| \chapter{Finite scalar histories and exact realization} |
| \section{The algebraic observation model} |
| This chapter isolates the finite-dimensional reconstruction algebra from the |
| geometry. Let $A_1,\ldots,A_q\in\Orth(d)$ and let $u$ be a unit vector. |
| Letters are generator indices or their inverses. A word $w=a_1\cdots a_k$ |
| has length $k$ and operator $A_w=A_{a_1}\cdots A_{a_k}$. The empty word |
| has operator $\Id$. Inverting a word reverses its letters and inverts each |
| letter. These conventions ensure $A_{w^{-1}}=A_w^{\mathsf T}$. |
|
|
| The observed function is |
| \begin{equation}\label{eq:phiw} |
| \phi(w)=u^{\mathsf T}A_wu. |
| \end{equation} |
| It satisfies $\phi(w^{-1})=\phi(w)$, $\phi(e)=1$, and positive kernel |
| constraints: |
| \begin{equation}\label{eq:positivekernel} |
| \sum_{i,j}c_ic_j\phi(w_i^{-1}w_j) |
| =\left\|\sum_i c_iA_{w_i}u\right\|^2\ge0. |
| \end{equation} |
| The cyclic space is |
| \[ |
| E=\spanof\{A_wu:w\text{ a finite word}\},\qquad r=\dim E. |
| \] |
| It is invariant under every generator and inverse. Its orthogonal complement |
| is invariant too, so no echo can reveal the action on that complement. |
|
|
| This positive-kernel realization and its relation to state reconstruction |
| are standard mathematical ideas. The explicit formulas below are included |
| so that all subsequent noise and geometric statements have a declared finite |
| interface; they are not presented as a new invention of realization theory |
| \cite{HoKalman}. |
|
|
| \section{A finite basis horizon} |
| Let $E_k$ be the span of words of length at most $k$. Then |
| $E_0=\R u$, and $E_k\subseteq E_{k+1}$. |
|
|
| \begin{lemma}[Growth or stabilization]\label{lem:wordgrowth} |
| If $E_k=E_{k+1}$, then $E_k=E$. Consequently, there is a basis of $E$ |
| consisting of $r$ orbit vectors with words of length at most $r-1$. |
| \end{lemma} |
| \begin{proof} |
| For every letter $a$, multiplication by $A_a$ sends each spanning word of |
| $E_k$ into $E_{k+1}$. Under the assumed equality, $A_aE_k\subseteq E_k$. |
| This holds for inverse letters as well. Thus $E_k$ is invariant and contains |
| $u$, so it contains every orbit vector. Until this happens, the dimension |
| increases by at least one at every step. Starting from dimension one, at |
| most $r-1$ increases are possible. Select a basis from the resulting finite |
| spanning set, retaining $u$ as its first vector. |
| \end{proof} |
|
|
| Choose words $w_1=e,w_2,\ldots,w_r$ from this lemma and set |
| $V=[A_{w_1}u,\ldots,A_{w_r}u]$, viewed as a map $\R^r\to E$. |
| Define observable matrices |
| \begin{equation}\label{eq:realizationGC} |
| G_{ij}=\phi(w_i^{-1}w_j),\qquad |
| C^{(a)}_{ij}=\phi(w_i^{-1}aw_j). |
| \end{equation} |
| Then $G=V^{\mathsf T}V>0$ and $C^{(a)}=V^{\mathsf T}A_aV$. |
|
|
| \begin{theorem}[Finite exact realization]\label{thm:realization} |
| The matrices |
| \begin{equation}\label{eq:realizationT} |
| T_a=G^{-1}C^{(a)} |
| \end{equation} |
| represent all generators on the cyclic space and satisfy |
| \begin{equation}\label{eq:Gorthogonal} |
| T_a^{\mathsf T}GT_a=G. |
| \end{equation} |
| All their defining echoes have word length at most $2r-1$. The orthogonal |
| matrices |
| \begin{equation}\label{eq:realizationO} |
| O_a=G^{-1/2}C^{(a)}G^{-1/2} |
| \end{equation} |
| are the same representation in an orthonormal frame. It is uniquely |
| determined by the full echo function up to a common orthogonal change of |
| coordinates on $E$ carrying its marked vector to the other marked vector. |
| \end{theorem} |
| \begin{proof} |
| The coordinate identity $A_aV=VT_a$ follows by multiplying by |
| $V^{\mathsf T}$ and using that $V$ is a bijection onto $E$. Taking inner |
| products gives \eqref{eq:Gorthogonal}. The columns of $W=VG^{-1/2}$ are |
| orthonormal, and $W^{\mathsf T}A_aW$ is exactly \eqref{eq:realizationO}. |
| A mixed word has length at most $(r-1)+1+(r-1)=2r-1$. |
|
|
| For uniqueness, send a formal combination $\sum c_iA_{w_i}u$ to the |
| corresponding combination in a second model with the same echoes. Its norm |
| and all inner products are preserved by \eqref{eq:positivekernel}; a zero |
