Is the Ishraq or Duha prayer performed immediately upon the sun appearing over the mountains, or after it has risen by the measure of a spear, bearing in mind that the mountains obscure its appearance for a third to half an hour after its sunrise?

The time when prayer is forbidden ends approximately fifteen minutes after sunrise. It is not necessary to see the sun; what matters is its actual elevation, and relying on modern, accurate calendars is sufficient. Mountains are not considered a hindrance in this regard, and if the height of the mountain is about the length of a spear, then prayer is permissible even if the sun is still above the mountaintops.