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“Senpai, the definitions of a supremum and a maximum are completely different,” Miyuki Kazuya sighed. “I thought the examples we went through earlier had already made that clear.”
“Hnnngh... But for this one, this one’s supremum and maximum are equal! And these too... Aaaagh, am I seriously going to pass finals...?”
The senpai sitting across from him had his face all scrunched up, looking as though tears might spill out at any second.
Miyuki knew scenes like this had already happened countless times. He also knew Sawamura had, in fact, failed this course before. Even so, when his gaze landed on those golden-brown eyes misting over with tears, he paused.
“...I heard Furuya-senpai can consistently pass the practice exams now. Is continuing to panic going to help Sawamura-senpai beat him?”
“You little—!” Sawamura Eijun glared at him, the lingering tears at the corners of his eyes sparkling under the light. “Has anyone ever told you that you’ve got a pretty good face but an absolutely awful personality?!”
“Thanks for the compliment~”
Sawamura ducked his head and began angrily fiddling with his phone, muttering under his breath, “Okumura-senpai is ten thousand times better than this bastard four-eyes... If I begged him, he’d definitely teach me...”
“Okumura-senpai is busy working on his thesis lately. Do you really want to bother him?” Miyuki said casually.
“Sawamura-senpai already has me. We’re still in the middle of tutoring, so stop wasting time~” A thought suddenly flashed through Miyuki’s mind. Before Sawamura could respond, he continued, “A maximum has to belong to the set, but a supremum doesn’t. Well, that’s probably still too abstract for Senpai. But if I explain it using the shoujo manga you’re always reading...”
Miyuki deliberately drew out the last words, his gaze behind his lenses fixed squarely on the other man.
Sawamura had already put down his phone and was staring at him with complete concentration.
“Imagine a heart-flutter value that starts at zero. As the protagonists go through different events, it rises and falls, just like a function.”
Miyuki reached over, pulled Sawamura’s heavily marked-up scratch paper toward himself, and drew a coordinate system in an empty space. His pen began at the origin and moved to the right.
“First, they go from strangers to friends.”
—Three months ago—
For the very first class of the very first course in his university major, Miyuki Kazuya arrived twenty minutes early.
He had already previewed the Mathematical Analysis textbook over summer break, and with online resources on top of that, self-study probably wouldn’t be a problem. If the professor turned out to be bad at teaching, he could sleep in a little longer from then on... Or, actually, could he just skip class altogether?
Miyuki took a seat in a corner near the back, bag slung over one shoulder. As his thoughts wandered, he looked around at the handful of other students who had already arrived.
One, two, three.
Not many.
Two girls dropped their bags around the middle of the classroom, then walked back out with breakfast in hand, laughing together. That left only one guy sitting dead center in the front row, holding a book and reading with intense concentration.
Fluffy brown hair at the back of his head. A simple T-shirt neatly tucked into jeans, a narrow black leather belt outlining a surprisingly trim waist.
From this angle, he looked like nothing more than an ordinary male college student with a decent build and decent taste in clothes.
Miyuki leaned slightly to the left and narrowed his eyes at the upright book in the guy’s hands.
It wasn’t the blue cover of their textbook. Looked pink, maybe? And it was ridiculously thick... Were those pictures inside, or what?
Hmm. Too far away to make out the details.
Just as Miyuki was about to look away and get back to reviewing, he heard a faint sniffle.
The guy smoothly pulled an enormous pack of tissues out of his bag and began wiping his tears while crying quietly. Before long, the used tissues had piled into a small mountain.
More and more students filtered into the classroom, and the room gradually grew noisy. Miyuki had his textbook open on the desk, along with a notebook covered in dense handwriting, but his attention was somewhere else entirely.
Every now and then, almost without meaning to, he glanced toward the guy in the center of the front row from the corner of his eye.
He saw the lingering redness around the guy’s eyes when he stood up and waved both arms enthusiastically to call over a classmate. He saw the energy with which he talked to the sleepy-looking person who sat beside him. He saw the professor come over before class and deliberately flick him on the forehead.
Ah.
He’s interesting.
————
The pen drew a straight line rising diagonally from the origin.
