Membrane de servofrein qui fuit

The diaphragm inside the brake booster split or perished, so the booster can no longer hold the vacuum it uses to multiply pedal effort. It is the booster's internal failure, as distinct from a leak in the hose feeding it — the hose is external, cheap and checked first, and the diaphragm means the booster itself.

It is a large vacuum leak into the intake as well as a braking fault, and that is what brings it into the engine's codes. A failed diaphragm admits unmetered air directly to the manifold, so the engine idles roughly and runs lean while the pedal goes hard. A hard brake pedal together with a lean idle is the pairing that identifies it and is far more specific than either symptom alone.

Test by isolating it. Clamp the vacuum hose at the booster with the engine running: an idle that settles immediately puts the leak in the booster, and one that does not moves the search elsewhere in the intake. A booster that cannot hold vacuum for several pedal applications after the engine is stopped is the direct check, and both take a minute.