I'm not a mathematician, but it appears to me that "emergent" properties are being defined as those which do not appear in a minor form below a threshold. Howe

I'm not a mathematician, but it appears to me that "emergent" properties are being defined as those which do not appear in a minor form below a threshold. However, many natural phenomena that are fully explainable from first principles show this property, giving rise to sigmoidal "S-curves", as shown in Figure 1.