Chapter 322 - 176: Finally (3)

Chapter 322: Chapter 176: Finally (3)

Wu Tao looked up, his eyes bloodshot.

"The boundary of an isolated subnet is empty. If we force a mapping into the homology group, the ranks won’t match, and the value will just explode. We’re back where we started."

A brief silence fell over the office.

Li Jianming stood up, walked over to Wu Tao, and looked down at the lines of equations. His brow furrowed as well.

It was a serious flaw.

The theoretical algebraic space is perfect, but they were trying to solve a practical network topology model. Those irregular, isolated edge nodes were like cracks in perfect porcelain, easily capable of tearing the entire mapping system apart.

"Could we add an auxiliary theorem to isolate the solitary nodes and calculate them separately?"

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

"No."

Li Jianming shot it down immediately.

"Calculating them separately would destroy the purity of the global invariants. When we merge them back in at the end, the error terms will pop up again. We already learned that lesson the hard way with calculus; we can’t make the same mistake twice."

Both men turned to look at Chen Zhuo.

Chen Zhuo was leaning against the blackboard.

He was looking at the matrix that was causing the error on Wu Tao’s scratch paper.

He didn’t say anything.

After about a minute, Chen Zhuo tilted his head slightly. The corners of his mouth curved into a faint smile. It held a hint of gentle inquiry, but upon closer inspection, it revealed a sliver of almost cunning confidence.

"Senior Brother Wu."

Chen Zhuo’s voice wasn’t loud, but it was perfectly clear in the quiet office.

"When you were calculating the distance for this matrix, you used the Euclidean metric, right?"

Wu Tao froze for a second.

"Yeah, for measuring node distance, the Euclidean metric is the most standard..."

"That’s habit."

Chen Zhuo cut him off.

Chen Zhuo walked to the desk, picked up a pencil, and lightly drew a few dots on the edge of Wu Tao’s formula-covered scratch paper. He connected them with a line, intentionally leaving a gap in the middle.

Chen Zhuo looked at Wu Tao’s somewhat stunned face, his words cutting straight to the heart of the matter.

"This is a discrete space. Don’t use a ruler for continuous spaces to measure discrete distances."

"Throw out the Euclidean metric. Replace it with the shortest-path metric from graph theory."

Chen Zhuo straightened up and placed the pencil back on the desk.

"Under the shortest-path metric, the distance from an isolated node to any other node is infinite. They won’t cause a dimensional collapse. They’ll automatically generate a new homology class."

Chen Zhuo turned and walked back to the blackboard, drawing a white line across it with a piece of chalk.

"It won’t explode. It will be precisely equal to the number of connected components."

"Go on, calculate it."

Wu Tao sat stunned in his chair.

The Euclidean metric.

He looked down at his scratch paper.

It was true what they said: old habits die hard. When dealing with geometry problems, subconsciously defaulting to the Euclidean metric was practically a reflex for him.

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

No wonder the dimension was collapsing.

Wu Tao took a deep breath. Without a word, he crumpled the incorrect draft into a ball and threw it on the floor.

He took out a new sheet of paper.

He introduced the shortest-path metric matrix and plugged it back into the calculations.

「Fifteen minutes later.」

Wu Tao let out a long, heavy breath, the hand holding his pen trembling slightly.

"It aligns. The ranks match perfectly. 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 expression complex.

In the presence of this young man, nearly a full twelve-year cycle his junior, the solid foundation he took so much pride in as a PhD student sometimes felt more like a mental shackle. Chen Zhuo’s intuition, completely unbound by traditional frameworks, was terrifyingly suffocating.

"Continue."

Li Jianming’s voice sounded from the side. The old professor’s gaze never left the scratch paper, as if he was completely unconcerned by the small interruption.

The derivation continued.

「Two in the morning.」

Even the stray cats on campus were asleep. In the entire Mathematics Building, only the light in their office remained on.

It was an utterly tedious mental meat grinder.

There was none of the breathtaking feeling of a sudden burst of inspiration. All that remained was mechanical, tedious calculation and verification that allowed for no errors.

The blackboard was full.

Chen Zhuo picked up the eraser, wiped away the verified primary mappings on the left side, and began writing out the high-dimensional derivation anew.

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

Wu Tao had long since unbuttoned the top two buttons of his shirt. He was no longer sitting, but kneeling on his chair with one leg, half-sprawled over the desk as he calculated.

The crumpled balls of paper on the floor had already formed a small mound.

"How do we handle the torsion subgroup?"

At half past three in the morning, they hit the second major obstacle.

Wu Tao’s voice was as hoarse as if it had been scraped with sandpaper.

"A torsion component has appeared in the homology group. This part of the space is twisted. It doesn’t provide a free dimension. If we don’t eliminate it, we’ll be left with an extra term on both sides of the final equation."

Wu Tao threw his pen down and rubbed his temples.

"And there’s no way to just cut this term off. It’s entangled with the main free abelian group."

Li Jianming stood by the window, gazing out at the pitch-black night.

"Force a quotient?"

the old professor suggested in a low voice.

"We can’t. If we take the quotient group, we can no longer guarantee the surjectivity of the mapping. The short exact sequence we established earlier will break."

Wu Tao clutched his hair in agony.

Only the sound of heavy breathing filled the office.

Chen Zhuo leaned against the blackboard and closed his eyes.

His mind was racing, like an overclocked processor, searching through the vast expanse of algebraic structures for the one tool that could unravel the tangle.