R-layer Mode Theory (RLMT) XCIII: Dynamic Layer Geometry and Effective Multi-System Tensor Fields in Layered Expansion Cosmology

This volume extends the static layer formalism of Appendix U to a fully dynamic and observer-dependent framework suitable for rotating, binary, and multi-system configurations in Layered Expansion Cosmology (RLMT). As stated in the manuscript, “the tension field is promoted to a time-dependent scalar T(x,t), and the layer window functions become time-dependent as well.” We develop spherical-harmonic multipole expansions for rotating systems, construct effective multi-system tension fields for binaries and larger configurations, and introduce observer-dependent observational kernels W(x,t;uμ). These tools allow the definition of effective layer averages in realistic astrophysical environments where geometry, motion, and observational perspective all vary with time. An effective energy-momentum tensor Teff is constructed by coarse-graining over high/mid/deep layers, and its mathematical properties—boundedness, continuity, and stability—are analyzed using functional-analytic methods. This provides a robust foundation for an effective Einstein equation in dynamic layered settings. The resulting formalism demonstrates that even in highly complex astrophysical systems, the relevant observables can be captured by a small number of effective quantities. This framework is designed to support future satellite missions such as LAYER‑X and to connect RLMT predictions with real observational data.