Interdentals in English: One or two phonemes?

The (inter)dental sounds dh and th in English are distinguished by all speakers, but simple rules with few exceptions can accurately predict which sound is used in any given word. Because grammatical category and some etymology are needed for the rules, this may not be enough reason to conclude that there is only one combined th-dh phoneme in English.
Yet there are very few minimally differing pairs.

The rules for prediction of th/dh, with examples, are as follows:

In initial position:

In medial position:

In final position:

Note 1:

Among the very few cases in which the difference between [D] and [T] distinguishes two otherwise identical words are:

loath loathe
wreath wreathe
sheath sheathe
thus (incense) thus
ether either (Am. pronunciation)
thigh thy
sooth soothe
mouth (noun) mouth (verb)
thou (short for 1000) thou (pronoun)
teeth teethe

All of these are, however, according to the rules mentioned above.

Note 2:

The rules for final [D]/[T] in nouns, adjectives and verbs also apply for [s]/[z] and [f]/[v] in some cases:

use (noun) use (verb)
loose (adjective) lose (verb)
life (noun) live (verb)

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