Black Holes as Critical Degeneracy of Geometric Projectability: AJoint ERP–Thales Framework

Abstract: This work proposes a structural relation between two frameworks addressing the critical structure of black-hole regimes from complementary perspectives. The Effective Relational Projectability (ERP) approach interprets spacetime geometry not as fundamentalbut as a consistent projection of an underlying correlational organization. Within this view,black-hole formation corresponds to a degeneration of geometric projectability. Independently, the Thales cascade provides a bounded quantitative diagnostic derived within general relativity, based on the geometry of constrained sectoral partitions satisfying x +y = 1.Weshowthat these two constructions can be connected through a nontrivial structuralcorrespondence. Under a normalized sector mapping, the Thales deficit defines an exactlocal bridge to the additive ERP departure from balance. While this does not establish a fullformal equivalence between the two frameworks, it shows that both select the same equilibrium point and admit a local translation between multiplicative and additive measuresof departure from criticality.When the cascade threshold is derived from the Einstein field equations, it yields aparameter-free numerical localization of an ERP-type projectability boundary. Illustrativecases—including binary black-hole inspiral, M87*, and TON 618—show that the combinedERP–Thales framework provides a physically interpretable and empirically accessible description of approach to strong-gravity critical regimes