ABSTRACTThe Bonding Principle established in Smith (2026c) proposes that the universe is atotality of bonding relationships and that bond sign — whether a relationship expands orcontracts possibility space for its participants — is a functional category, not a moralone. This paper formalises the mathematics of bond sign management: themechanisms by which coherent systems maintain, restore, and optimise the sign of theirbonding relationships to sustain homeostasis. Three mathematical objects areintroduced and developed: the bond sign function s (Bᵢj, t) taking values in -1, 0, +1;the basin coherence metric C (Ω) as the normalised weighted sum of bond signcontributions across a system; and the shade function Iₑffective = gₖ (Iᵣaw), describing how mediating structures convert potentially negative bonds into positivecontributions by managing interaction intensity. A fourth object — the state of play S (t) — formalises the set of all possible future bonds given current basin configuration, withlife defined as the process of maintaining and expanding |S (t) | and death as |S (t) | = 0. Second-order bonding — bonds that manage other bonds — is identified as theuniversal mechanism of homeostatic regulation, with the immune system, predator-preydynamics, the ozone layer, and the SHAI Veto Gate as instances across scales. TheHomeostatic Boundary Integral (HBI) is proposed as a unification target — a candidatefor a substrate-independent scalar characterising homeostatic stability — explicitlyframed as an open research question rather than a completed formalism. Cross-domainvalidation is provided across seven domains including a novel application todevelopmental attachment and allostatic load. Falsification conditions are explicitlystated. This paper is the fourth in the SymbioMind Bonding Relationships Series (2026).
Smith et al. (Sun,) studied this question.