Chapter 1: Chalk and Shadow 粉筆と影

  Numbers, the bedrock of truth and the foundation of civilizations. Numbers support claims. The grades that define a future, the investments that secure a life, the numerical representation in a researcher's paper to a politician's promise. Numbers can also be deceptive.

  Consider Fermat's Little Theorem: if p is prime, then

  a^(p?1)≡1(mod p)

  However, the reverse is a dangerous assumption. Assuming any ?? that passes this test is prime,

  a^(?n?1) ≡1(mod n)

  We invite composite impostors. Fermat pseudoprimes, these elegant lies, slip past our simplest checks.

  Even perfect logic, when wielded carelessly, becomes the ideal weapon for betrayal.

  My shoes clicked softly against Yūgen Academy’s stone path as I indulged myself with historic and modern architecture in the campus. Polished Hinoki wood panels structure in contrast with the cutting-edge modern buildings, blending timeless craftsmanship and innovative design, a harmonious blend of past and future.

  As the afternoon bell tolled from a nearby temple, its deep, resonant chime echoed gently through the Kyoto spring air, gently pushing me forward.

  I inhaled deeply, savoring the freshness of spring and silence. As I slowly approached the teacher’s room. Still captivated by the echoes of history in every corner, a familiar elegant and tall figure stepped from the arched corridor ahead – Hoshino-sensei.

  Close call there…I was almost late

  Her signature pixie hair-cut framed beautiful features. A tiny beauty mark beneath her lip, and frameless glasses giving her a disciplined charm. The lab coat was a modern counterpoint to her traditional grace. On her left wrist, a fashionable bracelet glinted subtly in the light.

  “Welcome to the Yūgen Academy Kanzaki”

  The unforgettable gentle tone, her hand rested on my shoulder.

  “How was your first week at school, getting used to everything ?”

  “Yea, thank you very much, Hoshino-sensei. I’m honored to be here, especially with your recommendation and scholarship letter."

  Shaking her hand with a grateful smile.

  “The classes and materials are exceptional, library resources are plentiful. Never a dull moment…”

  Admission was fiercely competitive, with acceptance rates around 3-5%. The meticulous selection process that favored all the qualities, academic excellence, refinement, legacy, leadership and social grace. Scholarships were rare and often symbolic. I had assumed that such an environment would naturally foster excellence in all who walked its halls…

  Well, most students I have met come from very privileged and affluent backgrounds, some of them display dismissive attitudes and of course, there is a culture of subtle toxicity and arrogance. It's Yūgen Academy afterall.

  I did something absurd at the wine tasting club a few days ago. I started listening to the wine(non-alcoholic). Yeah, I held my glass up to my ear and said, “I think it’s whispering notes of blackberries and mystery.” To my surprise, people actually followed and nodded like I’d just uncovered the secret language of grapes.

  After that, I realized you don’t need to know anything about things, just say something confidently weird, and people will follow you.

  “Oh, drop the formality please”

  She grinned sharply as if saw through my mind.

  “The academy will provide networking opportunities and exposure to influential social circles”

  “Learning to navigate in these environments can build valuable social skills and confidence that benefit your future academic and career ahead, Kanzaki. Just don’t overdo it, rest is important too, alright?” Her smile was warm.

  We strolled through the hallways, where delicate calligraphy decorated the walls. Her lab coat fluttered freely as she excitedly showed me around. The library is packed with ancient books, research materials, a dojo filled with disciplined precision and laboratories where students fully immerse themselves in programming.

  Passing through the courtyard, I noticed a student hunched over a Math textbook, fingers raking through his hair in frustration. While nearby, two others exchanged worried whispers about upcoming exams and family expectations. The air filled with a tension that contrasted sharply with the peaceful cherry blossoms outside the zen garden.

  As we continued, she spoke proudly of the Math Research Club’s commitment to excellence and its unique role.

  “What is it about the Math Research Club that sets it apart from the rest?”, curiosity stirring.

  Her eyes gleamed with amusement, framed by a knowing, cheeky smile.

