\chapter{An exact closed-loop rotation writer}
\section{A short path built from a rotating commutator}
Work first in an embedded copy of $\SO(3)$ in $\SO(d)$. Choose elementary
rotations $X,Z$ sharing one coordinate direction and satisfying
$\norm X_Q=\norm Z_Q=1$, $Q(X,Z)=0$. Conjugation by $\exp(rX)$ rotates
$Z$ in a two-dimensional plane of the Lie algebra. In particular,
\begin{equation}\label{eq:adjointcircle}
 Q\bigl(Z,\Ad_{\exp(rX)}Z\bigr)=\cos r.
\end{equation}
For $0<r<\pi$, define the based loop
\begin{equation}\label{eq:circleloop}
 \gamma_r(t)=a\exp(tZ)a^{-1}\exp(-tZ),\qquad
 a=\exp(rX),\quad 0\le t\le2\pi.
\end{equation}
The path begins and ends at the identity because $\exp(2\pi Z)=\Id$.
Although its coordinate formula contains factors making full rotations, the
actual curve $\gamma_r(t)$ is short when $r$ is small. The geometric budget
is the length of this curve, not the sum of lengths of its displayed factors
viewed as separate sequential operations.

Direct differentiation gives
\begin{equation}\label{eq:circlebody}
 \gamma_r^{-1}\dot\gamma_r
 =\Ad_{\exp(tZ)}C,\qquad C=\Ad_aZ-Z.
\end{equation}
The $Q$-norm of the right side is independent of $t$. By
\eqref{eq:adjointcircle},
\begin{equation}\label{eq:circlelength}
 \norm C_Q^2=2-2\cos r=4\sin^2(r/2),\qquad
 \ell(\gamma_r)=4\pi\sin(r/2).
\end{equation}
This is an exact length formula in the fixed soul metric.

\section{Solving the transport exactly}
The transport equation along the curve is
\[
 \dot U=-\lambda\exp(tZ)C\exp(-tZ)U.
\]
Set $U(t)=\exp(tZ)V(t)$. The transformed equation has constant coefficient:
\[
 \dot V=-(Z+\lambda C)V.
\]
At $t=2\pi$, the first factor returns to the identity. Therefore
\begin{equation}\label{eq:circleH}
 H_\lambda(\gamma_r)=\exp(-2\pi W),\qquad
 W=(1-\lambda)Z+\lambda\Ad_aZ.
\end{equation}
Within the active three-dimensional subspace, $W$ is a simple skew matrix.
Its $Q$-norm is
\begin{equation}\label{eq:Wnorm}
 s=\norm W_Q=\sqrt{1-4\coup\sin^2(r/2)}.
\end{equation}
For $0<r<\pi$ and $0<\lambda<1$, this is positive. Write $J=W/s$;
then $J$ is a unit elementary-type planar generator and $\exp(2\pi J)=\Id$.
Consequently
\begin{equation}\label{eq:positiveangle}
 H_\lambda(\gamma_r)=\exp(\beta J),\qquad
 \beta=2\pi(1-s).
\end{equation}

\begin{theorem}[Exact circle writer]\label{thm:writer}
Let $d\ge3$, $0<\lambda<1$, and $0<L<4\pi$. There is an absolutely
continuous, in fact smooth, based loop of length exactly $L$ whose normal
holonomy is a planar rotation through
\begin{equation}\label{eq:writer}
 \beta_\lambda(L)=2\pi\left(1-
      \sqrt{1-\frac{\coup L^2}{4\pi^2}}\right).
\end{equation}
The oriented output plane can be chosen arbitrarily. On the principal
one-circle branch, the inverse length formula is
\begin{equation}\label{eq:writerinverse}
 L=\sqrt{\frac{\beta(4\pi-\beta)}{\coup}},
 \qquad 0<\beta<4\pi\min(\lambda,1-\lambda).
\end{equation}
\end{theorem}
\begin{proof}
Choose $r=2\arcsin(L/(4\pi))$ in \eqref{eq:circleloop}. Equations
\eqref{eq:circlelength}--\eqref{eq:positiveangle} give the angle. Every unit
simple skew matrix in $\so(d)$ is conjugate under $\SO(d)$ to any prescribed
unit planar generator. Conjugating the entire base loop by the same group
element preserves its length and conjugates its holonomy, proving the plane
assertion. Squaring the relation between $\beta$ and $s$ gives
$\coup L^2=\beta(4\pi-\beta)$. The stated branch is exactly the image of
$0<L<4\pi$ before its boundary.
\end{proof}

The inverse formula is a constructive cost, not a universal isoholonomic
minimum at arbitrary angles. In particular, when $\lambda$ is very small,
a fixed non-small rotation may not lie on the one-circle branch. Repeating
circles is possible, but that repetition must be charged its full length.
It is not legitimate to extrapolate a small-loop square-root cost to an
arbitrary fixed target while taking $\lambda$ to zero.

