Re: Republic not South!
No, you're missing the point Navan Man.
The State we live in, pending the ending of partition, is Ireland/Eire, described as the Republic of Ireland, as stated in those two documents I quoted above. The 1916 proclamation has no basis in law, nor is anyone suggesting it does. Ireland/Eire and the Republic of Ireland are based on the foundation document of the State, Bunreacht na hEireann 1937, and on the 1948 Act.
Your insistence on the 1916 proclamation is a red herring.