My IQ has been increasing year by year.

Chapter 178 Finally

The corridors of the Mathematics Building were quiet in the evening.

The office door was ajar, and the lights were off.

The air inside the room seemed to stand still.

Li Jianming leaned back in the somewhat old rattan chair, his reading glasses perched on his nose. He held a copy of last month's "Annals of Mathematics" in his hands, his gaze fixed on the second paragraph on the left page, having not moved down a single line for a full ten minutes.

Across from the desk, Wu Tao was watering several potted green plants on the windowsill with a spray bottle.

"Wu Tao".

Li Jianming suddenly spoke, his voice sounding somewhat abrupt in the quiet room.

"You've ruined the roots of that pothos plant."

Wu Tao paused for a moment.

He glanced down and saw that there was indeed a layer of water in the saucer at the bottom of the flowerpot.

He silently put down the spray bottle, grabbed a tissue, and bent down to wipe the water that had spilled from the edge of the tray.

"I'm sorry, teacher, I was daydreaming."

Wu Tao threw the tissue into the trash can, straightened up, and subconsciously turned his head to look at the large blackboard on the side of the office.

The bottom right corner of the blackboard, covered with dense calculus formulas, remains exactly as it was a few days ago.

Because it hadn't been touched for several days, the edges of the white chalk writing had become somewhat blurred, and a thin layer of dust had settled in the blackboard slot.

That was a dead end they couldn't cross.

The integral divergence term at the boundary of that continuous domain is like a thorn stuck in the eye.

These past few days, Li Jianming has given a strict order not to touch the research topic, so Wu Tao really hasn't dared to write down a single related formula on paper.

But his mind never stopped racing.

He was thinking about it while eating, while walking, and even in his dream last night, he dreamt that he was trapped in an infinitely vibrating Riemannian manifold, unable to find a way out.

"Stop looking."

Li Jianming closed the journal in his hand, took off his reading glasses and threw them on the table, then pinched the bridge of his nose.

"If you stretch your brain too tightly, it's easy to break. If a path doesn't work, just leave it for now. You can't get inspiration from math by just grinding it out."

Wu Tao nodded without saying anything. He walked to the water dispenser, took a disposable paper cup, and prepared to fill it with water.

Suddenly, footsteps echoed down the corridor.

It wasn't the slow, hesitant gait of students passing by; it was a crisp, quick stride.

The sound stopped abruptly at the office door.

The door was pushed open.

Li Jianming looked up, and Wu Tao turned around with an empty paper cup in his hand.

Chen Zhuo stood at the door, his forehead covered in sweat.

This is not Chen Zhuo as usual.

In Li Jianming and Wu Tao's memory, this genius from the gifted youth program always remained calm, composed, and gentle, no matter how complicated the topology problem was.

But at this moment, Chen Zhuo's eyes shone with an astonishing light.

He strode into the office.

He held a crumpled piece of printing paper in his left hand, with a dark red oil stain on the edge, as if he had accidentally smeared some vegetable soup on it.

"Teacher Li, Senior Brother Wu."

Without exchanging any pleasantries, Chen Zhuo walked to Li Jianming's desk and laid the oil-stained scrap of paper flat on the surface.

He spoke much faster than usual, but his pronunciation was exceptionally clear.

"I've found the way."

Wu Tao paused for a moment, then took a step forward with the paper cup in his hand.

"What do you mean?"

Chen Zhuo did not answer Wu Tao. He turned around and walked straight to the blackboard.

He picked up the dusty blackboard eraser from the blackboard tray.

Under the watchful eyes of Li Jianming and Wu Tao, Chen Zhuo did not hesitate at all and directly pressed the blackboard eraser onto the densely packed continuous domain derivation in the lower right corner.

Chalk dust fell down in a flurry.

The formulas that had taken them most of the month to work on, and that they had tried countless scaling methods to smooth the boundaries, were wiped clean by Chen Zhuo in a few strokes.

The previously oppressive blackboard suddenly had a large empty space.

"Chen Zhuo!"

Wu Tao called out instinctively.

Chen Zhuo threw the blackboard eraser back into the trough, dusted off his hands, and picked up half a piece of chalk.

He turned around and looked at Wu Tao.

"Senior Brother Wu, what is the premise of calculus? It is the continuity of the manifold and the local smoothness."

