\chapter{Algorithms and reproducibility}
\section{A stable formula is part of a correct implementation}
The direct expression $1-\sqrt{1-z}$ loses relative accuracy when $z$ is
much smaller than one. Rationalizing gives
\[
 1-\sqrt{1-z}=\frac z{1+\sqrt{1-z}},\qquad
 z=\frac{\coup L^2}{4\pi^2}.
\]
The implementation uses the rationalized angle and computes the deficit as
$2\sin^2(\beta/2)$ rather than subtracting $\cos\beta$ from one.
Both changes preserve small positive signals in floating arithmetic much
longer than the naive formulas. They do not remove underflow at arbitrarily
small parameters or turn ordinary arithmetic into interval arithmetic.

The public function checks $0<\lambda<1$, positive finite $L$, and the
one-circle branch $L<4\pi$. Its inverse checks the corresponding range of
$\beta$. Values outside that range are rejected rather than continued onto
an unproved optimal-control branch. A caller needing repeated circles must
construct that protocol and account for its accumulated length explicitly.

\section{Constructing the simplex without a random basis}
A deterministic ambient formula avoids numerical orthogonal-completion
choices. Set
\[
 u=\frac{\mathbf1}{\sqrt d},\qquad
 Z=\sqrt{\frac d{d-1}}(\Id-uu^{\mathsf T}),\qquad
 V=\cos\beta\,u\mathbf1^{\mathsf T}+\sin\beta\,Z.
\]
The columns of $Z$ have unit norm, lie in $u^\perp$, and have the required
simplex pairwise inner products. This formula uses $d$-dimensional ambient
coordinates but only a $(d-1)$-dimensional tangent simplex. It supplies the
exact known frame for the numerical verification suite.

Let $P_1=\mathbf1\mathbf1^{\mathsf T}/d$. The Gram matrix and its inverse
are
\begin{align}
 G&=d\cos^2\beta\,P_1+
       \frac d{d-1}\sin^2\beta(\Id-P_1),\\
 G^{-1}&=\frac1{d\cos^2\beta}P_1+
       \frac{d-1}{d\sin^2\beta}(\Id-P_1),
\end{align}
whenever both trigonometric factors are nonzero. The numerical code compares
the eigenvalues of the actual frame with this formula; it does not construct
both sides by the same eigensolver call.

\section{A coordinate basis for the exterior sensor}
For each pair $i<j$, define $J_{ij}$ by
$(J_{ij})_{ji}=1$, $(J_{ij})_{ij}=-1$, and zero elsewhere. In this convention,
the coordinate of a skew matrix on $J_{ij}$ is $K_{ji}$. The map from
coordinates to a skew matrix is an isometry from ordinary Euclidean norm to
$\norm\cdot_\wedge$.

The sensor matrix is built column by column: for each input $J_{ij}$,
compute $V^{\mathsf T}J_{ij}V$ and extract its lower-triangular skew
coordinates. Its ordinary singular values can therefore be compared directly
with the predicted products $s_is_j$. Counting both the upper and lower
entries without changing the norm would create an erroneous $\sqrt2$
factor. The reference code uses one coordinate per independent plane.

If $C=V^{\mathsf T}KV$ is known exactly, reconstruction is
$K=V^{-\mathsf T}CV^{-1}$. The implementation uses linear solves rather
than explicitly forming both inverse matrices. For a severely ill-conditioned
frame this is still an ill-conditioned mathematical problem; a numerically
stable solver cannot improve its information-theoretic inverse radius.

\section{Independent computation of the quotient metric}
The metric test deliberately follows two different routes. One evaluates
$\norm X_Q^2+\zeta^{\mathsf T}B_v^{-1}\zeta$ directly. The other writes
$Z$ in the elementary skew basis and minimizes the original strictly convex
quadratic expression
\[
 a\norm{X+Z}_Q^2+b\norm Z_Q^2+|W-Zv|^2.
\]
The linear system for its minimizer is assembled from the vectors $J_{ij}v$.
Agreement catches normalization errors in $Q$, the choice of $a,b$, and the
adjoint $L_v^*$. It cannot by itself prove that a global quotient is complete
or that its zero section is totally convex; those properties require the
separate geometric arguments.

\section{Independent circle transport integration}
The exact answer $\exp(-2\pi W)$ is compared to an ODE solver for
\[
 \dot U(t)=-\lambda\Ad_{e^{tZ}}(\Ad_aZ-Z)U(t),\quad U(0)=\Id.
\]
This integration does not use the constant-coefficient gauge reduction when
advancing the state. The final matrix is compared both to the exact
exponential and to the reduced-angle rotation. The loop length is independently
checked from the body velocity norm. Errors are recorded in the JSON output,
not hidden behind a single aggregate pass count.

\section{Closed polygonal loops for a global-bound check}
The suite also constructs piecewise-geodesic base loops. If the first pieces
have body increments $X_1,\ldots,X_m$, their endpoint is
$g=\exp X_1\cdots\exp X_m$. A final short logarithmic closing piece
returns the base to the identity. Transport on one constant-body segment is
left multiplication by $\exp(-\lambda X_j)$. Thus transport and the base
product have different multiplication placements, as required by their
ODEs. Confusing them would incorrectly make every connection a pure gauge.

The total length is the sum of the $Q$-norms of the segment increments.
The recorded endpoint error confirms base closure. The transport norm is
then checked against the global quadratic inequality. Random tests are
finite consistency checks only: the proof covers all absolutely continuous
loops without a random-distribution assumption.

\section{Running and rebuilding the release}
The release has three computational entry points. The main library contains
validated functions for the construction and inverse calculations. The test
script exercises the theorem identities. The example generator writes the
tables. The supplied Windows batch file and POSIX shell script run the latter
two without uploading data or modifying any external account.

The manuscript can be rebuilt with
\begin{verbatim}
latexmk -pdf -interaction=nonstopmode \
  AEQORIA_Sharp_Holonomy_Observation.tex
\end{verbatim}
The source inputs are relative to the release directory. The code listings
are read from the actual executable files, so changing a source file and
rebuilding the PDF changes its printed listing too. This avoids distributing
one algorithm while typesetting a different one.

The pinned environment file records the versions used for the executed run.
Other sufficiently recent compatible versions may work, but their results
should be regenerated rather than assumed identical. A mismatch in a last
floating-point digit is not necessarily a theorem failure; a failed declared
inequality or a changed branch convention requires investigation.

\section{Evidence that is intentionally not claimed}
The reference suite is not an exhaustive finite-field enumeration, a machine
proof of all real inequalities, a validated integration over every path, or a
benchmark of asymptotic runtime. The 80-digit calculations improve precision
but do not provide directed-rounding enclosures. The exact-symbolic tests
certify the listed symbolic equalities in the executed algebra system; the
manuscript's analytic proofs are the justification for the general theorems.

A release integrity hash only establishes that a file has not changed since
the hash was computed. It says nothing by itself about mathematical truth.
For that reason the theorem ledger records proof locations, numerical support,
and external dependencies separately from file checksums.