“Because of outside factors—or simply because one of the protagonists becomes curious about the other—their affection gradually rises,” Miyuki explained smoothly. “Obviously, real human emotions aren’t simple numbers. I’m just abstracting them into a continuous variable to make this easier to understand.”
“If we stop at any arbitrary point in time”—the tip of his pen paused, extra ink bleeding faintly into the paper—“then the heart-flutter value at that moment is fixed. Since we’re looking at a continuous function over the closed interval from the origin to the present moment, it attains a maximum there. So in this case, the maximum is equal to the supremum.”
Sawamura’s eyes glazed over somewhere around the last two sentences.
“...Let’s try it another way. Say the heart-flutter value is a continuous variable from zero to one hundred. Forty means friends, eighty means best friends, and one hundred means lovers.” Miyuki tapped the pen against the page. “Let’s say we’re currently at twenty. You with me so far?”
Sawamura nodded.
“For the maximum, you only need to consider all the values they’ve actually experienced between zero and twenty—in other words, the whole process from feeling nothing toward each other to developing some initial interest. The highest affection value they’ve reached so far is twenty, so the current maximum is twenty.”
Sawamura nodded hesitantly.
“For the supremum, you also have to think about upper bounds. For the values they’ve experienced from zero to twenty, twenty, 20.001, 20.01, 20.1, twenty-one, twenty-two, one hundred, one thousand—they’re all greater than or equal to every value in the set. So every number greater than or equal to twenty is an upper bound in the current situation.”
Sawamura nodded very slowly.
“The supremum is the least upper bound. Here, that’s twenty itself. Which means the maximum and supremum are equal in the current state—”
Sawamura really wasn’t getting it, so he interrupted. “What does any of this have to do with shoujo manga or heart-flutter values?”
“...It gets more obvious later. Let me reorganize this.”
Miyuki took a deep breath and picked up where he had left off. “Twenty means they’ve developed some initial interest in each other. But before they can actually reach forty and become friends, things have to happen. For example, one protagonist might hear things about the other from someone else.”
—Three months ago, later that same week—
For the second Mathematical Analysis class of the first week, Miyuki once again arrived twenty minutes early.
This time, he sat slightly left of center in the third row—a spot where he could see the board clearly while also getting a perfect view of the front row.
Purely incidentally, of course, it also let him see whether that Sawamura-senpai was going to cry again.
Yes. In the span of only two days, Miyuki had been lucky enough to match the guy’s face to a name in a post on the university’s official social media account.
Sawamura Eijun. Second year. Major: Literary Studies and Criticism. Multiple-time winner of essay contests, poetry competitions, and other writing awards. Together with Furuya Satoru, known as one of the School of Literature’s “Twin Stars.”
So why was someone from the School of Literature taking Mathematical Analysis instead of ordinary college calculus?
As luck would have it, the course-selection group chat Miyuki’s roommate had dragged him into contained the answer.
It had started with a non-math major asking whether someone who happened to like mathematics could take Mathematical Analysis. A flood of people immediately tried to talk them out of it, and the example that came up most often was the real-life cautionary tale of the School of Literature’s Twin Stars.
【apparently sawamura heard that high school math and university math are totally different systems, so he dragged furuya into mathematical analysis yelling “WE HAVE TO CHALLENGE OURSELVES!”】
【result: both failed analysis I and II. they should be retaking it with you guys now】
【wait, furuya failed too?】
【YEP. so if you’re not a math major DO NOT TAKE MATHEMATICAL ANALYSIS AAAAAA YOU’LL GET TRAPPED IN AN ENDLESS CYCLE OF FAILING IT AAAAAAA】
That explained the professor’s attitude toward Sawamura during the last class.
And considering he had the nerve to sit dead center in the first row, maybe Sawamura was determined to pass his retake this time?
“Furuya, did you understand the continuity stuff Professor talked about last class?”
The energetic voice carried through the layers of surrounding students and landed precisely in Miyuki’s ears.
“Furuya, do you know how to do the sequence-convergence problems in the homework?”
“Furuya, why are there so many theorems about function limits? I never know which one I’m supposed to use for the proofs...”
