I Know What Science Doesn’t

Chapter 007

The lab went quiet.

Everyone was looking at Daniel Hayes now. The expressions around the room varied from curiosity to open disbelief, but they all seemed to carry the same unspoken question.

Did he have any idea what he was saying?

A physics PhD student claiming he had produced a meaningful result in computational work on a partial differential equation was already a stretch for most people in the room. Claiming that the result involved the third-order Kawahara-type equation pushed the whole thing into the realm of absurdity. It sounded as out of place as hearing that the quietest engineering kid on campus had somehow aced a constitutional-law exam without ever taking the class.

Of course, nobody would deny that mathematics and physics were closely related. They had always fed into one another. But when people talked about that overlap, they usually meant theoretical physics, not the day-to-day life of an experimental researcher. And Daniel Hayes was clearly on the experimental track. He had come up under Kevin Shaw, which meant laboratory work, materials, instrumentation, and practical results, not deep original theory on advanced differential equations.

The first person to break the silence was Chris Fly.

He turned to Kevin Shaw with a half-smile that carried more ridicule than warmth. "Your student is impressive," he said. "A physics PhD student making a breakthrough in computational mathematics, and on the third-order Kawahara-type equation of all things."

Then he added, "Did you teach him that?"

The tone made his real meaning obvious.

Several others wore the same expression. Even if they were willing to entertain the idea that a physics doctoral student might produce a mathematical insight, the specific equation made the claim hard to swallow. Research on the third-order Kawahara-type equation touched multiple application areas, and scholars across several mathematical specialties had studied it for years. Most published results only found particular cases where the equation could be simplified enough to handle. Those were useful in narrow settings, but they were nowhere near broadly applicable methods.

And this result was supposed to have come from Daniel Hayes?

That same Daniel Hayes who had not even officially joined the project yet? That same student who still carried the shadow of a data-fabrication controversy around his name, regardless of Kevin Shaw's insistence that the experimental data had been genuinely measured?

The room had reasons to doubt him.

Kevin Shaw looked over with a frown and leaned slightly closer. "You're not joking?"

"Professor Shaw, I wouldn't joke about something like this."

Daniel Hayes answered seriously, without trying to soften the statement or dress it up with a show of confidence. If anything, his straightforward tone made the words hit harder.

"Then explain it," Chris Fly said with casual indifference. "Let's hear whether it actually makes sense."

The words had barely settled when there was a knock at the door.

A thin, balding middle-aged man was standing outside. Thick lenses rested on his face, giving him the unmistakable look of someone who had spent a large portion of his life staring down complicated symbols and refusing to compromise with them.

"Kevin Shaw? Professor Shaw?"

The man called out Kevin Shaw's name with the urgency of someone who had not come for small talk.

Kevin Shaw rose and walked over. It took him a moment to place the visitor before he managed, somewhat awkwardly, "Professor Monroe..."

They were not close. George Monroe was a professor from the College of Mathematics, while Kevin Shaw belonged to a different school altogether. In half a year, they had probably crossed paths only once or twice. Remembering a surname was already respectable.

"Good, good." George Monroe stepped forward and shook his hand immediately. Then he asked, "Are you taking on a doctoral student named Daniel Hayes?"

Kevin Shaw turned instinctively and glanced toward Daniel Hayes, then nodded with obvious confusion. "He starts in March..."

"Then it's him. Do you have his contact information?"

When he saw the strange look on Kevin Shaw's face, George Monroe explained at once, "Here's the situation. I need to ask him about a mathematics problem. I've been thinking about it since yesterday. I've kept turning it over in my head and I still can't make sense of it."

That answer only made Kevin Shaw more stunned.

He looked back at Daniel Hayes, then at George Monroe, as if hoping the logic of the scene would reveal itself if he checked both ends of it again. It felt backward. Completely backward.

George Monroe was not just any mathematics professor. His work in partial differential equations had real weight, the sort that reached beyond a campus reputation and mattered internationally. A scholar at that level had shown up to ask a mathematics question.

And the person he wanted to ask was Daniel Hayes.

