m} \cdot (-1)^{m(m+1)/2} = (-1)^{m(m+3)/2}.\]

\end{definition}

\begin{definition}[Permuted Loop]\label{definition:D5-2}
 Let \(\tau\) be a permutation
of \(\{1, \ldots, k\}\). We set
\[Q_k^{(\tau)} = \widetilde U_{\tau(1)} \, \widetilde U_{\tau(2)} \cdots \widetilde U_{\tau(k)}, \qquad \Theta_k(\tau) = P_k \, Q_k^{(\tau)},\]
so that \(\Theta_k(\mathrm{id}) = \Theta_k\) and
\(\Theta_k(\tau_{\mathrm{rev}})\) for the reverse permutation
\(\tau_{\mathrm{rev}}(j) = k+1-j\) gives the trivial product
\(P_k \widetilde{P_k} = 1\) from Chapter 1.1. For
\(S \subseteq \{1, \ldots, k\}\) let
\[\iota(\tau, S) := \#\{(i, j) \in S \times S : i < j,\ \tau^{-1}(i) > \tau^{-1}(j)\}\]
be the \emph{inversion count} of \(\tau\) on \(S\).

\end{definition}

Let \(\mathcal{C}\) be a real Clifford algebra and let
\(B_1, \ldots, B_k \in \mathcal{C}\) be bivectors with
\[B_i^2 = -1, \qquad B_i B_j = -B_j B_i \quad (i \neq j).\] For real
parameters \(\theta_1, \ldots, \theta_k\) we set
\[U_i = \cos\frac{\theta_i}{2} + B_i \sin\frac{\theta_i}{2}, \qquad \widetilde U_i = \cos\frac{\theta_i}{2} - B_i \sin\frac{\theta_i}{2},\]
where \(\widetilde{\;\cdot\;}\) denotes reversion. Each \(U_i\) is a
rotor: \(U_i \widetilde U_i = 1\).

We consider the \emph{ordered chain} \[
P_k = U_1 U_2 \cdots U_k
\] and, with the same index order, \[
Q_k = \widetilde U_1 \widetilde U_2 \cdots \widetilde U_k,
\] as well as their product
\[\Theta_k = P_k \, Q_k = (U_1 \cdots U_k)(\widetilde U_1 \cdots \widetilde U_k).\]

Scope

Anticommuting conjugations eliminate nonempty scalar components except possible odd full support. The balanced exterior word has even parity, so its odd top coefficient vanishes and cancels that remaining aggregate. The statement includes zero trigonometric coefficients and all source signatures. It does not classify star families.

The literature screen in issue #22266 found no settlement in its successfully searched scope. Completeness is ASSUMED-UNVERIFIED: inaccessible and unindexed work is outside that screen.

Finite experiment

The independent exact checker is in the-omega-institute/trureturing-experiments, commit 8bb94a407557328c74589fd7b9dd31ed49b061e0, at docs/reports/permuted-rotor-loop/check.py, SHA-256 314438c29013e5ae8ec9dde252f9b29c526854c6c591d11d9c661242b055a759. The preregistration records command python3 docs/reports/permuted-rotor-loop/check.py, exit 0 and final ALL_CHECKS_OK. Its finite cases support the mechanism and do not prove the universal identity.