Sharp edge recurrence in lune-based beta-skeletons under anisotropic stretching

We study the presence of a fixed edge in a planar lune-based beta-skeleton under the common deformation (x,y) -> (sqrt(t) x,y), t > 0. For n distinct points and fixed beta >= 1, the presence set has at most n-1 connected components; for beta = 1 (Gabriel), it is an interval. For every real beta > 1, integer n >= 3 and epsilon in (0,1), we construct rational points attaining n-1 positive-length components, with all n-2 absence intervals inside (1-epsilon,1+epsilon) and with pairwise disjoint closures. The same edge remains a Gabriel edge and a convex-hull edge for every t > 0. The upper bounds follow from elementary affine inequalities. The focus is the localized sharp construction with simultaneous Gabriel and hull persistence. A directed literature search, including the internal Zenodo search, located no equivalent construction in the material reviewed; bibliographic priority is not certified. The construction is adversarial and does not establish typical instability or the total event complexity of the entire graph. The reproducibility package contains a Python standard-library exact-arithmetic verifier and its archived certificates. These finite checks support the separate mathematical proof; they are not proof-assistant verification. AI assistance: OpenAI Codex was used extensively for mathematical exploration, proof drafting and checking, code generation, literature search, and editorial preparation. Internal checks were AI-assisted and do not constitute independent human peer review. Responsibility for the final manuscript and submission rests with the human author.