\chapter{Finite-radius distances without an artificial noise floor}
\section{From a normal vector to an actual point}
A scalar normal echo is a tangent-space quantity. To connect it with a
Riemannian distance, fix $x$ in the soul and use the normal exponential map.
In the global coordinates of Chapter~2,
\[
 \exp_x(tu)=(x,tu),\qquad
 \exp_x(tHu)=(x,tHu).
\]
Define the normalized squared-distance signal
\begin{equation}\label{eq:radiusdef}
 \mathcal D_t(u,H)=
 \frac{d_{M_\lambda}((x,tu),(x,tHu))^2}{2t^2},\qquad t>0.
\end{equation}
For a general smooth metric one expects its limit to be $1-u^{\mathsf T}Hu$.
The explicit quotient family permits a multiplicative bound, uniform even
when this echo deficit is very small.

\section{The normal fibers are totally geodesic}
Recall the metric
\[
 g_\lambda=Q+ h_v(\mathrm dv+\lambda\theta v,
                         \mathrm dv+\lambda\theta v),
 \qquad
 h_v(W,W)=\ip W{B_v^{-1}W}.
\]
The fiber metric $h$ is invariant under the standard $\SO(d)$ action. The
horizontal flow of the connection acts on the fiber by such rotations and
therefore preserves its metric.

\begin{lemma}[Totally geodesic fibers]\label{lem:geodesicfibers}
Every fiber $\{x\}\times\R^d$ is totally geodesic in $M_\lambda$.
\end{lemma}
\begin{proof}
Take vertical vector fields $V,W$ and a horizontal field $X$. The Koszul
formula expresses $2g(\nabla_VW,X)$ as the negative horizontal derivative
of $g(V,W)$ plus the two correction terms involving the vertical parts of
$[X,V]$ and $[X,W]$. Their sum is the negative Lie derivative of the fiber
metric along the horizontal flow. That derivative is zero because the flow
is a fiber isometry. The other Koszul terms vanish by horizontal--vertical
orthogonality and because $[V,W]$ is vertical. Hence the horizontal part of
$\nabla_VW$ is zero. This is exactly the vanishing of the second fundamental
form of the fiber.
\end{proof}

Total geodesy alone does not say that arbitrary distant fiber points are
joined by ambient minimizing geodesics staying in the fiber. We use it only
inside a sufficiently small common convex neighborhood of the zero section.
For such points the fiber geodesic is an ambient geodesic and, by local
uniqueness, is the ambient minimizer.

\section{Uniform local distance comparison}
In polar fiber coordinates,
\[
 h=\mathrm dr^2+\frac{r^2}{1+\coup r^2}\,g_{\mathbb S^{d-1}}.
\]
Consequently, on the Euclidean fiber ball $|v|\le t$,
\begin{equation}\label{eq:fibercomparison}
 \frac1{1+\coup t^2}g_{\rm Eucl}\le h\le g_{\rm Eucl}.
\end{equation}
The function $r^2$ is geodesically convex for this fiber metric. For example,
the radial Hessian is positive and the angular Hessian has the sign of
$f'(r)$ for $f(r)=r/\sqrt{1+\coup r^2}$; here $f'(r)>0$. Thus a
minimizing fiber segment with endpoints of radius $t$ remains in that ball.

\begin{theorem}[Multiplicative finite-radius calibration]\label{thm:radius}
There is a radius $t_0(d)>0$ such that, for $0<t<t_0(d)$, every unit $u$,
every $H\in\SO(d)$, and every $0\le\lambda\le1$ satisfy
\begin{equation}\label{eq:radiusbound}
 \frac{1-u^{\mathsf T}Hu}{1+\coup t^2}
 \le \mathcal D_t(u,H)\le 1-u^{\mathsf T}Hu.
\end{equation}
At the pure-gauge metric endpoints, interpret the metric by its smooth
extension. In particular, at $\lambda=0$ it is the product metric and both
inequalities are equalities.
\end{theorem}
\begin{proof}
On a compact tube the metric coefficients depend smoothly on
$\lambda\in[0,1]$, and remain uniformly positive definite. The zero section
is compact and homogeneous. Standard local existence and uniqueness for
the geodesic equation, together with a finite cover of the zero section and
the compact parameter interval, therefore provide a common sufficiently
small convex neighborhood radius. Choose $t_0$ inside it. By
Lemma~\ref{lem:geodesicfibers}, the short fiber geodesic between the two
points is also the ambient minimizing geodesic.

For a lower bound, this segment stays within radius $t$ by convexity. Its
length is at least the Euclidean chord length divided by
$\sqrt{1+\coup t^2}$, by \eqref{eq:fibercomparison}. For an upper bound,
the Euclidean straight chord remains in the same fiber ball and its
$h$-length is at most its Euclidean length. Squaring and using
$|tu-tHu|^2=2t^2(1-u^{\mathsf T}Hu)$ proves the claim.
\end{proof}

The bound is relative to the true deficit. It does not introduce an absolute
bias of order $t^2$ when the signal itself is much smaller. In this family,
a very weak echo remains proportionally weak in the finite-radius geometry:
\[
 0\le(1-\phi)-\mathcal D_t
 \le\frac{\coup t^2}{1+\coup t^2}(1-\phi).
\]
This matters in the joint weak-connection and short-loop limit.

\section{A direct curvature check inside the fibers}
For completeness, the radial and tangential sectional curvatures of the
fiber warped metric can be read from $f$:
\begin{align}
 K_{\rm radial}(r)&=-\frac{f''(r)}{f(r)}
                  =\frac{3\coup}{(1+\coup r^2)^2},\label{eq:radialcurv}\\
 K_{\rm tangential}(r)&=\frac{1-f'(r)^2}{f(r)^2}
 =\frac{\coup(3+3\coup r^2+\coup^2r^4)}{(1+\coup r^2)^2}.
 \label{eq:tangentcurv}
\end{align}
The limits at $r=0$ are both $3\coup$. These equations verify the
nonnegative fiber curvature directly. They are not a computation of every
ambient mixed sectional curvature; the ambient nonnegative-curvature proof
is the Riemannian quotient argument.

\section{Detector noise still has to be charged}
Suppose an actual instrument measures the raw squared distance with additive
standard deviation $\sigma_D$. Dividing by $2t^2$ gives normalized signal
noise $\sigma_D/(2t^2)$. Shrinking $t$ improves the multiplicative geometric
bias but amplifies that raw detector noise. The Gaussian theorem of
Chapter~10 assumes a stated normalized noise level $\sigma$; it does not
claim that $\sigma$ is independent of the physical radius in every detector.

A practical radius choice therefore needs both a detector model and a bias
budget. The present result supplies the exact geometric side of that
tradeoff. No universally optimal radius is asserted without the missing
measurement law. In particular, a mathematical tangent limit is not a method
for obtaining infinitely precise curvature from a finite apparatus.