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.
Tsuyoshi Tohi (Fri,) studied this question.