Chen Zhuo's voice echoed in the office.

"But the network structure we are facing now is not smooth at all on a very small scale."

Chen Zhuo pointed to the oil paper on the table.

"In the cafeteria this afternoon, I overheard my friend talking about machining. An aluminum rod is fitted inside a steel pipe. When heated, it expands. Aluminum has a large coefficient of expansion, while steel has a small coefficient of expansion. At the microscopic level, the atoms inside them are frantically squeezing and competing with each other."

Wu Tao frowned.

"What does this have to do with our topic?"

"It's a very important matter."

Chen Zhuo turned around and made a heavy tap with the chalk on the clean blackboard, drawing a circle to represent a node.

"If you use continuous domain calculus to calculate the boundary forces of this metal, you will never be able to calculate it correctly, because the positive and negative errors inside are constantly diverging and oscillating infinitely."

Chen Zhuo flicked his wrist and wrote a symbol next to the circle.

"But the physical reality is that no matter how much pressure is applied inside, as long as the outermost frame is locked, the metal will not move at all on a macroscopic level. Why? Because the divergent forces inside cancel each other out."

Li Jianming's gaze sharpened slightly. He didn't speak, but his body unconsciously leaned forward.

Chen Zhuo's chalk moved quickly across the blackboard.

"Since the continuous domain approach is not feasible, we will abandon it."

"Project the entire network onto a discrete algebraic topological space."

Rows of algebraic symbols began to appear on the blackboard.

It is no longer an integral sign, no longer a limit, but a group, a homomorphism, a mapping.

"We introduce homology groups."

Chen Zhuo wrote and spoke at the same time.

"Consider each node in the network as a zero-dimensional simplex, and the connections between them as a one-dimensional simplex."

"Those divergent oscillations at the internal nodes, those non-convergent errors, are like the struggle between aluminum and steel. In algebraic topology, what are they?"

Chen Zhuo wrote down Zn and Bn.

"They are closed chains, and also boundaries."

"Senior Brother Wu, when a chain is a boundary, what is the result under the homology group mapping?"

Wu Tao stood by the water dispenser, the paper cup in his hand already unconsciously crushed.

His lips moved.

"It's zero."

"right."

Chen Zhuo drew an arrow pointing to a capital zero.

"No matter how chaotic or divergent things are internally, it doesn't matter. Under the effect of the closed chain, they cancel each other out, and the result is always zero. This is what Dayong meant when he said that the two forces cancel each other out."

Chen Zhuo's movements became faster and faster, and the sound of chalk hitting the blackboard became a dense white noise.

"We don't need to prove that damn integral convergence; we just need to construct this global topological invariant."

Chen Zhuo wrote a number in the very center of the blackboard.

"Once this invariant is established, the macroscopic boundary is absolutely locked. The more divergent the microscopic world is, the more topologically conserved the macroscopic world becomes."

Chen Zhuo put down his pen.

He turned around and looked at Wu Tao, who was completely stunned, and Li Jianming, who was sitting in the rattan chair.

Chalk dust floated slowly in the halo of the setting sun filtering through the window.

"This path is not at the bottom of the cliff of calculus, it is in the sky of algebra."

A hush fell over the office; the occasional sounds of students chatting and laughing at the end of the corridor seemed so distant and unreal.

Wu Tao dropped the paper cup in his hand to the ground.

He didn't pick it up; he stared intently at the new mapping path on the blackboard, composed of simplex and homology group.

excellent.

Exquisite and unparalleled.

They can even be completely unreasonable.

Li Jianming did not stand up.

He remained seated in the wicker chair, staring at the blackboard for a long time.

Then, his gaze slowly moved down and landed on the piece of oiled paper that Chen Zhuo had slapped on the table.

There was a rough closed loop symbol drawn on the paper, with a bit of cafeteria oil stain next to it.

Li Jianming reached out, picked up the paper, and examined it carefully.

No one spoke in the office.

Half a minute later, Li Jianming opened the bottom drawer on the right side of the desk.

That was the drawer where he usually kept his most important documents.

He took out a large stack of brand-new A4 draft paper and placed it neatly on the table.

The old professor looked up at Wu Tao.

His voice was flat, without a trace of excitement, and one could not even discern joy or anger.

