R-layer Mode Theory VI: Internal Bound States on Domain Walls

This is Version 2 of Volume VI 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 VI develops the microscopic structure of domain walls formed by the mode field. By linearizing fluctuations around kink solutions, the volume derives a Schrödinger‑type equation whose bound states correspond to internal modes localized on the wall. The basic Pöschl–Teller potential yields two bound states, while extended potentials naturally produce exactly three bound states, providing a geometric origin for the three generations and their mass hierarchy. The volume further analyzes overlap integrals, Yukawa‑like couplings, and the universality of bound‑state structures across the cosmic wall network. These results establish a bridge between cosmology, gravity, and particle physics within the RLMT framework. This version preserves all scientific results and improves documentation clarity and transparency regarding AI‑assisted writing.