Left barycentre is not a term I was familiar with.
PS (Parti Socialiste) has always been a center-left party. To make it short, center economically speaking and left when it comes to social policies (mostly). The core of the PS is what we call social-democrats which have accepted economical globalization and mostly try to navigate under its umbrella. Even though, on average the PS is a center-left party, there are several movements within it. Every 3 or 4 years they have congresses so they could elect the first secretary (the president of the party) and then they choose a motion which sets their policies until next congress.
In every congress, there are usually more or less 5 "motions" which navigate between those two main poles :
- the social-liberals (libéral in the French sense as in capitalist) which are those who are center with sprinkles of left in them ie Valls, Macron (when he was in it), Strauss-Khan, Hollande and them).
- the leftists which still question capitalism (Emmanuelli, Filipetti, Hamon) though kinda accept the overall idea of it to be honest.
Social-liberals usually win (though those who win are not the most hardcore but those who are in between the two poles) and that's the reason why the PS is currently losing its leftist base to Mélenchon (which we will talk about later).
What I meant with barycentre was like in the mathematical sense, a center of mass. Hamon regardless of all the cons he had as a candidate had at least the redeeming quality to never have swayed from his currant (whether it was for his party or as a deputy in the European parliament). In 2008, in the Reims congress, he led the most leftist motion in which he had a now known support which was named Mélenchon. He also consistently supported leftist motions in most of the congresses. And his position and similar ones of others had the importance of keeping the most leftist supporters in PS by keeping mass towards the left of the party.
The 2008 congress is the reason Mélenchon left the PS to successfully create his own party. Hamon led the motion he supported but he finished 4th out of 6 and the first 3 motions (Aubry, Royal and Delanoé) were all motions swaying towards the center. Also Hamon was dead last of the 4 main candidates as the first 3 were between 25 and 30% and he was under 20. That accelerated Méluche's process of quitting the party because when he was in the PS, he was always on the leftist pole and always battled for a "true" socialist PS instead of the social-democrat one. When he didn't led one, he always supported the leftist motion of the leftist candidate for presidency of the party. It was only logical that seeing the PS going consistently the centrist way, he who started in the PS in the 1970s had enough of it and then created his own party.
Maybe that's why Hamon will end up doing after a while
