Sorry, but you are wildly in error with regard to the Albigenses. Certainly lies were told about their practices and beliefs, but the facts are sufficient to show that they were not Christians. Perhaps the most obvious being their beliefs that the God of the OT was actually Satan and that the material world and everything in it is inherently evil. It seems fairly obvious that they were heavily influenced, perhaps indirectly thru Manichaenism, by eastern religions such as Buddhism.
Possibly you are confusing them with the Waldenses and other groups in the same general area at the time and later who were indeed forerunners of Protestantism. These groups were also persecuted violently by the Church.
“the facts are sufficient to show that they were not Christians.”
No facts that I know of. How about you? If they were not Christians, then their exemplary lives would not have been a threat to the Vatican’s stronghold on religion and society itself. Their communites were bright spots of learning, productivity, morality and loyalty to the civil authorities.
But because that was true in a Europe of immorality and superstition, and because the Albigenses were spreading their doctrine through missionary efforts, the Popes felt that their establishment system was threatened.