Is a circumambulation valid for someone who enters through the door of the Hijr and does not complete the circumambulation from outside it?

Entering the Ka'bah from the door of the Hijr and exiting from the other door is a grave error, because one is not considered to have performed Tawaf (circumambulation) around the House. The Prophet, peace and blessings be upon him, performed Tawaf around the House from behind the Hijr. So, if a person performs Tawaf from inside the Hijr, it is not considered Tawaf around the House, and his Tawaf is not valid. The Ka'bah must be to the left of the circumambulator. Also, naming the Hijr as the Hijr of Ishmael is incorrect, as Ishmael has no known connection to it.