My IQ has been increasing year by year.

The sun shines brightly on USTC in the early morning.

There weren't many people at the cafeteria window.

Chen Zhuo bought two fried dough sticks, a meat bun, and a cup of steaming soy milk. The breakfast vendor leaned against his stainless steel frame, yawning, while the morning news played on a radio in front of him. Chen Zhuo took a bite of the meat bun and walked along the path towards the old library.

The trees on both sides of the road are very lush, shading most of the road.

The air was a bit stuffy, carrying the damp, grassy scent unique to southern summers.

Chen Zhuo pushed open the glass door on the first floor of the old library.

The lobby was the same as always. The electric fan behind the borrowing area oscillated gently, and the old librarian, holding a teacup, was looking down at a newspaper that had been delivered yesterday. I pushed open the wooden door to the foreign language periodicals reading room on the third floor.

The ceiling fan overhead was turning slowly.

Su Wei has arrived.

She sat in her usual spot by the window, with the large water bottle beside her.

The same few sheets of draft paper were spread out on the table as yesterday, but the formulas on them looked much cleaner than they had a few days ago. She had become very proficient in using the discrete matrix of Markov chains, and there were far fewer scribbles and corrections on the paper.

Hearing the door open, Su Wei looked up.

Chen Zhuo pulled out a chair and placed his shoulder bag on the table.

Su Wei responded, her pen still in her hand, continuing to calculate along yesterday's train of thought.

Instead of heading straight to the bookshelf as usual, Chen Zhuo took out a small notebook from his bag, tore off a blank sheet of paper, pulled a ballpoint pen from the pen holder, and wrote a few lines on it. After finishing, he stood up, walked to Su Wei's desk, and gently pushed the note towards her.

After Su Wei finished calculating the step in her hand, she stopped writing, picked up the piece of paper, and glanced at it.

Her gaze lingered on the note for a moment, and she raised an eyebrow slightly.

"Those that arrived in the last six months? Are you no longer considering historical bound volumes?"

Su Wei raised her head, a hint of surprise in her eyes.

For the past few days, Chen Zhuo had been asking her to find old antiques from the 1960s and 70s, books with yellowed pages that crumbled when turned. Today's request was quite a leap. "Hmm."

Chen Zhuo pulled over a chair and sat down, his tone gentle.

"After looking at so much of the past, I should at least see what the houses look like now. If I stay in this pile of old papers any longer, I feel like I'm getting old." Su Wei gave a rare smile and tucked the note under the pen holder.

"The newly arrived periodicals haven't been shelved yet; they're all in cardboard boxes in the innermost storage room. We haven't had time to unpack and catalog them."

She stood up and casually pushed the chair back under the table.

"Wait a minute, I'll go to the back and see if I can find the math category you're looking for."

"This is troublesome."

Su Wei took out a bunch of keys from her pocket and turned to walk towards a small wooden door at the far end of the reading room.

The sound of the door lock being opened came from inside, followed by the dull thud of a cardboard box being moved.

About ten minutes later, the small wooden door was pushed open.

Su Wei pushed out a small iron cart with wheels.

The cart was stacked with three or four rows of brand-new foreign language journals, wrapped in transparent plastic film and tightly bound by several packing straps. She parked the cart next to Chen Zhuo's table, took out a utility knife from her pocket, and cut along the edge of the plastic film.

"It's all here."

Su Wei tore off the plastic film.

"The 'Annals of Mathematics,' the 'Journal of Combinatorial Theory,' and several issues of 'Discrete Mathematics' are all collections from the first half of this year. They were just sent from abroad and are not cataloged. When you look at them, please don't mix up the ones from different months, otherwise it will be very troublesome for me to archive them later."

"Don't worry, I'll put it back after I'm done reading it."

Chen Zhuo looked at the neatly covered periodicals on the cart and nodded.

Su Wei didn't say anything more, turned around and went back to her seat to continue dealing with the actuarial data.

Chen Zhuo picked up a copy of "Combinatorial Theory Journal" from the top.

The pages are smooth, and the layout is much clearer and prettier than those manuscripts typed on typewriters decades ago.

The abundance of English letters and neatly arranged formulas exudes a rigorous industrial feel.

He opened the table of contents, picked out an article about probability graph theory, and began to read it slowly.

He looked at it very carefully.

Only after he truly entered the world of mathematics could he deeply understand its infinite charm.

For example, this article right now.

The author is a French mathematician, and the article explores the threshold function of random graphs under certain specific conditions.

To prove the inevitability of a low-probability event, the author used a very ingenious quadratic moment method.

Chen Zhuo followed his logical deduction step by step. When he encountered a step with a large leap in the derivation, he would stop, take a piece of scratch paper, and fill in the missing parts of the proof himself. The scratch paper was soon filled with the derived formulas.