For a moment Kevin Shaw simply could not process it.

By then Daniel Hayes had already walked over. After catching enough of the exchange to understand the direction of it, he asked, "Professor Monroe, were you looking for me?"

"He is?" George Monroe looked toward Kevin Shaw.

"You don't know him?" Kevin Shaw stared for half a beat, then said quickly, "He's Daniel Hayes."

Recognition flashed across George Monroe's face at once. "So it really is you. Excellent, excellent."

He pulled a stack of papers from his bag and practically thrust them forward. "Take a look. Is this your analysis?"

Daniel Hayes accepted the pages and read them carefully. They were the notes he had written for Melissa Ken's equation analysis.

He nodded. "Yes. I wrote this."

"Then I found the right person." George Monroe let out a breath and immediately pointed to a place on the second page. "This substitution-conversion here, and there are a few later passages that look similar. They should be using the same method."

"I checked the transformations. They're correct. But I can't see how you did it."

That was the heart of it.

After Melissa Ken had come to him the previous day, George Monroe had spent nearly all of his time trying to understand the substitution-conversion steps. He had thought about them in his office. He had thought about them after going home. He had apparently gone to bed still thinking about them. What initially seemed like a small question had refused to yield anything at all.

He had gotten stuck, and the failure had clearly irritated him.

That was why he had gone so far as to check the incoming doctoral-student list for March, discovered that Daniel Hayes would be studying under Kevin Shaw, and come directly to the lab building in person.

Once the pieces clicked into place, Daniel Hayes understood immediately.

He looked around. The doorway to the main office was not the place to discuss higher mathematics in any serious way. Too many eyes, too much noise, too many distractions.

"Let's find somewhere quiet," he said.

"Yes, yes. Somewhere quiet."

Kevin Shaw pointed across the hall. "Use that small office."

So George Monroe and Daniel Hayes crossed into the smaller room opposite the main office.

The large office behind them sank into an odd mood. People looked at each other, then away, then back again. A few wanted to resume discussing the experiment, but there was no real momentum left in that conversation. The obvious issues had already been raised. The possible fixes had already been proposed. They were all stuck in the same place, which was why some of them had started to suspect that the problem might lie not in the new experimental attempts, but in the conclusions drawn from the precursor study.

Kevin Shaw was one of the people who believed that.

So were Brandon Brooks and several of the associate researchers, though not all of them were equally willing to say it aloud. The difficulty was that confidence in the precursor study remained high precisely because the underlying data had already been checked multiple times, and every recheck had produced the same result. If they recalculated it all over again, odds were they would simply land on the same answer once more.

After a short silence, Brandon Brooks said thoughtfully, "What Daniel Hayes said just now was that his research could simplify the coverage-level computational burden for the third-order Kawahara-type equation, right?"

Everyone nodded.

When he had first said it, the claim had sounded like fantasy. But now a professor like George Monroe had come to ask him a mathematics question in person.

That changed the atmosphere.

Could it be real?

"I'm going to take a look," Kevin Shaw said at last. He could no longer sit still.

He stood and went toward the smaller office.

Behind him, Brandon Brooks turned to Chris Fly. "What do you think?"

"It can't be that easy," Chris Fly said. But his tone was less certain than before. "The third-order Kawahara-type equation is too complicated. Too many people have worked on it already. However good Daniel Hayes is, he's still a physics doctoral student."

He paused, then admitted, "If it really works, though, that would be something."

Inside the smaller office, George Monroe's question turned out to be exactly the issue Melissa Ken had failed to understand: how the approximate substitution-conversion had been carried out within the equation analysis.

He could not see the mechanism.

Daniel Hayes was not surprised. He might be only a physics PhD student, but even he knew how difficult the problem was. Even now, what he had truly found was not a proof, only a pattern governing the approximate substitution-conversion. The correctness of that pattern had not been formally established.

When he had first worked through the algebraic changes, he had been guided by that sharpened internal sense that kept steering him toward what was right.

"I don't really know why the substitution has to be done this way," he said plainly. "But I found the pattern."

