Spectral Energy Regularization in Lorentzian Spacetimes

We formulate a covariant spectral regularization of the energy–momentumtensor in Lorentzian spacetime and state it explicitly as an effective meanfield diagnostic of structural admissibility, not as a completed fundamentalvariational theory of gravity. Given the mixed endomorphism Tµν = gµαTαν,the regularized endomorphism is defined by functional calculus,Tstruct = Ξε(T) = fε(T), fε(λ) = λ2 +ε2, ε >0.This gives the spectral lower bound spec(Tstruct) ⊂ [ε,∞) and converts negativeor null spectral modes into positive spectral data. We prove that the covarianttensor gµαfε(T)αν is symmetric whenever the original mixed tensor is g-selfadjoint and the functional calculus is admissible. The induced exchange currentQν =∇µTstructµν is obtained from the Frechet derivative, with a holomorphicresolvent form used to cover non-semisimple Jordan sectors and to state precisecontinuity assumptions across spectral degeneracies. In the TTN interpretation,Qν is treated as a diagnosed source for a structural balance sector containinga coherence field Φ, a nodal-pressure field K, and a massive balance one-formBν. The cutoff is tied to nodal pressure by ε2(x) = ε2 0 +γK2(x), and a localfixed-point condition gives a simple safeguard against runaway feedback. Theconstruction is therefore a falsifiable admissibility test: a regularized source isaccepted only when the structural balance equations admit regular finite-energyfields