Gadādhara Paṇḍita, the fourth branch, is described as an incarnation of the pleasure potency of Śrī Kṛṣṇa. No one, therefore, can equal him. In the Gaura-gaṇoddeśa-dīpikā (147–53) it is stated, “The pleasure potency of Śrī Kṛṣṇa formerly known as Vṛndāvaneśvarī is now personified in the form of Śrī Gadādhara Paṇḍita in the pastimes of Lord Caitanya Mahāprabhu.” Śrī Svarūpa Dāmodara Gosvāmī has pointed out that in the shape of Lakṣmī, the pleasure potency of Kṛṣṇa, she was formerly very dear to the Lord as Śyāmasundara-vallabhā. The same Śyāmasundara-vallabhā was present in Lord Caitanya’s pastimes as Gadādhara Paṇḍita. Formerly, as Lalitā-sakhī, she was always devoted to Śrīmatī Rādhārāṇī. Thus Gadādhara Paṇḍita is simultaneously an incarnation of Śrīmatī Rādhārāṇī and Lalitā-sakhī. In the Twelfth Chapter of this part of the Caitanya-caritāmṛta there is a description of the descendants or disciplic succession of Gadādhara Paṇḍita.
(Sri Caitanya Caritamrta---1:10:15---translation and purport).

In his Anubhāṣya, Śrī Bhaktisiddhānta Sarasvatī Ṭhākura, describing the truth about the Pañca-tattva, explains that we should understand that Lord Śrī Caitanya Mahāprabhu is the supreme predominator and that Nityānanda Prabhu and Advaita Prabhu are His subordinates but are also predominators. Lord Śrī Caitanya Mahāprabhu is the Supreme Lord, and Nityānanda Prabhu and Advaita Prabhu are manifestations of the Supreme Lord. All of Them are viṣṇu-tattva, the Supreme, and are therefore worshipable by the living entities. Although the other two tattvas within the category of Pañca-tattva—namely, śakti-tattva and jīva-tattva, represented by Gadādhara and Śrīvāsa—are worshipers of the Supreme Lord, they are in the same category because they eternally engage in the transcendental loving service of the Lord.
(Sri Caitanya Caritamrta---1:7:15---- purport).

There is no difference between the energy and the energetic in regard to the Lord's appearance as Śrī Caitanya Mahāprabhu and His four associates—Nityānanda Prabhu, Advaita Prabhu, Gadādhara, and Śrīvāsa. Amongst these five diverse manifestations of the Supreme Lord (as the Lord Himself and His incarnation, expansion and energies) there is no spiritual difference. They are five in one Absolute Truth. For the sake of relishing transcendental flavors in the Absolute Truth, there are five diverse manifestations. These are called the form of a devotee, the identity of a devotee, the incarnation of a devotee, the pure devotee and devotional energy.
Out of the five diversities in the Absolute Truth, the form of Lord Caitanya is that of the original Personality of Godhead, Kṛṣṇa. Lord Nityānanda is the manifestation of the first expansion of the Supreme Lord. Similarly, Advaita Prabhu is an incarnation of the Supreme Lord. These three—Caitanya, Nityānanda and Advaita—belong to the category of Viṣṇu-tattva, or the Supreme Absolute Truth. Śrīvāsa represents the pure devotee, and Gadādhara represents the internal energy of the Lord for the advancement of pure devotion. Therefore Gadādhara and Śrīvāsa, although included in Viṣṇu-tattva, are dependent, diverse energies of the Supreme Lord. In other words, they are not different from the energetic, but they are manifest diversely for the sake of relishing transcendental relationships. The whole process of devotional service involves a transcendental reciprocation in the relationship between the worshiper and the worshiped. Without such a diverse exchange of transcendental flavors, devotional service has no meaning.
(Teachings of Lord Caitanya).