  “I would say It’s a haven for intellectual exploration and discovery”

  “In the clubroom, students engage in mathematics competitions at both national and international levels, while also undertaking research that spans diverse fields of mathematics. Kisaragi Akari is the current president”

  I see, basically a room full of anti-socials

  “She’s been through a lot… but she has a gentle and kind heart. A medalist in and an assistant coach who pours herself into this club. As per our agreement, you will be joining the Math Research Club, and I will provide the necessary documents for your enrollment”

  “We can discuss the finer details of the arrangement at a later time. I believe that with your talents and passion, you will illuminate the club and breathe new life into its spirit. And of course, you’ll excel here”

  She added, “students are also encouraged to ask even the wildest STEM questions, and develop problem solving skills. We host a tutorial section everyday after school for assignment and lab assistance. Sounds fun right?”

  She ended with a sly smile.

  Kisaragi Akari had long been hailed as Japan’s prodigy, her name appearing in news features and math training camps. Confident and articulate, her eyes seemed to glow whenever she discussed complex mathematical theories.

  The Kisaragi family name was renowned throughout Kyoto and beyond, synonymous with influence, tradition, and excellence. She was undeniably a striking beauty, someone who naturally drew attention without even trying.

  Though I had never truly interacted with her, whispers of rumors followed her everywhere, through the halls of the school and beyond. Scandal or truth, I cared little for the gossip.

  “Sounds great, this is nothing short of a sanctuary for a devoted math believer like me.” I said with a grin.

  Yes, I’m using Leonardo Da Vinci as my phone wallpaper

  “New energy into the club you say?”

  “I am sure that the essence of my words will eventually unfold before your eyes.”

  “Kanzaki-kun, I also hope to gain your valued perspective as well.”

  “C’mon, I’m just a nobody from an old-town. Without your guidance, I might have ended up lost in the mire,” I laughed

  “Hey, I am the early investor type, and I see a gem waiting to shine”

  Her praise flustered me and her beauty caught me off guard.

  Our walk ended at a heavy wooden door, its dark grain worn smooth by years of eager hands.

  “Anyway, this is the clubroom, Kanzaki.” Hoshino-sensei said with a playful wave. “Get settled, I’ll join you shortly, ciao !”

  I pushed it open, stepping into a stillness that contrasted the academy’s lively pulse outside the building, broken only by the soft scritch-scratch of chalk on the blackboard.

  —

  The late afternoon sunlight poured gently through the towering glass windows of the clubroom, casting a warm, golden glow that shimmered across the desks where four figures sat in quiet anticipation. Their notebooks lay open, pencils poised delicately above blank pages.

  I slipped into a seat near the back, my gaze drifting past the desks to a well-organized filled trophy case and archive at the back. Faded photos showed two girls clutching awards. Hoshino-sensei beaming beside them. And vast amounts of prestigious international gold awards, gleamed with Kisaragi’s name alone, like a private podium. Though, none of them were team-based tournaments.

  Oh my gosh, more shiny gold trophies? Seriously, another Kinkaku-ji trying to steal the spotlight. I’m pretty sure the math would keep the tourist away though…

  IMO, AMGO, EGMO, AMO, JMO, Anything you can imagine.

  My gaze then settled on the slender figure at the front, Kisaragi Akari. Her silhouette stood sharp against the blackboard, where an integral sprawled like a silent hymn to mathematical elegance, its curves and a delicate dance of logic and beauty.

  Unauthorized tale usage: if you spot this story on Amazon, report the violation.

  She paused, brushing a lock of dark hair behind her ear with a gesture both precise and unhurried, then turned to face us. Her eyes, framed by long lashes, cut through the room, hovering behind a guarded wall.

  Kisaragi’s most striking feature was her long, jet-black hair, styled in low twin-tails, it framed her face perfectly. She exuded a cold and disciplined composure. The uniform was pristine, tailored to perfection. Carrying a disciplined presence but heavy, as if there’s an invisible weight, a quiet burden beneath her composed exterior.

  It’s been quite some time since the last training camp, but looking at Kisaragi now, it’s clear that she hasn’t changed much, at least not on the surface. She’s definitely grown a lot taller though, there’s a new distance in her eyes. She seems more detached, almost emotionless

  “Alright”

  Her voice is steady and resonant

  “This isn’t just about finding the solution. It’s about understanding why it works. Break it into pieces, see the structure first before you chase the answer.”

  “Consider the Gamma function. It extends the factorial, normally defined only for natural numbers, to the entire continuum of real and complex values, all through a single elegant integral. Much like this problem, it challenges us to grasp the deeper purpose behind the formula, not just the computation.”

  I leaned forward, intrigued.

