けむ

⟨ke₁mu⟩ → */kʲemu/ → /kemu/

From Old Japanese けむ. Attested in the Kojiki of 712^[1] and the Man'yōshū of 759 and earlier.^[1][2]

Various theories exist regarding the ultimate derivation of this auxiliary verb suffix.^[1][2][3]

• Compound of け (ke, a hypothetical 未然形 (mizenkei, “irrealis or imperfective form”) of past-recollective auxiliary き (ki), usually found instead as せ (se)) +‎ む (mu, suppositional auxiliary suffix).^[1][3]

The ke conjugation for past-recollective き (ki) is only hypothetically considered to be the irrealis form. This ke is not found outside of a small handful of suffixes such as this けむ (kemu), its derivative けまく (kemaku), and separate けく (keku), and it remains largely unexplained. That said, the initial /s-/ ↔ /k-/ alternation exhibited in the better-documented conjugated forms is also largely unexplained.

• Compound of 来 (ki, “coming”, the 連用形 (ren'yōkei, “continuative or stem form”) of verb 来る (kuru, “to come”)) +‎ 経 (he, “passing through, as of a process or time or place”, the 連用形 (ren'yōkei, “continuative or stem form”) of verb 経る (heru, “to pass through”)) +‎ む (mu, suppositional auxiliary suffix).^[1]

*/ki pe mu/ → */kifemu/? → */kiemu/ → /kemu/

Due to the presence of initial /p-/ in the medial Old Japanese 経 (pe, continuative stem, terminal form was pu), the phonological shifts required for this derivation ultimately make this theory seem quite unlikely: linguists generally reconstruct initial /p-/ for Old Japanese and hold that this did not undergo lenition until the late Nara period (710–794) or early Heian period (794–1185), while ⟨ke₁mu⟩ → /kʲemu/ is already attested in the Kojiki of 712. See also below.

• Compound of き (ki, past-recollective auxiliary suffix) +‎ あむ (amu, hypothetical earlier form of suppositional auxiliary suffix む (mu)).^[1][2]

*/ki amu/ → */kiamu/ → /kʲemu/ → /kemu/

The parsing of the 未然形 (mizenkei, “irrealis form”) form and -mu as -am- for the suppositional suffix is favored by linguists such as Bjarke Frellesvig and Alexander Vovin.

One potential issue with this is that past recollective auxiliary suffix き (ki) is generally considered by Japanese references as not having any 連用形 (ren'yōkei, “continuative or stem form”), the conjugational form required to append onto a following verb.^[4][5]

That said, the け (ke) form attested as the 未然形 (mizenkei, “irrealis form”) of auxiliary き (ki), which conjugational form would be required for the traditional analysis of suppositional suffix む (mu), might itself be a phantom obscuring that ki may have indeed been the continuative or stem form, with the ke form simply reflecting the result of this /i a/ fusion into /e/, much as we see with the development of suffix けり (keri) as a fusion of き (ki) + あり (ari, copula).

• けん (-ken)

けむ • (-kemu) 

1. [from 712] (Classical Japanese) used to make the past conjectural form of verbs
   1. presents a supposition about a past event
   2. presents a supposition about the cause of a past event
   3. presents hearsay about a past event

Attaches to the 連用形 (ren'yōkei, “continuative or stem form”).

This word is morphologically an inflectional suffix. It is classified as 助動詞 (jodōshi, “auxiliary verb”) in traditional Japanese grammar.

• 〜たろう (~tarō)
• 〜ただろう (~ta darō), だっただろう (datta darō)