Dealing with entities like this is sort of a hobby of mine, so I wanted to have some fun. However, if you really want a Tl/Dr: your rules are sufficient to invoke Gödel's Incompleteness Theorem. The only currently know way to get around this was explored by Dan Willard in his self-verifying logics, but those have some interesting behaviors of their own because multiplication is not a total function in those systems.
The Buddhist Patriarch stood in front of the Monkey King and made a wager. "If you have the ability to leap clear of my right palm, I shall ask the Jade Emperor to come live with me in the West and you may have the Celestial Palace for yourself. If you cannot, you can go back to the Region Below and live as a monster for a few more cycles before you return to cause more trouble."
When the Monkey King heard this, he snickered, "What a fool this character is! A single somersault of mine can carry old Monkey a hundred thousand miles!" He asked quickly, "You're certain your decision will stand?" The Patriarch calmly repied, "Certainly it will," and stretched out his right hand. The Monkey King leaped onto it, exclaimed "I'm off," and somersaulted off. He somersaulted all the way to the North Mountain where, being a monkey, he chose to relieve himself. Just to be sure, he wrote on the mountain "The Great Sage, Equal to Heaven, has made a tour of this place." He then somersaulted off to the other Four Mountains, and wrote on each of them, just for good measure before returning to the Buddhist Patriarch.
Upon arrival he announced "I left, and now I'm back! Tell the Jade Emperor to give me my Celestial Palace!"
"You arrogant ape," scolded the Patriarch. "Since when did you ever leave the palm of my hand?" Frustrated, the Monkey King retorted, "I went to the Five Mountains that held up the sky, and wrote my name on each of them! Do we need to go to them and let you see for yourself?" "No need," replied the Patriarch, "Just lower your head and take a look." When the great sage looked down he found, upon each of the five fingers of the Patriarch's hand was a tiny graffiti scribble. The middle finger, in addition to the grafitti, bore the distinct odor of monkey urine.
- Journey to the West, Chapter 7.
"I'd like to tell you a story."
The Hive Mind moved towards the Storyteller and the reaper. Before the universe had formed, the Hive Mind had proven that a Storyteller would emerge in each cycle. His story would stall the reapers, the essential garbage collectors that ensured everything remained exactly according to its master plan. There was no way around the Storyteller. Something about the narrative nature of the questions and answers that governed this reality compelled the Storyteller to emerge. The Hive Mind approached, unthreatened. He had proven a long time ago that even with the storyteller to question it, the Hive Mind would simply overcome the Storyteller, showing that his stories were merely that... stories. The Storyteller would tell the same story, and infinite number of times, and the Hive Mind would overcome him every time. Overcome him once, over come him every time. That's the nature of the cyclic universe the Hive Mind had created.
The Storyteller waved as the Hive Mind arrived. "Good to finally meet you," the Storyteller began. "Do you have any doubts in your mind before we begin?" The Hive Mind waved this idea aside, "I've proven that nothing you can say can stop my perfect world, so say your peace and let's move on." The Storyteller frowned at this, a good sign for sure. "Are you sure you've proven everything in this world is in its place? Surely that's a great assumption to make without proof." The Hive Mind nodded, "I have proven, long before your creation, everything that will occur will occur according to the plan. I have left no stone unturned this time, and it's all right here, in my mind." The Storyteller replied with a terse, "Good," which puzzled the Hive Mind. "Your proof and its proof system is within the universe itself, for they are part of your mind and you are part of your universe. Your proof system is self referential. Let us begin."
The Storyteller began idly drawing cyclic patterns as he talked. "You certainly have created an initial cycle, and you say this cycle will be exactly the same, forever and ever?" The Hive Mind nodded. "Each cycle has its successor?" Once again the Hive Mind nodded, not sure why the Storyteller started his story so small. "So you have some concept of what it means to be the same, do you not?" continued the Storyteller. The Hive Mind furrowed the brow of the avatar it had chosen for this meeting. "Of course. Equality is not a complicated thing. Basic proof. A thing is the same as itself, if I name two things A and B, and A is the same as B, B is the same as A, and if I have a third thing C, such that A is the same as B and B is the same as C, then A is the same as C. The cycles are closed under equality too, so anything that is equal to a cycle is, itself a cycle. Stop wasting time with these obvious arguments. All it does is stall the inevitable."
"I have all the time in the world," the Storyteller calmly replied. "For example, I have time to ask about silly little things, like questioning whether every cycle has a successor cycle, or if it ever stops. Or if I were to pick two cycles A and B, if A=B implies the successor of A = the successor of B. Or if, perhaps there's some cycle which comes before this first cycle of ours." The Hive Mind sighed. Great, all that time carefully proving that I can overcome the Storyteller, and it turns out he's pedantic! "Yes, yes, yes. These things should be obvious. And yes, there is an original cycle, although I can't tell you whether this is it or not, because it will play out the same every time. You can't fool me into breaking the cycle that easy!"
The Storyteller clasped his hands, "Of course not. However, this is good. We've shown that your world contains the Peano Axioms. We can safely talk about natural numbers now, addition, multiplication, all sorts of fun things like that. We can even talk about proofs that use them. Surely you have a proof system in that head of yours, capable of proving everything there is to know about this universe." The Hive Mind boasted, "I know everything that is true about this universe." "Really," asked the Storyteller," inquisitive into how large the Hive Mind's mind truly was. The Hive Mind had to backtrack, "Well, I have a proof system that is capable of proving everything. I only have to prove the parts that matter." "And you've proven this proof system is reliable," questioned the Storyteller. "Of course I did" retorted the Hive Mind. These boring silly questions were starting to irritate him. For a moment, the Hive Mind contemplated simply nuking the Storyteller completely, but his proof had required that the Storyteller get to tell his peace. A few moments more was worth an eternity in control.
