UCET–8: Universal Curvature Equivalence on Noncompact, Bounded, and Lorentzian Geometries (Eighth research paper in UCET Series)

In this eighth installment of the Universal Curvature Equivalence Theorem (UCET) program,

I extend the theory of estimator–independent curvature convergence to

\emph{noncompact manifolds}, \emph{manifolds with boundary}, and

\emph{Lorentzian ambient geometries}. 

While UCET–1 through UCET–7 established local and global equivalence for compact

Riemannian settings under deterministic, stochastic, and dynamic conditions, these results did not cover open, bounded, or indefinite–signature domains—precisely the geometries of most analytical and physical interest.  UCET–8 fills this gap by introducing weighted Sobolev and decay structures that guarantee curvature–estimator equivalence on noncompact manifolds, boundary–regularized curvature flux formulas ensuring consistency at interfaces, and a Lorentzian generalization of the equivalence theorem that preserves causal signature and geometric flux conservation across spacelike hypersurfaces.  The resulting framework demonstrates that curvature equivalence is not merely a numerical property but a universal conservation principle linking discrete, stochastic, and physical models of geometry and topology.  I show that, under mild regularity and decay assumptions, all admissible curvature estimators converge to the same geometric invariants, and the corresponding integrated fluxes remain estimator–independent even in indefinite metric spaces.  This establishes UCET–8 as a unifying bridge between computational geometry, geometric analysis, and relativistic field theory.

For clarity, throughout this paper I denote by \(h>0\) the intrinsic neighborhood or bandwidth parameter controlling estimator locality, and by \(\eta\) the Lorentzian metric tensor of signature \((-+\dots+)\) used for curvature and flux computations.