"Wu Tao".

Wu Tao shuddered and immediately stood up straight.

Stop spacing out.

Li Jianming pointed to the draft paper on the table.

"Bring a chair over, and get some paper and a pen."

Wu Tao took a deep breath, strode to the table, dragged over a chair and sat down, took a pen from the pen holder and spread a stack of draft paper in front of him.

Li Jianming turned to Chen Zhuo.

"The intuition is good, and the approach is right, but this is just a framework."

The old professor set the oiled paper aside, his gaze sharpening once more.

"In algebraic topological isomorphisms, if you miss a single dimension or a single symbol, the entire logical chain will collapse. This blackboard cannot contain a complete proof."

Li Jianming leaned back in his chair, his hands clasped in front of him.

"Chen Zhuo, explain your thought process and break down the mapping process."

"Wu Tao, you do the underlying verification, making sure that the matrix transformations at each step are absolutely aligned in dimensions. If you can't get it right, stop and fix it. Don't skip steps."

"I'll oversee the overall logical loop."

The old professor glanced at the clock on the wall.

7:10 PM.

"Let's begin."

The sky grew completely dark.

Wu Tao got up and turned on the overhead light in the office. The stark white incandescent light filled the entire room, dispelling the shadows in the corners.

The content on the blackboard began to increase rapidly.

Chen Zhuo stood in front of the blackboard, his shirt sleeves rolled up to his elbows. He held a piece of chalk in his hand and wrote while speaking in a low voice.

"Define a simple complex K, and construct a chain group for any dimension p."

"Boundary operator, Senior Brother Wu, verify its universality under this specific network structure."

Wu Tao buried himself in the draft paper and wrote quickly.

The sound of pen nib scratching against paper became the only melody in the room.

"Substituting the first set of node matrices, the mapping is normal."

Wu Tao responded without looking up.

"The quotient group structure of the core is clear, and no twist groups appear."

Time passed by, second by second.

10 PM.

The hot water in the water dispenser kept turning on and off.

The tea in Li Jianming's cup had long since lost its heat, and the tea leaves had turned white and sunk to the bottom of the cup.

The amount of wads of waste paper on the ground began to increase.

"This isn't right."

Wu Tao stopped writing. He stared at the matrix he had just calculated on the paper, his brows furrowed tightly.

"Chen Zhuo, wait a moment."

Chen Zhuo turned around, the chalk in his hand hovering in mid-air.

"When dealing with isolated nodes at the edge in the third-dimensional mapping, dimensional collapse occurs."

Wu Tao tapped the draft paper heavily with the tip of his pen.

"The isomorphic mapping you just defined assumes that all nodes are in the same connected component, but in real network models, there must be isolated subnets."

Wu Tao raised his head, his eyes bloodshot.

"The boundary of the isolated subnet is empty. If we force it to be mapped into the homology group, the rank will not match, and the value of this will explode. We have gone back to square one."

There was a brief pause in the office.

Li Jianming stood up, walked behind Wu Tao, looked down at the few lines of calculations, and frowned.

It is indeed a major flaw.

Theoretically, algebraic spaces are perfect, but what they need to solve are real-world network topology models, those irregular, isolated edge nodes, which are like cracks in perfect porcelain, easily tearing the entire mapping system apart.

"Could we add an auxiliary theorem to calculate isolated nodes separately?"

Wu Tao rubbed his sore wrist and tentatively proposed a conventional solution.

"no."

Li Jianming directly vetoed it.

"Separating the calculations destroys the purity of the global invariants, and when they are finally combined, the error term will reappear. This is where calculus falls short, and we can't afford to make the same mistake again."

The two of them looked at Chen Zhuo at the same time.

Chen Zhuo leaned against the blackboard.

He looked at the error matrix on Wu Tao's draft paper.

He didn't speak.

After about a minute, Chen Zhuo tilted his head slightly, a faint smile curving his lips. The smile carried a gentle, inquisitive tone, but upon closer inspection, it also revealed a hint of almost cunning confidence.

"Senior Brother Wu."

Chen Zhuo's voice wasn't loud, but it could be heard clearly in the quiet office.

"When you calculated this matrix distance, you used Euclidean metric, right?"

Wu Tao was stunned for a moment.

"Yes, Euclidean metric is the most standard for measuring node distances."

