| \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. |
|
|
