Early in the morning, Chen Zhuo slung his backpack on and went to the School of Mathematics.
The School of Mathematics building was old, and the light in the hallways was always dim.
Chen Zhuo took the stairs to the third floor and pushed open the door to the large conference room at the end of the hall.
The heating in the old building hadn’t been turned on yet. The space was large, and a cold draft snaked its way down his collar.
Wu Tao was standing in front of the massive blackboard that took up an entire wall.
He was bundled up in a thick, dark blue down jacket, zipped all the way to his chin. A cigarette dangled from his mouth, its glowing tip bobbing up and down as he clenched his jaw.
The blackboard was densely packed with derived formulas. The left side had been erased and written on, the right side written on and erased, leaving behind layers of white chalk dust residue.
A messy pile of scratch paper was scattered across the conference table. Pinned beneath it all in the very center was the five-page outline on homology mapping that Chen Zhuo had brought over a few days ago.
Hearing the door open, Wu Tao turned his head.
His eyes were bloodshot, and the bags under them were so heavy he looked like he hadn’t had a good night’s sleep in days.
Seeing Chen Zhuo walk in, Wu Tao tossed the half-stick of chalk in his hand into the blackboard tray, clapped the dust from his hands, and took the cigarette from his mouth, pinching it between his fingers.
"You’re here."
Wu Tao’s voice was a bit hoarse. He pulled out a chair beside the conference table and sat down.
Chen Zhuo walked over to him, placed his backpack on the table, and pulled out another chair to sit down.
He glanced at the board full of formulas, then at the scratch paper on the table covered in crossed-out marks, and asked in a casual tone.
"Where are you stuck?"
Wu Tao gave a bitter smile and tapped the third page of Chen Zhuo’s five-page outline.
"Going from this lemma to the next step, the isomorphic group."
Wu Tao drew a line between two rows of formulas with his fingernail.
"The leap between them is too big."
Chen Zhuo looked where he was pointing.
"If you directly substitute the boundary conditions of the continuous domain, it should logically transition," Chen said.
"It works in theory, but it can’t be calculated."
Wu Tao stood up, walked back to the blackboard, picked up a new piece of chalk, and quickly wrote a long string of equations with integral signs in the blank space at the bottom right.
The chalk tapped crisply against the board.
"See? When we substitute the boundary conditions and try to expand the integral in the third dimension, the result diverges."
Wu Tao stopped writing and turned to look at Chen Zhuo.
"In a topological space, once the integral diverges, the subsequent isomorphic group doesn’t hold."
Chen Zhuo didn’t answer immediately.
He sat in his chair, his gaze fixed on the divergent integral on the blackboard, his mind rapidly reviewing his original derivation process.
In his original conception, the continuity here was supposed to be a given.
Wu Tao sighed, tossed the cigarette in his hand into the trash can, then pulled a lollipop from his jacket pocket, unwrapped it, and stuck it in his mouth.
"Liu Ming and I have tried four different integration paths these past few days. They all diverge near the Singularity."
Wu Tao mumbled, his words muffled by the candy.
"If we can’t get over this hurdle, everything that follows will have to be scrapped and redone."
Chen Zhuo took a pen from his backpack and pulled over a blank sheet of scratch paper.
"Since the integral of the continuous domain diverges at the Singularity, let’s not use the continuous domain."
As he spoke, Chen Zhuo quickly wrote a set of matrix equations on the paper.
Wu Tao leaned in to take a look.
"At this fault line, we introduce a discrete algebraic matrix."
Chen Zhuo’s pen glided across the paper, writing out clear lines of algebraic expressions.
"We forcibly sever this continuous singular surface, use discrete eigenvalues to contain the divergent tendency, and then remap it back to the continuous domain at the next safe node."
Wu Tao stared at the matrix on the scratch paper for over ten seconds.
He didn’t object immediately. Instead, he pulled over a chair, sat down, took the pen from Chen Zhuo’s hand, and began to quickly calculate next to the matrix.
The conference room fell silent, leaving only the soft SCRATCH of the pen against the paper.
After about five or six minutes, Wu Tao stopped writing and shook his head.
"It doesn’t work."
Wu Tao pushed the scratch paper back in front of Chen Zhuo and circled the last line of the equation with the tip of his pen.
"Using a Discrete Matrix to sever the Singularity... the idea is wild, and it does get around the divergence, but you missed a step."
Wu Tao’s tone was certain, backed by the intuition of someone who’d long been doing foundational derivations.
"After the matrix closes, its Jacobian determinant is not equal to zero."
Chen Zhuo looked down at Wu Tao’s calculations.
"In a system of topological invariants, if the Jacobian determinant is non-zero, it means the space has become distorted."
Wu Tao leaned back in his chair and pointed to the blackboard.
"Then the subsequent isomorphic group doesn’t hold. That’s not right."
Chen Zhuo stared at the non-zero result, his fingers tapping lightly on the tabletop.
There was nothing wrong with Wu Tao’s calculations.
When it came to concrete, practical calculations, the doctoral student had an extremely sharp and solid grasp of the fundamentals, able to see the rejection reaction of the algebraic matrix within the topological space at a glance.
A non-zero Jacobian determinant means a distorted space.
Chen Zhuo’s mind raced. The continuous domain was a dead end, and discrete severing led to distortion.
’Then the only option is to manipulate the severing process itself.’
"Senior Brother Wu."
Chen Zhuo abruptly stopped tapping his fingers on the table and looked up at Wu Tao.
"The Jacobian determinant is non-zero because the matrix closes directly within the current dimension, right?"
Wu Tao nodded.
"Right. And that’s why the space gets distorted."
"Then what if we don’t let it close immediately?"

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