| \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. |
|
|
