What is the nature of Dyothelitism according to the Council of Chalcedon? What is the nature of Dyothelitism according to the Council of Chalcedon?
At the Council of Chalcedon, the church taught that the Incarnate Lord Jesus Christ had two natures: divine and human, united in his one person (Hypostatic union), without mixing or blending them into one nature (Monophysitism). In this Christology, Jesus had two minds: divine mind and human mind (Dyothelitism).
My question is specifically asking about the nature of the Dyothelitism. The following sub questions are really helpful in navigating for an answer:

What is the relationship of the divine mind with the human mind?
When Jesus was incarnated, did he use his human mind primarily?
If so, how did Jesus access his divine mind? Is it like a memory being remembered?
Was Jesus operating in both human mind and divine mind simultaneously when he was on earth?

I am seeking answer from authoritative sources regardless of denomination. Preferrably those who believe in the Council of Chalcedon and Diothelitism. A Biblical answer is also acceptable as long as it can explain logically the nature of Dyothelitism.
The Definition produced by the Council of Chalcedon in 451 is an interesting text because it wasn't intended to stand alone; it was produced by people who knew well that they were part of a series of church councils stretching back over a century. So the Chalcedonian Definition endorses the 325 Nicene Creed, and it's 381 Constantinopolitan revision, as well as endorsing the 431 Council of Ephesus. This last council forbade any departures from the Nicene Creed, and so the Council of Chalcedon was careful not to say that it was producing a new creed, but instead a definition or statement that merely clarified the Nicene Creed. The Council of Chalcedon was called in order to unite a divided Christendom, so in addition to endorsing the previous Creeds, it also endorsed the letters of Cyril of Alexandria and the Tome of (Pope) Leo, in order to bring together the three major camps of Christianity of the era: Western, Antiochene, and Alexandrian.
The Tome of Leo is of particular interest to this question because it gives great clarity to the Chalcedonian Definition in a way that I hope will answer your questions. This is the relevant section:

There is nothing unreal about this oneness, since both the lowliness of the man and the grandeur of the divinity are in mutual relation. As God is not changed by showing mercy, neither is humanity devoured by the dignity received. The activity of each form is what is proper to it in communion with the other: that is, the Word performs what belongs to the Word, and the flesh accomplishes what belongs to the flesh. One of these performs brilliant miracles; the other sustains acts of violence. [EWTN]

The Tome of Leo rules out monothelitism because rather than each nature acting and performing what is proper to itself, the divine will would be acting for both the divine and human. (Note that mono-/dyothelitism strictly refers to one/two wills, but the theological principles that lead us to understand that Christ had two wills apply equally to the mind.)
So on to your questions.


What is the relationship of the divine mind with the human mind?


So from the Chalcedonian Definition we understand that, as they are faculties of the natures, they must be &quot;unconfused, unchanged, undivided, inseparable&quot;. From the Tome of Leo, we know that each nature exercises its faculties in communion with the other. And from the Third Council of Constantinople (which rejected monothelitism), we get a clearer picture of just how the two natures relate in general:

And the two natural wills not in opposition, as the impious heretics said, far from it, but his human will following, and not resisting or struggling, rather in fact subject to his divine and all powerful will. (Source)



When Jesus was incarnated, did he use his human mind primarily?
Was Jesus operating in both human mind and divine mind simultaneously when he was on earth?


The person we call Jesus is the Son of God who exists now in two natures, so we mustn't make the mistake of saying that &quot;Jesus&quot; refers just to the human nature. So the answer to question 2 is no and the answer to question 4 is yes: Jesus didn't primarily use his human mind; his divine nature thought exclusively with his divine mind, and his human nature thought exclusively with his human mind, both natures were united and acting together. Even if we think of the human mind as following and submitting to the divine mind, we cannot think of him therefore predominantly using his divine mind. Jesus was not a comatose body animated by the divine mind of God, but a complete living human who was the incarnate Son of God.


If so, how did Jesus access his divine mind? Is it like a memory being remembered?


I think this is where any answer will get more speculative, for I don't think the mechanisms of the communion of the two natures has been revealed to us with any great clarity. But there are several instances in the Gospels which show us some of the ways in which his natures worked together.
Several times the Gospels say that Jesus knew what other people were thinking (Matthew 9:4, 12:25, Luke 6:8, 9:47, 11:17). This kind of knowledge is outside the capabilities of a human, and so in these times we see, somehow, the communication of knowledge from the divine mind to the human mind.
There is at least one major case where the divine mind's knowledge is kept from Christ's human mind: the time of his return (Matthew 24:36, Mark 13:32). More on that at this question/answer.
Another interesting case is when Jesus healed the woman who had been bleeding for twelve years (Matthew 9:20-22, Mark 5:25-34, Luke 8:43-48). I think his question &quot;Who touched me?&quot; is usually thought of as a rhetorical question, but is that necessarily true? Mark says Jesus &quot;kept looking around&quot; and Luke has Jesus quite adamantly respond to Peter that he knew he had been touched and power had gone out from him. I think it would be better to read this not as a rhetorical question from Christ's human nature which had been given omniscience by his divine nature, but instead the genuine question of a human man who for whatever reason was not at this time being given insight from his divine nature.
Lastly are the times when the human nature of Christ expressed emotions that are foreign to the divine nature: exhaustion (John 4:6), grief (John 11:33-35), anxiety and dread (Matthew 26:36-39, Luke 22:40-44). Perhaps even frustration (Matthew 16:5-11), if we think of the divine nature as having unlimited patience. In all of these times the divine nature could have intervened, to re-energise his human nature, to give comfort, to give patience. And no doubt the divine nature did just that at many times to various extents, but in these times the human nature was seemingly left without divine intervention to experience the troubles of human existence which he shares with us. But we must not think of the natures acting in discordance even in these times: even these times when divine power was not communicated to the human nature were times when the two natures of Christ were operating together with singular purpose, no doubt to reassure us that he can be our high priest because he is truly one of us (Hebrews 2:17-18.)