Bhashya
अथेदानीं ब्रह्मविद्यार्थस्य मधुकाण्डस्य वंशः स्तुत्यर्थोब्रह्मविद्यायाः ।मन्त्रश्चायं स्वाध्याय्यार्थो जपार्थश्च ।तत्रवंश इव वंशः- यथा वेणुः वंशः पर्वणः पर्वणो हि भिद्यतेतद्वत् अग्रात्प्रभृति आ मूलप्राप्तेः अयं वंशः; अध्यायचतुष्ट-यस्य आचार्यपरम्पराक्रमो वंश इत्युच्यते; तत्र प्रथमान्तःशिष्यः पञ्चम्यन्त आचार्यः ; परमेष्ठी विराट् ; ब्रह्मणो हिरण्यगर्भात् ; ततः परम् आचार्यपरम्परा नास्ति ।यत्पुनर्ब्रह्म, तन्नित्यं स्वयंभु, तस्मै ब्रह्मणे स्वयंभुवे नमः ॥
In plain words
Now we come to the lineage-list closing the Madhu section, the section devoted to the knowledge of brahman — and its whole purpose is simply to praise that knowledge. This list is itself a mantra, meant both for daily recitation and for silent repetition. It is called a vaṃśa, a word that literally means 'bamboo,' because it works the same way a bamboo stalk does: just as a bamboo stalk is marked off joint by joint, this list runs, joint by joint, from its very top all the way down to its root. The unbroken succession of teachers across the four preceding chapters is what earns it that name. Reading it is simple: whenever a name appears in the nominative, that is the disciple; whenever a name appears in the ablative, that is the teacher. Parameṣṭhin, in this list, is Virāj, and Virāj comes from Brahman — that is, from Hiraṇyagarbha. Beyond that point, there is no teacher standing before another; the succession simply ends there. What remains is brahman itself, eternal, needing no origin outside itself. To that self-existent brahman, homage.
Close to the Sanskrit
Now, at this point, the lineage-list of the Madhu section, whose purpose is the knowledge of brahman, has as its purpose the praise of the knowledge of brahman. And this mantra also serves the purpose of daily recitation and the purpose of silent repetition. Here, it is called a vaṃśa because it is like a bamboo-stalk: just as a bamboo-stalk is divided joint from joint, in that same way, from the top down to the reaching of the root, this is a vaṃśa; the sequence of the succession of teachers belonging to the four chapters is called the vaṃśa. In it, the name ending in the nominative is the disciple; the name ending in the ablative is the teacher. Parameṣṭhin is Virāj; he comes from Brahman, that is, from Hiraṇyagarbha; beyond that there is no further succession of teachers. But that brahman which is eternal, self-existent — to that self-existent brahman, salutation.