Stop asking Furuya. He’s already asleep. Have you seriously not noticed?
Sitting directly behind Sawamura, Miyuki kept his eyes on his own notes while quietly listening in.
The bell rang.
Sawamura packed at lightning speed, grabbed Furuya, and charged out of the classroom.
Miyuki leisurely stacked his printed materials and put them into his bag. As he turned to leave, he overheard two girls whispering behind him.
“Sawamura-senpai has so much energy! I saw his social media this morning. He said he stayed up all night writing because he barely managed to submit something before the deadline, but he still wasn’t late for a nine a.m. class.”
“Maybe he’s just trying to make an impression on the professor? He grabs the center seat in the first row and acts super enthusiastic, but his quiz and homework grades are both awful. Maybe he’s hoping the professor will go easy on him and let him pass.”
Miyuki’s step slowed.
He thought back to the extremely easy homework and even easier quiz.
And suddenly, an enormous curiosity toward the person called Sawamura Eijun rose inside him.
I wonder what he looks like when he begs someone for help.
————
Miyuki pulled himself out of the memory, his pen pausing lightly against the paper.
“While gathering information, they might hear good things or bad things about the other person. So naturally, the corresponding heart-flutter value rises and falls.”
The black line continued from the end of the previous segment, bending upward and downward in a wavering curve.
“Of course, since this is shoujo manga, the author is going to make them friends eventually. So in our model, we can assume the heart-flutter value eventually crosses the threshold of forty.”
Sawamura frowned in thought, seemingly trying to remember something. His expression cycled repeatedly through confusion, contemplation, and sudden realization.
Miyuki didn’t wait for him.
His pen shot straight upward, drawing a line parallel to the vertical axis and piercing directly through the mark representing forty.
“The scene where they become friends might be cheerful, or it might begin with a conflict. Either way—”
“As long as they do successfully become friends, their heart-flutter value undergoes a qualitative jump.”
—Two months ago—
Miyuki Kazuya unbuttoned his coat and settled into his usual seat at the back of the classroom.
Most of the other students arrived in groups of two or three, but Miyuki had always been used to keeping to himself. Besides, there weren’t many first-years in the mathematics department to begin with, and the odds that any of them would happen to pick the same literature-oriented general education course were almost nonexistent.
At some point, the lone seat in the back corner had quietly become Miyuki’s personal spot.
The course was called Film Appreciation and Analysis. Miyuki had originally chosen it simply because the time slot was convenient.
Once classes started, he discovered it was unexpectedly easy. Most days consisted of everyone watching whatever film the professor had selected, writing a response, and then discussing it in class.
The professor used all sorts of interactive exercises. Since the beginning of the semester, they had already done mock screenwriting, improvisational acting, and several other activities.
Participation at the front wasn’t mandatory, so every time the interactive portion began, Miyuki lowered his head and flipped through his notes, doing his best not to attract attention.
So far, the professor had never called on him.
The lights went completely dark.
The title of the film slowly appeared on-screen.
At that exact moment, a dark silhouette suddenly bent low and hurried across beneath the projector screen, accompanied by the faint clinking and jingling of something metallic.
The sound drew closer and closer before stopping hesitantly beside the empty seat next to Miyuki.
“Hey, is anyone sitting here?”
It took Miyuki two seconds to realize the question was directed at him.
He instinctively looked up.
Golden-brown eyes gleamed like flowing honey. A thin layer of sweat stood out along the other man’s profile beneath the warm light from the screen, even tinting his cheeks faintly red.
“No. Go ahead.”
Miyuki mechanically bent down, moved his shoulder bag onto the floor, then sat up again with inexplicably stiff limbs and stared straight at the screen.
What was that movie about?
Miyuki Kazuya couldn’t remember.
What he remembered was the way Sawamura-senpai’s eyes, visible at the edge of his vision, constantly shifted focus with every turn of the plot.
And the silver necklace resting against his collarbone, catching the light.
The hour-long film ended quickly, and then came the usual interactive segment.
Everything that happened afterward felt vaguely dreamlike.
First, the professor announced that they would be reenacting classic scenes.