  Ah, they’re working on Euler’s Gamma function, a brilliant extension of factorials to continuous values. It’s a powerful tool not just in pure math but also in physics and computer graphics. Truly, Euler’s genius continues to amaze. Hats off to the great Euler-sama!

  A first-year near the back, raised a tentative hand. “A general question, but what if we can’t figure it out?” he asked, his voice tinged with worry. “There are a lot of times where I don’t even know where to start?”

  Kisaragi’s usual composure softened, a flicker of kindness breaking through.

  “Confusion is often the first hurdle for all who seek understanding, it comes to us naturally. Please, do not let it overwhelm you.”

  “Step back, reassess the problem with fresh eyes. Do not start the problem by trying to solve the problem, start the problem by understanding it first. Diagram the problem, discern patterns, or consider particular cases.”

  Her tone remained measured and patient, though a subtle tremor betrayed the weight of her concern.

  The boy’s pencil sprang to life, scribbling eagerly, and I nodded to myself.

  Solid advice, small and specific cases are amazingly useful. One should never really run out of things to try, try to prove it and maybe fail, pick another approach or maybe try to prove a stronger result that you conjectured, find out that your conjecture was false, move on to another approach.

  A burst of energy came in the form of a girl sidled up beside me. With a conspiratorial whisper, she introduced herself with a smile.

  “Hey! I’m Uchida Minami, a member of the club. We have a rare guest here ! Are you here with a math question, or just seeking inspiration?”

  Flashing what I hoped was a sincere smile

  “Pleased to meet you, I am Kanzaki Seiya. I have been recruited as a member of the math research club and I’m here both to learn and to contribute” returning her smile.

  “Ohh! Kanzaki-kun ! Welcome aboard ! Hoshino-sensei raved about you. We’re excited to have you! ”

  “Just a guy who would rather wrestle equations than people,” I shrugged.

  “Numbers don’t judge and punch back...”

  Uchida looked surprised and raised an eyebrow with a smile. “That’s the spirit, we’re happy to have you here ! less punching, more scribbling.”

  Kisaragi stepped to the board, her writing unfolding with a meticulous grace, each stroke of chalk deliberate, each answer boxed precisely with some unique quirky hand-writings, QED, boxed proofs, a signature as unique as her eyes. I couldn’t help but mimic her hand-writings on my notepad. She continues to explain the importance of gamma function and the properties of it.

  She continued “Let’s briefly apply what we’ve learned with a quick exercise”

  “Prove that (32) = 2”

  The noise of pencil scribbling and the noise of the chalk echoing the room. Some puzzled, others stared blankly, drained.

  Math books lied about happy kids, didn’t they?

  I tilted my head, recognizing the Gamma function’s form.

  Integration by parts would work. Should I say something? Nah, let them wrestle with it a bit. It’s a rather straightforward problem.

  As I glanced around the room, our eyes met, and suddenly the noise around us seemed to fade. It was as if time slowed, She approached, low twin-tail swaying like a pendulum, her faint smile warm yet measured.

  “Hello, It seems you have found the solution… ?”

  “Welcome to the club, pleased to meet you. I am Kisaragi Akari” She initiated the ice-break and the bow was perfect, nothing like the distant star I expected.

  Damn, she has perfected the art of greeting

  Every movement she makes is a study in graceful calculation, leaving me unable to fully relax, mindful of my own manners in her presence.

  I must admit, Kisaragi’s eyelashes are exquisitely long...

  My brain stalled and paralyzed, caught in her striking elegance, her aura has completely engulfed me.

  “Kanzaki Seiya,” I stammered, my bow a total mess.

  “Eee…Super excited to join, looking forward to learning from you.”

  “So,” I said, scrambling, “the club doesn’t compete much anymore. What happened?”

  Crap, I got a bit nervous and asked something stupid…

  Kisaragi’s hand tightened briefly around the chalk, a faint tremor passing through her fingers before she steadied it. Her gaze flickered, a distant shadow crossing her eyes. Meanwhile, Uchida looked shocked and paused.

  “Coaching is my focus now,” she said, her voice calm but measured, “helping others find their spark and sharing the joy of mathematics.” She turned slightly, as if to hide the quick breath she took to compose herself. She paused, then added softly, “What motivates you, Kanzaki-kun?”

  “I’m here for the puzzles, studies and contribution…I guess?”

  Uchida piped up beside me and followed up, “Yeah, we figured winning is overrated ! Teaching and helping others are more fulfilling…hahaha”

  “Good thing I’m great at losing spectacularly.”

