I Know What Science Doesn’t

Chapter 006

The third-order Kawahara-type equation really was brutal.

Even when a result was already known in outline, writing out the intermediate steps took time. Writing them out carefully enough to compare them, test substitutions, and look for a hidden pattern took even longer. It was the sort of problem that punished impatience.

Daniel Hayes stayed in the study room until nine that night. He only realized how late it was when hunger stopped being background noise and started demanding attention. His shoulders were stiff, his fingers were stained with ink, and the stack of draft paper beside him had grown thick enough to look like a minor dissertation. He packed up, stepped out into the evening air, and found a roadside stall to deal with dinner. The food was cheap, hot, and greasy in exactly the way a long day of mental labor required.

After that, he headed back to the budget hostel where he had been staying. The room was cramped, the desk was barely more than a narrow board pushed against the wall, and the lighting was not kind to the eyes, but it was enough. He spread out his papers, uncapped his pen again, and kept working until three in the morning.

He woke at eight.

One cup of coffee later, the world felt newly assembled.

He flexed his fingers, rolled his neck, and let out a quiet breath. If this had been his previous life, one all-nighter would have wrecked the next day, and trying to catch up on sleep would only have left him more sluggish. But now he was in his early twenties, at that reckless peak when the body forgave almost anything.

That thought did not reassure him. It made him more cautious.

A healthy body was still the foundation of everything. If he ran himself into the ground, none of the rest would matter. Even the system reflected that reality. There was a physical-stat category sitting plainly on the interface, and the next upgrade required a score of seventy. Right now he was only at fifty-nine.

When he thought back over the last few days, the number was almost generous. He had nearly drowned, spent hours on the road, slept too little, and pushed his brain like he was trying to wring the last drop of value from every minute. The fact that his condition had not gotten worse was probably the real miracle.

By morning he was back in the study room, sitting in the same spot as if he had never left. He pulled out a fresh spread of densely covered papers and resumed the work without ceremony.

What he was studying was one specific phase in the equation analysis, a kind of approximate substitution-and-conversion step. The substitution process itself had gone surprisingly smoothly, but only because his judgment kept steering him toward the right options. It had not come from a standard textbook derivation. He had been working with something more intuitive, almost like a sharpened internal sense for which path was viable and which path was trash.

So the papers in front of him were full of alternatives. One candidate term after another. One transformation after another. He had marked the ones that felt correct, then lined them up against the original equation and the broader analytical process, trying to determine the rule hiding underneath all of it.

Three hours later, he found it.

Not a perfect theorem. Not a polished proof. But a pattern.

The pattern was not guaranteed to be universally correct, yet every one of the dozen-plus substitution items he had written out checked against the equation. More importantly, those items were not all variations of the same trick. They fell into seven distinct transformation forms. If one pattern could account for all seven, then he was no longer looking at coincidence.

That was why he could say the rule was probably right.

Probably, not certainly.

The difference mattered. Discovering a pattern and proving a pattern were not remotely the same thing. The gap between the two could be a lifetime of work.

It reminded him of the Goldbach conjecture in number theory. Goldbach had observed that every even integer greater than two could be expressed as the sum of two primes. Nearly three centuries had passed, and a formal proof still had not arrived. But every verified case supported it. Outside the narrowest, most rigorous corner of pure mathematics, most people were comfortable treating it as functionally true.

His substitution-conversion rule felt similar. He could see how the substitution should be done. He could see that the transformed equation remained extremely close to the original one. What he could not do, at least not on his own, was rigorously establish the mathematical principle behind the conversion or quantify the precise degree of approximation with full theoretical elegance. That was a different level of problem entirely.

He wrote the seven transformation forms onto a clean sheet and compared them to the original equation again, line by line, refusing to trust excitement alone. He stared at the page until the symbols blurred, then forced himself to recheck everything from a colder angle.

The conclusion he reached was simple and slightly irritating.

The proof problem was beyond him.

If he wanted that part solved properly, he would need someone near the top of the mathematical food chain, the kind of professor who lived and breathed partial differential equations and spoke comfortably in the language of functional analysis and group theory. Daniel Hayes was a physics PhD candidate. His main work had always been experimental electromagnetics and materials. He was not a professional mathematician, and he had never spent years specializing in those branches of theory.

Still, even without a proof, this was not a trivial outcome.

If the substitution-conversion rule held, then the transformed form of the original equation reduced the computational burden enough to make broad machine-led coverage feasible. In practical terms, that meant a computer could be used to generate an extremely close solution set rather than relying on laborious, hand-guided reduction. For actual applications, that mattered a great deal.

It was not just elegant. It was useful.

He leaned back and let the implication settle.

This equation showed up constantly in electromagnetics experiments and applied research. If his method really worked, then he had the skeleton of a paper already. Whether the field classified it as applied mathematics or computational mathematics was almost secondary. What mattered was that it had practical leverage, and practical leverage was something researchers respected even when the underlying proof remained incomplete.

He thought of one person immediately.