Then he immediately spotted Sawamura hiding among the students, dragged him out, and told him to choose a partner for the performance.
Sawamura looked around.
His handsome face immediately attracted plenty of enthusiastic attention. Just as several guys and girls were preparing to volunteer themselves, Sawamura abruptly turned around, looked down at Miyuki, and asked whether he wanted to partner up.
Of course Miyuki couldn’t refuse a senpai’s invitation.
So he went onstage and delivered his own lines with enough awkward stumbling to perfectly showcase Sawamura’s emotionally vivid performance. Finally, under the professor’s laughing encouragement, he returned to his seat.
What exactly had just happened?
Miyuki Kazuya couldn’t remember.
What he remembered was that when Sawamura invited him, and when Sawamura performed onstage—
for those brief moments, Sawamura-senpai’s attention had rested entirely on him.
Unexpectedly difficult to look away from.
The moment Sawamura stood up to leave after class, Miyuki did something bold.
He caught the hem of Sawamura’s shirt and looked up at him.
“Sawamura-senpai, your performance just now really impressed me. Would you mind explaining it to me a little more?”
Sawamura stared at him.
Just as Miyuki was about to apologize for being rude, Sawamura planted both hands on his hips and burst into loud laughter.
“Ahahahahaha! I didn’t expect you to recognize the great Sawamura’s talent so thoroughly! Do you have time now? We should talk while it’s still fresh! That’ll work best! Ahahahahaha!”
Immediately afterward, Sawamura grabbed Miyuki by the wrist with one hand and thundered out of the classroom.
Through the crowd.
Up the stairs.
All the way to the rooftop terrace of the humanities building.
“Hardly anyone comes up here! We can act however we want and nobody’ll really see us! Ahahahaha!”
Sawamura-senpai waved his arms around energetically as he explained his interpretation of the character’s state of mind and the body language he had deliberately designed for the scene, occasionally punctuating everything with strange little sound effects.
What exactly had Sawamura-senpai said?
Miyuki Kazuya couldn’t remember.
What he remembered was complimenting Senpai a few times, then watching suspicion grow stronger and stronger in those eyes until Miyuki finally made his crucial proposal.
“I really don’t know much about this subject, so I’d like Senpai to help me understand it better. Of course, I wouldn’t ask you to help me for nothing.”
Sawamura frowned and thought about it, a trace of wariness appearing in his eyes.
“I’m fairly confident in Mathematical Analysis. So far, I’ve gotten full marks on every homework assignment and quiz. If Senpai ever needs help, you can ask me anytime.”
Sawamura’s brow instantly smoothed out.
His whole expression brightened, and as if terrified Miyuki might change his mind, he immediately said, “Deal! I need your help right now!”
“I have something else to do later. How about we find an empty classroom this weekend and go over it?”
“Hnnngh... I know somewhere better. Forget that for now—let’s add each other first so we can stay in touch!”
Saturday.
Sawamura took Miyuki by train to a quiet café.
The moment they walked in, Miyuki met the eyes of Furuya Satoru behind the register.
Furuya gave him a slight nod in greeting, his gaze lingering on Miyuki for an extra second before he lowered his head and resumed wiping a glass.
“Furuya! This is the super nice kouhai I told you about! He’s amazing at Mathematical Analysis! I really think we’re gonna pass this time!”
“Senpai, don’t make me sound that impressive...”
“Miyuki, have more confidence in yourself! I asked the professor, and he said you’re the only person in the entire class with perfect scores on everything! Teaching us is definitely going to be easy for you~”
“Wait. When Senpai says us, do you mean I’m tutoring Furuya-senpai too?”
“...Teaching one person is teaching. Teaching two people is also teaching...” Sawamura’s voice gradually became guiltier. “But look! This café belongs to Furuya’s grandpa, so he’s providing us with the absolute best possible study environment...”
“Friends are supposed to help each other, right?”
Golden-brown eyes filled with watery pleading as they gazed at Miyuki.
“...Friends... Right...”
For a moment, Miyuki’s mind went completely blank.
Then, almost unconsciously, he agreed.
“Yes!”
“Yesss! Miyuki, you’re the best! Furuya, say something too!”
