The gray-haired girl silently returned to the door of Ms. Tsukuyomi Komoe's apartment. She didn't knock, but simply touched the lock with her finger. The old door slid open a crack as if turned by an invisible key, making a soft "click" sound.

She pushed the door open and went in, her movements so quiet that she made no sound.

The room was almost unchanged from when she left. The air was stuffy, the old fan whirred monotonously, and two figures on the tatami mat were struggling with some kind of "disaster."

Kamijou Touma was lying on the low table, with a thick notebook open in front of him. The mechanical pencil in his right hand was already dented from being chewed, while his left hand was painfully scratching his spiky hair.

The draft paper in front of him was covered with messy calculations, but most of them had been roughly crossed out.

"Damn it... Why are there two answers to this summer homework?! Does the teacher have a split personality?!" Touma Kamijou let out a desperate wail, slamming his head heavily on the table with a dull thud.

Opposite him, Index had changed back into her pure white monastic robe, which had been re-secured with safety pins and was still a bit drafty but at least wearable.

She knelt, her bright green eyes fixed on a piece of writing paper, looking at its contents with a puzzled expression. In her other hand, she held a dictionary, flipping through the pages.

"Touma... what does this circle mean?" Index showed the paper, which was covered with Touma's scribbles, written in a frantic manner as he rushed to finish his homework.

"Yes, yes, Sister, you're right..." Touma Kamijou replied weakly, staring blankly at the ceiling. "But I'm just an ordinary high school student who wants to get through summer vacation. Why do I have to endure this kind of literary criticism..."

The gray-haired girl watched this scene quietly. Sunlight streamed through the gaps in the curtains, falling precisely on Touma's frantic face and Index's focused profile, while tiny dust particles floated in the air. This atmosphere of bickering over trivial matters, yet strangely harmonious, was a comforting place for her.

She walked to the low table and sat down in the empty seat between the two of them. Her movements were so natural, as if she had always been there.

Touma Kamijou and Index both looked up at her at the same time.

"You're back?" Touma Kamijou breathed a sigh of relief. "You suddenly went out just now and startled me. Are you alright?"

The girl shook her head, her gaze falling on the math problem set covered in corrections in front of Kamijou. "Need any help?"

"Help?" Touma's eyes lit up, but then dimmed again. "You've only just learned Japanese...can you solve math problems?"

The girl didn't answer, but simply reached out and took the textbook from Touma Kamijou, flipping through it from beginning to end. Then Touma witnessed a scene he would never forget: the gray-haired girl immediately began working on the problems. Her handwriting was initially a little clumsy, but quickly became neat and fluent. Without complicated calculations, she directly wrote down the answers to each problem, along with the key solution steps, concise and clear.

For that factorization problem that had driven Touma crazy, she even wrote tiny notes on the side explaining the applicable conditions of the solution and the possible traps the question setter might have set. In less than five minutes, she pushed the problem set back to Touma. The blank spaces in the second half of the workbook were filled with clear and accurate answers and key points.

Touma Kamijou's eyes widened as he picked up the notebook and flipped through it, his mouth agape as if he could fit an egg in it. "This...how is this possible?! You've never even attended school!"

The gray-haired girl tilted her head, seemingly not understanding why they were so surprised. "The solutions to the problems are already in the book... and the people introduced have already condensed the math into formulas, so you can just call them up when you need them..."

To her, it seemed as natural as breathing. The rules formed by those numbers and symbols were clearly discernible in her eyes, and deriving the answer was simply a matter of following those rules.

"A pattern..." Touma Kamijou looked at the neat answer, then at his own scribbles, and slumped back onto the table, utterly dejected. "Unfortunate... Is my IQ really lower than that of a girl lacking common sense...? Touma-san is a high school student..."

Index approached Touma Kamijou, interlacing her fingers, seemingly radiating a holy light. "Touma, don't be discouraged! Everyone has strengths and weaknesses! Look, even though your Japanese homework isn't great, at least your handwriting is much better than mine was when I first started learning Japanese!"

This consolation only made it more heartbreaking.

With the help of the gray-haired girl, the math homework was cleared... uh, finished at an astonishing speed. Kamijou Touma, meanwhile, lamented as he corrected it. Time slipped away amidst the scratching of pens and Touma's occasional screams, creating an atmosphere that was both lazy and frantic, typical of summer vacation.

Around noon, the sound of a key turning came from the entryway, followed by the rustling of a plastic bag and light footsteps.

"I'm back~" Ms. Tsukuyomi Komoe squeezed through the door, carrying several full shopping bags. Her short pink hair was slightly damp with sweat, clinging to her forehead, but a satisfied smile graced her face. "Oh dear, it's so hot... I bought beef, eggs, vegetables, and watermelon! I'll make you all a nutritious feast today!"

Her gaze swept across the room: Touma Kamijou was frowning as he looked at his revised book report, while Index was checking his English homework, which was relevant to his major.

The gray-haired girl sat quietly to one side, looking at a beautifully illustrated botanical guide that she had taken from the teacher's bookshelf. Her homework was piled up to the side; she had miraculously finished her math assignment.

Teacher Xiaomeng's eyes curved into crescent moons. "Looks like everyone's been working really hard! How's your homework going?"

"Thanks to her, I've sorted out math..." Touma Kamijou said listlessly, pointing at the gray-haired girl. "But the rest... are still lingering in hell."

