14 COS-EMT - Collapsing-Structure Energy-Momentum Tensors: Discrete and Continuum Derivations, Quantum Extensions, and Matching to the Alena Framework)

The COS–EMT module provides a unified reference framework for defining and using the energy–momentum tensor (EMT) across the COS ecosystem. In the continuum, the gravitational source is defined as the Hilbert EMT via metric variation, and the relationships among the canonical (Noether) tensor, the Belinfante–Rosenfeld symmetrization, and the superpotential equivalence class are organized in consistent conventions. For (approximately) scale-invariant sectors, the module reviews the Callan–Coleman–Jackiw (CCJ) improvement in flat and curved spacetime, including its trace and conformal implications. Closed-form EMT expressions are provided for scalar, electromagnetic, non-Abelian Yang–Mills, and fermionic sectors. On the COS lattice, the module formulates an EMT using a variational definition together with Ward-identity constraints, and shows that—with suitable superpotentials and renormalization constants—the renormalized lattice tensor approaches the Hilbert EMT up to O (a²) discretization effects. In the quantum setting, the manuscript summarizes locally covariant Hadamard point-splitting renormalization, preservation of Ward identities, the trace anomaly, and the semiclassical Einstein equations including curvature counterterms. Finally, for an FRW background it records the perfect-fluid reduction and gauge-invariant linear perturbations, and outlines a compatibility map to the Alena framework. Keywords: energy–momentum tensor; Hilbert tensor; Noether tensor; Belinfante–Rosenfeld; superpotential; CCJ improvement; Ward identities; lattice EMT; renormalization; trace anomaly; Hadamard renormalization; semiclassical gravity; FRW perturbations; COS ecosystem.