No more integral signs, no more limits. Now, it was about groups, homomorphisms, and mappings.
"We introduce homology groups."
Chen Zhuo said as he wrote.
"We treat each node in the network as a zero-dimensional simplex, and the connections between them as one-dimensional simplices."
"Those divergent oscillations from the internal nodes, those errors that fail to converge... they’re like a tug-of-war between aluminum and steel. In algebraic topology, what are they?"
Chen Zhuo wrote down a Z_n and a B_n.
"They are cycles, and they are also boundaries."
"Senior Brother Wu, when a chain is a boundary, what is its image under the homology group mapping?"
Wu Tao stood by the water cooler, the paper cup in his hand long since crushed without him realizing it.
His lips moved.
"Zero."
"Correct."
Chen Zhuo drew an arrow pointing to a large zero.
"It doesn’t matter how chaotic the inside is or how much it diverges. Under the action of the cycles, they cancel each other out. The result is always zero. This is what Dayong meant when he said the two forces canceled each other out."
Chen Zhuo’s movements grew faster and faster. The tapping of the chalk on the blackboard merged into a dense wash of white noise.
"We don’t need to prove that damn integral converges. We just need to construct this global topological invariant."
In the very center of the blackboard, Chen Zhuo wrote a number.
"As long as this invariant is established, the macroscopic boundary is absolutely locked in. The more the microscopic level diverges, the more the macroscopic level is conserved in a topological sense."
Chen Zhuo stopped writing.
He turned around, looking at the utterly stunned Wu Tao and at Li Jianming, who was seated in the wicker chair.
Chalk dust floated slowly in the halo of evening sunlight filtering through the window.
"This path doesn’t lie at the bottom of a cliff in calculus. It’s in the heavens of algebra."
The office fell silent. The occasional sound of students laughing and talking at the end of the hallway seemed distant and unreal.
The paper cup in Wu Tao’s hand fell to the floor.
He didn’t bend to pick it up. He stared intently at the brand-new mapping path on the blackboard, constructed from simplices and homology groups.
’Brilliant.’
’Perfectly elegant.’
’Downright unfair.’
Li Jianming did not stand up.
He remained in his wicker chair, looking at the blackboard for a long time.
Then, his gaze slowly shifted downward, landing on the piece of greasy paper Chen Zhuo had slapped onto the desk.
A crude closed-loop symbol was drawn on the paper, a small grease stain from the cafeteria still visible next to it.
Li Jianming reached out, picked up the paper, and held it before his eyes to study it carefully.
No one in the office spoke.
After half a minute, Li Jianming pulled open the bottom drawer on the right side of his desk.
It was the drawer where he usually kept his most important documents.
He took out a large stack of new A4 scratch paper and placed it neatly on the desktop.
The old professor looked up at Wu Tao.
His voice was calm, without a trace of excitement, his expression unreadable.
"Wu Tao."
Wu Tao flinched and immediately stood up straight.
"Snap out of it."
Li Jianming pointed to the scratch paper on the desk.
"Pull up a chair. Get some paper, get a pen."
Wu Tao took a deep breath, walked quickly to the desk, dragged a chair over, and sat down. He pulled a pen from the holder and took a stack of scratch paper, spreading it out 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 greasy paper aside, his gaze growing sharp again.
"With isomorphic mappings in algebraic topology, be off by a single dimension or a single sign, and the entire logical chain collapses. There isn’t enough space on this blackboard for the complete proof."
Li Jianming leaned back in his chair, folding his hands in front of him.
"Chen Zhuo, you lay out the idea and break down the mapping process."
"Wu Tao, you run the low-level calculations. Ensure every matrix transformation is perfectly aligned dimensionally. If something doesn’t compute, we stop and solve it. No skipping steps."
"I will oversee the overall logical closure."
The old professor glanced at the wall clock.
「Seven-ten in the evening.」
"Let’s begin."
The sky had grown completely dark.
Wu Tao got up and turned on the office’s overhead light. A stark white fluorescent glow flooded the room, chasing away the shadows in the corners.
The content on the blackboard began to grow rapidly.
Chen Zhuo stood before the blackboard, the sleeves of his shirt rolled up to his elbows. He held a piece of chalk, speaking in a low voice as he wrote.
"Define the simplicial complex K. For any dimension p, construct the chain groups."
"The boundary operator. Senior Brother Wu, verify its universality for this specific network structure."
Wu Tao was hunched over his scratch paper, writing furiously.
The scratching of his pen against the paper became the only melody in the room.
"First set of node matrices substituted. Mapping is normal."
Wu Tao replied without looking up.
"The quotient group structure of the kernel is clear. No torsion subgroups have appeared."
Time ticked by, minute by minute.
「Ten o’clock at night.」
The water in the cooler boiled, then went quiet, then boiled again.
The tea in Li Jianming’s cup had long gone cold. The leaves, steeped pale, had settled at the bottom.
The number of crumpled paper balls on the floor began to grow.
"Something’s wrong here."
The pen in Wu Tao’s hand stopped. He stared at a matrix he had just calculated, his brow furrowed tightly.
"Chen Zhuo, hold on."
Chen Zhuo turned, his chalk hovering in mid-air.
"The third-dimension mapping. When processing peripheral, isolated nodes, a dimensional collapse occurs."
Wu Tao tapped the tip of his pen heavily on the scratch paper.
"That isomorphic mapping you just defined assumes all nodes are in the same connected component. But in a real-world network model, there are bound to be isolated sub-networks."

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