“Thank you, Miyuki-kouhai.”
Ah.
So we’re friends already.
————
The line that had pierced straight upward through forty continued rising toward the right in a smooth, steep curve.
It climbed steadily—
until the moment it reached seventy.
Then it plunged sharply downward, stopping just above forty.
“I know!” Sawamura exclaimed. “That’s because the protagonists have some kind of fight or conflict!”
“Bingo~” Miyuki nodded. “From the origin to the current state, can Senpai tell me the maximum and supremum?”
“Uh... The maximum is the biggest value they’ve actually reached so far...” Sawamura frowned. “Hnnngh... Seventy?”
Miyuki nodded in satisfaction.
“Correct! And the supremum?”
“I remember... uh... you said earlier that the supremum is... the least upper bound? What was an upper bound again, aaagh...”
“It’s not a complete definition, but for now Sawamura-senpai can just remember that, when there’s a maximum, every number greater than or equal to that maximum is an upper bound~”
“...That sounds vaguely suspicious, but I’ll trust you... So the upper bounds are all the numbers greater than or equal to seventy, which means the supremum is... seventy?”
“Not bad! Looks like you might actually catch up to Furuya-senpai~”
Miyuki ignored the flames practically shooting out of Sawamura’s eyes and continued.
“Shoujo manga has all sorts of plot developments, so the heart-flutter value will keep rising and falling whenever the protagonists have conflicts. But the author is obviously going to make them lovers eventually. Which means after that, the heart-flutter curve is going to look like this...”
—Half a month ago—
“Sawamura-senpai, there’s something I’ve been wondering about for a while.”
During a break in tutoring, Miyuki took a sip of his Americano and brought it up as though it were nothing.
Sawamura currently had a piece of chocolate cake stuffed into his mouth. A faint smear of light-brown cream clung to one corner of his lips as he raised his head and gave Miyuki a questioning look.
“That necklace.” Miyuki pointed at his own collarbone. “I think the design looks pretty nice. Would you mind telling me where you bought it?”
“Mmph... This?” Sawamura lifted the silver pendant hanging around his neck and smiled. “I made it with one of my senpai. It’s one of a kind!”
He stood up and leaned closer to Miyuki, sliding the top silver plate aside to reveal the lettering underneath.
“Look! It has my name on it!”
Senpai?
Miyuki barely managed to keep his expression steady.
He reached out and carefully rubbed a thumb over the silver plate engraved with Sawamura’s name, asking casually, “Which senpai?”
“Okumura Koshu! He’s in your department too. He’s insanely good!!!”
Sawamura apparently saw absolutely nothing wrong with their current position. He remained leaning forward while praise for Okumura poured endlessly from his mouth.
Miyuki listened patiently and silently summarized the information into three points.
One: Okumura had gone to the same high school as Sawamura and belonged to the same club.
Two: when Sawamura had been preparing for university entrance exams, Okumura had tutored him and successfully rescued his failing mathematics grade.
Three: their dorms were close together, and they often went out together.
“...But Okumura-senpai seems really busy lately. I haven’t seen him in forever, and whenever I go to his dorm he’s never there.” Sawamura sighed, his eyes unconsciously drifting toward Miyuki’s face. “If both of you tutored us, it’d be super effective! Okumura-senpai could teach me at this table, and Miyuki could teach Furuya at the table next to us. Hehe...”
Something tightened abruptly around his neck.
The next second, Miyuki had stood up.
His left hand landed on Sawamura’s shoulder while his right caught hold of the silver necklace, almost pulling Sawamura halfway across the table and into his arms.
“Let go, let go! This is really uncomfortable!”
The hands restraining him slowly loosened.
Sawamura retreated to the opposite side of the table and carefully straightened his collar.
“Why’d you suddenly get so mad?”
“I just think it’s a little frustrating that I met Sawamura-senpai so late,” Miyuki replied flatly.
For the rest of the tutoring session, Sawamura could clearly tell that Miyuki was in a bad mood.
Still, he explained every problem carefully and remained perfectly patient.
Sawamura thought and thought.
