Emergence occurs when there is a conceptual discontinuity between two descriptions targeting the same phenomenon. This does not mean that emergence is a purely subjective phenomenon — only that scientific ‘double coverage’ may be a good place to look for emergent phenomena.
For example, in the case of starling murmuration, there is an aggregate description of individual birds, and a description of the flock as a unified entity. The latter phenomenon invites description in terms of concepts from fluid dynamics, but descriptions of individual birds, however detailed, typically do not.
In the case of phase transitions in physics, the description of one phase of matter, such as gas, does not fully map onto descriptions of the other phases. Surface tension, for example, is not defined for gases, since gases do not have surfaces. In the transition from gas to liquid, a qualitatively new attribute not only emerges, it becomes a defining feature of the post-transition system. From a different perspective we can say that it is the emergent qualitative property that enables us to determine that the transition has occurred in the first place. Quantitative readings of some control variable (such as temperature or pressure) cannot themselves be used to mark out ‘events’ — they can only be used to index them.
A common type of theoretical disjunction involves mismatch between descriptions of parts and wholes. A description of micro-level constituents in terms of atomic properties does not lead in any smooth way to descriptions in terms of thermodynamics, hydrodynamics, or solid state physics. In physics, the lack of smoothness in transitioning from one theoretical domain of discourse to another is not always apparent, since the two domains are often well specified mathematically, while the (often ad hoc) linking assumptions enabling the transition are neglected in popular (and even introductory textbook-level) explanations.