[14] COS-EMT - Collapsing-Structure Energy-Momentum Tensors: Discrete and continuum derivation, quantum extensions, compatibility fittings

The COS-EMT module provides the energy-momentum tensor reference and compatibility framework for the Collapsing-Structure (COS) program. It does not introduce a new dynamical theory, but fixes a unified vocabulary and set of conventions for using energy-momentum tensors across continuum, discrete, quantum, semiclassical, and cosmological COS applications. The manuscript systematizes the relationship between the Hilbert energy-momentum tensor, canonical/Noether tensor, Belinfante-Rosenfeld symmetrization, superpotential equivalence classes, and CCJ-type improvement terms. It gives standard sector-level formulae for scalar, electromagnetic, non-Abelian Yang-Mills, fermionic, and related effective matter sectors, while clarifying which tensor arises from metric variation, Ward identities, or improvement choices. On the discrete side, COS-EMT treats the lattice description as a local regulator and fitting layer for COS shell/simplicial patches, not as a claim that the full fluctuating COS geometry is globally a hypercubic lattice. The module formulates variational and Ward-identity-based definitions of the discrete EMT and records the conditions under which the renormalized discrete tensor is expected to match the Hilbert EMT up to controlled discretization and improvement errors. The quantum and semiclassical sections summarize locally covariant renormalization, Hadamard point splitting, Ward identities, trace anomalies, curvature counterterms, expectation values, correlators, and the domain of semiclassical Einstein-equation use. These sections provide conventions and consistency checks, not a complete independent semiclassical closure. For cosmological applications, the module records the perfect-fluid reduction, conservation equations, gauge-invariant perturbative variables, and the interface to COS-C. It also outlines a compatibility mapping to the Alena framework, while keeping the standard Hilbert/Belinfante forms as the reference definitions. COS-EMT therefore serves as a citable EMT vocabulary, fitting, and verification layer for COS-FRMW, COS-CL, COS-SM, COS-C, COS-NUM, and related modules.