稽 appears in formal/classical compounds: 稽查 (to inspect/investigate), 稽考 (to verify historically), 无稽之谈 (unfounded claim). It also appears in 滑稽 (huájī, comical/funny) where the pronunciation shifts to jī.
Left grain radical — the indexing radical, pictograph of a grain plant with drooping ripe head. Original meaning of 稽 was 'to delay, hold up,' the way standing grain awaits harvest. From there came 'to examine carefully, investigate' (taking time over something) — the dominant modern senses: 稽查 (inspect), 无稽 (groundless).
Top-right 尤 supplies the sound — yóu shifted to jī through an onset alternation. Same phonetic appears in 就 (right then), 优 (excellent), all sharing the y/j wobble. Sits stacked above 旨 to form the right-side phono-semantic compound.
Bottom-right 旨 — a spoon over a mouth, originally 'tasty' and by extension 'the gist, the intended meaning.' Contributes the 'examining the point of something' flavor that pulls 稽 toward 'inquire, investigate.' Most famous in the bowing formula 稽首 (touch head to ground in deepest respect).
