Second, sitting here in my office, I'm not likely to come up with ANY "primary source" material from the 900s. If you can dig up that, you either are living in a castle in France, or are a fraud.
Third, White specifically did NOT say that the Europeans "had" the stirrup first---but that they first used it in shock combat, so apparently you haven't read his "critics" right or you haven't read him. I think Victor Hanson accepts White's interpretation of the Frankish use of the stirrup in this regard.
Fourth, I don't need to provide ANYTHING. The necessity for the "Peace of God" and "Truce of God" (or do you need "primary source documents" on those, too?) alone are sufficient evidence that there was widespread violence in common European cities, and that the Bishops of Aquitaine, among others, thought it necessary to do something about it from the ecclesiastical perspective.