"I've caught up! ... Huh?"
After Gwendolyn fell for the turtle's trap. The other students also began to move.
"If speed is the problem, then I'll show you extreme speed!"
A senior mage condensed flames in both hands, then detonated them behind his back. With a roar, his body shot forward like a rocket. Another student tried to manipulate the track itself.
"Rise, O earth!"
At his cry, the marble floor rose up and pushed him forward like a wave. All sorts of outlandish attempts ran rampant, from magic that reduced air resistance to spells that eliminated friction with the ground. But soon, they realized.
This race wasn't a battle of speed.
"What is this...!"
"I can't pass it!"
If you take one step closer, the turtle moves forward by 1/10 of that distance. If you then close that 1/10 of the distance, the turtle moves forward another 1/100. A chain of an infinite process. This place was a prison woven from 'logic'.
"Dammit! Why isn't it working?"
A senior running beside him screamed. But there was no one to answer his cry.
"Isaac Upperhills."
Just then, Gwendolyn gave up on the race and came over to my side.
"You know something, don't you?"
"... What makes you think that?"
"You're the only one not running."
"Aha."
So that's why I drew her attention. I immediately started running. No matter how much I ran, the turtle in front of me was always ahead.
One step, half a step, a quarter of a step... I could experience the strange sensation of time being continuously partitioned. I said to her.
"How about this?"
"... It's too late. It doesn't feel sincere at all."
"Dammit..."
I stopped short. I stopped running.
Slowly, steadily.
The turtle in front of us was still crawling along just fine. Gwendolyn opened her mouth.
"Come to think of it, during the first exam, you also just stood there quietly before suddenly disappearing and passing." ...
Did I?
"If you're going to disappear, could you at least give me a hint?"
"... Alright. I'll tell you."
At my words, the gazes of the other students in the race all turned to me.
"What? What did he just say?"
"Acting like he knows something when he can't even get out himself..."
The senior who had shot off like a rocket scoffed while frozen in mid-air.
"Hey. Stop wasting time and just run."
I said to that senior.
"This paradox exists because 'infinity' is vaguely defined. It makes you mistakenly believe that an infinite process is the same as infinite time."
"Mistaken? You're saying that after seeing the state I'm in? This is reality."
He was flying toward the turtle, having become a fire-breathing rocket.
Slowly, steadily.
Of course, he couldn't pass the turtle. 'How do I explain this...?' I think I need to use the 'epsilon-delta argument', the most basic of calculus basics...
But that 'basic' thing is harder to explain with words than I thought.
"... Explaining it with words will be a bit difficult. I'll show you directly."
I gently closed my eyes. In that moment, the space and time of the world grew distant, and my mind connected to the core of the magic. -Achilles can never catch the turtle- Beneath the most important proposition were several corollaries. -Discontinuity of space, -Partitioning of time, -Error of infinite sequences and so on... The preliminary analysis is done.
Now, I just need to overwrite it with new magic. Beyond my fading consciousness, I heard Gwendolyn's shout.
"... Don't tell me you're directly modifying the magic? It's dangerous!"
The other students reacted to Gwendolyn's words.
"What? Direct modification? Is he insane?"
"Quick, stop the freshman! He's going to die!"
"You crazy bastard!"
It must be for the same reason as when I modified Liam's magical device.
Rebound.
A mistake could cause a mana backlash and kill him. But there's no need to worry. 'I've studied calculus for years, you think I'd get something like this wrong?'
The students, including Gwendolyn, rushed at me all at once to stop me. But they couldn't reach me.
Which meant.
The space between me and them began to be infinitely partitioned. When they took one step closer, half a step remained. When they closed that half step, a quarter of a step remained.
They had become trapped in the very 'prison of logic' they had just experienced.
"You son of a...!"
"No way... he's already mastered the magic?"
Just then, the crystal turtle that had been crawling along came to a halt. It turned its head and, for the first time, glanced at me. Is it intrigued?
'It's still too early for that...' This alone isn't enough to pass the turtle. To pass it, I must go beyond understanding and establish a new law.
First. t→c, the definition of a point of convergence. I defined the specific point in time when the turtle and I would meet as the variable c. My mana vibrated subtly, fixing a single solution, one this barrier had ignored, like a coordinate in the void of logic. Second. lim(t→c) f(t) = L, the limit.
I declared that as time 't' 'approaches infinitely close' to 'c', the distance function f(t) between me and the turtle will converge to the limit L = 0. I overwrote the concept of 'infinity' that formed the barrier with the counter-logic of 'convergence'. At that, with a jolt, the turtle glared at me.
"So this isn't enough either, huh?"
I can't break this magic with vague words like 'approaches infinitely close'. Then, the third. ∀ε>0, ∃δ>0 s.t. ∀x∈D, (0<|x-c|<δ ⇒ |f(x)-L|<ε) The epsilon-delta argument. I attempt 'a rigorous proof of the limit'.
"How about this?"
Shiver, shiver.
The turtle's legs were trembling. It pained me to feel like I was bullying a cute turtle, but it couldn't be helped. With this, all calculations and adjustments were complete.
Law construction complete.
Proof.