| combination is therefore sent to zero. This defines a well-defined isometry |
| on the cyclic span. It intertwines all generators because extending a word |
| by a letter commutes with the construction. Surjectivity onto the second |
| cyclic span follows in the same way. |
| \end{proof} |
|
|
| The marked coordinate vector is $e_1$ in the nonorthogonal basis. Hence |
| arbitrary future echoes are recovered by |
| \begin{equation}\label{eq:replay} |
| \phi(a_1\cdots a_k) |
| =e_1^{\mathsf T}GT_{a_1}\cdots T_{a_k}e_1. |
| \end{equation} |
| No finite-precision guarantee follows from the horizon alone. A positive |
| Gram eigenvalue can be arbitrarily small, as the fixed-soul construction |
| makes explicit. |
|
|
| \section{Discovering a basis through scalar tests} |
| Suppose a partial independent frame has Gram matrix $G_k$. For a candidate |
| word $w$, let $g_i=\phi(w_i^{-1}w)$. The squared residual outside its span |
| is the Schur complement |
| \begin{equation}\label{eq:schurresidual} |
| \rho(w)=1-g^{\mathsf T}G_k^{-1}g. |
| \end{equation} |
| Indeed, $G_k^{-1}g$ gives the orthogonal projection coefficients and the |
| candidate has norm one. In exact arithmetic, a positive residual means the |
| candidate adds a direction. Exploring extensions of retained words by all |
| letters is a finite breadth-first procedure. Once every such extension has |
| zero residual, the retained span is invariant and complete by |
| Lemma~\ref{lem:wordgrowth}. |
|
|
| With noisy data, the sign of a small residual is not a certified rank test. |
| One must propagate an error interval through both $G_k^{-1}$ and $g$, or |
| assume a separation margin for nonzero residuals. A stopping rule that treats |
| floating-point zero as mathematical zero can silently discard weak geometric |
| directions. Chapter~9 gives a quantitative calibration threshold but does not |
| pretend that unknown rank has a universal noise-independent threshold. |
|
|
| \section{Recovering signed transformation order} |
| For arbitrary words $a,b$ with operators $A,B$, define |
| \[ |
| \Delta_{ij}=\phi(w_i^{-1}ab w_j)-\phi(w_i^{-1}ba w_j). |
| \] |
| Then |
| \begin{equation}\label{eq:orderdefect} |
| \Delta=V^{\mathsf T}(AB-BA)V. |
| \end{equation} |
| The physical defect in the orthonormal cyclic frame is |
| $G^{-1/2}\Delta G^{-1/2}$. Thus its Frobenius norm, singular values, and |
| rank are recovered exactly. Its entries in an external laboratory frame |
| remain undetermined unless that frame has also been calibrated. |
|
|
| A different scalar protocol measures the total defect strength. Put |
| $C=(AB)^{-1}BA$ and |
| \[ |
| \zeta_i=1-\phi(w_i^{-1}Cw_i) |
| =\tfrac12\|(C-\Id)v_i\|^2. |
| \] |
| Because $AB$ is orthogonal, |
| $\|(C-\Id)\|_\Frob=\|BA-AB\|_\Frob$ on $E$. Therefore |
| \begin{equation}\label{eq:energyframe} |
| \frac{\lambda_{\min}(G)}2\|AB-BA\|_\Frob^2 |
| \le\sum_i\zeta_i\le |
| \frac{\lambda_{\max}(G)}2\|AB-BA\|_\Frob^2. |
| \end{equation} |
| To prove this, write the sum as |
| $\frac12\tr((C-\Id)^{\mathsf T}(C-\Id)VV^{\mathsf T})$ and use the |
| spectral bounds on $VV^{\mathsf T}$. If $G=\Id$, the equality is exact: |
| $2\sum_i\zeta_i=\|AB-BA\|_\Frob^2$. |
|
|
| The energy protocol uses fewer distinct scalar values but squares the |
| small-defect amplitude. In a small-loop curvature experiment, a signed order |
| signal can have order $\eps^4$, whereas this nonnegative energy signal has |
| order $\eps^8$. Counting observations without counting this sensitivity |
| loss is not a valid comparison. |
|
|
| \section{What the finite horizon does and does not say} |
| The horizon $2r-1$ is an algebraic word bound. It is not a Riemannian length |
| bound. If the chosen generators have lengths at most $\ell_0$, it supplies |
| the sufficient length bound $(2r-1)\ell_0$ by concatenation, but different |
| word representations can have different geometric costs. Conversely, |
| continuous controls can describe a short base curve by a formula containing |
| large intermediate group factors, as in the circle writer. The length of |
| the actual curve, not the typography of its group expression, is the relevant |
| resource in the geometric theorems. |
|
|
