The auxiliary べき is inflected like a Classical -i adjective. Or I guess, more historically accurately, as a Classical -ki adjective.
(Note: This was introduced to me using the adnominal -ki form. Other materials might start from the terminal / dictionary form ending in -shi instead.)
活用形【かつようけい】 |
Inflection |
Form |
未然形【みぜんけい】 |
Irrealis / Incomplete |
--- |
連用形【れんようけい】 |
Continuative / Adverbial |
べく |
終止形【しゅうしけい】 |
Terminal / Predicative (also "Dictionary Form") |
べし |
連体形【れんたいけい】 |
Adnominal |
べき |
已然形【いぜんけい】 |
Realis / Conditional |
べけれ |
命令形【めいれいけい】 |
Imperative |
--- |
- Note: Many Japanese grammars include a mizenkei for adjectives, as this is the canonical conjugation for verbs for constructing the negative. However, for adjectives, the negative simply comes after the adverbial, and moreover it is clearly separable as you can insert a particle between the adverbial ~く and the following ない.
Since べく is clearly the adverbial conjugation, yes, such phrases are syntactically and grammatically adverbial.
PS: Can anyone help file a bug report for how the furigana are not working when included within a table?