  “Akari-chan is a great teacher !” Uchida added, glancing at Kisaragi, who stayed silent.

  The door creaked open, and Hoshino-sensei slipped in, a stack of papers cradled in her arms. Her sharp eyes lingered on Kisaragi a heartbeat longer than on the others, a crease forming as Kisaragi brushed hair aside with too-precise grace.

  “Oh, you seem particularly loquacious today, that’s wonderful ! How’s it going?”

  Her tone is gentle yet tentative, as if carefully testing the atmosphere around her student.

  “Hoshino sensei… Welcome back” Kisaragi said while bowing, voice relived.

  “Slow but steady, like an exponential curve steadily climbing its path.”

  Hoshino-sensei smiled. “Every summit starts with a step ! Great job everyone.”

  Kisaragi and Uchida moved among the desks, their voices low as they guided struggling students. I joined them, peering over a student’s notebook. His page was a mess of crossed-out integrals, his brow furrowed in defeat.

  “Try a substitution,” I whispered, pointing to the Gamma function problem.

  “Let’s call this part u, It simplifies the exponent, giving the problem a new shape that’s easier to handle.” His eyes lit up, and he scribbled furiously.

  Uchida caught my eye, grinning in awe. “Not bad Kanzaki kun. You’re so quick ! ”

  “Just happened to work on a similar problem before.”

  Kisaragi glanced over, with an interested look.

  Uchida nodded “Bet you want to use Akari’s Hagoromo chalk, Math god’s blessing.”

  “Hagoromo?” I laughed. “That’s worth more than my grade points.”

  These chalks, Hagoromo, are important cultural and religious artifacts. Why aren't they displayed in Daitoku-ji yet ? They deserve a special place in the treasure room.

  Really, one stroke, you’ll fall in love instantly. And regret that you’ve done so…

  Uchida winked, passing me the Hagoromo, math nerds’ gold. And I stepped to the board and started writing.

  Recall we want to find the value of a special function called the Gamma function at one and a half. This function is defined by an integral, a kind of infinite sum, where we multiply a power of a variable by an exponential decay and add up the results from zero to infinity.

  To solve this, we use a technique that’s like breaking the problem into parts, called integration by parts. Imagine you have two pieces: one that’s a bit complicated, and another that’s simpler to handle. We choose the complicated part to be the variable raised to the half power, and the simpler part to be the exponential decay.

  By applying the technique, the original integral splits into two terms. The first term involves evaluating the product of these two pieces at the boundaries, zero and infinity.

  Luckily, the term disappears because at zero, the variable raised to the half power is zero, and at infinity, the exponential decay shrinks to nothing faster than the variable grows.

  What remains is a simpler integral, very similar to the original but with the variable raised to a different power. Specifically, to the negative half power. This integral is actually a well-known value, it equals the square root of pi, a famous mathematical constant

  Putting it all together, the value of the Gamma function at one and a half turns out to be half the square root of pi.

  Uchida and the fellow members gave me an approving nod but Kisaragi is deep in thought.

  “Perhaps there is a neater way to express this,” her chalk gliding effortlessly.

  The Gamma function has a neat property called the recurrence relation, which is like a shortcut connecting the value of the function at one number to its value at a smaller number. Think of it as a stepping-down rule: to find the Gamma function at a certain point, you can use the value at a point just one less, multiplied by a simple factor.

  The Gamma at 1.5 equals (1.5 minus 1) times the Gamma at 0.5. Which you will get the square root of pi.

  A murmur swept through the room, mingling awe with quiet confusion. I took a step back, half-awed, half-challenged. She made it look like a dance, while I’d been hauling bricks. Kisaragi’s recurrence relation showcased her elegant, insightful approach. It was mathematically superior, on a different level entirely.

  “That’s... art,” I said. “The fact that the recurrence relation, also derived using integration by parts, lets you calculate Gamma values for non-integers recursively, it’s beautiful.”

  Her glance lingered, curiosity sparking in her eyes. “It’s nothing extraordinary, just the quiet work behind the scenes,” she said softly, revealing a crack in her armor.

  She turned math into something alive, almost graceful. My way worked, sure, but it was like digging a trench with a spoon. Kisaragi’s method ? A literal straight line cutting through the complexity.

  I couldn’t tell if I was more impressed or more determined to catch up, but either way, a fire of challenge sparked within me.

  Uchida looked confused “Recurrence relation derived using integration ? Huh ?”

