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

Abstract:
 This work proposes a structural relation between two frameworks addressing the crit
ical structure of black-hole regimes from complementary perspectives. The Effective Re
lational Projectability (ERP) approach interprets spacetime geometry not as fundamental
but as a consistent projection of an underlying correlational organization. Within this view,
black-hole formation corresponds to a degeneration of geometric projectability. Indepen
dently, the Thales cascade provides a bounded quantitative diagnostic derived within gen
eral relativity, based on the geometry of constrained sectoral partitions satisfying x +y = 1.
Weshowthat these two constructions can be connected through a nontrivial structural
correspondence. Under a normalized sector mapping, the Thales deficit defines an exact
local bridge to the additive ERP departure from balance. While this does not establish a full
formal equivalence between the two frameworks, it shows that both select the same equi
librium point and admit a local translation between multiplicative and additive measures
of departure from criticality.
When the cascade threshold is derived from the Einstein field equations, it yields a
parameter-free numerical localization of an ERP-type projectability boundary. Illustrative
cases—including binary black-hole inspiral, M87*, and TON 618—show that the combined
ERP–Thales framework provides a physically interpretable and empirically accessible de
scription of approach to strong-gravity critical regimes