| \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. |
|
|
