In Systemic Functional Grammatics, delicacy is the scale from the most general (grammatical) features to the most specific (lexical).
The notion of lexis as most delicate grammar is exemplified by Halliday & Matthiessen (1999: 198-9) as follows:
… we can differentiate both processes and participants into finer and finer subcategories, until we reach a degree of differentiation that is associated with the choice of words (lexical items). Note that it is not (usually) the lexical items themselves that figure as terms of the systems in the network. Rather, the systems are systems of features, and the lexical items come in as the synthetic realisation of particular feature combinations. Thus lexis (vocabulary) is part of a unified lexicogrammar; there is no need to postulate a separate “lexicon” as a pre-existing entity on which the grammar is made to operate.
The process of instantiation is the selection of features and the activation of their realisation statements in a system network, from the most general (grammatical) systems to the most delicate (lexical) systems.
The cline of instantiation is the scale from the system, as potential, to the instance — to the features and activated realisation statements of an actual text.
The instantiation of lexis, therefore, is the instantiation of the most delicate systems of the lexicogrammar. Halliday & Matthiessen (1999: 327):
Instantiation is the relation between the system and the instance. When we shift attention along this scale, we are moving between the potential that is embodied in any stratum and the deployment of that potential in instances of the same stratum … this move can be made at any degree of delicacy.The notion of lexis as most local context (Fontaine ISFC 2017 Keynote) is, of course, nonsense.