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Consider an archiphoneme N that can be realized as n, ng, or as a nasal on a vowel depending on the context.
Is this representation, below, standard i.e. with the archiphoneme as a capital letter on the phonetic level?
If so, how does on typically refer to an archiphoneme when providing the phonemic inventory?
An archiphoneme is employed when a surface phone (which has a definite phonetic value) could derive from a number of underlying sounds /x,y,z/ and there is no contrast between these segments in that environment. For example the question suffix of Turkish appears on the surface as [mi, mu, my, mɨ], and rather than arbitrarily select a specific phoneme as underlying the variable vowel, archiphonemicists posit an archiphoneme /I/ – you would have to read the specific ontological claims made by a proponent of the practice, to know exactly what that means. In underspecification theory, it simply refers to a vowel that is high, but not specified for backness or rounding.
The reason for not reducing the Turkish alternating vowel to allophony is that each of /u, i, y, ɨ/ are phonemes, and each allophone can only be assigned to a single phoneme, that is, allophony does not neutralize distinctions. That's the job of archiphonemes and morphophonemes.
Archiphonemes generally do not exist "on the surface", which is populated only with phones. An exception is that some people interpret "surface" as being the result of all of the phonological rules, but not including the results of phonetic interpretation. Some people of that persuasion allow the possibility that a given segment may not have full specification of all features, in fact privative feature theories extensively make that claim (that e.g. [u] is round and [i] is not specified for roundness). Those who adhere to privative feature theory at the surface do so also for pre-surface levels; in which case, all segments are "archiphonemes" (in lacking some feature specification), and in privative theory, the notion of "archiphoneme" doesn't make much sense.
Orthographically, a capital letter is the standard way of indicating an archiphoneme; it's hard to say from your examples whether that would be a standard archiphonemic analysis. Archiphonemes are not phonemes, they are things that stand for classes of phonemes, so would not strictly be included in a list of phonemes. But if you view "phoneme" as simply being "any segment in underlying forms", then it should be included in a list of phonemes. So it depends on what you mean by "phoneme".
An archiphoneme is really not a phonetic entity. Trubetzkoy distinguished his archiphoneme from its surface manifestation, which he called the archiphoneme representative (Vertreter). However, there is a natural phonetic representation available for the typical case where the phonemes which are neutralized have all the possible values for some phonetic features: just give a feature representation for the archiphoneme which omits those features which differ among the neutralized sounds.
