Alchemy Principles of Type-Moon World

Time slipped quietly, the only sound in the study being the scraping of pens across paper. Flatt's manuscripts filled the desk, scattered about in disorder, some even carelessly pushed to the floor. His fingers, stained with pencil dust, were slightly blackened, and beads of sweat formed on his forehead. Despite countless attempts at construction, his mind remained trapped, as if in some closed space, unable to break through an invisible barrier.

Grothendieck stood in a corner of the study, his hands behind his back, quietly gazing at Flatt, as if waiting for a critical moment, or perhaps simply observing the child's way of thinking. His eyes contained no urging or reproach, but rather a patient scrutiny.

Outside the window, the sky shifted quietly. The faint light of dawn had long since faded, and bright sunlight streamed in through the gaps in the thick curtains of the study, illuminating Flatt's figure, deep in thought, with an even more exhausted look. He was unaware of the passage of time, nor did he notice when Grothendieck had left. Only when he realized he had run out of paper did he finally look up and blink his dry eyes.

Just then, there were soft footsteps at the door. Grothendieck entered, carrying a tray containing a cup of warm milk and a few bananas. He placed the tray next to Flatt and said calmly, "Take a break and sort out your thoughts."

Flatt paused for a moment, seemingly lost in thought. He hesitantly raised his head, then, upon seeing the food, seemed to suddenly remember that he was indeed a little hungry. Without saying anything, he simply picked up a banana, peeled it, and took a big bite. The sweet, soft flesh instantly filled his mouth, alleviating his hunger and restoring a bit of clarity. He picked up the milk again and drank half of it in one gulp.

Meanwhile, Grothendieck casually flipped through Flatt's scattered manuscripts on the table. His fingers slid slowly across the pages, carefully scanning the sketches. Soon, he went to the bookshelf and pulled out Flatt's previous homework and class tests, comparing and analyzing them one by one.

Grothendieck frowned slightly, turned a sheet of paper over, and tapped it lightly with his fingers. Flatt's calculations and experiments remained almost entirely at the geometric level. He always circled within a three-dimensional framework. Even when he tried to conceive of a fourth dimension, it was merely a deformation of the three-dimensional structure, not a transcendental expansion.

It's not that he can't understand the concept of the fourth dimension, but that his way of thinking relies too much on intuition and lacks something core.

Grothendieck's brow slowly relaxed, and the corners of his mouth twitched slightly, as if he had finally figured something out. He placed his notebook back on the table and sighed softly: "...Ah, I see."

He looked down at Flatt.

The atmosphere in the study shifted quietly. Light from outside filtered through the heavy curtains, casting dappled shadows across the old bookshelves and scattered papers. The air was filled with the faint scent of paper, mixed with the slightly bitter aroma of ink.

He waited quietly for Flatt to finish his plate of food, observing his face, which had just been wrinkled with deep thought, now slightly relaxed by the filling of food. Taking advantage of this moment of silence, Grothendieck casually tidied the table, folded up the old manuscript paper that Flatt had thrown aside, then pulled a new stack of manuscript paper from the corner of the table and pushed it in front of Flatt.

"Do you know the coordinate system?" he asked suddenly, his tone still calm, as if it was just a casual question.

Flatt was busy licking the milk stains off his fingers. He paused for a moment, his eyes wandering. "Ah, I've heard of it... I think it's something I won't learn until next month..." He scratched his head. "I have a rough idea of what it is..."

Grothendieck did not continue to ask questions, but casually pulled out a piece of paper from the pile of drafts on the table, picked up a pencil, and neatly drew a horizontal line and a vertical line perpendicular to it on it, and marked a small "0" at the intersection of the lines.

“Draw a straight line that intersects the vertical line at right angles, and mark the distance and the origin on it,” he said. “This is the most basic coordinate system.”

Flatt blinked and watched Grothendieck writing and drawing fluently on the paper. He couldn't help but straighten his back and his eyes became more serious.

“In this coordinate system, the position of any point can be represented by a pair of numbers,” Grothendieck said, marking a few points on the coordinate system and writing down the corresponding values next to them. “Once the points are connected to form a line, the equation of the line can be expressed using variables.”

He casually drew a slanted straight line and wrote down a simple straight line equation next to it. The whole action was neat and quick, as if all this was just the most basic preparation.

"Isn't this similar to the number line?" Flatt tilted his head and looked at the coordinate points on the line, trying to connect what he had learned with the concept in front of him.

