The part of a sentence 'stands at the core' is correct and usable in written English.
You can use it when you are trying to emphasize something as the main or fundamental element in a situation. For example: The economic stability of the country stands at the core of all public policies.
The archbishop of Canterbury stands at the core of the British establishment, by protocol outranking even the prime minister.
The basic question of whether "Redskins" is offensive stands at the core of support for keeping the name.
This experiment stands at the core of the scientific debate over whether large piers are ecologically beneficial or detrimental to the harbor's environment.
Preserving white privilege stands at the core of the Jena case, whether or not the laws of the state match up with the facts of the case.
Khatami said he took deep offense at that insistence and contended that defining the question as one of freedom amounted to brushing aside the sacred dignity that stands at the core of any religion.
His office constitutionally stands at the core of the Islamic Republic and its primary role is to assure the sustenance and stability of the Islamic Republic even at the time of transition".
It's a precarious existence for the class that would normally stand at the core of a healthy economy.
Such questions have long stood at the core of the study of images, from Walter Benjamin on to Susan Sontag and many others now.
In the future, much as in the past, the United States and Europe will have to stand at the core of such cooperative efforts.
Other residents continue to assert loyalty to a company that they say once stood at the core of the town's identity.
His words reflected a strained moment in a German-American relationship that has stood at the core of Germany's postwar growth into a European power.
