I’m presently combing modern Thomist interpretations of the Eucharist in an effort to find similarities between St. Thomas and St. Calvin (as one professor here calls him). I’ve realized that Thomas’ commentary on the Gospel of John is a good place to start. See this quote, for instance:
What our Lord said about eating his flesh is interpreted in a material way when it is understood in its superficial meaning, and as pertaining to the nature of flesh. And it was in this way that the Jews understood them. But our Lord said that he would give himself to them as spiritual food, not as though the true flesh of Christ is not present in this sacrament of the altar, but because it is eaten in a certain spiritual and divine way. Thus, the correct meaning of these words is spiritual, not material. So he says, The words that I have spoken to you, about eating my flesh, are spirit and life, that is, they have a spiritual meaning, and understood in this way they give life. (Commentary on John 6, p. 42)
I know there are definite disagreements, but I think the idea of spiritually partaking of Christ through faith could be the Archimedean point I’m looking for. There is also some parallel in the objective nature of the offering.