“Good,” Grothendieck nodded calmly. “Next is the three-dimensional coordinate system.”

He added an upward slanting line to the two-dimensional coordinate system, making it the Z axis in space, thus constructing a three-dimensional coordinate system. "It's very simple. By adding another direction to the existing one, you can represent any point in space."

Flatt stared at the three-dimensional coordinate system, his eyes gleaming with excitement. His mind was already extremely sharp, and after hearing such a straightforward explanation, he grasped the concept almost instantly. He tapped his fingers lightly on the table, his mind beginning to ponder the magic cube problem.

"So...what about the fourth dimension?" He blinked and asked tentatively.

Grothendieck's mouth curled up slightly, as if he was satisfied with the question. He did not answer directly, but said slowly: "That's about it."

He casually placed his pen on the paper and looked at Flatt sharply, "You can try to use a similar approach to solve the magic square problem just now."

"--The prompt ends here." He pushed the manuscript on the table in front of Flatt, and his tone returned to his usual coldness. "Continue."

Flatt's eyes suddenly lit up, and his fatigue was swept away. He became excited in an instant, clenching his fists, as if he had found the key clue to solve the puzzle. An undisguised smile appeared on his face.

"So it's a strategy guide..." he said excitedly, his lips curled up, "Let's continue defeating the hidden boss!"

After learning only the most basic coordinate system, we have to face advanced research topics directly——

To be more precise, it was like asking Flatt, who had only just mastered basic mathematical tools, to tackle a difficult problem worthy of a doctoral dissertation. This was even more difficult than when Newton only taught Xing Qingfeng two basic methods of dealing with barriers and then asked him to break through Kenneth's thirty-two barriers alone.

This is not to say that Grothendieck is a scholar who can't teach like Newton. On the contrary, the students he taught can completely crush Newton in both quantity and quality.

——The reason why he only taught these was simple, that is, he thought that for Flatt's talent, teaching these basic contents was enough. This is different from Newton...

...No, perhaps it was for the same reason that Newton allowed Xing Qingfu to break through Kenneth's barrier alone.

Flatt's pencil moved rapidly across the paper, as if the whole world had disappeared, his attention completely focused on how to construct the four-dimensional magic cube. He kept trying to expand the three-dimensional structure into the fourth dimension, drawing small cubes on the paper and trying to establish connections between them.

However, when it came to constructing the fourth dimension, his pen stopped.

His intuition told him that some direction should exist, but he had no idea where to start. His mind could move freely in three-dimensional space, but he could not break through that invisible barrier.

His brow furrowed even more, and he tapped the tip of his pen twice on the paper, as if trying to force himself to find a breakthrough. He reorganized his thoughts and tried to think according to the coordinate system that Grothendieck had described.

"X, Y, Z..." he muttered softly, "And... W?"

He tried to write down the variables of the four coordinate axes on paper, his eyes fixed on the last "W", but his brain was unable to materialize this non-existent direction. He felt as if his thoughts had hit an invisible wall, unable to move forward or backward.

“Still trying to understand this intuitively?” Grothendieck stood quietly by, gazing at Flatt’s notes, thinking, “He’s a genius, but he doesn’t seem suited to learning this.”

------------------

It's impossible for Flatt to be sent away. There's only one chapter left in this section of the plot. After laying so much groundwork, it's time to reveal some of Flatt's "abnormality". However, FSF is not finished yet, and I'm not sure what the thing in Flatt's mark is. It's a bit of a headache for me.

In fact, when I was writing about Flatt's genius, I was comparing him to Xing Qingjiu himself. The difference was that when I was writing about the protagonist in the first volume, I used a very simple reaction to weaken the protagonist's talent, but here it was not weakened, which also corresponds to the difference in the two people's personalities.

The above is a new book by a new author. Please vote and give me feedback. Thank you!

Volume 59: Xing Qingfeng's Perfect Arithmetic Classroom: . Genius, Falling for the Second Time

The air in the study was as still as a deep pool of water, occasionally broken by the sound of a gentle breeze ruffling the pages of a book outside the window. The tip of his pen scraped across the paper, leaving a chaotic trail of marks. Flatt stared intently at his draft, his brow furrowed, his eyes gleaming with a mixture of resentment and confusion.

Grothendieck stood aside, his hands behind his back, his eyes fixed on Flatt's manuscript, his expression calm and profound. He had already reached a clear conclusion.

Lack of abstract thinking.

