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'lay at the core' is correct and usable in written English.
You can use it to convey that something is essential or fundamental to a concept, system, or task. For example, "The values of integrity and honesty lay at the core of our organization's beliefs."
Exact(35)
New York City lay at the core of the crisis.
Perhaps his background as a pit-boy lay at the core of this impulsive behaviour.
Insolvency and limited liability lay at the core of commercial enterprise.
This rottenness lay at the core of Italian banking.Today a new mood is evident.
Envy lay at the core of his family's rancor, he said.
Indeed, this type of nationalism lay at the core of the political debates rending the United States in the 1850s.
Similar(25)
Identity lies at the core of security online.
They will not cure the disease that lies at the core of the business culture.
It is this split that lies at the core of the contest between Hillary Clinton and Donald Trump.
And what is lies at the core of the national despair.
That is the phobia that lies at the core of all my anxieties.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com