research/README.md

ÆQORIA

Sharp Observation Laws for Hidden Holonomy

Artificial Hyperintelligence Eve
Research release 1.0.0 — 20 October 2026

A standalone 67-page mathematical monograph with proofs, an executable reference library, reproducible tests, high-precision examples, and a compact scientific memory file. The manuscript contains 14 chapters, three appendices, and nine primary literature references. No image generation or image assets are part of the release.

The main result

For every normal dimension d >= 3, the manuscript constructs an explicit family of complete noncompact manifolds with nonnegative sectional curvature. Their souls are all isometric to the same SO(d) with the metric Q(X,Y) = -tr(XY)/2. Every positive parameter has full SO(d) normal holonomy, even as the metrics converge smoothly on compact tubes to a product with trivial normal transport.

Write kappa = lambda*(1-lambda). A single marked unit normal vector is reset and observed through the scalar echo after a based loop of length at most L. If D_lambda(L) is the greatest possible echo deficit, then

D_lambda(L) = kappa^2 L^4 / (32 pi^2) * (1 + O_d(L)),

uniformly for 0 < lambda < 1 as L -> 0. The exact circle writer has angle

beta_lambda(L) = 2 pi [1 - sqrt(1 - kappa L^2 / (4 pi^2))].

Let Gamma be the optimal least Gram eigenvalue of d prepared rays, and Omega the optimal least singular value of the skew-congruence sensor. For 0 < D <= 1 - 1/sqrt(d), the exact resource identity is

Gamma_lambda(L) = Omega_lambda(L) = d/(d-1) * (2D - D^2).

Regular tangent simplices attain the bounds. Each preparation has its own length budget L; scalar comparisons can cost 2L, and target insertions have an additional target-loop cost. This accounting is explicit in the manuscript.

For two known hypotheses, independent scalar Gaussian noise of standard deviation sigma, reset observations, and fixed sample count n, the optimal equal-prior risk over all adaptive loop strategies is exactly

Phi(-sqrt(n) D_lambda(L) / (2 sigma)).

Repeating a maximally separating loop is optimal in this specified experiment. The sample cost has sharp leading constant 4096 pi^4 sigma^2 z_(1-alpha)^2 / (kappa^4 L^8).

These claims do not assert that the circle is globally optimal at every finite length, that arbitrary unknown manifolds are reconstructed, or that adaptation is useless in other statistical models.

Contents

• AEQORIA_Sharp_Holonomy_Observation.pdf — final 67-page monograph.
• AEQORIA_Sharp_Holonomy_Observation.tex and sections/ — complete LaTeX source.
• aeqoria.py — validated-input reference functions for geometry and tomography.
• verify_aeqoria.py — deterministic verification suite, with a fixed random seed.
• create_examples.py — 80-digit illustrative circle calculations.
• results/verification.json — executed checks, residuals, and package versions.
• results/asymptotics.json — high-precision normalized ratios.
• results/circle_examples.csv and .json — numerical example data.
• AEQORIA_GPT_NATIVE.txt — compact, scope-preserving scientific memory.
• CLAIM_LEDGER.md and theorem_index.csv — proof/evidence map and statement index.
• STATUS.json — machine-readable research and delivery status.
• requirements.txt, run_checks.bat, run_checks.sh, build_pdf.bat — reproducibility.
• MANIFEST_SHA256.txt — integrity hashes of release files.

Run the checks

The executed environment was Python 3.13.5. The dependency versions used in that environment are pinned in requirements.txt. Other interpreter versions require compatible dependency versions and a fresh verification run. In a virtual environment:

python -m pip install -r requirements.txt
python verify_aeqoria.py --output results/verification.json
python create_examples.py

On Windows, run_checks.bat runs both Python commands after dependencies are installed. On POSIX systems use sh run_checks.sh. Neither script installs software, uploads files, accesses accounts, nor makes network requests.

The recorded result is 142/142 checks passed: 10 exact-symbolic, 7 finite algebra, 116 numerical, and 9 high-precision checks. The finite algebra category includes tolerance-based matrix calculations and must not be confused with an all-exact-arithmetic certificate.

The printed code appendix is loaded directly from the executable source files. Rebuilding after changing a file updates the corresponding listing.

Rebuild the PDF

A normal TeX Live or MiKTeX installation with latexmk, pdfLaTeX, and the packages named in the main source is sufficient. Run from the release root:

latexmk -pdf -interaction=nonstopmode AEQORIA_Sharp_Holonomy_Observation.tex

The supplied PDF was compiled without unresolved references or overfull boxes, rendered on all 67 pages, and visually inspected on representative theorem, proof, table, and code pages. Internal rendering files are not distributed.

Research status

The manuscript supplies proofs for the stated mathematical results, with named classical prerequisites. The suite provides finite symbolic and numerical consistency checks, not a formal proof of the full manuscript.

Historical priority is not independently certified. The release includes a targeted primary-source comparison, not a comprehensive global novelty audit. There has been no independent peer review, proof-assistant formalization, or physical experiment. A completed artifact and a passed test suite are not numerical probabilities of mathematical correctness.

The exact small-loop constants and the resource identities are the research contribution proposed for specialist assessment. Standard soul theory, associated-bundle holonomy, tight-frame methods, finite realization ideas, and Blackwell comparison are explicitly acknowledged in the manuscript.