research/sections/13_examples.tex

\chapter{Worked examples and the executed verification record}
\section{A three-dimensional normal space}
Take $d=3$ and $\lambda=1/2$, so $\coup=1/4$. The soul is the fixed
three-dimensional Lie group $(\SO(3),Q)$ and the total manifold has
dimension six. With individual preparation length $L=0.5$, the circle
writer gives, to the displayed precision,
\[
 \beta=0.0049756\ldots,\qquad
 D_{\rm circle}=1.237808\times10^{-5}.
\]
The displayed angle is rounded; the files store substantially more digits.
The exact formula, rather than this rounded value, is used in all
calculations. The regular-simplex frame has
\[
 \lambda_{\min}(G)=3.713402\times10^{-5},\qquad
 \operatorname{cond}_2(G)=8.078645\times10^4.
\]
Its skew space has dimension three. Two sensor singular values are
$3\cos\beta\sin\beta/\sqrt2$, while the third is
$3\sin^2\beta/2$. Thus the weakest rotation is much less observable than
the other two, even though the prepared vectors are linearly independent.

In the Gaussian experiment with normalized $\sigma=10^{-4}$ and target
risk $\alpha=0.05$, repeating this particular circle requires an
illustrative integer threshold of $707$ shots. This is the exact formula
for the selected loop evaluated numerically. It is not a certified statement
that $707$ is the globally optimal finite-$L$ threshold over every possible
loop. The geometric theorem proves equality of the leading asymptotic
constant, not finite-length optimality of the circle.

\section{Weak coupling at fixed soul and loop length}
Table~\ref{tab:examples} holds the normal dimension at three and uses the
same normalized detector noise and target risk. Only the connection parameter
or the length of the chosen circle changes. Every positive-parameter row has
full normal holonomy and the same soul metric.

\begin{table}[htbp]
\centering\small
\begin{tabular}{rrrrr}
\toprule
$\lambda$&$L$&Circle deficit&Least Gram eigenvalue&Shots\\
\midrule
0.5&0.5&$1.237808\,10^{-5}$&$3.713402\,10^{-5}$&707\\
0.1&0.5&$1.603389\,10^{-6}$&$4.810164\,10^{-6}$&42,096\\
0.01&0.5&$1.939609\,10^{-8}$&$5.818828\,10^{-8}$&287,664,265\\
0.5&0.2&$3.166688\,10^{-7}$&$9.500062\,10^{-7}$&1,079,206\\
0.5&0.1&$1.978992\,10^{-8}$&$5.936976\,10^{-8}$&276,328,970\\
\bottomrule
\end{tabular}
\caption{Illustrative circle-design values, evaluated with 80 decimal digits
before rounding. Noise is $\sigma=10^{-4}$, and the equal-prior target risk
is $0.05$. Shots are numerical evaluations of the ceiling in the selected-loop
Gaussian formula, not interval-certified global finite-length optima.}
\label{tab:examples}
\end{table}

The size of these numbers is a consequence of the declared scalar signal
and independent noise model. It is not an estimate for a photonic device,
a biological sensor, or an unspecified geometric computer. A detector with
a different observable or radius-dependent variance would require a separate
calibration before these numbers could be interpreted physically.

The table nevertheless illustrates a rigorous qualitative separation. The
holonomy group does not change as $\lambda$ decreases, whereas the weakest
Gram direction and the scalar gap shrink. Increasing the number of exact
algebraic directions does not force a favorable noise scale.