The moment I completed the final logic, the world around me twisted. My magical state had become different from the other students'. In their world, distance would still be a concept that could be infinitely divided.
But in my world, distance is subject to the law of the 'limit'. Before my eyes, a single line appeared, one that others could not see. It was the point of convergence where the turtle and I would meet.
Step, step.
I walked toward the turtle. Slowly, steadily, the turtle tried to run desperately, but its end was already decided. Finally, my foot crossed the 'point of convergence' and passed the turtle.
When I looked back, the turtle was still floundering on the track. Slowly, steadily, the turtle huffed and puffed, taking one step at a time toward me. It looked pitiful, sweating profusely. ...
Is this perhaps animal abuse?
When I came to my senses, I was already back in the classroom. I looked around and saw only Valerius and me. It's a little awkward.
"..."
After a moment of silence, Valerius spoke.
"I let them out. Since you, disciple of Doloren, passed the turtle."
"I see."
"Nothing more to say?"
"No, sir."
"..."
Valerius's brow furrowed.
"Then I'll be the one to ask. How on earth did you do it?"
"... I gave a rigorous definition for the limit."
"The limit?"
"Yes."
"..."
Valerius was speechless for a while.
"The limit... It's not that I haven't thought of it."
Valerius handed me a document.
[The Use of Infinitesimals] Infinitesimals...
I can guess the content just from the title.
"I thought this might solve it somehow, but it didn't work."
I glanced over Valerius's document. 'As I thought.' It's just as I expected.
The document described a method to solve the 'Achilles and the Tortoise' problem using the concept of infinitesimals. Valerius's mouth opened.
"I thought I had researched it enough, but I got stuck here. Can you point out what's wrong with this paper?"
"..."
I could, if I wanted to. But is there any need to tell him? It'll obviously become a hassle if I do...?
"..."
Valerius's gaze was prodding my face.
"I will explain why that paper is flawed."
"...!"
"On one condition."
"A condition?"
"Yes. I want you to publish the concepts I will teach you as a Magic Tower paper, 'without leaving anything out'."
"... Without leaving anything out?"
Valerius's eyes widened.
"... You want me to teach such a secret spell to all mages? Why on earth?"
I didn't answer. The knowledge I would release in the future was mountainous. Rather than explaining the basics from scratch every time, it's better to just raise the world's average level of understanding.
I just looked at him silently. Valerius seemed to read something in my silence and soon nodded.
"Very well. I accept."
We exchanged satisfied smiles. The contract was made. I promised to create the 'Principles of Calculus' for Valerius by next week.
I bowed my head deeply.
"Then, I'll take my leave."
"... Before you go. Haven't you forgotten something, disciple of Doloren?"
"Sir...?"
Forgotten something?
"You were the first one to break through. Make a wish."
"Ah."
I'd forgotten, preoccupied with thoughts of organizing the calculus notes back at my dorm. I took a breath and voiced the true purpose for my being here.
"My wish is..."
After Isaac left.
Hehehe.
Valerius found it hard to suppress his laughter. To see the 'Achilles and the Tortoise' problem solved within his lifetime. And to think he would learn the solution by next week.
"One week..."
The time felt too long. What if I hit my head hard and woke up next week? That way, I could get my hands on the principles of calculus in just one day.
But Valerius would do no such thing. He wasn't an idiot. Still, the laughter kept bubbling up.
How long had it been since he'd felt this sensation? The thrill he'd felt when he met the young Doloren decades ago. This far surpassed it.
'That child... he's the real deal.' A being who could become a Great Mage, a Dean, or anything he wanted if he just set his mind to it. 'The problem is, he doesn't seem to want to...'
Valerius recalled the wish Isaac had made at the end. -"From now on, please guarantee my safety above all else. And I also want the right to publish all my research findings under your name whenever I wish." He had never imagined someone would make such a wish.
Not fame, not wealth, not power, but only 'safety' and 'anonymity'. When he'd absentmindedly agreed, the boy had even presented him with a contract. Valerius looked down at the contract on his desk, the ink not yet dry.
[Mutual Cooperation and Confidentiality Agreement] Article 1 (Purpose): This agreement is for the purpose of ensuring a safe research environment for 'Party A' (Isaac Upperhills) and sharing academic achievements with 'Party B' (Master Valerius).
Article 2...
Article 3 (Confidentiality Obligation): 'Party B' shall guarantee the personal safety of 'Party A' as the top priority, and shall not disclose his identity or research content to any third party without the explicit permission of 'Party A'.
"... What an audacious boy."
Valerius let out a dry laugh. But at the end of the laugh, his expression turned serious again. Why did this child make such a strange request?
What was making him so anxious? 'Could it be that something is happening in the Magic Tower?' If there was a problem in the Magic Tower, especially one that could threaten the safety of a freshman, the Dean would surely know.
'Which means... the Dean, that pointy-eared bastard, is plotting something.' All the puzzle pieces began to fit together, forming an ominous picture.
"I'll have to look into it."
Meanwhile, inside the miniature racetrack model. "Um... how long do we have to stay here...?" A student muttered in a dazed voice. They were still chasing the turtle they could never reach.