Wu Tao frowned.
"How are we supposed to use the Gauss-Bonnet Theorem? Without it, there’s no way to connect the curvature and genus of the entire topological network. Doesn’t that make this problem a dead end?"
Li Jianming leaned back in his rattan chair and let out a long sigh.
"Xiaowu’s right."
The old professor stood up, walked to the window, and watched the students occasionally passing by below.
"Chen Zhuo, your approach using a Discrete Matrix was a brilliant move. But math is math. Math doesn’t trust intuition; it only recognizes airtight logic. If the boundary fails to converge, everything that came before is for nothing. If we can’t find a tool to replace the Gauss-Bonnet Theorem..."
Li Jianming turned around, his gaze sweeping over the wall covered in draft papers and the blackboard. His voice was a little heavy.
"I’m afraid this path has led us into another dead end."
The office fell silent once more.
The atmosphere was even heavier than before.
Chen Zhuo didn’t argue.
He knew Li Jianming was telling the truth.
He could propose a discrete perspective, he could introduce an imaginary time variable, but if he couldn’t provide a logically self-consistent conclusion in the final stages, this paper—good enough to make a run at a top international journal—would be nothing more than a pile of worthless scrap.
He stared at the oscillating error term on his draft paper.
His mind was racing.
’If calculus doesn’t work, what will?’
’In physics, when a system’s boundary experiences unpredictable oscillations, how do physicists handle it?’
’No matter how it oscillates locally, some things are always conserved.’
’Energy, momentum, electric charge...’
’What about in math?’
Chen Zhuo’s brow furrowed slightly as well.
He felt a paper-thin barrier in his mind.
The feeling was unbearable, like having a name on the tip of his tongue that he just couldn’t recall.
He had a fleeting grasp of a clue, one completely different from the perspective of analysis, but the thread was too faint. Before he could get a clear look, it slipped away again.
It had been a long time since Chen Zhuo had felt this stuck.
Even when he was deriving the imaginary time variable, the solution had come to him as a matter of course.
But today, on this problem of boundary closure, he had hit a solid wall.
Seeing Chen Zhuo frowning and silent, Wu Tao felt a strange sense of balance settle in his heart.
’Even this freak is stumped.’
’Looks like it’s not that I’m stupid; this problem is just a total beast.’
"What if we try to construct an auxiliary function from a different angle?"
Still unwilling to give up, Wu Tao walked to the blackboard, picked up a piece of chalk, and tried to add something next to the oscillating term.
"If we introduce a weighted decay factor at the boundary to forcibly suppress the amplitude of the oscillation..."
"That’s useless."
Chen Zhuo cut him off.
Chen Zhuo stood up and stretched his stiff neck. He walked over to Wu Tao and looked at the formula.
"Senior Brother Wu, it’s like a balloon about to pop. Instead of finding a sturdier net to contain it, you’re trying to pinch the spot that’s about to burst. If you pinch the left side, the right side will just bulge out. The error term can’t be suppressed by a decay factor. It will just continue to diverge in a higher-dimensional space you can’t see."
Chen Zhuo’s voice was calm. He turned his head and looked at the dark circles under Wu Tao’s eyes.
"Let it go, Senior Brother Wu. If you keep at it, you’re going to erase a hole in this blackboard."
Chen Zhuo patted Wu Tao’s shoulder.
"If the direction is wrong, the harder you work, the more hopeless it becomes."
Wu Tao’s hand, holding the chalk, froze in mid-air, unsure where to write.
’Yeah, the direction is wrong.’
’But what other path is there to take?’
Li Jianming walked back to his desk.
He glanced at the wall clock. It was already five-thirty in the afternoon.
The sunlight was no longer harsh, but had turned a pale yellow, casting long shadows across the office.
"Alright."
Li Jianming’s voice rang out in the quiet room, carrying the decisiveness and helplessness of a veteran scholar.
"That’s enough for today."
He gathered the stack of calculation papers on his desk and unceremoniously stuffed them into a drawer.
"Research isn’t manual labor. When your brain won’t work, banging your head against a wall is just a waste of time."
Li Jianming looked at the two young men before him: one, his own student in whom he had placed great hope, and the other, a prodigy from the gifted youth program.
Right now, the exhaustion on both their faces was plain to see.
"Both of you, go home. For the next few days, I forbid either of you from touching this topic. Clear your minds completely. Get some sleep, eat proper meals, and to hell with this boundary oscillation."
The old professor waved his hand, as if shooing them away.
"But, Professor, what about our progress...?"
Wu Tao grew anxious.
"The progress isn’t going to grow legs and run away!"
Li Jianming shot him a glare and couldn’t stop a curse from slipping out.
"Look at the state of you! Your eyes are so damn red they look like they’re about to bleed. Even if inspiration struck right this second, your brain is too fried to catch it."
Wu Tao opened his mouth to speak, but in the end, just lowered his head in resignation.
"I understand, Professor."
He turned and began to slowly pack up the books and materials on his desk, his movements mechanical and sluggish.
Chen Zhuo, on the other hand, didn’t have as much emotional baggage as Wu Tao.
He was still young, after all. Getting stuck on a problem of this level for weeks, or even months, was par for the course. Besides, if every difficult problem could be solved in an afternoon, the seven Millennium Prize Problems would have been solved long ago.
He walked over to the sofa, neatly stacked his dozen or so pages of scratch paper, and folded them in half.

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