\section{An explicit dimension-four frame}
For $d=4$, choose $u=\mathbf1/2$ and let
\[
 Z=\sqrt{\frac43}(\Id-uu^{\mathsf T}).
\]
Its four columns are the unit vertices of a regular simplex in $u^\perp$.
Set $V=\cos\beta\,u\mathbf1^{\mathsf T}+\sin\beta\,Z$.
Then the Gram spectrum is
\[
 4\cos^2\beta,\quad
 \tfrac43\sin^2\beta,\quad
 \tfrac43\sin^2\beta,\quad
 \tfrac43\sin^2\beta.
\]
The six-dimensional skew sensor has three singular values
$4\cos\beta\sin\beta/\sqrt3$ and three singular values
$4\sin^2\beta/3$. This is the smallest normal dimension in which the
strong and weak skew layers have equal multiplicities. The formula is exact
for all angles in the cap regime and supplies a direct, small matrix check
of the resource law.

\section{Exact-symbolic tests}
The executable suite contains ten symbolic equality checks performed by
SymPy. They verify the fixed-soul metric coefficient, the connection
parameter, the curvature factor, the leading circle-angle coefficient,
the inverse writer relation, fiber-curvature normalization, simplex spectral
identities, and the mean and variance identities used by the Gaussian
comparison kernel. These tests simplify expressions to zero or compare
exact expressions. Their scope is the stated algebraic identities, not the
entire differential-geometric proof.

The finite algebra category contains seven checks. It includes generation
of the skew Lie algebra and scalar-history reconstruction in selected
dimensions. Some use floating matrix arithmetic and a stated numerical
tolerance. The label ``algebra'' indicates what is being tested; it does
not imply that every test in that category is exact arithmetic.

\section{Numerical and high-precision tests}
The numerical category contains $116$ tests. These cover the quotient
minimization against the explicit metric, exact circle transport against an
independently integrated ODE, arbitrary closed polygonal loop bounds,
frame spectra, exterior sensor singular values, cap inequalities, inverse
worst-direction attainment, Gaussian thresholds, noisy Gram calibration,
and orthogonal completion of skew contractions.

Six independent circle ODE integrations compare the exact transport with
a high-accuracy adaptive integration. In the executed run the largest
operator-norm discrepancy is below $1.3\times10^{-12}$. This is numerical
agreement in those six instances; it is not an interval enclosure of all
solutions or a proof of the exact writer theorem.

Nine additional checks use 80-digit arithmetic for joint small-loop and
weak-connection examples. They evaluate the normalized ratios
\[
 \frac{\beta}{\coup L^2/(4\pi)},\qquad
 \frac{D_{\rm circle}}{\coup^2L^4/(32\pi^2)},\qquad
 \left(\frac{\sin\beta}{\coup L^2/(4\pi)}\right)^2
\]
for $\lambda\in\{0.5,0.01,10^{-8}\}$ and
$L\in\{0.1,0.01,0.001\}$. These checks prevent ordinary floating-point
cancellation from being mistaken for a vanishing geometric signal.

\begin{table}[htbp]
\centering
\begin{tabular}{lrr}
\toprule
Executed category&Checks&Passed\\
\midrule
Exact-symbolic identities&10&10\\
Finite algebra checks&7&7\\
Numerical consistency tests&116&116\\
High-precision asymptotic checks&9&9\\
\midrule
Total&142&142\\
\bottomrule
\end{tabular}
\caption{The executed reference suite. Passing these finite tests is distinct
from formal proof verification and independent expert review.}
\end{table}

\section{Reproducing the record}
The recorded environment is Python 3.13.5, NumPy 2.3.5, SciPy 1.17.0,
SymPy 1.14.0, and mpmath 1.3.0. The deterministic random seed is
$20261020$. Run
\begin{verbatim}
python verify_aeqoria.py --output results/verification.json
python create_examples.py
\end{verbatim}
from the release directory. The first command fails immediately when an
assertion fails and writes the machine-readable record only after completing
the checks. The second regenerates the illustrative circle tables with
high-precision arithmetic. Run times depend on the environment and are not
part of a mathematical performance claim.

The JSON record includes each check's category, pass status, relevant
residuals, and package versions. The accompanying CSV and JSON example files
retain more numerical precision than the typeset table. No downloaded data,
remote service, model checkpoint, or image generation is required to run
the mathematical tests.