Chapter 9:The Geometry of Integration

In the cold 5.am morning,”God this is the worst… Can we please take a… break…,” Kenji was doing push ups.His legs were feeling about to break but he felt that each training felt less painful.

“This is your warmup,now for today’s training,we’ll be doing physical exercises to build up endurance and fluid dynamics.”

“Fluid dynamics?” Kenji wheezed,his breath barely keeping him alive.”Rei,I'm a physics student,not a… professional swimmer!My muscular-skeletal system is currently screaming for a break!”

Rei stood over him,his breath visible in the frosty morning air.He wasn’t wearing his hoodie today,he’d left it with Hiyori and the cold seemed to have no effect on him.”In a fight,your body is like water,Kenji.If you’re not visioning yourself with the flow,your rhythm is not clear to you.If you vision yourself with the flow,your rhythm will be clearer.Now,get up. We’re moving to the riverbank.”

The Kiso River was a grey and silver reflection at this hour. Rei led Kenji to a flat stretch of stones where the current was steady.

“Stand in the shallows,I don’t want you to drown,” Rei instructed.”I want you to make a stance and do multiple punching combinations a hundred times,each with different combinations but let’s start with Jab and Cross,Cross,Lead Hook.After you’re done with that,I want you to shadowbox,but please remember don’t fight the water.Feel the resistance of the current against your shins,let your body match the flow with the current.If you move too fast and too stiff,the water creates drag.You need to move with ‘Laminar Flow’ to minimize the distraction.”

Then Rei said, "Be like water,"

Kenji stepped into the icy cold water,his teeth chattering.”Laminar…Flow… right.If the Reynolds number stays low… the fluid particles move in smooth layers…” He threw a punch,but the current immediately threw him off balance.

“Don’t calculate it with your head,Kenji.Calculate it with your center of gravity,”Rei said,stepping into the water with him. Rei stood in a stance,he let his mind in peace and feel the flow with his body.With a sudden flow,Rei moved his hands slicing through the surface with barely a ripple.It was a terrifying display of self-control.”Yesterday,I told Hiyori the truth.About our former school.About my past.”

Kenji stopped mid-punch,the water swirling around his knees.”The whole truth?Even about the Shadow of Reign?”

“Everything,” Rei muttered,looking at the moving water that reflected back to Rei.”She didn’t change her entire thought about me.She didn’t even flinch at the truth.She just… looked at me.”

“That’s Hiyori for you,” Kenji said,finally catching his breath.He adjusted his fogged-up glasses.”Most people see the monster behind Rank 2 because that's the high-contrast version.She sees the resolution behind it.”

Rei nodded slowly,then his face returned to its neutral mask.”Less talking,focus on the training. If you can finish this training without any issues,I'll pay for breakfast.”

The school day brought no different.As Rei walked through the hallways of Inuyama Academy,the usual “Red Sea” effect where students parted in fear seemed to be changing. Rei noticed some students greeted him.

He arrived at the Class 2B room and found his desk was covered in architectural sketches.Hiyori was already there,but she wasn’t alone. Mio was leaning over the desk,her analytic eyes scanning the structural plans for the festival.

"The torsion on these joints is a mess," Mio stated, tapping a wooden ruler against the paper. "If we use the traditional interlocking method, it’ll hold. If we use nails, the cedar will split under the humidity."

She looked up as Rei approached. She didn't glare this time; she just gave him a sharp, evaluative nod. "I checked the wood shipment, Koga. It’s high quality. You and the 'Data Nerd' did good work selecting it.”

“He’s Kenji,not ‘Data Nerd’” Rei sighed.

Hiyori looked up,beaming.She was wearing her own school blazer today,”By the way,your hoodie is right over there.” she said while pointing to her desk.On top of her desk was a neatly folded charcoal hoodie.”Rei! Mio was just saying we could actually finish the gate by the end of the week if we start the ‘Integration’ phase today.”

“Integration?” Rei questioned,sitting down.

“The whole class is joining in,” Hiyori explained excitedly.” Since you and Mio proved the structure is safe,the others aren’t as scared to help out.Look!”

