\chapter{The observation problem}
\section{Geometry that is present but poorly observed}
A holonomy group records which linear transformations can be produced by
parallel transport around closed loops. It does not record how long those
loops must be, how accurately the resulting transformation can be measured, or
how many scalar observations are required to distinguish it from the identity.
These are different questions. In particular, a full normal holonomy group
need not provide a well-conditioned finite experiment.

We distinguish three levels. At the algebraic level, a collection of
transformations may generate a cyclic orbit of one vector. At the geometric
level, only the part of that orbit reachable within a specified loop budget
is available. At the statistical level, the accessible scalar differences
must be compared with a specified noise distribution. Confusing these levels
would make a rank theorem look like a stability theorem, or a stability theorem
look like a statement about experimental signal strength.

The setting of nonnegative sectional curvature is especially useful because
it supplies compact souls and strong geometric structure without making every
normal connection trivial. The soul theorem and Perelman's rigidity theorem
are the classical background \cite{cheeger-gromoll,perelman}. The construction
below does not attempt to strengthen their topological conclusions. Instead,
it isolates an observation problem within their geometric category.

\section{The marked-ray contract}
Fix $d\ge3$, the Lie group $G=\SO(d)$, and the inner product
\begin{equation}\label{eq:Q}
 Q(X,Y)=-\frac12\tr(XY),\qquad X,Y\in\so(d).
\end{equation}
Every elementary plane rotation has $Q$-norm one. The corresponding
bi-invariant Riemannian metric on $G$ will remain fixed throughout the family.
A loop means an absolutely continuous path
$\gamma:[0,1]\to G$ with $\gamma(0)=\gamma(1)=\Id$. Its length is
\begin{equation}\label{eq:length}
 \ell(\gamma)=\int_0^1\norm{\gamma(t)^{-1}\dot\gamma(t)}_Q\dd t.
\end{equation}
Piecewise smooth loops give the same suprema used below. Absolute continuity
is convenient for compactness and attainment statements.

Choose a marked unit vector $u\in\R^d$. The observation returned after
transport around $\gamma$ is
\begin{equation}\label{eq:echo}
 \phi_\lambda(\gamma)=\ip{u}{H_\lambda(\gamma)u}.
\end{equation}
The experiment knows the chosen loop and can reset the vector to $u$ before
the next observation. It can concatenate loops and reverse them. It does not
receive the coordinates of $H_\lambda(\gamma)u$, a transverse physical
reference, or an independent measurement of the ambient metric. These excluded
observations could change the inverse problem.

The scalar deficit is
\begin{equation}\label{eq:deficit}
 d_\lambda(\gamma)=1-\phi_\lambda(\gamma)
   =\frac12\norm{H_\lambda(\gamma)u-u}^2.
\end{equation}
Orthogonality of transport makes it nonnegative. The geometric parameter
$\lambda$ is not itself a measured echo. In the statistical comparison of two
specified hypotheses, its alternative value is assumed known to the design
procedure; estimating an arbitrary unknown parameter is a different problem.

\begin{definition}[Three extremal observation quantities]
For $0<\lambda<1$ and $L>0$, set
\begin{align}
 D_\lambda(L)&=\sup_{\ell(\gamma)\le L}d_\lambda(\gamma),\label{eq:Ddef}\\
 \Gamma_\lambda(L)&=\sup_{\ell(\gamma_i)\le L}
          \lambda_{\min}(V^{\mathsf T}V),\label{eq:Gammadef}\\
 \Omega_\lambda(L)&=\sup_{\ell(\gamma_i)\le L}
          \sigma_{\min}(\sensor_V),\label{eq:Omegadef}
\end{align}
where $i=1,\ldots,d$, $v_i=H_\lambda(\gamma_i)u$,
$V=[v_1\ \cdots\ v_d]$, and
$\sensor_V(K)=V^{\mathsf T}KV$ on $\so(d)$. The norm on this last space is
$\norm K_\wedge=\norm K_\Frob/\sqrt2$.
\end{definition}
The $d$ preparation loops have separate budgets $L$. Their total preparation
length is at most $dL$, not $L$. A scalar echo used to compare two prepared
rays generally traverses both preparation loops, so it has length at most
$2L$. A target-inserted measurement has an additional target-loop cost.

