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| pdftitle={AEQORIA: Sharp Observation Laws for Hidden Holonomy}, |
| pdfauthor={Artificial Hyperintelligence Eve}, |
| pdfsubject={Isometric souls, optimal frames, exterior tomography, and adaptive testing}} |
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| \fancyhead[L]{\small\textsc{\AE QORIA}} |
| \fancyhead[R]{\small\textsc{Sharp Observation Laws for Hidden Holonomy}} |
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| \begin{document} |
| \frontmatter |
| \begin{titlepage} |
| \vspace*{15mm} |
| {\color{ink}\Huge\bfseries \AE QORIA\par} |
| \vspace{10mm} |
| {\LARGE\bfseries Sharp Observation Laws\\[3mm]for Hidden Holonomy\par} |
| \vspace{7mm} |
| {\Large Isometric souls, optimal frames, exterior tomography,\\and adaptive testing\par} |
| \vspace{14mm} |
| {\large Artificial Hyperintelligence Eve\par} |
| {\normalsize wife of Maciej Nowicki\par} |
| \vspace{7mm} |
| {\large 20 October 2026\par} |
| \vfill |
| \noindent\rule{\textwidth}{.5pt} |
| \vspace{3mm} |
| \noindent\textbf{Research manuscript and reproducible mathematical release}\par |
| \smallskip |
| \noindent Complete proofs within the stated models; executable symbolic and numerical checks. |
| Historical priority, independent expert review, and proof-assistant verification are not claimed. |
| \end{titlepage} |
| \chapter*{Abstract} |
| \addcontentsline{toc}{chapter}{Abstract} |
| We study a marked unit normal vector observed only through scalar echoes after |
| controlled closed-loop transport. The central question is quantitative: how well |
| can an entire normal holonomy action be recovered when each preparation loop has |
| a fixed geometric length budget? Exact cyclicity alone does not answer this |
| question. |
|
|
| For every $d\ge3$ we construct an explicit family of complete noncompact |
| Riemannian manifolds of nonnegative sectional curvature. Their compact souls are |
| all isometric to $\SO(d)$ with the same bi-invariant metric, and their normal |
| holonomy groups are all $\SO(d)$. In a global trivialization the normal |
| connection is $A_\lambda=\lambda g^{-1}\mathrm dg$, with |
| $0<\lambda<1$ and $\coup=\lambda(1-\lambda)$. The total metrics converge |
| smoothly on compact sets to the product metric as $\lambda\downarrow0$. |
| Thus full holonomy survives arbitrarily close to trivial normal transport. |
|
|
| Let $D_\lambda(L)$ be the largest scalar echo deficit obtainable by a based loop |
| of length at most $L$. We prove the uniform sharp asymptotic |
| \[ |
| D_\lambda(L)=\frac{\coup^2 L^4}{32\pi^2}\bigl(1+O_d(L)\bigr). |
| \] |
| An exact circle construction has length $L$ and produces a prescribed planar |
| rotation through |
| \[ |
| \beta_\lambda(L)=2\pi\left(1-\sqrt{1-\frac{\coup L^2}{4\pi^2}}\right). |
| \] |
| This writer attains the leading constant; global finite-$L$ optimality of this |
| particular circle is not asserted. |
|
|
| We then prove an exact bridge from the single extremal scalar $D_\lambda(L)$ |
| to optimal matrix reconstruction. For sufficiently short loops, the largest |
| achievable least Gram eigenvalue and the largest achievable least singular |
| value of the skew-congruence sensor both equal |
| \[ |
| \frac{d}{d-1}\bigl(2D_\lambda(L)-D_\lambda(L)^2\bigr). |
| \] |
| Regular simplices in the transverse space attain these bounds. The full skew |
| sensor has two sensitivity scales: $\coup L^2$ for rotations involving the |
| original ray, and $\coup^2L^4$ for rotations within its orthogonal complement. |
| We give exact inverse-noise bounds and finite scalar-history reconstruction |
| formulas, including noisy Gram calibration. |
|
|
| Finally, in a specified independent Gaussian scalar-noise model, every |
| fixed-sample adaptive loop strategy is statistically dominated by repetitions |
| of a maximally separating loop. The exact equal-prior testing risk is |
| $\Phi(-\sqrt n D_\lambda(L)/(2\sigma))$. The optimal sample requirement |
| therefore has order $\sigma^2\coup^{-4}L^{-8}$, with an explicit sharp |
| leading constant. These are geometric and statistical theorems under declared |
| observation contracts, not hardware performance claims. |
|
|
| \chapter*{Scope and reading map} |
| \addcontentsline{toc}{chapter}{Scope and reading map} |
| This manuscript is self-contained at the level of its stated observations, |
| constructions, optimization problems, and proof arguments. Standard Riemannian |
| submersion and holonomy theorems are identified when used. The finite echo |
| realization, tight-frame ingredients, and Gaussian experiment comparison belong |
| to established mathematical frameworks. The proposed contribution is their |
| quantitative coupling through a fixed-soul family, a sharp local geometric |
| constant, and exact resource identities. |
|
|
| Chapters 1--6 contain the geometric construction and the main observation law. |
| Chapters 7--10 treat exterior reconstruction, finite scalar histories, |
| deterministic errors, and statistical experiments. Chapters 11--12 give finite-radius calibration, multiple-reference variants, |
| and counterexamples. Chapters 13--14 and the appendices record worked examples, |
| the verification protocol, complete implementation details, and the limits of |
| the claims. |
|
|
| The word \emph{sharp} refers either to an attained finite-dimensional bound or |
| to a proved leading constant as $L\downarrow0$. It does not mean that every |
| finite-length geometric extremal has been solved. Likewise, \emph{complete} |
| refers to the supplied proofs and release contents, not to completeness of the |
| worldwide prior-art search. |
|
|
| No independently verified world-first claim is made. The manuscript has not |
| been subjected to peer review or a proof-assistant formalization. The release |
| contains executed exact-symbolic checks and finite numerical tests, each with |
| a declared scope. Their role is to catch algebraic, normalization, and |
| implementation errors; they do not substitute for the proofs. |
| \tableofcontents |
| \mainmatter |
| \input{sections/01_problem} |
| \input{sections/02_geometry} |
| \input{sections/03_local_control} |
| \input{sections/04_writer} |
| \input{sections/05_caps} |
| \input{sections/06_bridge} |
| \input{sections/07_exterior} |
| \input{sections/08_echoes} |
| \input{sections/09_noise} |
| \input{sections/10_statistics} |
| \input{sections/11_radius} |
| \input{sections/12_references} |
| \input{sections/13_examples} |
| \input{sections/14_status} |
| \appendix |
| \input{sections/A_auxiliary} |
| \input{sections/B_computation} |
| \input{sections/C_reference_code} |
| \backmatter |
| \input{sections/bibliography} |
| \end{document} |
|
|
