To possess a language core is to have achieved a level where you have developed the capacity of understanding how a language works, by learning its essential vocabulary and how the elements can be combined together to convey any thought or idea. But a language core is not only that. On a psychological level. you have started developing a “feel” for the language and a side of your identity in it. You have started living your life or a part of your life in it. So you have developed a core when you have only only figured out the patterns of a language, but also once you have started living it, by interacting with native speakers and getting exposed to authentic content.
For example, a person who knows thousands of words and has read dozens of books may be skilled in the language, but without speaking with native speakers, they will never develop a proper “core”. By contrast, someone who has a lesser vocabulary, but hours and hours of direct experience living and interacting with native speakers will have such a core.
You can watch Luca talking about it with David in the following two YouTube videos:
https://www.youtube.com/watch?v=5I8sytPJhdU (Luca and David talking about language core)
https://www.youtube.com/watch?v=FOKD6i_z5LY (Language Core Q & A)
And you can read more on this article of the blog www.thepolyglotdream.com