That made George Monroe pause. He looked back down at the substitution-conversion steps on the page, and then his expression shifted. As a scholar who specialized in partial differential equations, he knew perfectly well that mathematics was full of discoveries like this. People found a rule first. The proof came later, if it came at all.

That was one of the oldest shapes in the discipline:

spot the pattern first, then spend the rest of the time trying to prove it.

Daniel Hayes was about to keep explaining when George Monroe interrupted him with a different question.

"Have you submitted a paper?"

"Not yet. I'm preparing one now. It should be soon."

"Then do it quickly. Finish the manuscript and get a preliminary submission in first. You need to lock the result down under your own name."

He shook his head, almost reluctantly. "In that case, don't explain the full pattern to me yet."

There was a reason for that caution. The pattern itself was already a meaningful result. The third-order Kawahara-type equation had broad practical use, and in most industrial or experimental applications, people relied on substitution-based numerical methods to approximate solutions. A method that could slash the computational load and allow wider coverage-based calculation could lead to more accurate solution sets. The value of research was never measured only by how difficult it was. Application mattered too, sometimes more than people liked to admit.

And this was the kind of thing with real application value.

Daniel Hayes understood what he meant. He smiled easily. "That's fine, Professor Monroe. I'll organize everything tonight and upload the core result first."

"Good. As soon as possible."

Only then did George Monroe settle down and listen in earnest.

In truth, Daniel Hayes was not especially concerned. What he was explaining represented only part of the full pattern. There were seven transformation forms in the approximate substitution-conversion rule, and only two of them were relevant to Melissa Ken's equation. The other forms that applied to different equations could still be worked out by comparison, but doing so would take time.

That was enough time to get a preliminary submission filed.

When Kevin Shaw stepped into the office, he found Daniel Hayes in the middle of a calm, methodical explanation.

Across from him, George Monroe was listening with complete seriousness, nodding every so often.

The rule behind the approximate substitution-conversion was somewhat intricate, but because the audience was a scholar in partial differential equations, many sections only needed brief explanation. Shared background cut away a great deal of effort. The two transformation forms tied to Melissa Ken's equation were covered quickly.

When Daniel Hayes finished, George Monroe did not stop there. He immediately began applying the pattern himself, carrying out the approximate substitution-conversion and checking the results on paper.

Standing to one side, Kevin Shaw could not help asking, "You're talking about that research of yours?"

"Yes."

Daniel Hayes gave him a brief explanation.

Kevin Shaw followed the general idea, but the shock in his chest did not go away. Simplifying computation for the third-order Kawahara-type equation was not the sort of thing anyone expected from a physics PhD student, much less one who was supposed to be heading into experimental work.

He looked toward George Monroe, who was still focused entirely on verification, and decided it was better not to interrupt again.

Before long, Brandon Brooks and Chris Fly also came over. They pulled Kevin Shaw aside, and he gave them the same condensed explanation.

They were still talking in low voices when George Monroe suddenly stopped writing.

He looked up, visibly excited.

"The pattern is correct."

He tapped the paper in front of him. "Three different equations. The same approximate substitution-conversion method. And every verification comes out extremely close."

Then his excitement surged into something less restrained.

"Excellent. This is excellent. This rule is really something."

He stood so quickly his chair scraped the floor.

Turning toward Daniel Hayes, he asked, "You haven't officially enrolled yet, correct?"

That question made Daniel Hayes blink.

In the next moment George Monroe grabbed his hand, his whole face lit with conviction.

"A person who can discover a rule like this is absolutely a genius. A real mathematical genius."

"A talent like yours belongs in the Department of Mathematics. Why are you studying physics? Experimental physics, no less. That's wasting talent."

"Ivystate University allows students with a physics master's degree to apply across disciplines for a mathematics PhD. How about it? I'll file a special request myself. Come study under me."

"Once you enroll, we can work on this rule together and prove it properly."

"With a result like this, you could easily become one of the top doctoral students in your cohort. You might even win an award. Later on, staying at the university would be no problem at all..."

Daniel Hayes froze for a beat.

Off to the side, Kevin Shaw's face darkened on the spot.