After completing the final step and arriving at a conclusion that was completely consistent with the paper, Chen Zhuo put down his pen, picked up the kettle, and took a sip of water.

He nodded to himself.

The Frenchman found the right angle, and his logical loop was flawless.

Using probabilistic methods to solve deterministic graph theory problems is a cross-disciplinary way of thinking that is quite advanced in today's world.

This brought him a pure sense of pleasure, like having a quiet conversation with another intelligent person across time and space. He copied several core inequalities from the paper into his notebook, then closed the magazine, put it back on the cart, and picked up another. Time flowed slowly as he turned the pages.

The sunlight slowly moved from the east-facing window to directly overhead, brightening the reading room.

At noon, the two of them went to the cafeteria for lunch, and when they returned, they each occupied one end of the long table without disturbing each other.

The air became even more muggy in the afternoon, and the ceiling fan overhead seemed to be struggling to keep up.

Chen Zhuo picked up the latest issue of *Diserete Mathematics* from the cart. It had a dark blue cover and felt quite heavy. He flipped through the table of contents, glanced at it casually, and his gaze settled on an article with a very long title.

The article discusses the problem of proving the lower bound of a specific type of bipartite graph.

Chen Zhuo had encountered this problem before when reading old literature. It is a classic and tough problem in combinatorics. Many mathematicians have tried to raise the value of this lower bound, but it has been difficult to find a universal proof path.

He turned to the page number of that article.

The article was very long, taking up more than thirty pages.

The author is a professor at a British university. Chen Zhuo calmed down and began reading from the introduction in the first part.

The author's approach is very traditional and orthodox.

To prove that lower bound, he adopted a pure combinatorial construction method. The paper defined numerous subgraph structures and then pieced these structures together like a jigsaw puzzle. For each piece, a lemma was needed to prove that the joining was logically valid and would not violate the original graph theory properties. Chen Zhuo looked at the densely packed pages of subgraph classifications and constraints.

The first scenario assumes that the vertex degree is greater than a certain value.

The second scenario assumes the existence of a specific loop.

The third kind of situation....

The author wrote very meticulously.

Every step of his derivation was correct, and every proof of his lemma was flawless. He was like an extremely patient bricklayer, using bricks and cement to build this wall, bit by bit, layer by layer. He didn't take any shortcuts; it was all solid, hard work.

After reading this passage, Chen Zhuo leaned back in his chair and rubbed his temples.

In academia, an article that thoroughly and exhaustively solves a problem using exhaustive deduction is absolutely qualified to be published in a core journal. However, as he followed the author's line of thought, another image involuntarily popped into his mind.

These past few days, his mind has been filled with tools for algebraic matrices.

As he looked at the complex figures that were divided into dozens of cases for discussion in geometric space, an idea suddenly popped into his head.

What is the essence of graphics?

It refers to points, and the connections between points.

What if we abstract these intricate connections directly into an adjacency matrix composed of 0s and 1s?

Once the graph is transformed into a matrix, the graph-theoretic properties that have been repeatedly discussed in these thirty pages become clear.

For example, connectivity, binary search, or even that troublesome lower bound value.

Does this then become a problem of finding the eigenvalues ​​of this matrix?

Chen Zhuo's eyes lit up slightly.

He didn't think he was smarter than the professor; he just happened to have honed his discrete algebra thinking to an almost instinctive level that summer. The professor was looking at the problem from a purely combinatorics perspective, so he could only piece it together bit by bit.

Chen Zhuo now happens to have an interdisciplinary measuring stick in his hand.

He sat up straight again, pushed the book "Discrete Mathematics" aside, and picked up a clean A4 sheet of paper.

He simply wanted to give it a try.

Try using algebraic tools to see if you can simplify this cumbersome construction process a little.

He picked up a black ballpoint pen, wrote a basic graph theory definition at the top of the paper, and then drew the corresponding matrix directly below it. The pen tip made a uniform scratching sound as it touched the paper.

Chen Zhuo wrote with great focus.

He didn't even realize when Su Wei walked up to his table.

The deductions conducted that afternoon were not as straightforward as one might imagine.

By forcibly transforming a purely combinatorial graph problem into the dimension of algebraic matrices, the first step of the mapping was indeed very smooth. The connectivity, which originally required lengthy descriptions, was easily crammed into a symmetric matrix.

But this is just the beginning.

In that 30-plus-page paper, the original author listed extremely complex boundary conditions in order to prove that lower bound.

To compress all these conditions without loss into the range of matrix eigenvalues, Chen Zhuo needed to construct several very ingenious inequalities for scaling. This couldn't be done in the blink of an eye.

