刻
Meanings
historical time period; two-hour zodiac hour
Historical timekeeping term: one 刻 was typically about two modern hours and was associated with the twelve zodiac signs, as in 子の刻 or 丑の刻. Commonly encountered in historical writing, period drama, and set expressions rather than modern scheduling.
At the hour of the Ox, the castle gate was closed.
The hour of the Rat corresponds to around midnight today.
carving; engraving; cutting finely
Only when written 刻
Used mainly as a kanji or bound Sino-Japanese element for cutting, carving, or engraving. The English gloss also covers the related idea of cutting into small pieces, seen more naturally in forms such as 刻む.
The 刻 in words such as 彫刻 and 篆刻 expresses carving or engraving.
The verb 刻む, used for finely cutting vegetables in cooking, also belongs to this meaning of 刻.
Only when written 剋
Rare as an independent noun. The spelling 剋 is mainly seen in compounds with the sense of defeating, overcoming, or prevailing over someone or something.
The 剋 in 下剋上 expresses someone lower in rank overcoming someone above them.
A limited and old-fashioned morphemic use meaning harsh, severe, or cruel. It is more safely understood through compounds such as 刻薄 than as a freely used modern standalone noun.
The 刻 in 刻薄 is used with the meaning of being mercilessly harsh.