Ayutasiddhi in Hinduism is a term used to express an internal relationship in Hindu logic. As per Hindu religion, the word ‘ayuta’ comes from the root ‘ya’ meaning to join, or to separate two mutually opposite concepts. Ayutasiddha in the present context means accomplishment or proof of complete union. The concept was postulated by logicians to uphold the realistic pluralism.
Later, however, Ramanujacharya (1017 – 1137 AD, who established the Vishishta Advaita school of Vedanta named this relationship as a prthak-sambandha identity-and-difference, as an internal relationship.
Ayutasiddhi is the inherent relationship existing between an object substratum and a quality content, part and whole, universal and individual, action and agent, and between an extern object and its special characteristic termed vishesha.
Hindu logic accepts that the two components are indivisible.
The 9th century philosopher Vachaspati Misra reasons out by describing them as always experienced as mutually joined.
A part cannot be conceived as independent and the whole is unthinkable without the components; they are ayutasiddha.
This relationship is more popularly known as samavaya.
External objects comprise every atom of the five physical objects, namely, earth, water, fire, air and ether (akasha) and the four eternal objects, namely time, space, soul and mind.
Some have subsumed samavaya under the later concept of svarupa sambandha, that is individual nature; but Vaiseshika school rejects this idea.
Samavaya being one, objections have been raised leading to several discrepancies, but these also have been explained away.
One of the ayutasiddha relata is subordinate to or dependent on the other, and both occupy the same space. The relata are always mutually heterogeneous. A quality cannot exist apart from the substance in which it inheres, whereas the substance can exist independently. The relation is external in that dependence is not mutual.
Source - Encyclopedia of Hinduism published by IHRF - Volume 2 - page 244