Part III: Conditional Rigidity and Spectral Threshold Mechanisms in the Riemann Hypothesis Framework

This paper is the final part of our research series. Building upon the Non-circularity Principle and the topological constraints established in Part I, and the inverse categorical mapping alongside the repulsion potential introduced in Part II, we propose a rigorous analytical paradigm to bridge the gap between "infinite approximation" and "strict equality." We hypothesize a critical threshold p_c of non-trivial zeros strictly located on the critical line. By constructing a "Rigid Threshold Structure" and an "Oscillatory Energy Divergence Mechanism," we demonstrate that when the ratio of on-line zeros reaches p_c, the system undergoes an analytical phase transition equivalent to a topological collapse. Any microscopic deviation from the critical line triggers strict divergence in the local energy functional due to GUE oscillations, acting as a geometric driving force. This inevitable rigidity collapse forces all remaining non-trivial zeros to be strictly locked onto Re(s)=1/2, while simultaneously satisfying the non-degeneracy condition through the rigid connection between the real and imaginary parts. We provide a logically self-consistent, analytical phase-transition framework for the Riemann Hypothesis.