Martin (1992: 102):
As far as participant identification is concerned the central oppositions have to do with reminding and relevance phoricity. With nominal groups, redundancy phoricity has to do with recovering experiential meaning, not participant identity, and so can be set aside here. The core reference paradigm is thus:
[presenting] [presuming] [comparison] a smaller frog the smaller frog [–] a frog the frog
Formulated systemically as in Fig. 3.2 this gives the simultaneous systems [presenting/presuming] and [comparison/–]. Presenting reference signals that the identity of the participant in question cannot be recovered from the context; presuming reference signals that it can. Presenting reference is thus strongly associated with first mention and presuming reference categorically associated with second mention. The [comparison/–] system makes reference to the identity of related participants optional.
 To be clear, "reminding and relevance phoricity" are Martin's rebrandings of Halliday's co-reference and comparative reference, respectively.
 To be clear, "redundancy phoricity" is Martin's rebranding of Halliday's ellipsis–&–substitution, misunderstood as a subtype of reference.
 To be clear, (the recovery of) the identity of participants is (the recovery of) experiential meaning. Given this confusion, and his dependence on Du Bois (1980), Martin may be locating 'the identity of participants' outside language; that is, treating meaning as transcendent of semiotic systems, contra the epistemological foundation on which SFL is based.
 Having mistaken ellipsis–&–substitution for a type of reference, Martin now excludes it from his model of 'reference as semantic choice' — on mistaken criteria.
 This again confuses the referent — in this case, a potential reference point — with the reference item. Halliday (1994: 309):
A participant or circumstantial element introduced at place in the text can be taken as a reference point for something that follows.
To be clear, in SFL theory, "presenting reference" is not reference, by definition, if it there is no marking of an identity as recoverable elsewhere.
 To be clear, "presuming reference", whether [comparison] (comparative reference) or [–] (personal or demonstrative co-reference) can also be cataphoric, and so, contrary to Martin's bare assertion, also "associated with first mention".
 To be clear, given that "presenting reference" is not reference, the [comparison/–] system is another rebranding of Halliday's distinction between comparative reference and co-reference (personal and demonstrative).
 To be clear, of the four examples, only the frog appears in the text ([3:88]) on which 'most of the examples are based' (p99). If a smaller frog and the smaller frog are interpreted as including cohesive reference items, then, in terms of SFL theory, the former makes comparative anaphoric reference (smaller), and the latter makes both demonstrative anaphoric co-reference (the) and comparative anaphoric reference (smaller).
Importantly, because Martin mistakes the nominal group in which a reference item appears for the reference item itself, he only attributes one (cross-classified) referential function to each nominal group. As a consequence, the two reference markers in the smaller frog — the and smaller — are conflated into one function, and the one (genuine) reference marker in a smaller frog — smaller — is miscategorised, in terms of Martin's own definition, as "presenting" ('not recoverable from context') rather than "presuming".