Geometric Necessary Conditions for Holographic Correspondence: A Thales Framework Diagnostic for AdS/CFT Structure

In this framework, the Thales construction is not interpreted as a literal geometric embedding of spacetime, but as a functional geometry encoding constraint structure, channel competition, and regime transitions. This technical note introduces a geometric diagnostic framework based on Thales’ theorem to analyze structural conditions associated with holographic correspondences. By representing the bulk–boundary relationship as a right triangle inscribed in a semicircle, the framework identifies a unique “apex” configuration that maximizes geometric quantities such as altitude-to-hypotenuse ratio and symmetric projection. This configuration aligns with key structural features of the AdS/CFT correspondence, including symmetric source–response structure, a well-defined radial direction, and maximal information accessibility. Configurations away from the apex exhibit progressive geometric degeneration, corresponding to known limitations of flat-space and de Sitter holography. In particular, the degenerate limit retains only global transition amplitudes, consistent with the appearance of meta-observables in de Sitter quantum gravity. The framework does not derive holography from first principles and introduces no new dynamical mechanisms; rather, it provides a geometric lens for organizing and diagnosing known holographic structures. The results highlight the structural rigidity of AdS/CFT and offer an intuitive geometric perspective on why full holographic dictionaries arise uniquely in anti-de Sitter spacetime.