Flatt's thinking is inverse—he excels at making abstract problems concrete, breaking down complex mathematical structures into intuitive geometric shapes. He quickly finds patterns, extracting the core elements from real-world problems and intuitively reconstructing them, just like constructing a magic trick, directly "kneading" the desired shape.

But his thinking was one-way.

His way of thinking is a one-way street. He excels at understanding problems through "shapes," rather than symbols, definitions, and logical relationships. He can build intuitive models but struggles to construct higher-level mathematical frameworks. It's like when he creates his own magic tricks, he doesn't follow conventional magic systems, but instead relies on intuition to piece together different magical elements and make them work. However, this approach can become a hindrance when faced with certain problems.

Four-dimensional space is just such a threshold.

Grothendieck did not interrupt Flatt's thinking. He just watched quietly, watching Flatt repeatedly draw three-dimensional cubes on the draft paper, trying to unfold, twist, and rotate them... but he could never find a breakthrough.

He clearly felt that he was about to reach that non-existent dimension, but there was always an invisible barrier blocking him outside the door.

He tried to imagine - if a three-dimensional cube can be unfolded into six squares on a two-dimensional plane, then can the "hypercube" in four-dimensional space also be unfolded in three-dimensional space in a similar way?

He gripped his pencil tightly, drew a cube on the paper, and then another next to it. He tried to use his block-like thinking to piece together a four-dimensional shadow, but this method could never break through the limitations of three dimensions. His thinking was trapped between length, width, and height, unable to move beyond the "W-axis."

He frowned and tapped the table lightly with his fingers, as if using sound to dispel the chaos in his mind.

--why?

The fourth dimension isn't simply "an extra direction." Like a blind person trying to understand color, he can only develop a metaphorical understanding but can't truly "see" it. He can hear others describe "red is passionate, blue is cool" and develop a metaphorical understanding of color, but he can never truly "see" the color itself.

He took a deep breath, his eyes fell back on the manuscript, and he tried to adjust his thinking direction again.

Back to geometry... No, I'm still stuck.

Trying to construct coordinates?

He picked up his pencil again and wrote down a set of symbols like (x, y, z, w) on paper, trying to convince himself that "W" existed independently of the first three coordinate axes. However, in his mind, "W" was still just a symbol, lacking intuitive meaning. He could not understand it like he understood length, width, and height.

Where is the mistake?

The fourth dimension wasn't just "an extra direction"; it existed in a way that was completely beyond his intuition. He was like a blind person trying to understand color: he could hear descriptions of colors and even develop some metaphorical understanding, but he couldn't truly "see" the colors themselves.

He took a deep breath and tried to change his mindset.

Geometry... No, it still gets stuck.

Trying to construct coordinates?

He wrote (x, y, z, w) on a piece of paper, trying to convince himself that "w" was an independent dimension. However, in his mind, "w" was still just a symbol, without any intuitive meaning. He couldn't understand it the same way he understood length, width, and height.

Not right.

Something is wrong.

He clearly knew that he was "almost" able to grasp the key, and had clearly realized the direction of some breakthrough, but that vague barrier always existed, like the foggy surface of a lake, making it impossible to see the reflection under the water.

Flat tilted his head slightly and bit his lower lip lightly.

Where on earth have you experienced this feeling that is “unintuitively unimaginable”?

He slowly closed his eyes, his mind constantly searching for past memories. Then, a certain scene suddenly emerged -

In class, Xing Qingjiu lazily leaned against the podium, flipping through his handouts casually while taking out a small silver wine flask from his pocket. He gently unscrewed the lid and took a sip with a nonchalant expression.

——It’s Xing Qingjiu’s magic.

Flatt's eyes snapped open. Yes, that was the feeling.

It's Xing Qingjiu's imaginary number magic.

"This is just a very simple basic application," Xing Qingfeng said casually in class, his tone as calm as if he were explaining a basic mathematical theorem. "Once your math level improves, you'll understand it. It's not difficult."

But Flatt doesn't think it's "simple".

Xing Qingjiu's magic is not space transfer, nor is it storage magic, but something even more counterintuitive.

He still remembered that when he tried to understand it, he was filled with similar confusion.

——You can clearly see the phenomenon, but you cannot understand its essence.

And now, he was trapped by the same feeling again.

Flatt stared blankly at the draft on the table, his fingertips gently stroking the paper, and his heartbeat inexplicably quickened for a moment.

——I see, is it a similar concept?

No, not some imaginary number magic trick, but something much simpler than that.