"Very good, very good! Let's rest first, eating is the most important thing!" Teacher Xiaomeng energetically carried the bag into the narrow space that doubled as a kitchen and started getting busy. The sounds of washing vegetables, chopping meat, and beating eggs rang out, and soon, an enticing aroma filled the entire room.

To Touma's surprise, Komoe-sensei didn't press him about the promise she made that morning to "explain things." She simply continued cooking as usual, humming a song, occasionally peeking out to ask, "Do you like your beef tender or well-done?" or "Can Miss Index handle spicy food?" It was as if last night's magical ritual, the wound on her back, and the mysterious pursuit were merely insignificant interludes in the summer, already turned over a new page.

This "normal" attitude actually helped Touma Kamijou relax a little; perhaps the teacher was expressing her trust in this way.

Lunch was plentiful, with tender beef over rice, golden omelets, refreshing vegetable salad, and sliced, chilled watermelon.

The four sat around a low table, enjoying a simple yet heartfelt meal in the warm breeze from an old electric fan. Index ate with particular gusto, devouring almost the entire table, her eyes narrowed, temporarily forgetting her pain and worries.

The gray-haired girl carefully observed everyone's actions, then imitated them, using chopsticks to put food into her mouth. A faint glimmer occasionally flashed in her gray eyes.

After the meal, while clearing away the dishes, Teacher Xiaomeng suddenly spoke up as if she had just remembered something: "Oh, right, the water heater in the bathroom of this apartment building seems to be broken. The building manager said it won't be fixed until tomorrow. In this hot weather, we can't go without taking a shower. I know a nice public bathhouse nearby; it's cheap and the water is hot. Do you want to go together? It'll be like a post-meal walk to digest."

A public bathhouse? Touma Kamijou instantly conjured up an image of a traditional Japanese bathhouse: a huge tub, billowing steam, and… separate entrances for men and women. He glanced at Index, then at the gray-haired girl. Taking these two girls to a public bathhouse? The thought itself was… a bit…

"Great!" Index responded enthusiastically, her bright green eyes sparkling. "I've never been to a place like this where everyone bathes together in England! The books say it's a very important cultural experience in Japan!"

"...Okay." Touma Kamijou rubbed his forehead, a bad feeling creeping in, but it was probably just his imagination.

As evening approached, the summer heat subsided slightly. Touma Kamijou, Index, and the gray-haired girl, carrying towels and toiletries prepared by Ms. Komoe, walked out of the apartment.

Index seemed to be full of anticipation for the public bathhouse. She walked briskly beside Kamijou, carrying a washbasin, and even hummed a strange song that she had heard somewhere.

"Hmph hmph hmph~ Hmph hmph hmph~"

"So you care about taking a bath that much? What does it matter if you smell?" Touma Kamijou teased casually, trying to dispel the awkward feeling in his heart.

"You like the smell of sweat?" Index turned her head and looked at him with an innocent yet slightly provocative gaze.

That's not what I meant!

"Touma, Touma." Index suddenly strode up to Touma, turned around, and continued to call softly with a smile, "Touma, Touma."

"What's wrong?"

"It's nothing, I just wanted to call you..."

Index thought for a moment, then asked him in a muffled voice, "In Japanese public bathhouses, do you get coffee with milk? Like a cappuccino?"

"Public bathhouses aren't that fancy!" Touma Kamijou said, half-laughing and half-crying. "But such a big bathhouse must have surprised you, right? Most bathhouses in England are like those tiny bathtubs in business hotels, aren't they?"

"Hmm!" Index nodded vigorously, then her eyes became longing as she thought of something. "But, going to the bathhouse with... friends seems like a lot of fun." As she said this, she secretly glanced at the gray-haired girl walking quietly to the side.

The girl seemed to have received her gaze, looked back at her, and nodded slightly.

Touma scratched his head and continued walking at the very back. Watching the two girls walk together from behind, he felt unusually relaxed. He gazed at the night sky, where the stars were obscured by the lamplight, and then thought about the bathroom he was about to go to.

"Ah...did Index buy a child's ticket?" Touma Kamijou suddenly remembered this question and asked it instinctively. He immediately realized something was wrong.

"Touma..." Index stopped in her tracks, lowering the plastic basin she had been holding high. She exuded a powerful aura of pressure.

"Ah, sorry..." Touma Kamijou quickly apologized, while preparing to explain his meaning. "You misunderstood, Miss Index? I meant that if you go in with that white-haired girl, they might mistake you for a child... or something?"

"Touma...!" Index lunged at Touma Kamijou, attacking his hair with her teeth. Then, she actually grabbed a washbasin, turned around, and strode quickly toward the bathhouse that Komoe-sensei had indicated in her memory, her long silver hair swinging behind her, clearly showing that she was really angry.

"Hey! Index! Wait!" Touma Kamijou hurriedly tried to chase after her, but Index turned around and glared at him, her eyes clearly saying, "Don't follow me!" and then walked even faster.

He sighed, slowed his pace, and decided to let Index calm down first.

Just then, the gray-haired girl who had been quietly following looked at Index, who was walking ahead in a huff, and then at Kamijou Touma, who was looking dejected.

Then, without saying a word, she simply started walking and followed Index, walking alongside her. She even reached out and gently tugged at Index's sleeve.

Index paused for a moment, then turned to see it was her. Her angry expression softened slightly, but her lips were still pouting. The gray-haired girl didn't say anything, just walked quietly beside her.

The two girls' figures gradually faded into the distance, disappearing around the street corner.

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