The evidence of the birth of any one is either witness from two trusted persons, or witness of trusted persons who saw him alive.
We have thousands of witnesses of trusted persons who witnessed his birth or / and saw Imam Al-Mahdi (AS).
Many Sunni scholars have also recorded his birth in their books like
1. Al-Dhahabi in Siyar A'lam al-Nubala 13: 119.
2. Mohammad Ibn Talha al-Shafi'ee in Matabilb al-Sa'ool .
3. Ibn al-Sabbagh al-Maliki in al-Fosool al-Muhimmah 237.
4. Sibt Ibn al-Jawzi in Tathkirat al-Khawaas.
5. Ibn Hajar al-Haythami in Al-Sawaa'eq.
6. Al-Shabraawi al-Shafi'ee in al-Et'haaf.
7. Al-Qundoozi al-Hanafi in Yanabee' al-Mawaddah .
8. Al-Shiblianji al-Shafi'ee in Noor al-Absaar.
9. Ibn Khallakaan in Wafayaat al-A'yaan .
10. Ibn Toloon and many others.