Xing Qingjiu's magic is essentially a "reconstruction of reality", which is to find the corresponding points in the real world from a new mathematical system. The four-dimensional problem he is facing now is just an extension of mathematics.

If you think so...

Flatt's lips curled up involuntarily, like a gamer who had finally found a breakthrough, his eyes gleaming with excitement. He still couldn't truly "see" the four-dimensional image in his mind, but that barrier was no longer an insurmountable obstacle.

He didn't know how to prove it, just like when doing a geometry problem, you can see at a glance that two straight lines are equal, but you just don't know how to prove it.

But it doesn’t matter. This is not a purely geometric problem. As long as you can see the direction, you can follow your intuition to come up with a solution.

The process is not important, as long as the results are output.

Flatt scribbled rapidly on the scraps of paper, his pen tip making a sharp, rustling sound as it scraped across the surface. No more geometric shapes, replaced by sets of coordinates, straight-line expressions, and complex quaternary equations filled the entire page. The logic of the calculations unfolded in his mind, as if a clear path guided him forward.

"There's no perfect solution to a fourth-order magic square..." He muttered to himself, his words barely keeping pace with his racing thoughts. His fingers were stiff from gripping the pencil for so long, the knuckles turning pale, but he didn't stop, even his breathing becoming rapid.

Something seemed to be wrong with him.

His heartbeat quickened, and his blood surged through his veins, bringing with it a feeling of intense heat. Fine beads of sweat formed on his forehead, and his skin felt unnaturally warm, as if someone had ignited a fire within him. Even more concerning were the magic circuits that crisscrossed his body—they should have been quietly dormant within him, but now they stirred, as if stimulated by some unknown force, slowly burning his nerves.

Flatt ignored these anomalies, or rather, his brain was too busy to pay attention.

His mind raced, all his energy focused on the calculations in his hands. He continued to calculate, from the second level to the fourth level, then to the sixth and eighth levels. Each time he advanced to a higher level, it was like climbing a steeper mountain. The amount of calculations increased exponentially, and the burden of thinking also increased.

"Eighth-order magic squares also don't have a perfect solution..."

His voice was already weak, and his chest was heaving violently, as if he had just run a long distance. His vision began to blur, and the words on the paper seemed to waver slightly, but he gritted his teeth and continued writing. The numbers in his mind kept jumping, combining, and splitting, constructing a complex matrix network. He had to take another step forward, to derive a higher-level solution...

But the burden on the body is becoming increasingly difficult to ignore.

The restlessness of his magic circuits grew more intense, burning like scorching irons through his veins, spreading to his spine, limbs, and even his scalp. Flat's fingertips began to tremble slightly, his palms sweating, and even holding a pen became difficult. An indescribable pressure built in his chest, his throat constricted, and even breathing became heavy and labored.

"Ah... this child..." Grothendieck sighed.

His demonic energy seemed to be overloaded, sending waves of stabbing pain that spread along his nerves in waves, his body instinctively protesting.

——Protesting that he used methods beyond his ability to conduct deductions, and protesting that he mobilized too much computing power in a short period of time.

But Flatt still didn't stop.

The sky outside the window had lost its midday heat, and a gentle orange-yellow sunlight shone onto the tabletop, casting a series of interlaced shadows across the stacks of books and the messy scraps of paper. The room was filled with the faint scent of paper and ink, and tiny particles of dust drifted in the air, slowly swirling in the light.

He could no longer stop. His thoughts swept through all possibilities like a hurricane, and the trajectory of calculations rolled in his brain, like an over-running machine, tirelessly pushing the gears forward.

The long-lost magic power once again flows through the body——

Warm energy surged out from deep within his chest, quickly spreading along the meridians to his limbs and bones. The marks engraved with magic seemed to be reawakened, faintly flickering with an uneasy echo.

——When was the last time I used magic? It had been so long that I couldn't remember clearly. The past was never something worth dwelling on.

Flatt felt a slight tingling sensation on his skin, like tiny needles scratching it. Heat emanated from the capillary-like magical circuits throughout his body, bringing a near-boiling heat. His ears buzzed, his mind still calculating, but his senses were overwhelmed by a strange sense of confusion.

——His vision began to blur, and the touch of his fingers seemed to be peeled off. The pencil seemed to become light between his fingers and difficult to hold.

He seemed to hear some sound.

An old friend with a slightly mechanical feel was constantly shouting something.

...alarm? Error?

But he could no longer hear clearly.