The Instrumental -3rd case endings after ātmān are not elided when an Ordinal Numeral follows.,
Thus आत्मनापंचमः, आत्मनाषष्ठः ॥ The Instrumental case here takes place under the Vartika तृतीयाविधाने प्रकृत्यादीनामुपसंख्यानं (II. 3. 18 Vartika). And compounding takes place by 2.1.30, by separating तृतीया of that aphorism and making it a separate sutra or in this way :-आत्मना कृतः पंचमः = आत्मनापंचमः ॥ How do you explain the form आत्मचतुर्थ in जनार्दनस्त्वात्मचतुर्थ एव? It is a Bahuvrihi compound = आत्मा चतुर्थोऽस्य ॥ The word पूरणे is a later addition of the Vartikakara.,