\section{The main theorem}
The next statement gathers the principal conclusions. Each assertion is proved
in the chapters indicated by its content; none is based on a numerical fit.

\begin{theorem}[Sharp observation law]\label{thm:main}
For every $d\ge3$ there is a family $(M_\lambda,g_\lambda)$,
$0<\lambda<1$, with the following properties.

\textup{(i)} Each manifold is complete, noncompact, and has sectional
curvature at least zero. Its zero section is a compact soul isometric to
$(\SO(d),Q)$, and its normal connection is
$A_\lambda=\lambda g^{-1}\mathrm dg$. Its normal holonomy group is
$\SO(d)$.

\textup{(ii)} Write $\coup=\lambda(1-\lambda)$. Every based loop obeys
\begin{equation}\label{eq:globalmain}
 \norm{H_\lambda(\gamma)-\Id}_\op
       \le \frac{\coup}{2}\ell(\gamma)^2.
\end{equation}
There is $L_0(d)>0$ such that, uniformly for $0<\lambda<1$,
\begin{equation}\label{eq:Dmain}
 D_\lambda(L)=\frac{\coup^2L^4}{32\pi^2}
                \bigl(1+O_d(L)\bigr),\qquad 0<L<L_0(d).
\end{equation}

\textup{(iii)} For sufficiently small $L$, again uniformly in $\lambda$,
\begin{equation}\label{eq:bridge-main}
 \Gamma_\lambda(L)=\Omega_\lambda(L)
 =\frac{d}{d-1}\left(2D_\lambda(L)-D_\lambda(L)^2\right).
\end{equation}
Consequently
\begin{equation}\label{eq:Gammamain}
 \Gamma_\lambda(L)=\Omega_\lambda(L)
 =\frac{d}{16\pi^2(d-1)}\coup^2L^4\bigl(1+O_d(L)\bigr).
\end{equation}

\textup{(iv)} In the fixed-sample independent Gaussian echo experiment with
variance $\sigma^2$, the optimal equal-prior risk for distinguishing the
product connection from the specified connection $A_\lambda$ is
\begin{equation}\label{eq:riskmain}
 \risk(n,\lambda,L,\sigma)
 =\PhiN\left(-\frac{\sqrt n D_\lambda(L)}{2\sigma}\right).
\end{equation}
Allowing the next loop to depend on all earlier observations does not improve
this risk.
\end{theorem}

The theorem is not a general classification of nonnegative-curvature metrics.
It is an explicit counterexample to uniform quantitative observability from
qualitative holonomy data, together with sharp local resource laws in the same
family. In particular, the base metric, base dimension, normal dimension, and
normal holonomy group can all remain unchanged while
$\Gamma_\lambda(L)$ tends to zero.

\section{An exact constructive witness}
The upper bound in \eqref{eq:Dmain} is attained to leading order by a closed
circle construction. For $0<L<4\pi$, it has exact holonomy angle
\begin{equation}\label{eq:writer-intro}
 \beta_\lambda(L)=2\pi\left(1-
       \sqrt{1-\frac{\coup L^2}{4\pi^2}}\right).
\end{equation}
The output plane can be prescribed by conjugating the loop. For a plane
containing $u$, its deficit is $1-\cos\beta_\lambda(L)$. For any regular
simplex $z_1,\ldots,z_d$ of unit vectors in $u^\perp$, copies of this loop
produce
\begin{equation}\label{eq:writer-frame-intro}
 v_i=\cos\beta_\lambda(L)\,u+\sin\beta_\lambda(L)\,z_i.
\end{equation}
The frame spectrum is then known exactly. This is useful both as an attainment
construction and as an executable normalization test. The fact that it has a
closed formula does not establish its optimality at arbitrary finite $L$.

\section{What is and is not being inferred}
A scalar echo determines an inner product, not a complete vector. Several
histories can synthesize a coordinate frame algebraically, but that frame is
only determined up to a common orthogonal change of coordinates. All operator
reconstruction statements respect this gauge freedom. No claim recovers an
unmarked laboratory orientation from scalar data alone.

The main family has full normal holonomy, but any chosen finite collection of
loops could still fail to be cyclic. The geometric design results select loops
that are cyclic. The finite realization results later explain exactly what can
be reconstructed from an arbitrary selected collection.

Finally, the noise model applies to the normalized scalar echo. Additive noise
in a squared-distance instrument before division by a radial scale is not the
same model. Chapter~11 distinguishes those quantities and supplies a separate
finite-radius geometric estimate.