The Geometric Resolution of Forces: Discrete φ-Fractal Harmonic Stability in the King’s College Fan Vault

The Late Gothic fan vault of King’s College Chapel (Cambridge, UK) presents a mechanical paradox: a high-mass masonry shell (span 12.7m) supported by slender lateral walls that feature extensive glazing and minimal cross-sectional area. Conventional elastic continuum analysis predicts significant lateral thrust that should compromise the glazed enclosure. However, despite a documented global settlement of >120mm, the lateral walls remain intact. This paper proposes a new structural model: the vault functions as a φ-fractal harmonic NESS structure, where geometric tuning minimizes hoop forces at the wall interface. By combining high-resolution geometric survey data with Discrete Element Modeling (DEM) and Thrust Network Analysis (TNA), we identify a specific “Zero-Hoop” elevation (h0) decoupled from the lateral walls. We demonstrate that the structure maintains stability not through rigid static equilibrium, but through a Dissipative Non-Equilibrium Steady State (NESS), permitting kinematic adjustments via mortar-stuffed fissures and dry joints.