Do you believe that the word perpetual somehow means that there should ALWAYS be a pope at every moment of time? If that's what you believe, then I can't help you.
Why not? Because of interregnums. To state the following is not heretical:
Pope St. John I was elected on 13 August, 523, after an interregnum of seven days. For that week, the Church was without a pope, but did not cease to exist.
Pope Innocent XI was elected after an interregnum of two months.
Pope Boniface III was elected on 19 February, 607, after an interregnum of 3 days less than a full year.
Pope Innocent IV ascended the papal throne on 25 June, 1243, after an interregnum of 1 year, 7 months, and 15 days. For nearly two years, the Church had no earthly head, but it did not cease to exist.
In 29 Novemeber 1268, Pope Clement IV died, and there was no successor to the papal throne until September 1, 1271, when Pope Gregory was finally elected. That means that there was no pope for almost three years. But there was a Church.
Nicholas IV died in 4 April 1292, and his successor, Celestine V was not elected until July 7, 1294, another period of over two years, where the Holy Catholic Church had no pope, but still existed.
Benedict XV revealed himself as a public heretic, innovator of Tradition and enemy of Faith and Church unity in 1914, and the Church has been in a long and bitter interregnum since that time.