Mathematical Foundations: The Coastal Survey — Grounding the Framework's Formal Objects in Existing Mathematics

This paper systematically attempts to ground the Bonding Relationships framework's five formal mathematical objects — the bond sign function, basin coherence metric, shade function, state of play, and homeostatic boundary integral — in existing mathematical frameworks. Four candidate frameworks are engaged: information geometry (Amari & Nagaoka, 2000), signed spectral graph theory (Harary, 1953; Kunegis et al., 2010), Lyapunov stability and nonlinear control theory (Khalil, 2002), and topological data analysis including persistent homology (Edelsbrunner & Harer, 2010). For each object, the connection attempt is made with rigour, the points of success are identified, and the points of failure are documented with precision. The results are: the shade function connects successfully to Lyapunov stability theory with appropriate care; the bond sign function and basin coherence metric connect partially to information geometry and signed spectral graph theory respectively; the homeostatic boundary integral connects partially through discrete differential geometry; and the state of play fails to connect adequately to persistent homology due to the forward-looking temporal character of potential bond space. These documented failures are not presented as deficiencies of the framework. They are presented as the precise mathematical argument for Paper 8: the minimum mathematical extension necessary to fully formalise what the framework describes.