\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.