Final answer:
No, you cannot sign off on a student's solo cross-country flight if you are not their instructor.
Step-by-step explanation:
No, you cannot sign off on a student's solo cross-country flight if you are not their instructor. According to 61.93(c)(3) of the Federal Aviation Regulations, only the student's instructor, who meets the requirements outlined in the regulation, can sign off on their solo flight. Since you have never flown with this student and are not their instructor, you do not have the authority to sign off on their flight.