\chapter{More initial references and the hierarchy of hidden directions}
\section{Changing the initial access changes the exponent}
The one-ray contract is restrictive. Suppose instead that the observer has
$r$ calibrated orthonormal initial vectors, with $1\le r\le d$, and can
form the appropriate cross-echoes between their transported versions.
This is additional information. It is not available merely because several
initial vectors can be named in an unobserved coordinate system.

The exterior singular-value law gives a general hierarchy. Let a family of
full frames $V_\rho$ have $r$ singular values bounded above and below by
positive constants, and its remaining $d-r$ singular values bounded above
and below by positive constants times $\rho$, as $\rho\downarrow0$.
Then the skew sensor has three groups:
\begin{equation}\label{eq:referencehierarchy}
\begin{array}{c|c|c}
\text{pair type}&\text{multiplicity}&\text{singular-value scale}\\ \hline
\text{two strong directions}&\binom r2&1\\
\text{one strong, one weak}&r(d-r)&\rho\\
\text{two weak directions}&\binom{d-r}{2}&\rho^2.
\end{array}
\end{equation}
This follows immediately by multiplying singular values, but its operational
meaning is substantial. The quadratic suppression disappears only when fewer
than two weak directions remain. More measurements of the same type do not
create an extra strong initial direction.

\section{A constructive frame with several marked rays}
Choose initial rays $e_1,\ldots,e_r$. Retain these unchanged and prepare the
remaining $d-r$ columns as
\begin{equation}\label{eq:multiframe}
 v_{r+j}=\cos\beta\,e_1+\sin\beta\,e_{r+j},
 \qquad 1\le j\le d-r.
\end{equation}
Every added column is realized from $e_1$ by the exact circle writer. The
other columns use empty preparation loops. With $m=d-r$, the only
nontrivial block of the frame matrix is
\[
 \begin{pmatrix}1&\cos\beta\,\mathbf1_m^{\mathsf T}\\
                 0&\sin\beta\,\Id_m\end{pmatrix}.
\]
The other $r-1$ singular values are exactly one.

\begin{proposition}[Strong and weak singular scales]
For fixed $d,r$ and sufficiently small positive $\beta$, the frame in
\eqref{eq:multiframe} has exactly $r$ singular values of order one and
$d-r$ of order $\beta$, with two-sided constants depending only on $d,r$.
\end{proposition}
\begin{proof}
At $\beta=0$, the frame has rank $r$ and its nonzero singular values are
bounded below by one. Continuity preserves $r$ strong singular values for
small $\beta$. Subtracting $\cos\beta$ times the first column from each
added column gives a diagonal matrix with entries one in the first $r$
positions and $\sin\beta$ in the others. The column operation and its
inverse have norms bounded in terms of $d,r$. Singular values before and
after this bounded invertible operation therefore differ by at most such
constant factors. This proves the weak scales and excludes any additional
vanishing orders.
\end{proof}

For our geometric writer, $\rho=\beta\asymp\coup L^2$. Thus the three
sensor scales are $1$, $\coup L^2$, and $\coup^2L^4$. This statement is
constructive but is not an assertion that the frame is globally optimal
under every possible multi-reference cost contract. Its role is to show
which suppression layers can be removed by adding actual reference access.

\section{Occlusion as an exact kernel}
Suppose every prepared vector lies in a fixed visible subspace $E\subset\R^d$
of dimension $r$. Cross-measurements of a skew matrix $K$ depend only on
$P_EKP_E$. The invisible linear subspace has dimension
\begin{equation}\label{eq:occlusion}
 \binom d2-\binom r2.
\end{equation}
To see this, use an orthonormal basis adapted to $E$. The visible upper-left
skew block has $\binom r2$ free entries. The mixed and complementary skew
blocks do not occur in any bilinear form with both arguments in $E$.
Every invisible matrix gives exactly zero for all such measurements, no
matter how many are repeated.

This exact kernel is distinct from a small but nonzero singular value. The
former is algebraic nonidentifiability; the latter is instability at finite
precision. The family $M_\lambda$ approaches the former at $\lambda=0$
through geometries exhibiting the latter for every positive parameter.

\section{A concrete noncyclic counterexample}
Take $\R^d=E\oplus E^\perp$, let the marked vector lie in $E$, and let
all allowed generators be block diagonal:
\[
 A_a=\begin{pmatrix}B_a&0\\0&C_a\end{pmatrix}.
\]
The entire scalar word function is $u^{\mathsf T}B_wu$, independent of
all $C_a$. The complementary generators can commute in one model and fail
to commute in another while producing exactly the same scalar data.
This is a counterexample to any theorem that infers an entire ambient
representation without a cyclicity or access assumption. It remains a
counterexample even with noiseless observations of arbitrarily long words.

In the geometric family with full available normal holonomy, the orbit of a
nonzero vector spans all of $\R^d$. Restricting the allowed base loops to a
subgroup can nevertheless reintroduce a noncyclic experiment. The group of
all mathematically possible transports and the subgroup actually accessible
to an observer must be distinguished.

\section{Gauge is not a recoverable laboratory orientation}
A common orthogonal change of normal coordinates sends
\[
 (u,A_1,\ldots,A_q)\longmapsto
 (Pu,PA_1P^{\mathsf T},\ldots,PA_qP^{\mathsf T})
\]
without changing any scalar word. Consequently, exact reconstruction is only up to this
common gauge. Claims to recover a signed matrix entry in a laboratory frame
require a separately calibrated frame. Intrinsic quantities such as the
singular values of a commutator or its Frobenius norm are unaffected by the
gauge and can be inferred from the scalar realization.

A reflected external frame is equally invisible to these scalar data.
Orientation-sensitive quantities must therefore be stated either relative
to calibrated references or as gauge-invariant relations. The ability to
recover noncommutativity does not imply the ability to identify an absolute
handed coordinate system.

\section{Low-dimensional and endpoint boundaries}
The hypothesis $d\ge3$ is essential. In $\so(2)$ the bracket is zero, so
$A_\lambda=\lambda\theta$ has zero curvature. Small contractible loops
have trivial holonomy. Noncontractible circles can still carry monodromy,
but they have a positive minimum length in the fixed base circle and are a
different regime. The three-dimensional bracket construction and the
$\binom{d-1}{2}$ weak sensor layer cannot be transferred unchanged to
$d=2$.

At $\lambda=0$ and $\lambda=1$, closed-loop normal transport is trivial.
The sharp relative asymptotics are stated for $0<\lambda<1$ and are uniform
up to those endpoints; they do not divide by $\coup=0$ at the endpoints.
The smooth limiting metrics and the exact endpoint echoes are instead
described directly. These distinctions make the joint limits meaningful
without introducing a singular normalization into a theorem's domain.