Neoryaë ∿ Aegis: Rotation-Amplified Lower Bounds for Opaque Disk Barriers, Robust Coverage Inequalities, and Universal Finite-Direction Realization

Neoryaë ∿ Aegis presents a mathematical investigation of opaque barriers, geometric line coverage, and the relationship between finite directional verification and complete opacity. An opaque barrier for a planar convex body must intersect every entire straight line meeting that body. The manuscript studies actual geometric segment sets and conventional rectifiable barriers, allowing unrestricted locations and numbers of components unless additional assumptions are stated. The principal result is a rotation-amplified lower bound for opaque barriers of the unit disk: L > π + 8.4450517 × 10⁻⁶. This bound applies to conventional rectifiable opaque barriers. The argument combines orientation-dependent exterior incidence waste, allocation of disjoint material among four boundary caps, localized projection redundancy, and continuous rotation averaging. A sharp incidence-reuse inequality controls how often the same geometric intersections contribute to the rotated estimates. The numerical constants are supported by an exact rational interval certificate using integer-square-root enclosures, Machin’s identity for π, and alternating Taylor bounds. A robust extension applies to every finite segment set, including sets that miss some required lines: L + 0.500388302 U > π + 8.4450517 × 10⁻⁶. Here L is geometric union length, and U is the unnormalized measure of missed disk-intersecting lines in dθ dp coordinates, with θ in [0, π) and p in [−1, 1]. The complete target-line space has measure 2π. This inequality gives an explicit tradeoff between material length and incomplete coverage. It also implies a positive gap above π for the lower limit of lengths of any sequence whose missed-line measure tends to zero, without imposing a uniform component count, segment count, or radius bound. The underlying material-allocation theorem is formulated independently of the disk geometry. In an abstract four-pool model, feasible disjoint pair allocations have a sharp quadratic minimax reward of 5m²/9, where m is the mass of each pool. A convex-reward extension supports the geometric application. The sharpness statement concerns this allocation model; it does not assert that the resulting disk-barrier bound is optimal. The manuscript also establishes a universal finite-direction realization mechanism. For every compact planar convex body with nonempty interior, every finite set of tested normal directions can be covered by actual finite segments inside the body at length arbitrarily close to half its perimeter. For any prescribed increasing sequence of finite directional audits, deterministic realizations can simultaneously have length measures converging to half the boundary arclength, Hausdorff limit equal to the whole boundary, and limiting generic line-hit probabilities (1/4, 1/2, 1/4) for zero, one, and two intersections. The line-hit convergence holds against every fixed integrable line observable. Thus increasingly detailed finite audits and boundary convergence can coexist with persistent missed lines. A complementary obstruction identifies information lost by additive incidence models. An explicit diffuse measure, retaining both position and tangent direction, attains value exactly π in the natural spatial-and-directional Crofton incidence relaxation. This shows why that relaxation alone cannot certify a strict improvement over π. The result concerns the specified incidence relaxation, rather than every possible linear formulation with additional geometric constraints. Further results address exact component and penalty frontiers for regular disk audits, quantitative replacement of finite auditors by fully opaque networks, sharp asymptotic costs for retained-circle augmentation, and computation of the optimum under a fixed component budget. An analytic ceiling is proved for the particular rotated opposed-envelope certificate family, distinguishing limits of that formula from limits of the underlying opaque-barrier problem. The release includes the 61-page manuscript, editable LaTeX source, Python implementations, scientific figures, and 17 JSON certificate and verification reports. Exact arithmetic checks are distinguished from high-precision and floating-point diagnostics. The computational checks support numerical hypotheses and finite identities; the mathematical arguments supply the universal geometric statements. They are not formal proof-assistant verifications. The work provides lower bounds, robust inequalities, realization theorems, and explicit limitations of selected verification and relaxation methods. It does not determine the exact unrestricted shortest opaque barrier for the disk or square, and exhaustive historical priority is not established. Classical overlap, disk-cap geometry, Steiner-tree, and compactness inputs are credited in the manuscript. The Stellar Blade inspiration supplies a blade-and-shield interpretation; the mathematical model concerns complete straight lines and idealized zero-width barriers.