""

"It's a habit."

Chen Zhuo interrupted him.

Chen Zhuo walked to the table, picked up a pencil, and lightly drew a few dots on the edge of Wu Tao's draft paper filled with formulas. Then he connected them with a line, deliberately leaving a gap in the middle.

Chen Zhuo looked at Wu Tao's somewhat bewildered face and spoke directly to the core issue.

"This is a discrete space; don't use the ruler of continuous space to measure discrete distances."

"Ditch the Euclidean metric and replace it with the shortest path metric from graph theory."

Chen Zhuo straightened up and put the pencil back on the table.

"Under the shortest path metric, the distance between isolated nodes and other nodes is infinite. They do not cause dimensional collapse, and they automatically generate a new homology class."

Chen Zhuo turned around and walked back to the blackboard, drawing a white line on it with chalk.

"It won't explode; it will only be exactly equal to the number of connected components."

"I guess so."

Wu Tao sat blankly in the chair.

Euclidean metric.

He looked down at his draft paper.

Habits really do kill people; when dealing with geometry problems, subconsciously substituting Euclidean measures is almost an instinctive reflex for him.

He actually tried to use a ruler meant for measuring continuous, smooth curves to measure a pile of discrete sand.

No wonder there was a dimensional collapse.

Wu Tao took a deep breath, and without saying a word, crumpled the incorrectly written draft paper into a ball and threw it on the ground.

A new sheet of paper has been used.

Introduce the shortest path metric matrix and substitute it back in.

Fifteen minutes later.

Wu Tao exhaled a heavy breath, his hand holding the pen even trembling slightly.

"It's aligned, the rank is perfectly matched, and the Betti number of the connected components has been calculated precisely."

Wu Tao looked up at Chen Zhuo, who was leaning against the blackboard, his eyes filled with complex emotions.

Standing before this young man who was almost a decade younger than him, his solid foundation, which he, as a doctoral student, was so proud of, sometimes became a shackle on his thinking. Chen Zhuo's intuition, which was completely unbound by traditional frameworks, was simply terrifying and suffocating.

"continue."

Li Jianming's voice rang out from the side, but the old professor's eyes never left the draft paper, seemingly unconcerned about the little incident that had just occurred.

The derivation continues.

Two o'clock in the morning.

Even the stray cats on campus were asleep; in the entire math building, only the office lights were still on.

This is an extremely tedious mental grinder.

There was no sense of wonder or inspiration; all that remained was mechanical, tedious calculation and verification that allowed no room for error.

The blackboard was full of writing.

Chen Zhuo picked up the blackboard eraser, wiped away the primary mapping that had already been verified on the left, and started writing the high-dimensional derivation again.

Chalk dust settled on his white shirt cuffs and on his black hair.

Wu Tao had already unbuttoned two buttons on his shirt. He was no longer sitting, but kneeling on one knee on the chair and half-lying on the table to do calculations.

The wads of waste paper on the ground had piled up into a small hill.

How to deal with the "twisting group" (a term used to describe a group of people arguing or making mistakes)?

At 3:30 a.m., we encountered the second major hurdle.

Wu Tao's voice was hoarse, as if it had been sanded on sandpaper.

"The torsion group component appears in the homology group, and this part of the space is twisted. It does not provide a free dimension. If it is not removed, there will be an extra tail on both sides of the final equation."

Wu Tao threw down his pen and rubbed his temples.

"This tail can't be cut off at all; it's entangled with the mainstream free abel group."

Li Jianming stood by the window, looking at the pitch-black night outside.

"Forced sale?"

The old professor suggested in a low voice.

"If the quotient cannot be removed, the surjectivity of the mapping cannot be guaranteed once the quotient group is processed, and the preceding short positive sequence will break."

Wu Tao scratched his hair in anguish.

Only heavy breathing could be heard in the office.

Chen Zhuo leaned against the blackboard and closed his eyes.

His brain was working at high speed, like an overclocked processor, searching through the vast algebraic structure for the tool that could untangle the knots.

Time seemed to stand still.

A full ten minutes passed.

Chen Zhuo opened his eyes.

Instead of going to the blackboard, he went to the water dispenser, filled a half-cup with cold water, and gulped it down.

