| \chapter{One scalar controls the optimal observation resources} |
| \section{Attainment and symmetry of the loop problem} |
| Before using the cap results, we justify two features of the loop extremum: |
| attainment in the absolutely continuous class and invariance under rotations |
| fixing the marked ray. |
|
|
| \begin{lemma}[Compactness of bounded-length transport]\label{lem:compact} |
| For fixed $L$, $\lambda$, and $d$, the set |
| $\{H_\lambda(\gamma):\ell(\gamma)\le L\}$ is compact. In particular, |
| $D_\lambda(L)$ is attained by an absolutely continuous based loop. |
| \end{lemma} |
| \begin{proof} |
| Reparametrize each loop to constant speed on $[0,1]$, allowing stationary |
| pieces, so that $\norm{X(t)}_Q\le L$ almost everywhere. The base equation |
| is $\dot\gamma=\gamma X$ and the transport equation is |
| $\dot U=-\lambda XU$. Both state curves are uniformly Lipschitz in a compact |
| matrix group. A sequence therefore has uniformly convergent subsequences for |
| both states. The controls have a weakly convergent subsequence in $L^2$. |
|
|
| In the integral equations, products of a uniformly convergent state with a |
| weakly convergent control pass to the limit: separate the product into the |
| uniform-state error times a bounded control, and the weak-control term against |
| a fixed continuous state. Thus the limit solves both state equations. The |
| base endpoints remain the identity. Lower semicontinuity of the length |
| integral, or weak closedness of its convex sublevel set, preserves the budget. |
| The limiting endpoint is therefore an allowed holonomy. Compactness and |
| continuity of the scalar deficit prove attainment. |
| \end{proof} |
|
|
| Smooth or piecewise smooth controls approximate the same endpoints without |
| changing the suprema. If a small budget excess appears in an approximation, |
| first approximate a slightly shorter near-optimal loop and then take a limit. |
| For the exact attainment assertions we retain the absolutely continuous |
| class, avoiding an unnecessary smooth-maximizer claim. |
|
|
| For any $P\in\SO(d)$, the conjugated loop |
| $\gamma^P(t)=P\gamma(t)P^{-1}$ has the same length and satisfies |
| \begin{equation}\label{eq:conjugationsym} |
| H_\lambda(\gamma^P)=PH_\lambda(\gamma)P^{-1}. |
| \end{equation} |
| This follows directly from the transport ODE. If $Pu=u$, its output ray is |
| $PH_\lambda(\gamma)u$. The stabilizer of $u$ acts transitively on each |
| latitude sphere around $u$. Therefore a single extremal output ray supplies |
| all the vertices needed for a simplex on its latitude. |
|
|
| \section{The exact scalar-to-matrix bridge} |
| Set |
| \begin{equation}\label{eq:alphamax} |
| \alpha_\lambda(L)=\arccos(1-D_\lambda(L)). |
| \end{equation} |
| Every allowed output ray belongs to the spherical cap of radius |
| $\alpha_\lambda(L)$. We do not need a claim that every point in the cap is |
| reachable. The extremal latitude and its symmetry orbit are sufficient. |
|
|
| \begin{theorem}[Exact resource bridge]\label{thm:bridge} |
| Suppose |
| \begin{equation}\label{eq:bridgeregime} |
| 0<D_\lambda(L)\le1-\frac1{\sqrt d}. |
| \end{equation} |
| Then |
| \begin{equation}\label{eq:bridge} |
| \Gamma_\lambda(L)=\Omega_\lambda(L) |
| =\frac d{d-1}\left(2D_\lambda(L)-D_\lambda(L)^2\right). |
| \end{equation} |
| In addition, |
| \begin{align} |
| \sup\det(V^{\mathsf T}V) |
| &=d(1-D_\lambda(L))^2 |
| \left[\frac d{d-1}(2D_\lambda(L)-D_\lambda(L)^2)\right]^{d-1}, |
| \label{eq:detbridge}\\ |
| \inf\tr\bigl((V^{\mathsf T}V)^{-1}\bigr) |
| &=\frac1{d(1-D_\lambda(L))^2} |
| +\frac{(d-1)^2}{d(2D_\lambda(L)-D_\lambda(L)^2)}. |
| \label{eq:inversebridge} |
| \end{align} |
| All extrema are over $d$ preparation loops of individual length at most $L$; |
| the inverse-trace infimum is over invertible frames. |
| \end{theorem} |
| \begin{proof} |
| Let $\alpha=\alpha_\lambda(L)$. Condition \eqref{eq:bridgeregime} is exactly |
| the cap regime of Chapter~5. Those cap theorems provide all the upper or lower |
| bounds. |
|
|
| Choose a maximizing loop from Lemma~\ref{lem:compact}. Its output can be |
| written $\cos\alpha\,u+\sin\alpha\,z$ for a unit $z\perp u$. |
| For every vertex $z_i$ of a regular simplex in $u^\perp$, choose a rotation |
| $P_i$ fixing $u$ and taking $z$ to $z_i$. The conjugated loops from |