Finally, after watching Miyuki silently finish his Americano, he cautiously asked, “Do you... want a matching necklace with me? We could go back to that shop together sometime...”
“You already have a necklace. You don’t need to make another one just because of me.”
Sawamura frowned, making an indistinct humming noise under his breath.
Then suddenly, he slapped his hands together in realization.
“Oh! So you want to make some kind of accessory together with me! You should’ve just said so! That shop has really nice bracelets too. I’ve wanted one forever, but Okumura—”
“I’ll go with you. When?”
Sawamura stared at the suddenly serious Miyuki and answered reflexively.
“Tomorrow?”
“Okay.”
The atmosphere improved considerably after that.
Sawamura even got to watch Miyuki openly laugh at one of his proofs without a shred of mercy, but at least he was back to normal.
Therefore, the great Sawamura-sama magnanimously decided not to hold it against some insignificant Miyuki.
The next day.
“I’ll transfer Senpai the money for the bracelet along with the money for lunch today.”
“No need! Compared to how much you’ve helped us with Mathematical Analysis, the help Furuya and I give you with literary analysis really isn’t much. You can accept this with a clear conscience! Ahahahahaha!”
“All right, then. Thanks, Sawamura-senpai~ Oh... By the way, I noticed Senpai kept looking at those unfinished bracelet blanks before we left. Were you thinking of buying one for someone?”
“Hm... Sharp eyes! Yeah, I was thinking about it...”
“Okumura-senpai?”
“Nope. Furuya!”
Sawamura simply grabbed Miyuki and started pulling him back toward the shop.
“We made Furuya one of those necklaces too, so I can’t give him less just because we’re making bracelets this time!”
“Necklace... Furuya-senpai’s necklace was made by you and Okumura-senpai together?”
“Yeah! Furuya happened to be busy at the time. Whoa, now that you mention it, that’s exactly the same as today!”
The person Sawamura was pulling abruptly stopped.
Confused, Sawamura turned around.
The dim yellow streetlight fell softly across Sawamura-senpai’s handsome face.
To Miyuki, the confusion written there looked almost like a kind of innocent cruelty.
Sawamura-senpai’s words always sent his emotions rising and plunging like a roller coaster.
But it still wasn’t enough.
He wanted a little more.
And then more than that.
“...If I only wanted the two of us to have matching bracelets, would Sawamura-senpai agree?”
Miyuki’s voice had gone slightly hoarse.
He stared intently at every tiny change in Sawamura’s expression.
Sawamura frowned, looking even more confused.
Then he said—
————
“There’s always some moment when they think the other person has sent them a signal.”
Miyuki paused.
“A signal that says, I’m different from everyone else to them.”
His pen fell silent as he drew the curve into a smooth, elegant arc—
one that approached one hundred infinitely closely, but never actually reached it.
“And that changes the entire state of the relationship.”
“So reaching one hundred means the confession succeeds?” Sawamura’s eyes lit with understanding. “Oh... Okay, I get it! At this point, they’re in unrequited-love mode and they’re basically one step away from becoming lovers, right?”
“Mm-hm~ So, Sawamura-senpai. What are the maximum and supremum of this heart-flutter curve now?”
“If it gets infinitely close to one hundred without ever reaching it, then the maximum could be 99.9, 99.99, 99.999...” Sawamura’s face twisted in horror. “Aaaagh, that never ends!”
“Exactly. So in this case, a maximum doesn’t exist. But the supremum does—”
“Don’t tell me! Hnnngh... The upper bounds are... uh... all the numbers greater than or equal to one hundred? So the supremum is the least upper bound, which means... one hundred?”
“Completely correct~ Now does Sawamura-senpai understand the difference between a maximum and a supremum?”
“Miyuki Kazuya... you’re seriously amazing!!! But why do you understand shoujo manga better than Sawamura-sama when I only lent them to you, like, half a month ago?! Damn it!”
“Probably because I’m smarter than Sawamura-senpai~~”
“You bastard four-eyes! Don’t think you can disrespect your senpai just because you’re younger than meeeeeee!”
As they squabbled, the loose sleeves around their arms slipped downward.
Two similar silver bracelets were revealed beneath them, gleaming softly in the warm light.