The cool water flowed down his throat and into his stomach, clearing his somewhat muddled mind a little.

"Senior Brother Wu."

Chen Zhuo put down his water glass and turned his head.

"You don't need to cut it off."

Wu Tao raised his head.

"If we don't cut it off, how can we balance the equation?"

Why do we need to balance the equation?

Chen Zhuo walked back to the blackboard and picked up the chalk.

"We're here to find global invariants, not to do arithmetic problems."

"A homology group can be decomposed into a direct sum of a free abelian group and a torsion group."

Chen Zhuo wrote the formula on the blackboard.

"Torsion groups represent spatial twisting; they do exist, but they do not contribute to the number of holes in space."

"We only take its rank."

Chen Zhuo added the rank symbol to both sides of the equation.

"For any finitely generated abelian group, the rank of the torsion group is always zero."

The chalk drew an X on the blackboard, directly crossing out the part representing the scratch group.

"Let it be tangled up there. As long as we take the rank, it will become invisible in a mathematical sense."

"Only the rank of the free portion."

Wu Tao was stunned.

"sharp."

Wu Tao muttered to himself, picked up his pen again, and began to follow this line of thought.

Five o'clock in the morning.

A faint, cold gray light began to peek through the sky over Huizhou.

The air in the office was so stuffy that it made people feel a bit oxygen-deprived.

Li Jianming sat in a rattan chair, holding a thick stack of draft paper that was already filled with writing. The old professor's back was still straight, but his fingers holding the paper were trembling slightly.

That's not fatigue; it's an involuntary physiological reaction of the body after extreme excitement.

Wu Tao slumped in his chair, staring blankly at the ceiling. His brain had completely shut down; even if you asked him to calculate one plus one now, it would probably take him a long time to process it.

In front of the blackboard.

Chen Zhuo was left with only a piece of chalk he couldn't hold.

His wrists were a little sore, and his clothes were stuck to his back, soaked with sweat.

He was in the bottom right corner of the blackboard.

He wrote down the last equation of the entire derivation.

Through the mapping of discrete algebraic topology, the boundary oscillation term that originally diverged infinitely in the continuous domain is perfectly contained within a finite Betti number.

The left side of the equation represents the localized, complex micro-level changes in the network.

The right side of the equation is a concise topological invariant consisting of integers.

On both sides, an equal sign was drawn, indicating absolute equality.

Chen Zhuo released his hand.

The short piece of chalk, almost too short to hold, fell into the blackboard slot.

In this quiet office, the soft sound was like the final rivet being driven into a magnificent building.

finished.

Chen Zhuo took two steps back.

He looked at the formulas covering the entire wall, at the algebraic symbols arranged like stars.

A profound sense of emptiness, followed by a feeling of satisfaction, overwhelmed him like a tidal wave.

Ji Jianming tapped the draft paper in his hand on the table to align the edges.

The old professor took off his reading glasses and placed them on the table.

He didn't look at Chen Zhuo, nor at Wu Tao; he just stared at the blackboard.

"It's a closed loop."

Li Jianming's voice was soft and hoarse, but carried an indescribable weight.

"This road is now open."

Wu Tao was lying in the chair. When he heard this, the corner of his mouth twitched, as if he wanted to laugh, but his facial muscles were too stiff to make any expression.

He could only close his eyes and let out a long sigh.

A weariness that seemed to seep into one's very bones instantly overwhelmed everyone.

Chen Zhuo leaned against the wall next to the blackboard.

After the adrenaline subsided, a sudden wave of dizziness from low blood sugar hit me.

His face turned pale, and his stomach tightened in waves, as if a hand was twisting inside.

From the moment I ate a bite of rice in the cafeteria yesterday afternoon until now, my brainpower has been pushed to its limit.

Chen Zhuo rubbed his stomach and straightened up.

He patted the chalk dust off his clothes and looked at the two professors and senior students who were too tired to move.

"Teacher Li, Senior Brother Wu."

Chen Zhuo's voice was slightly unsteady due to exhaustion, but it conveyed a sense of relief and ease.

He took out his meal card from his pocket and waved it.

"I don't even have the strength to go downstairs now."

Chen Zhuo looked at Wu Tao.

"Senior, could you use this meal card to buy me two meat buns at the cafeteria? And two cups of soy milk with sugar, too?"