What is the noun of lā of absolute negation?

لَا رَجُلَ فِي الدَّارِ (lā rajula fī al-dāri) does not say “a man is not in the house”. It says there is no man in the house at all — the whole class denied together. That is why it is called لَا لِنَفْيِ الْجِنْسِ.

The noun immediately after it is its اِسْم, and it is manṣūb. But it is marked unusually: a single fatḥa, no tanwīn, and no الـ. رَجُلَ, not رَجُلًا.

The noun drops into nasab without tanwīn; the khabar reporting about it stays in rafa.

Two conditions must hold, and the classical wording states both: the لَا must come directly upon the indefinite noun, and it must not be repeated.

When its noun is a muḍāf, or resembles one, the fatḥa comes back with tanwīn: لَا غُلَامَ رَجُلٍ فِي الدَّارِ, there is no man's servant in the house; لَا عِشْرِينَ دِرْهَمًا فِي الْكِيسِ, there are not twenty dirhams in the purse.

The most familiar sentence in Islam is built on this: لَا إِلٰهَ إِلَّا اللهُ — إِلٰهَ carries the bare fatḥa of its ism.

Do not confuse this لَا with the لَا that resembles لَيْسَ.

لَا لِنَفْيِ الْجِنْسِ puts its noun in nasab — bare fatḥa, no tanwīn — and leaves its khabar marfū': لَا رَجُلَ حَاضِرٌ.

لَا الْمُشَبَّهَة بِلَيْسَ leaves its noun in rafa and puts its khabar in nasab: لَا رَجُلٌ حَاضِرًا.

The tanwīn is the giveaway. رَجُلَ with a bare fatḥa is the noun of absolute negation; رَجُلٌ with tanwīn is not.