Rei looked around the room.A group of students was painting the banners,and the class rep,Sato,was actually walking toward Rei with a box of brackets.He showed visible trembling but not too much,he didn’t run either.

“K-Koga?” Sato stammered. “Could you.. help us align the crossbeam? We can’t lift it high enough to check the level.”

Rei looked at Hiyori. She gave him a tiny,encouraging thumbs-up.

Rei stood up,his presence made Sato gulp,but Rei kept his movements slow and deliberate.”Show me the beam,” he said with a friendly smile,a tip from Hiyori to help Reizen to make people calm.

As Rei walked toward the back of the room to help,Mio leaned in close to Hiyori,whispering,”He’s actually doing it.He’s trying to look normal now.”

“He’s not trying,Mio,” Hiyori whispered back,picking her camera.”He’s just remembering his new purpose now.”

The sun was beating down on the Inuyama Academy track,the heat blazing off the red clay in waves. For Kenji,physical education was usually a scheduled period of public humiliation,but today,something was different.

“Alright,everyone on the line!” The couch barked. “Hundred-meter sprint.Let's see some hustle!”

Kenji found himself lined up against the school’s star athlete from soccer club and a track-and-field regular.Usually,he’d be nervous and exhausted from 3 seconds after running,but he lowered himself into a crouch,his mind didn’t go to his fears.It went back to his training.

“Laminar Flow” Rei’s voice echoed in his head,”Minimize the turbulence”.

“Runners… set!”

Kenji exhaled,letting his lungs sit at that “Still Point” Rei had described.He wasn’t thinking about the finish line,he was adapting and feeling the flow of the wind.

BANG

The starter pistol fired.

To the onlookers,Kenji didn’t just start running,he seemed to vanish into the air within 1 second.He felt the air friction against his skin,but instead of fighting it,he tilted his body by seven degrees,slicing through the atmosphere like a needle.

“I see it,” Kenji thought,his eyes wide behind his spectacles.The frame data of the world!I’m doing it! I’m moving at the speed of subsonic friction!”

In Kenji’s “Flow State,” he imagined he was disrupting the sound wave in his wake,his feet barely touching the ground as he achieved what he believed was “Subsonic Friction”.He felt like a god of kinetic energy,another potential Rank 2 in a tracksuit.

In reality,he wasn’t disrupting or breaking the sound barrier,but he was absolutely torching the competition.His legs,conditioned by thousands of dawn step-ups and flow style shadowboxing,moved with an efficiency the other students lacked.

“What the–?!” The soccer player gasped,watching a blur figure with glasses overtook his place.

Kenji crossed the finish line a full three seconds ahead of the all students,his momentum so great that he fell to the sandpit at the end.

He sat up,sand in his hair,his chest huffing down and up.”Did you see that,Rei?!” He shouted at the school. “I moved at Mach 0.5! The displacement was… it was statistically impossible!”

The couch stared at his stopwatch,then back at Kenji.”Sato? I’ve never seen you that happy in this class nor even finished the race.”

From the sidelines,Rei and Mio stood near the fence.Rei had been watching with a neutral expression,but a tiny glint of pride was in his eyes.It reminded Rei back in his past,how he taught his juniors in his former dojo,how his juniors eyes lit up after they have won their first match.

“He thinks he reached your level,” Mio said leaning against the fence with her arms crossed.”He’s completely convinced he’s at a level same as you”

“He’s fast,”Mio admitted,watching Kenji try to explain the physics of his sprint to a group of surprised classmates.”But look at him.He’s so delusional right now,he’s probably going to try and break a wall next.”

“Let him have it,” Rei muttered,turning to walk toward the next class. “He earned the ‘Apex’ feeling for today.”

The final support beam was hammered into the place with a satisfying,solid thud.The “Hidden Inuyama” gateway stood tall in the school courtyard,a masterpiece of traditional and modern structural integrity.Even the skeptical interior design club had stopped by to admire its marvel at the seamless fit of the cedar logs,a result of Rei’s strength and Kenji’s obsessive physics knowledge.

“It’s perfect,” Hiyori whispered,stepping back to capture the structure through her lens.The sun setting caused the wood,making it glow like amber. “We actually did it.”

