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Etymonline refers to the "An Etymological Dictionary of the French Language" by Auguste Brachet, translated by G.W. Kitchin, Oxford, 1878. Its entry for achever, on page 152 of 558, states:

For f = v, see § 145. [...]

Strangely, § 145 doesn't explain how « à cheF » evolved into « acheVer », but §142 (on pp xcii-xciii) claims to do so, but I see no explicit elucidation and so I am still confused.

[...] When v is not final, there is no longer any reason for this strengthening process, and it remains unchanged according to § 140. This is the reason why the feminine of adjectives in -if is -ive; and why we have bovem, bcuf, but bovarius, bout'ier ; navem, nef, but navirium * , navire , servum, serf, but servire, servir; salvum, sauf, but salvare*, sauver; nativum, naif, but nativitatem, naibelé. The same rule enables us to explain the relation between the primitive chef and the derivatives chevet, achever, and between such words as brefand brevet, relief and relever.

How does the same rule explain how à cheF evolved into acheVer?

PS: I don't extract from the more laconic French version; whose entry on 'achever' does not broach the phonological changes.

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A more modern source would probably use the terminology of word-final obstruent devoicing, a process that happened early in the history of Old French (likely due to Frankish influence) where voiced final obstruents become voiceless at the end of words.

Concerning the segment in question, the /p/ in Latin caput gained voicing in Western Romance as it was intervocalic (something like *kabut; compare Portuguese capo [kabu]). At one point in the history of most of Western Romance languages, voiced intervocalic stops became fricatives (something like *kavut; compare Spanish cabo [kaβo]). As the second syllable eroded away in the noun due to a number of phonological changes, it ended up as word-final (*kav), and was devoiced (*kaf) in environments where it was word-final, but not elsewhere.

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