Because the ca. 1797 framing was in place, to prove the marble fragment had been originally found in the hearth the fragment was simply placed on the surviving hearth framing.  The Team knew that if the fragment was level with the surrounding floorboards, which are original Madison elements, then the fragment would have almost certainly been installed in the hearth in ca. 1816.  Incredibly, the fragment fit perfectly and so the Restoration Team had their hearth material.