R-layer Mode Theory VII: Matter Fields, Gauge Structure, and Wall-Localized Modes

This is Version 2 of Volume VII in the R‑layer Mode Theory (RLMT) series. This update adds a formal declaration regarding the use of Microsoft Copilot for proofreading, text refinement, and LaTeX‑related structural assistance. Minor corrections were made to the reference list, while all theoretical content, equations, and physical interpretations remain unchanged. Volume VII develops the matter and gauge sector of RLMT by interpreting the internal bound states Un(z) localized on domain walls as effective 4D matter fields. Their spatial structure determines quantum numbers, coupling strengths, and generational labels. The connectivity of the wall network induces effective gauge symmetries, where loops correspond to U(1) phases and junctions generate SU(2) and SU(3) structures. Gauge bosons arise as collective excitations of the wall geometry rather than as fundamental fields. Yukawa couplings and mass hierarchies emerge naturally from overlap integrals of localized internal modes, providing a geometric explanation for fermion masses and mixing angles without introducing additional symmetries. The volume establishes the bridge between cosmology, gravity, and particle physics, forming the foundation for the effective field theory developed in Volume VIII. This version preserves all scientific results and improves documentation clarity and transparency regarding AI‑assisted writing.