if there was an error in the permit the city can pull it.

I’d check with a local land use lawyer, to make sure it’s all kosher, and
give the city some notice, that there is a complaint but your lawyer has checked it, and they have 30 days to review the file, and otherwise you will presume it is correct.

i’d also have the lawyer send a letter to the neighbor informing him that a permit has been pulled and the city allows this by right and according to code, and that if he has any concerns, he is welcome to hire a geotechnical engineer
and review things, but that construction is planned to begin with 30 days and that if he wishes to express any concerns that he should have the engineer contact the lawyer NLT X day or he will be considered to have waived his rights to comment