Cài Yuándìng 蔡元定 (1135–1198), zì Jìtōng 季通, hào Xīshān 西山 (also Mùtáng 牧堂), of Jiànyáng 建陽 (Fújiàn). CBDB id 10049. Sòngshǐ biography in juan 434, Rúlín zhuàn.
Zhū Xī’s principal scholarly collaborator and the most important xiàngshù-and-numerology specialist of the Zhū-school inner circle. Joint author with Zhū Xī of the Yì xué qǐméng 易學啟蒙 (foundational treatise on the xiàngshù basis of the Yì) and substantial collaborator on the Lǐ shū 律呂 (musical-pitch-pipe theory) sections of the Yí lǐ jīng zhuàn tōng jiě 儀禮經傳通解.
Father: Cài Fā 蔡發 (hào Mùtáng 牧堂; the elder Cài, Jiànyáng recluse-scholar — Yuándìng’s chief source for the inherited xiàngshù expertise). Sons: 蔡淵 (Cài Yuān, 1156–1236), Cài Mò 蔡沕, 蔡沈 (Cài Shěn, 1167–1230).
Caught in the Qìngyuán dǎngjìn 慶元黨禁 of 1196 (the Wěixué 偽學 / “False Learning” proscription) and exiled to Dàozhōu 道州 (modern Dao county, Hunan), where he died in 1198 — among the most distinguished Zhū-school casualties of the proscription. Posthumous title (later Yuán-period) Wénjié 文節.
Within the Kanripo corpus he is the joint author with Zhū Xī of the Yì xué qǐméng (recorded under Zhū Xī’s name in the catalog). His specific authorial contribution to the Qǐméng’s xiàngshù foundations is documented by Zhū Xī’s own correspondence and by Cài Yuándìng’s son Cài Yuān (蔡淵)‘s acknowledgments in KR1a0052 and KR1a0053.
Authored independently the Lǚ lǚ xīn shū 律呂新書 (a foundational Sòng treatise on musical pitch-pipe theory, in 2 juan; preserved in the Sìkù).
The Cài family of Jiànyáng — Cài Fā → Cài Yuándìng → Cài Yuān / Cài Mò / Cài Shěn / further-generation Cài Háng 蔡杭, Cài Mó 蔡模, etc. — is the most important late-Sòng Dàoxué scholastic dynasty after the ZhūXī family proper.