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

ABSTRACT 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 Kunegis et al., 2010), Lyapunov stability and nonlinear control theory (Khalil, 2002), and topological data analysis including persistent homology (Edelsbrunner 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. Keywords: information geometry, signed spectral graph theory, Lyapunov stability, persistent homology, bond sign function, basin coherence metric, shade function, state of play, homeostatic boundary integral, Life Mathematics