A General Formula for Non-Social Terminal State: Minimal Sufficient Conditions for the Dissolution of Social Dependency and a Formal Proof of NST

This paper derives a minimal canonical model for the Non-Social Terminal State (NST) within a sedimentation field dynamics framework. Prior work in this framework has established that social fields are cross-field shared-core structures formed through long-term coupling of individual fields, and that social dependency originates from individual capacity gaps. This paper further asks: what sustains social dependency, whether that sustaining structure can hold permanently, and if not, how NST is structurally extracted from the general formula. Defining social dependency through the functional capacity gap CapacityGap_j := max(0, GroupSupport_j − IndividualCapacity_j) as a necessary and sufficient condition, the paper extracts three minimal sufficient conditions through an eight-step interrogation chain. First, a finite need bound (NeedBound_j < ∞): individual functional needs are finite, implying a finite upper bound on group support. Second, a sufficient external extension path (PathSufficient_j): an extension path exists that is not capped by a single field’s internal capacity, and whose ability ceiling exceeds the need bound. Third, a non-permanently-zeroed toolization integral (T_j := ∫₀^∞ ToolizationRate_j dt = ∞): group functions are continuously toolized and absorbed into individual effective capacity, and cannot be permanently blocked on an infinite time horizon. When all three conditions hold jointly, the paper formally proves: lim(t→∞) CapacityGap_j(f,t) = 0 (∀j) ⟹ lim(t→∞) SocialDependency(f,t) = 0 ⟹ NST(f). The paper also proves the minimal sufficiency of the three conditions — NST fails if any one is absent — and provides three counterexamples and two domain boundary cases to complete the penetration argument. Five supplementary formulas are provided. Appendix 1 gives the complete formal definitions of the non-social expression set, including capacity gap, social dependency, non-social condition, and effective capacity decomposition. Appendix 2 gives the non-social limit formula, including time domain setup, limit sufficiency conditions, and limit rate bounds. Appendix 3 gives the non-social dissolution formula, including the three dissolution operators, dissolution dynamics equation, dissolution sufficiency conditions, and dissolution irreversibility conditions. Appendix 4 gives the non-social phase diagram, constructing a four-quadrant phase space with functional dependency (FD) and psychological dependency (PD) as axes, tracing the evolutionary trajectory from full social state (S1) through toolized social state (S2) and functional non-social state (S3) to full non-social state (S4). Appendix 5 gives the boundary absorption formula, defining the absorption-type non-social terminal state (NST_A) in which the individual field is absorbed into a total field and the capacity gap becomes undefined, and argues that the coloring circuit remains fully operative under NST_A — emotional experience remains complete, with only the external-source dependency of coloring imagery dissolved. The paper concludes by defining the complete set of non-social terminal states: NonSocialUltimate(f) ⟺ NST(f) ∨ NST_A(f), with the two pathways independent in structure and mechanism, both sufficient to reach non-social terminal states.