From Relational Gravity to Spectral Geometry: An EAS–NCG Correspondence

This paper is archived as a speculative research work.Entanglement–Algebraic Spacetime (EAS) formulates its Scalar Fields (SFs) in terms of reciprocal, phase-indexed binary relationships and scalar-bearing relational organizations, rather than an antecedent spacetime metric. The EAS gravitational series develops qualified source registrations from this algebraic foundation and connects them, under explicitly admitted gravitational-interface hypotheses, to isolated Schwarzschild geometry and nonlinear Brill–Lindquist and Bowen–York Einstein initial data. We establish the corresponding operator-theoretic formulation in the commutative spin sector of noncommutative geometry (NCG). The finite-source spatial manifolds admit complete nonunital Dirac spectral triples, the isolated Schwarzschild exterior admits a complete Euclidean spin representation, and the multiple-source initial data admit complete four-dimensional Riemannian extensions preserving their induced metrics and signed extrinsic curvatures on marked, cooriented hypersurfaces. Original source coefficients are recovered from the Euclidean Schwarzschild bolt area or from marked conformal puncture residues, independently of nonlinear geometric contributions and changes of gravitational state. The localized four-dimensional Dirac spectral action contains the Einstein–Hilbert curvature term with its standard heat-kernel coefficient and a positive effective coupling under an explicit spectral-moment condition; its weighted variation reduces to the Einstein-tensor variation on a localization plateau. These results establish a certified conditional, source-preserving correspondence from an independently specified relational source foundation, through selected general-relativistic representations, to spectral gravitational geometry. The novelty resides in the SF antecedent and its recoverable continuation across the physical interfaces; the geometric Dirac and spectral-action mathematics serve as the corresponding interface representation.