The FSSP releases a shockingly scathing statement on the SSPX, written by Fr. Martin Ramm
Here are some highlight quotes
"We wish to apply this petition... That God may lead all who erred back to unity of the Church - to the separated brethren (Vatican term for non-Catholic Churches) of the SSPX"
"A schism is not overcome by denying it.... The SSPX emphasizes that it recognizes the Pope because it prays for him. However, this recognition remains purely verbal. Praying for someone does not... actually acknowledge their authority. While the papacy is recognized “in itself,” the specific Pope has no binding authority over the actual actions of the Society."
"Besides the explicitly advocated dogmatic sedevacantism, there is a subtle practical sedevacantism. Acting... as if the Pope did not exist. This... is clearly recognizable in the SSPX. When they speak of “fidelity to Rome,” they mean an abstract “Eternal Rome,” not the concrete, present Church... They consciously distance themselves from what they pejoratively call the “institutional Church.” In this way, the Society effectively sets itself as the standard for ecclesiolatry."
"Despite all the differences from Protestant churches, striking structural parallels cannot be overlooked."