\section{The sharp deficit follows}
Conjugate the output plane to contain the marked ray $u$. Then
\begin{equation}\label{eq:writerdeficit}
 d_\lambda(\gamma)=1-\cos\beta_\lambda(L)
                 =2\sin^2\bigl(\beta_\lambda(L)/2\bigr).
\end{equation}
The stable algebraic form
\begin{equation}\label{eq:rationalizedbeta}
 \beta_\lambda(L)=
 \frac{\coup L^2}{2\pi\left(1+\sqrt{1-\coup L^2/(4\pi^2)}\right)}
\end{equation}
avoids cancellation in numerical evaluation. It also shows directly that
\begin{equation}\label{eq:betaexpansion}
 \beta_\lambda(L)=\frac{\coup L^2}{4\pi}
                 +O(\coup^2L^4)
\end{equation}
uniformly for $0<\lambda<1$ and $L$ in a fixed sufficiently small interval.
Squaring the sine expansion gives
\[
 d_\lambda(\gamma)=\frac{\coup^2L^4}{32\pi^2}
                          \bigl(1+O(L^2)\bigr).
\]
Together with \eqref{eq:Dupper}, this proves
\begin{theorem}[Sharp uniform scalar law]\label{thm:scalarsharp}
For fixed $d\ge3$,
\begin{equation}\label{eq:scalarsharp}
 \sup_{0<\lambda<1}
 \left|\frac{32\pi^2D_\lambda(L)}{\coup^2L^4}-1\right|
 \le C_dL
\end{equation}
for $0<L<L_0(d)$. In particular, the leading constant $1/(32\pi^2)$ is
optimal, including in joint limits where $\lambda$ approaches a pure-gauge
endpoint as $L$ shrinks.
\end{theorem}

\section{Exact regular-simplex preparation}
Let $z_1,\ldots,z_d\in u^\perp$ be unit vectors satisfying
\begin{equation}\label{eq:simplex}
 \sum_i z_i=0,\qquad
 \ip{z_i}{z_j}=-\frac1{d-1}\quad(i\ne j),\qquad
 \sum_i z_iz_i^{\mathsf T}=\frac d{d-1}P_{u^\perp}.
\end{equation}
Conjugate $d$ copies of the circle writer so that their outputs are
\begin{equation}\label{eq:exactframe}
 v_i=c\,u+s_\beta z_i,\qquad
 c=\cos\beta,\quad s_\beta=\sin\beta,
 \quad\beta=\beta_\lambda(L).
\end{equation}
Every preparation loop has length $L$. The Gram matrix is
\begin{equation}\label{eq:exactGram}
 G_V=\frac d{d-1}s_\beta^2\Id+
       \left(c^2-\frac{s_\beta^2}{d-1}\right)\mathbf1\mathbf1^{\mathsf T}.
\end{equation}
Its eigenvalues are
\begin{equation}\label{eq:exactGramspectrum}
 dc^2\quad\text{once},\qquad
 \frac d{d-1}s_\beta^2\quad\text{$d-1$ times}.
\end{equation}
No asymptotic approximation is needed for these formulas.

When $0<\beta\le\arccos(1/\sqrt d)$, the first eigenvalue is the largest,
and the Gram condition number is
\begin{equation}\label{eq:conditionexact}
 \operatorname{cond}_2(G_V)=(d-1)\cot^2\beta.
\end{equation}
At the cap threshold $\cos^2\beta=1/d$, the frame is orthonormal. Whether
that threshold belongs to the single-circle branch depends on $\lambda$;
it always does at $\lambda=1/2$. The local regime of this manuscript instead
has $\beta$ small, so the transverse directions are necessarily much weaker
than the common direction.

\section{Why the construction is useful beyond a numerical example}
The circle writer performs four independent tasks. It gives an exact
differential-geometric realization of small rotations, it identifies the
correct length normalization, it proves attainment of the sharp local
constant, and it produces a complete family of known Gram spectra for testing
inverse algorithms. These roles are logically separate. A code check of the
spectrum is not a proof that the quotient has nonnegative curvature; that
curvature statement came from the quotient construction. Conversely,
nonnegative curvature alone would not provide the exact angle formula.

The orientation conjugations used in an optimal design may depend on a known
marked model and on the connection parameter. The design theorem is an
optimization statement over available loops, not a claim that an uncalibrated
instrument automatically knows the best loop. Finite scalar-history discovery
and noisy calibration are treated separately in Chapters~8 and~9.