Quick summary of what we know so far:
Barca has secretly paid 7M€ to Negreira, the vice-president of the referee comitee for 18 year during 4 different Barca presidencies. Including Laporta who still holds the position at the moment.
The payments stopped the moment Negreira lost his position. Negreira blackmailed Barca to show the irregularities regarding the club he had witnessed first hand unless they continued the payments until the previously agreed end.
In his declaration to the Spanish tax agency, Negreira said he was being paid to make sure no refereering adverse decision was taken against Barca, so it was all neutral. He also declared there was no report or similar document to prove any service made to Barca.
Barca has not admitted this. In the official declaration Barca argues that an external consultant was contracted for and that they were provided with videos of low-category players in Spain. This very declaration is already refuted as we now know the internal Barca deliberations when the tax agency asked about these payments. They aknowledged not having any contract that justified the payments and wondered if they should say so or just play dumb.
Some people argue Barca paid for video reports of rederee actuations. This does not seem probable as Valverde and Tata Martino, two Barca coaches during this period of time, admitted to have never looked at any ref report during his time at Barca and not even knowing they existed.
The payments happened to a company created by Negreira, his son was the administrator of the company. Negreira´s company main client was FC Barcelona, he barely made any money outside of them. The company was closed 2 years after Barca stopped paying them. The payments also happened to a company created by Negreira´s son.
A poster very smartly already pointed to this:
You don't really need to buy the refs, you need only to buy someone that has power over the refs and can punish them in promotions, etc, if they don't follow your "guidelines".
And so far, it looks like this is what happened.
We know Negreira´s son acted as a personal coach and a psychologist for Spanish referees. Pérez Burrull, an ex-referee active while Negreira was the VP of the ref comitee, has admitted that, given the competition between refs, privetaly hiring the son of the ref VP could help them in their careers. He specifically referred to being promoted or relegated to the second tier.
Thanks to Andújar Oliver, a ref of the same period, we also know that Negreira´s son would call the refs in charge of the important games that happened in Camp Nou. He would ask them at which hotel they were staying and then would drive them personally to the stadium. He admits that, he being the refs VP son, he could not turn him down.
Video of Negreira's son accompanying refs to Camp Nou
We also learnt how on one of this trips, where Negreira´s son was driving the 4 refs (referee, two assistant referees and the fourth referee) to the game at Camp Nou he told them:
“You know: here, calm down. Barcelona has a lot at stake and you can´t hesitate. You are more than prepared. You know exactly what you have to do."
It is worth noting that, while doing all these, Negreira´s son admits being employed as an external consultant by Barca (as per his CV available on the net). It is worth remembering that he, and his father, got paid 7M€ by Barca during 18 years.
((I would be more than happy to produce links to any on these points if you want to read more))