R-layer Mode Theory Appendix X-II: Technical Foundation — Continuous Mode Tension, Topological Sectors, Particle Hypotheses, and Research Boundaries

This document presents Version of the technical foundation for Recursive Layered Mode Tension (RLMT). It consolidates the continuous Abelian-Higgs-type sector, BPS structure, zero-mode verification, moduli-space dynamics, and near-critical stability, together with conjectural particle-sector hypotheses and preliminary gravity/dark-layer interfaces. Verified numerical results, inconclusive methods, and open problems are recorded explicitly to preserve a transparent research boundary for future RLMT papers and updates.