  "Would you all like a full proof of the recurrence relation? As Kanzaki san suggested, it follows naturally from integration by parts and the evaluation of the boundary terms…”

  “Actually, I believe it’s a worthwhile exercise to leave for you all to explore, as the steps are trivial, I’ll leave it for you all…"

  Kisaragi took a measured sip from her water bottle, then made a small, practiced movement near her mouth. I didn’t look up, but the gesture was familiar enough.

  The quiet rustle of papers and soft footsteps drew my attention. Hoshino-sensei paused nearby, distributing some problem sets among the participants.

  “That’s the spirit, everyone. Well done,”

  “This club could use this fire, keep grinding.” Her gaze settled on me for a moment longer with a smile, as if silently weighing my place in her cryptic plan to revive the club.

  Hoshino-sensei definitely wants more Math-Chad

  —

  The late evening bell rang, shattering the room’s amusement. As the session drew to its close, the members gathered their belongings, their voices rising in a chorus of vitality that briefly dispelled the room’s stagnation.

  A few students approached Kisaragi, their faces bright with admiration.

  “Sorry for the intrusion Kisaragi senpai… can we take some selfies?” they asked shyly.

  She smiled warmly, nodding as they huddled around the phones, laughter bubbling up in the small group. For a moment, the room felt lighter, filled with youthful energy.

  As the laughter faded, Kisaragi turned to the board, her arm sweeping in broad, deliberate arcs to erase the day’s work.

  I looked at the trophy cabinet again, in one faded photo, Kisaragi’s eyes gleamed, unmistakable even in youth. But as I glanced at her now, I couldn’t shake the feeling, something in her stillness felt heavier than the room’s quiet.

  The way she’d flinched earlier, just for a split second, when I’d mentioned competitions… What was she carrying behind that perfect bow?

  Shifting my focus deeper into the archives, my eyes caught something unusual. A worn, red-stained book tucked away between volumes. It seemed out of place, almost hidden deliberately. My pulse quickened as I reached for it, fingers trembling slightly. Curiosity surged, and I carefully opened it.

  The pages were filled with dense mathematical proofs, all scrawled in vivid red ink that almost seemed to glow in the dim light. I noticed some symbols, strikingly familiar, matching Kisaragi’s signature handwriting.

  Is this a club quirky signature ?

  Beside certain proofs, names had been harshly crossed out, the original initials Y.Y. boldly replaced with S.H

  Before I could turn another page, a sudden voice shattered the silence.

  “Please don’t touch those without permission. What are you doing”

  Kisaragi’s voice was sharp, urgent, and tinged with something almost desperate.

  She appeared beside me like a shadow. Her hands trembled, and her eyes widened with a fear and anger swirling beneath her stoic exterior I hadn’t seen before. The room seemed to grow colder as her warning hung in the air. I quickly offered a sincere apology and carefully returned the notebook to its hidden place. Then, as silently as a shadow at dusk, she vanished from sight.

  What lay within those pages felt like a secret folded away, quietly waiting or quietly forgotten

  A moment later, a gentle tap landed on my head as Hoshino-sensei handed me a problem set.

  “I don’t remember Michelangelo ever sculpting a statue of Kanzaki from a block of marble.” she said with a knowing smile.

  “I would like to discuss our arrangement, perhaps over dinner later? Nothing formal, just a chat about… bringing back the club’s spark.”

  A soft light cutting through the chill lingering in my surroundings, but her eyes held a quiet intensity that hinted at deeper resolve.

  “Of course ! sure thing… I’m down for free food and big plans”

  Her chuckle was soft and confident. “Good. Bring that sharp mind of yours, meet me at the parking lot, we’ll head off campus for a bite”

  She stepped away and left the room, I couldn’t shake the feeling her ‘spark’ meant more than new trophies.

  As I gathered my things, I couldn’t shake the feeling I was stepping into something bigger than equations. A story of triumphs and fractures, etched in Kisaragi’s trembling walls and Hoshino-sensei’s quiet hopes. Her confidence outside was a mask; the real Kisaragi Akari was here, brilliant and broken.

  As I left the room and stepped beyond the threshold, the dark corridor stretched before me like a void. Just beyond the edge of my vision, I felt a static shadow linger — a cold, silent presence quietly watching from far behind.

  Swallowing the rising unease, I forced my feet forward, my heart pounding in the heavy silence that seemed to press in from all sides.