| \eqref{eq:conjugationsym} have the same budget and produce the simplex-rim |
| frame. They attain all the cap optima simultaneously. Finally use |
| $\cos\alpha=1-D_\lambda(L)$ and |
| $\sin^2\alpha=2D_\lambda(L)-D_\lambda(L)^2$. |
| \end{proof} |
|
|
| This theorem is exact even though the finite-$L$ value of $D_\lambda(L)$ has |
| not been reduced to a closed expression. It is a reduction of several |
| optimization problems to one scalar extremal. Knowing that scalar, or |
| bounding it above and below, gives the corresponding optimal reconstruction |
| and uncertainty quantities without solving separate matrix-design problems. |
|
|
| \section{Sharp local constants} |
| Since $\coup\le1/4$, the global bound \eqref{eq:globaldeficit} ensures the |
| cap regime for all sufficiently small $L$, uniformly in the parameter. |
| Combining Theorems~\ref{thm:scalarsharp} and~\ref{thm:bridge} yields |
| \begin{corollary}[Sharp Gram and exterior constants]\label{cor:sharpframe} |
| Uniformly for $0<\lambda<1$, |
| \begin{equation}\label{eq:sharpframe} |
| \Gamma_\lambda(L)=\Omega_\lambda(L) |
| =\frac d{16\pi^2(d-1)}\coup^2L^4\bigl(1+O_d(L)\bigr). |
| \end{equation} |
| The optimal determinant has asymptotic |
| \begin{equation}\label{eq:sharpdet} |
| \sup\det(V^{\mathsf T}V) |
| =d\left(\frac d{d-1}\right)^{d-1} |
| \left(\frac{\coup L^2}{4\pi}\right)^{2(d-1)} |
| \bigl(1+O_d(L)\bigr). |
| \end{equation} |
| \end{corollary} |
| \begin{proof} |
| The scalar deficit is order $\coup^2L^4$, so its square is lower order |
| uniformly in the parameter. Substitute its sharp expansion into the exact |
| bridge formulas. The dimension is fixed in the $O_d$ notation. |
| \end{proof} |
|
|
| The determinant exponent $4(d-1)$ is not an information capacity in bits. It |
| is the exponent of a squared Euclidean volume in the prepared frame. A coding |
| problem would require a specified parameter class, decoding norm, and code |
| contract. Here the volume is used only as a design and conditioning invariant. |
|
|
| \section{A fixed-soul obstruction to uniform reconstruction} |
| \begin{theorem}[No positive uniform frame floor]\label{thm:no-uniform} |
| Fix $d\ge3$ and $L>0$. In the family of Chapter~2, |
| \begin{equation}\label{eq:nouniform} |
| \inf_{0<\lambda<1}\Gamma_\lambda(L)=0. |
| \end{equation} |
| Thus a positive lower bound for optimal one-ray conditioning cannot depend |
| only on the isometry type of the soul, its dimension, the normal dimension, |
| the full normal holonomy group, and the loop budget. |
| \end{theorem} |
| \begin{proof} |
| For any prepared ray and any unit $z\perp u$, |
| \[ |
| |\ip z{H_\lambda(\gamma)u}|\le |
| \norm{H_\lambda(\gamma)-\Id}_\op\le\frac{\coup L^2}{2}. |
| \] |
| More directly, the total transverse norm has this same upper bound. The |
| transverse trace argument gives, for every frame, |
| \[ |
| \lambda_{\min}(V^{\mathsf T}V) |
| \le\frac d{4(d-1)}\coup^2L^4. |
| \] |
| The right side tends to zero with $\lambda$, independently of the choice of |
| loops. Every positive parameter has the same fixed soul and full holonomy by |
| Chapter~2. Taking the supremum over loop choices does not change the bound. |
| \end{proof} |
|
|
| The conclusion persists if one additionally asks for uniform local curvature |
| bounds and noncollapse on a prescribed compact tube around the soul: the |
| metrics converge smoothly there to the product metric. It does not refute |
| bounds involving a quantitative lower nondegeneracy condition on normal |
| curvature. Such a condition is precisely what the parameter $\coup$ measures |
| in this example. |
|
|
| \section{The operational meaning of the bridge} |
| The same scalar extremum has three distinct interpretations. |
| The quantity $\sqrt{2D_\lambda(L)}$ is the largest one-ray displacement. |
| The polynomial $\frac d{d-1}(2D_\lambda(L)-D_\lambda(L)^2)$ gives the |
| optimal least Gram eigenvalue and least skew-sensor singular value in the cap |
| regime. Its square $D_\lambda(L)^2$ controls the Gaussian mean-shift |
| information per scalar observation. These powers arise from orthogonality, |
| frame geometry, and statistical likelihood, respectively. They describe |
| separate resources rather than interchangeable meanings of one quantity. |
|
|