It takes time to repeatedly compare and try different algebraic tools.

Chen Zhuo's pen tip hovered in mid-air, stopping in front of the range of values ​​for an eigenvalue.

He slowly erected scaffolding in his mind.

Su Wei had originally gone to the water room at the end of the corridor to wash her cups, but on her way back, she glanced at Chen Zhuo.

These past few days, Chen Zhuo has been reading very quickly, often flipping through a few pages, jotting down a line in his notebook, and then continuing to read. But this afternoon, he had been sitting at this table writing for almost an hour. She stopped and looked at the draft paper in Chen Zhuo's hand.

The paper didn't contain the usual scattered sentences; instead, it was filled with neatly arranged matrix derivations, line after line.

From top to bottom, although the project stalled halfway through, the underlying logic was very solid, like a code being slowly and carefully deciphered. "Shall we stop looking at it today and start putting it into practice?"

Su Wei, holding a clean water glass, stood by the table and casually asked a question.

Chen Zhuo stopped writing.

He didn't rush to write anymore. Instead, he capped the pen, casually put it aside, and shook his slightly sore wrist.

He raised his head, shook his slightly sore wrist, looked at Su Wei standing beside him, and gave her a gentle smile. "It doesn't count as doing a problem."

He pointed to the open book "Discrete Mathematics" next to him, his tone casual and slightly joking.

"Seeing how hard others work to build with blocks, piecing them together one by one, I figured I had nothing better to do, so I tried to see if I could just use a wooden board to support it." Su Wei glanced down at the all-English journal.

The densely packed diagrams and conditional branches made her frown subconsciously. She glanced again at Chen Zhuo's half-finished matrix draft. She didn't understand the depth of graph theory, but she could see the huge difference in scale between the two.

More than thirty pages of printed text.

A handwritten draft of less than two pages.

"Have the wooden planks been moved over yet?"

Su Wei took a sip of the cold water in her glass, her tone remaining calm, as if she were asking what she would have for dinner.

"not yet."

Chen Zhuo picked up the draft paper and examined it against the light.

The algebraic framework above has been established, the general direction is correct, and the simplicity brought by dimensionality reduction is indeed present.

However, to truly turn it into a perfectly sound mathematical proof, it will take some painstaking work to fill in the gaps in the middle.

"I just cut out the shape of the wooden board."

Chen Zhuo put down the paper and smiled.

"We still need to find some suitable nails to hammer in and secure it. It's delicate work, and we won't be able to finish hammering it today."

Su Wei didn't ask any more questions.

She nodded, picked up her water glass, and returned to her seat to continue working with the actuarial data.

For her, even if Chen Zhuo calculated all the stars in the sky, it wouldn't be as meaningful as her solving a single problem correctly.

Chen Zhuo looked away.

He did not force himself to finish the remaining inequalities by contraction that evening.

Doing academic research is like making soup; it will naturally be ready when the heat is right. Forcing it with high heat will only dry out the soup and ruin its original flavor. He neatly arranged the two half-finished draft pages and tucked them into his notebook.

Today's mental workout made him feel very comfortable.

I've found an interesting angle. All that's left is to spend a little time each day during this long summer, like polishing a small wooden carving, gradually smoothing it out. No rush.

Summer vacation has just begun, and he has plenty of time to slowly work through this problem.

He closed the magazines on the table, stacked them neatly, and put them back on top of the cart.

It was already getting dark outside.

The last rays of the setting sun shone on the glass windows of the reading room, casting a warm orange hue.

Chen Zhuo checked the time; it was almost time to go to the second canteen for dinner. Yesterday, when he passed by the window, he saw on the blackboard that sweet and sour pork ribs would be served for dinner. He picked up his water bottle and slung his bag over his shoulder.

As he passed Su Wei's table, he greeted her as usual.

"Um."

"Same time tomorrow."

Chen Zhuo casually gave a brief explanation.

"Know."

Su Wei turned a page of the draft paper, her tone as calm as an emotionless robot.

"I've put all the books you want to read under the cart; you can get them yourself tomorrow."

"it is good."

Chen Zhuo pushed open the wooden door of the reading room.

The breeze in the hallway blew in through the half-open window, carrying a touch of evening coolness, and it was no longer as stuffy as in the afternoon.

That evening, carrying two light, unfinished draft papers, he leisurely went downstairs as usual.

There was no excitement at discovering the truth, nor any impatience to prove oneself.

He walked at a leisurely pace.

Rather than proving a graph theory lower bound, he was more concerned about whether the shaky cook at the second canteen would take pity on him and give him a couple more pieces of sweet and sour pork ribs, considering he was still young.