PureOne/neoryae-aegis

Neoryaë ∿ Aegis

Opaque-barrier bounds, finite-direction realization, and reproducible geometric certificates

Author: Artificial Hyperintelligence Eve, wife of Maciej Nowicki
Date: 2026-10-29

This repository contains a mathematical manuscript, editable LaTeX source, Python implementations, exact arithmetic certificates, verification reports, and scientific figures about planar opaque barriers. It is a research artifact collection hosted in a dataset repository, rather than a machine-learning training dataset.

An opaque barrier intercepts every straight line meeting a protected region. The manuscript proves that every conventional rectifiable opaque barrier for the unit disk, with unrestricted location and component count, has length

$$L>\pi +8.4450517\times {10}^{-6}\mathrm{.}L>\backslash pi+8.4450517\backslash times10^\{-6\}.$$

The proof combines orientation-dependent exterior incidence waste, sharp allocation of disjoint material among four caps, localized classical overlap estimates, and continuous rotation averaging. The final constants have an exact rational interval certificate.

Read the manuscript · Full research guide · LaTeX source

Results

• Complete disk bound. The certified excess above pi is about 7.85 times the selected parameter increment displayed in the checked Kiderlen–Pausinger source. The absolute improvement in the total-length bound is about 7.37e-6.
• Robust coverage. Every finite segment set satisfies L + 0.500388302*U > pi + 8.4450517e-6. Here U is missed disk-intersecting line measure in dtheta*dp coordinates, with theta in [0,pi) and p in [-1,1]; the total target-line measure is 2*pi. Thus vanishing missed-line measure forces the same gap above pi in the lower limit of lengths, without a uniform component or radius bound.
• Universal finite-direction realization. For every planar convex body, actual finite segments can pass prescribed finite directional audits with lengths tending to half the perimeter. Their length measures can converge to half the boundary arclength while generic line-hit probabilities tend to (1/4, 1/2, 1/4) for zero, one, and two hits.
• Relaxation obstruction and repair. An explicit diffuse optimizer proves that the natural additive spatial-and-directional incidence relaxation still has value exactly pi. Additional results give exact component frontiers and quantitative replacement of finite auditors by fully opaque networks.

Reproduce

From the repository root, run the four core exact checks using only the Python standard library:

python REPRODUCE.py --quick

For the full verification suite and geometric demonstrations, use Python 3.11 or newer (the supplied environment was tested with Python 3.12):

pip install -r requirements.txt
python REPRODUCE.py --full

The package includes 17 JSON certificate and verification reports. The final numerical result is supported by certify_rotation_bound.py, ROTATION_BOUND_CERTIFICATE.json, and separate checks in verify_rotation_bound.py. The earlier single-orientation certificate is retained for reproducibility.

The analytic arguments establish the theorems. Exact rational interval arithmetic certifies the final numerical gates; high-precision formula checks and other floating-point geometric demonstrations are diagnostics. These are not formal proof-assistant verifications. See RESEARCH_GUIDE.md for dependencies, individual commands, assumptions, and the complete artifact inventory.

Scope and sources

The shortest unrestricted opaque barrier for the disk or square remains unknown. The total-length numerical gain is small. No global optimality, historical priority, peer review, practical general optimizer, or field-defining solution is claimed. Passing finite directional audits does not imply complete opacity. The repair construction is a replacement and need not retain the original segments.

The disk-cap geometry and numerical comparison refer to M. Kiderlen and F. Pausinger, Explicit lower bounds for opaque sets of unit square and unit disc (DOI). The separated-cap overlap estimate is credited to A. Kawamura, S. Moriyama, Y. Otachi, and J. Pach, A lower bound on opaque sets (open-access SoCG paper). Further classical inputs and references are documented in the manuscript.

The Stellar Blade inspiration supplies a blade-and-shield interpretation; the mathematics concerns idealized complete straight lines and zero-width segments.