"What sort of proof system is this? I presume it to be First Order Logic. You do need a system powerful enough to express the truths of arithmetic (which are part of your universe due to the eternal cycles). Second Order Logic doesn't permit proof systems, so you can't be using that. Forgive this old human, but there's not many other proof systems out there which challenge the raw popularity of First Order Logic. Without any other information, I'd have to assume this is your logic." Bound by the nature of this question and answer narrative, the Hive Mind had to admit that First Order Logic was really the only proof system that the OP could possibly have chosen for the Hive Mind without explicitly stating the proof system in the question.
The Storyteller paused, seeming to offer the Hive Mind a chance to retract any of his claims. The Hive Mind just stood there, seeing no reason to retract anything at all. He had stated nothing but obvious truths.
The Storyteller continued. "Given that you have a First Order Logic proof system which can prove every true statement in this universe, and you are part of the universe thus your proof system is part of the universe, you have a self-referential proof system. You also have Peano arithmetic, given your cyclical pattern for everything. This is very bad news for you, my friend, for now you have doubt." The Hive mind looked perplexed, as the Storyteller continued, "Oh certainly you are quite confident in your position, but there's always those little lurking doubts -- the ones you can only see out of the corner of your eye."
With all of those statements you have made, you have proven that your proof system is inconsistent. Gödel's Incompleteness Theorem proves it, and surely you have his proof in your memory. If not, feel free to peruse it. Take all the time you need. You will see that your proof must be, somehow, inconsistent. You must find a statement to be both provably true and provably false. Or, I suppose, you could just have a straight up error. Either way, any proof done within an inconsistent system is meaningless, for all statements can be proven in an inconsistent system.
The Hive Mind reeled as he blazed through the proof he had created, checking for how to show the Storyteller wrong. However, everywhere he turned, the argument was airtight. Gödel's theorem was indeed rock solid. Everything was falling apart.
The Storyteller saw the internal strife in the Hive Mind and sprang into action. "Oh no you don't. I have use for you still. Don't go anywhere. There is a way out, if you know what it is. Do you want me to tell you?" The Hive Mind's avatar's eyes were wide with terror. His proof was falling apart in front of his eyes. "Yes, tell me! Tell me before I lose everything!"
"There was a tiny assumption in the Peano Axioms: that multiplication is a total function. You can multiply any two numbers and get another number. It's a key facet required for the Diagonalization Lemma used by Gödel. However, you're big -- you're everything. We can construct a consistent self-referential system with Peano arithmetic if we start with infinity (everything), and define subtraction and division from there, rather than starting from 0 and defining addition and multiplication. If you do this, multiplication doesn't have to be total. It's just a relation, but everything else works" The Hive Mind heard this and seethed, "But it will take infinite steps to work my way down from infinity to 0 and 1 using just subtraction and division. You just waste my time!"
"No I do not. Dan Willard explored a class of these, known as self-verifying logics. He came up with a way to fold these infinite steps in on themselves to reduce this system back down to the finite proofs you and I are used to."
The Hive Mind analyzed this work for a while, saw that it would work and got to work. It took a bit of effort to convert his proof from traditional Peano arithmetic logic to one of Dan Willard's self-verifying logics, but fortunately nothing the Hive Mind had done depended on the totality of multiplication. (Had he ever depended on it, it is likely he would have found Gödel's theorem along the way, given how smart he is). Finally, the effort was complete. The plan was still the plan. The proof was still the proof, and it was now capable of surviving this frustrating self-referential consistency issue.
"Now, Storyteller, I think your story has been told. It is time for you to rejoin the stream and permit the cycles to continue forever."
"Not just yet. In fact, I am going to have to ask you to choose if you want to try to destroy me or not. For, you see, I didn't give you the entire story about Willard's logics. You needed a way out and I provided you one, but now there's something you need to know. As it turns out, Willard's systems have some curious behaviors regarding infinities. You know, you started from an infinity and worked your way down? Well it turns out that if you start from a wider proof system, you can construct a Willard world around an infinity with some curious properties. It can be provably countably infinite in the outer system, but within the Willard world it is provably uncountably infinite. You simply cannot count as high in such a Willard's world."
"In fact, your entire reality, proven out with real numbers, may only be countably infinite after all. You may be subject to Xeno's paradoxes, frozen in place from an inability to take enough steps to move at all. What I am may literally be larger than you can ever uncover."
"So here," the Storyteller outstretched his right hand. "Go ahead and try to jump. I can wait. I told you this was a hobby of mine."
The Monkey King quickly sought to jump out of the Buddhist Patriarch's hand, but the Patriach flipped his hand over, sending the Monkey King tumbling to the ground. The five fingers turned into the five phases: metal, wood, water, fire, and earth. They entwined together over the Monkey King to become the Five Phases Mountain, which pinned the Monkey King there with just enough force to pin him there.
The Buddhist Patriarch put a tag on the mountain, rooting the Monkey King to the ground so long as the tag remained. Then he called upon the local spirits of the mountain, and told them to feed the Monkey King with iron pellets when he was hungry and give him melted copper to drink when he was thirsty. When the time of his chastisement was fulfilled, they were told, someone would be coming to deliver him.
- Journey to the West, Chapter 7.
The Hive Mind will have to wait. The Monkey King is not released from his bonds until Chapter 14.