Outside the window, the sky had grown completely dark.
The motion-activated lights in the hallway had flickered on for a moment, then went out silently as the footsteps of the last few students faded away.
The overhead lights in the office were off.
Li Jianming had only switched on the old eye-care lamp on his desk. Its white halo of light was diffuse, barely illuminating half of the blackboard.
The other half of the blackboard was lost in the dim shadows.
"This path is a dead end."
Li Jianming stopped the chalk in his hand.
He turned his head to look at the long string of Discrete Matrix transformation equations he had just written beside him.
"Isolating the Singularity was the right idea, but when we tried to wrap it in an algebraic cycle, the dimension of the complex manifold collapsed."
The old professor wiped the sweat from his forehead with the back of his chalk-dusted hand.
"The moment the dimension collapses, your first-order truncation is no longer just a simple cutoff of higher-order terms. It rips the entire topological space apart. It converges, yes, but in terms of mathematical logic, it becomes a stillbirth."
Chen Zhuo stood at the boundary between light and shadow.
He tilted his head back slightly, gazing at the dense array of symbols on the blackboard.
His cuffs were so smeared with chalk dust that their original color was hidden. He held a half-stick of chalk, unconsciously rolling it back and forth between his thumb and forefinger.
"If it’s a dimensional collapse..."
Chen Zhuo spoke, his voice still gentle but his pace slower than usual. The intense mental calculations had tinged his voice with a hint of fatigue.
"Professor, what if we don’t treat it as a complete topological space?"
Li Jianming froze for a second.
"What do you mean?"
Chen Zhuo took a step forward, into the halo of the desk lamp.
He raised his hand and tapped the chalk against the symbol on the blackboard representing the dimensional collapse, drawing a small X over it.
"Since it’s already torn, let’s follow the tear."
As Chen Zhuo spoke, he started a new line next to the X.
"We perform a dimension-reducing homomorphic mapping. We don’t require it to preserve the properties of the complex manifold globally, only that it aligns with the boundary of the truncation on the local rational algebraic closed chains."
Li Jianming stared at the half-line of the formula Chen Zhuo had written.
His eyes narrowed slightly.
"Dimension-reducing homomorphism..."
Li Jianming muttered the words, his mind rapidly running through the possibilities of this approach.
"If we only align it locally... then the Singularity is no longer a problem. It becomes a trivial term in a lower-dimensional space."
Li Jianming’s breathing grew heavier.
He snapped his head toward Chen Zhuo, a flicker of fire reigniting in his eyes.
"But if we do that, how do you close the outer integral? The dimensions don’t match. What will you use to fill the gap in the middle?"
Chen Zhuo thought for a moment.
"We’ll fill it with the intersection number of the algebraic variety."
Chen Zhuo handed the chalk to Li Jianming.
"You set the framework, and I’ll calculate the matrix for the intersection number. As long as the intersection number can be aligned on the field of rational numbers, we can forcibly fill the gap with an algebraic cycle."
Li Jianming didn’t take the chalk.
He looked at Chen Zhuo as if he were looking at an irrational monster.
Using the intersection number of an algebraic variety to fill a dimensional gap in a topological space.
The idea was utterly barbaric.
But logically, it seemed irrefutable.
Li Jianming turned around and picked a new piece of chalk from the box.
"I’ll take the left side, you take the right. We’ll meet in the middle, at the homology class."
Li Jianming’s voice carried an unquestionable decisiveness.
"Alright."
Chen Zhuo took back his half-stick of chalk and moved to the right side of the blackboard.
Wu Tao sat in a chair behind the sofa.
The black gel pen in his hand had run out of ink. He casually tossed the dead pen onto the coffee table, pulled a new one from a drawer, bit off the cap, and continued writing in his notebook.
His wrist was so sore it had gone numb.
He had already filled dozens of pages in his notebook, each one covered in the kind of derivations that would make one’s head spin just by looking at them.
He looked up and glanced at the blackboard.
The speed at which Li Jianming and Chen Zhuo worked was astonishing.
They barely exchanged another word.
As soon as one finished a step, the other would glance at it and immediately take up the next. Their rapport transcended language, becoming something akin to a purely mental synchronization.
The empty space on the blackboard dwindled.
Even the lamp’s halo of light seemed to dim amidst the intense derivation.
Li Jianming wrote down the final boundary condition for the complex manifold.
He stopped writing.
His chest heaved violently; his stamina was at its limit.
The old professor braced himself with his left hand on the wooden chalk rail at the bottom of the blackboard, his right hand still clutching the chalk as he turned his head to the right.
Chen Zhuo was still writing.
The chalk left trail after trail of white marks on the blackboard.
Under Chen Zhuo’s hand, the previously scattered and divergent Discrete Matrices began to converge toward the center, as if they were living things.
Finally, they coalesced into a single, concise algebraic cycle expression.
SNAP.
The half-stick of chalk in Chen Zhuo’s hand broke.
A small piece fell to the floor with a soft CLINK, rolling into the corner of the room.
Chen Zhuo paid it no mind.
Using the tiny stub that remained, he drew an equals sign at the end of the algebraic cycle expression.
Then, he connected the other side of the equals sign to the boundary condition Li Jianming had written.
On the left was profound and complex algebraic geometry.
On the right, the cold, precise Discrete Matrices.
On either side of the equals sign, they achieved a perfect, seamless union.
The first-order truncation that Li Jianming had once cursed as a "hoodlum’s trick" was now, enveloped by this massive algebraic cycle, completely closed.

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