No one knows if this third element has anything to do with the Catholic Church.
Dan Brown probably knows that the Catholic Church does not allow its members to be part of masonry, and if someone does they could be excommunicated.
The first papal document to prohibit membership in masonry was drafted in 1738, just 21 years after masonry was born. The Church has confirmed this position in 12 other official documents. The last one was signed by Cardinal Joseph Ratzinger in 1983.
The documents state that masonry and Catholicism are not compatible. Apart from the anti-clericalism that has characterized the masons, masonry considers absolute truths to be non-existent, and it puts all religions on the same level. To them, Jesus was a great teacher like Buddha or Zoroaster.