Meaning
Apṛthak-siddhi (Sanskrit: अपृथक्सिद्धि) — from a-pṛthak (not-separate) + siddhi (establishment, existence) — means 'existing only in non-separation' or 'inseparable existence.' It captures the ontological dependence of chit and achit on Brahman: they cannot be established (i.e., they cannot exist or be conceived of) apart from Him.
Contrast with Separate Existence
In Advaita, the world is māyā — an appearance superimposed on Brahman that has no independent reality. In Dvaita, world and souls are genuinely separate from God. Viśiṣṭādvaita's apṛthak-siddhi offers a middle position: chit and achit are genuinely real (not illusory) but genuinely inseparable from Brahman (not independently real). Like the color of a jewel — real, but incapable of existing apart from the jewel itself.
Analogy: Body and Soul
Rāmānuja repeatedly uses the analogy of body and soul: a body is real, it has its own form and properties — but it cannot exist without its soul. Similarly, chit and achit are real with their own natures, but they subsist only within and through Brahman. Remove Brahman and they literally cease to be.
Import for Devotion
Apṛthak-siddhi grounds the Śrī Vaiṣṇava devotee's confidence: the Lord cannot abandon the soul even if He wished to, because the soul is His own body. This is why prapatti — self-surrender to the Lord — is not a desperate gamble but a recognition of the ontological truth already in place.