"Physics doesn't lie," Kenji proclaimed, though he was currently leaning against a pile of leftover sawdust, looking like he might fall asleep standing up. "Mass plus labor equals... a very tired Kenji."

Mio wiped sweat from her brow with her forearm, looking at Rei with a rare, genuine smile of approval. "Not bad, Koga. It might actually survive the festival without collapsing on the guests. I’m heading to the dojo for a late session. Don't let the 'Data Nerd' or whoever he is,sleep in the dirt."

After Kenji had been poured into a bus home, Rei and Hiyori found themselves alone at the school gates. The air was filled with the scent of wood from the structure and the quiet hum of the town settling into evening.

“I’ll walk with you,” Rei said.It wasn’t a question,just a low ,protective statement.

“You don’t have to,but if you really want to,thank you Rei,” Hiyori replied with a slight red on her cheek.

They walked in a comfortable silence that felt light,a contrast to the heavy pressure that Rei was used to in his former school.Hiyori occasionally stopped to take a photo of a stray cat or the way streetlamps flickered to life. Rei watched her,realizing that for the first time,he wasn’t predicting an ambush.He was just… walking.

Hiyori grabbed a cat,”Here! Stroke its head.” Hiyori carrying the cat to Rei.

“Not a big fan of cute things,” which made Hiyori disappointed but she still persists to let him carry the cat.

“Fine,” Rei replied with a defeated tone.He carried the cat in which the cat’s paw touched Rei's cheek.He felt a soft and warm paw touching his face,he didn’t instinctively react to the cat’s paw nor did he have to.

“Maybe,I do like cute things after all,” Rei said after gently putting down the cat.

“Maybe you do but I'm mad that you didn’t stroke its head,” Hiyori responded ,her face pouting.

When they reached the small,ivy-covered gate of her house.Hiyori stopped and turned to him.She looked up at him,her face framed by the lingering warmth of the evening.

“Rei,” she started,then hesitated.her fingers playing with the strap of her camera. “You know the ‘Still Point’ we talked about… I think we found it today.”

Rei looked at the house,then back at her.”The friction is lower here,” he admitted.

Hiyori reached into her bag and pulled out her phone. She tapped the screen a few times and then held it out to him. "The festival starts on Monday. We’re going to need to coordinate... you know, for 'structural emergencies' and booth shifts. And maybe... just to talk."

Rei took his own phone out—a basic, rugged model that looked like it could survive a drop from a building. He took her phone and carefully typed in his digits, his large, scarred thumbs moving with surprising precision. When he handed it back, Hiyori’s fingers brushed against his, and for a second, his world felt electric.

"There," she said, her cheeks turning a soft pink as she looked at the new contact name: Rei Koga. "Now you can't disappear into the library whenever you feel like a ghost. I have a way to pull you back."

Rei felt a strange, fluttering sensation in his chest—a biological variable he couldn't quite calculate. "I'm not going anywhere, Hiyori."

"Good," she smiled, stepping back toward her door. "Goodnight, Rei.”

As she disappeared inside,Rei stood at the gate for a moment.He looked down at his phone.A new message had already arrived.

Hiyori:[Image Attached]

It was a photo from earlier,Rei carrying the cat,looking like someone who adores cats the most.

Rei pocketed the phone,a small,genuine smile finally breaking his neutral mask.He turned and started the walk back to the Sato household,his footsteps light on the pavement.

Back at the Hiyori’s household,Hiyori was drying her hair with a hair dryer.Suddenly,a message had arrived

Rei Koga:Hey,I know it sounds weird but are you free for tomorrow because you want to go out tomorrow?

Hiyori’s expression was shocked,she turned off her hair dryer.”Ekkkk! Rei asked me out!” Hiyori thought to herself,blushing.

Hiyori:Sure! What time do we meet up and where do we meet up?

Rei Koga:After I'm done with Kenji’s training,at the town centre.

Hiyori:Okay!

Author's note:

✍️ Fun Fact:

Rei telling Kenji to "be like water," is a reference to a famous martial artist's quote "Be like water, my friend" by Bruce Lee.The quote meaning to empty the mind, be formless, and flow with circumstances, like water, allowing you to be fluid, resilient, and powerful, whether flowing gently or crashing through obstacles.