Verb for "to be a cornerstone"

The word underpin is primarily used to mean supporting a building from underneath, but can also apply to organisational structures. Lexico has

underpin
_VERB

1 Support (a building or other structure) from below by laying a solid foundation below ground level or by substituting stronger for weaker materials.
At present, it is not thought that underpinning the foundations would be enough to save the building

2 Support, justify, or form the basis for.
The idea of fairness that underpins the democratic process is grounded in different ways in different theories.

So the sentence could be

The central idea underpinned the observed structure.

This conveys that the 'central idea' is essential for the 'observed structure' to exist, and the association with buildings and foundations gives it solidity.

The central idea upholds the observed structure.

uphold (v.)

To support or sustain physically; to keep from falling or sinking.

To support, sustain, maintain, by aid or assistance; to preserve unimpaired or intact. OED

Racial discrimination denies equality and thus undermines the structure which that cornerstone upholds. US Commission on Civil Rights; Housing in Washington

In a building the base upholds the pillars, and the pillars uphold the entire structure; this is what the church is to God. The Collected Works of Witness Lee, Vol. 1

To take this road, every member of the Jewish People must have a sense of historic destiny continuing through him; he must feel that he is the last remaining pillar which upholds the entire structure of Judaism. Irvin Bunim; Ethics from Sinai

Kapha has its main function in the structure, cohesion, and lubrication of the body. Kapha holds together the tissues, jonts, neck, chest, and head, to name a few. By and large, kapha upholds the entire body and gives you the structure and foundation you have. Varsha Khatri; Think Healthy, Choose Healthy.

The dam consists of the entire structure which upholds the pond, and extends to and includes the first lock. Wisconsin Supreme Court; Wisconsin Reports