His link to the actual letter of the Pope is very clear that he refers to laborers and wages and a Universal Basic Wage, not a Universal Basic Income given to all, including the idle. His criticism of your article is on point and warranted. Your response is not adequate, you merely restate your case that he has disproven using original documents.

It's possible that the Pope also supports UBI for the idle. His letter here does not support that claim though.