Four o’clock in the afternoon. In Li Jianming’s research lab, the only sound was the frantic tapping of chalk against the blackboard.
The sound was heavy, as if made through gritted teeth.
SNAP.
A half-stick of chalk, pressed too hard, had snapped clean against the board.
The broken piece fell to the floor, rolling into a thin layer of chalk dust.
Wu Tao stopped what he was doing.
He stared at the blackboard, his brow furrowed in a tight knot.
His hair, once neatly combed, was now a messy bird’s nest from him running his hands through it. His shirt was rolled up to his elbows, and his forearms were smeared with white chalk dust.
The blackboard was covered in a dense web of equations.
From left to right, top to bottom.
Continuous domain integrals, derivations of the Gauss-Bonnet Theorem, transformations of the Jacobian matrix... Some spots had been vigorously erased, only to be covered with new calculations on the semi-dry surface, leaving behind blurry white smudges.
Wu Tao’s chest heaved. He took two steps back, as if trying to view the blackboard from a more macroscopic perspective.
He stared for a full minute.
He took a deep breath, and the half-stick of chalk in his hand slipped weakly through his fingers, clattering into the chalk tray.
"It doesn’t work."
Wu Tao’s voice was hoarse.
Behind the desk, Li Jianming sat in an old rattan chair.
He wasn’t wearing the reading glasses he usually used for papers. His desk was piled with a thick stack of calculation sheets. The top page was a mess of pencil marks, covered in crossed-out calculus paths.
In the white porcelain cup at his elbow, the tea leaves had long since sunk to the bottom, the tea itself gone completely cold.
Hearing Wu Tao’s words, Li Jianming didn’t look up. He just tossed the pen in his hand onto the desk and pressed his fingers firmly against the bridge of his nose.
"It won’t converge, right."
Li Jianming’s voice was laced with deep exhaustion.
"Yeah."
Wu Tao turned and leaned dejectedly against the wall next to the blackboard.
"Locally, it’s completely fine. That imaginary time variable Chen Zhuo introduced really did smooth out the Singularity distorting the Jacobian determinant. The entry point with the Discrete Matrix was nothing short of beautiful, but..."
Wu Tao paused, his tone full of frustration and confusion.
"But... it’s like we cut out a failing heart, fixed it externally, and now we’re trying to put this discrete heart back into the body of a continuous network topology. The problem is with the sutures."
He turned his head, pointing to the long integral equation in the bottom-right corner of the blackboard.
"The error term at the boundary simply won’t converge."
Wu Tao strode back to the blackboard and jabbed a finger at the end of the equation.
"As soon as the number of network nodes, N, approaches infinity, this error term begins to oscillate endlessly. It’s like a lunatic, jumping from large to small. The traditional Gauss-Bonnet Theorem is completely useless here. If the boundary can’t be closed, then all our previous work cutting out the Singularity becomes a castle in the sky, mathematically speaking. Without convergence, the proof is invalid."
A heavy silence fell over the office.
This was the most despairing moment in a pure mathematical proof.
The fear wasn’t having no ideas from the start; it was thinking you’d scaled the highest mountain, only to find an uncrossable chasm just beyond the peak.
The method was clearly correct. Their intuition told them this path would work. But they were stuck on this final step: the proof of convergence.
And they just couldn’t get past it.
On a somewhat faded sofa in the corner, Chen Zhuo was curled up.
On the old coffee table in front of him, a dozen or so sheets of scratch paper, also covered in equations, were scattered about.
He was sunk deep into the cushions, a pencil pinched between his fingers, staring at the top sheet.
On the paper was the very same boundary oscillation term Wu Tao had just written on the board.
Hearing Wu Tao’s complaint, Chen Zhuo stopped unconsciously spinning the pencil in his hand.
He sat up straight, picked up the sheet of paper, and held it up to the light coming through the window.
"Senior Brother Wu is right."
Chen Zhuo spoke, his pace unhurried, with the calm of someone stating an objective fact.
"I’ve derived it using four different integration paths over here, and the result is the same. As long as you try to fit it back into the framework of continuous calculus, the high-frequency oscillation at the boundary can’t be eliminated."
He placed the scratch paper back on the coffee table and took a sip from the lukewarm water beside him.
"This doesn’t feel like we made a calculation error somewhere."
Chen Zhuo looked at the blackboard, a hint of resignation in his voice.
"It’s more like... there’s a problem with the tool itself."
Wu Tao froze for a second, then turned to look at Chen Zhuo.
"A problem with the tool? What do you mean?"
Chen Zhuo set down his glass, resting his clasped hands on his knees.
"Calculus deals with smooth, continuous manifolds. But what we’re facing now is a network boundary that’s been cut by a Discrete Matrix, with nodes approaching infinity. Senior Brother Wu, don’t you think it’s a bit like we’re using an incredibly sharp scalpel to cut a piece of modeling clay that’s constantly changing shape?"
He looked at Wu Tao, the corners of his mouth twitching into a subtle curve.
"No matter how precisely you cut it, the edge of the clay will always be rough. If you force the limit of a continuous domain onto it, of course it’s going to oscillate."
Wu Tao opened his mouth, wanting to argue.
As a conventionally trained pure mathematics Ph.D. student, his first reaction to a divergent integral was always to figure out how to smooth it or bound it using more refined analytical methods. It was an academic instinct carved into his very bones.
"But if we don’t fit it back into the continuous domain..."

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