Ask Kevin Shaw.

Even so, he did not rush straight over. By then he had learned that arriving too early, too eager, often made a result sound weaker instead of stronger. People trusted work more when the person presenting it seemed calm enough to live with it for a few extra hours.

So he packed up, walked around campus for a while, and let the morning tension leak out of his body. The sun had climbed high enough to flatten the shadows across the pavement. Students drifted in loose knots between buildings, moving at that half-serious university pace where everyone looked busy even when they were not. Daniel Hayes ate lunch, kept his thoughts in order, and waited until almost two in the afternoon before heading to the Applied Electromagnetics Lab.

Kevin Shaw was usually somewhere in the lab building.

When Daniel Hayes reached the office on the first floor, he found several people already inside. Chris Fly was there. Brandon Brooks was there. A few associate researchers from other labs had joined them as well. The room carried the stale density of a discussion that had been going in circles for too long. Coffee cups sat on desks. Printed graphs were spread across a corner table. Nobody looked particularly happy.

One of the associate researchers noticed Daniel Hayes at the door and nodded with a polite smile.

At that moment, Brandon Brooks was still talking through the latest experimental failure.

"It's the same result again. So where exactly is the problem?" he said, his voice dry with frustration. "We were more precise this time than we were a few days ago. The setup was tighter in every respect. In theory, even the worst-case outcome should have shown better conductivity in the product. Instead, the result is worse than elemental iron."

That left the office briefly silent.

Then the discussion resumed from another angle.

Kevin Shaw spotted Daniel Hayes by the doorway and gestured toward an empty seat. "Over here."

There was no formal hierarchy in how people were sitting, so Daniel Hayes crossed the room and sat down without fuss. No one paid him much attention after that. The others were still focused on the failed run.

It only took him a few sentences to understand what had happened. The team's new attempt had failed again, and the result was almost identical to the previous one.

Brandon Brooks suspected the issue lay in the substrate structure. The experiment used bismuth telluride as a base, and through a carefully controlled process its surface was coated with multiple layers of bismuth atoms. The level of precision required for that procedure was extremely high. A tiny deviation in execution, or even a subtle error in the calculations behind the design, could be enough to wreck the outcome.

Chris Fly, on the other hand, avoided a direct judgment on the core procedure. His suggestion was to alter the geometry of the substrate and raise the temperature of the vacuum-tin vaporization process to increase reactivity. In his view, that might improve the chances of obtaining the desired growth behavior.

Once the two professors had finished, the rest of the room offered their opinions in turn. Some leaned toward process refinement. Some favored recalibration and another round of verification. Others floated adjustments that sounded plausible but lacked conviction. It was obvious that everyone was circling the same uncertainty from different directions.

At last only Kevin Shaw remained.

He said, "I'm in charge of running the experiment. If you have ideas, say them. I'll do the work."

Brandon Brooks and Chris Fly exchanged a look. Neither of them responded immediately.

The division of labor within the project was fairly clear. Brandon Brooks was the overall lead and had final control over the broader research direction as well as the experimental line. Chris Fly's group handled the calculation-related work and the data analysis. Kevin Shaw handled experimental design and execution, but in practice the detailed procedures came out of collective discussion. Nobody in the room needed that spelled out.

But Kevin Shaw was not finished.

He turned toward Daniel Hayes and said with a smile, "He'll be joining the experiment soon. Let him talk too."

That got everyone's attention.

Daniel Hayes?

A PhD student?

Several expressions in the room shifted at once. Some were surprised. Some were skeptical. Some simply looked curious about whether this was mentorship, theater, or a genuine invitation. Daniel Hayes himself was taken aback for half a second, but the encouraging look in Kevin Shaw's eyes made the intention clear enough.

He stood.

"I haven't formally joined the project yet," he said, steady and direct, "but I do know the basic flow of the research."

"The conclusion from the precursor study was that a substrate covered with multiple layers of bismuth atoms can grow high-quality stanene. Personally, I think that conclusion still needs to be questioned."

"The verification has failed twice. If we can't identify a concrete flaw in the experiments themselves, then the right move is to go back through the precursor study from beginning to end and re-examine it carefully."

"At the very least, that would either expose the weak point or give us stronger confidence that the precursor study was actually sound."

That was exactly what Kevin Shaw had wanted said.

The reason he had not wanted to say it himself was obvious. Much of the precursor-study work had been done under Chris Fly's side of the project. If Kevin Shaw openly challenged the validity of those conclusions, the discussion could turn personal very quickly.

But Daniel Hayes was different. He was a doctoral student who had not even fully joined the project yet. If he said the wrong thing, it was easy to treat it as youthful overreach. If he happened to say the right thing, then the room would remember him. It was a safe gamble with real upside.

There was another layer too. Kevin Shaw had been relatively isolated inside the project group. He had no PhD students under him, and even the associate researchers assisting with experimental work were more familiar with Brandon Brooks and Chris Fly. He had largely been standing on his own. Now that Daniel Hayes was here, he finally had someone who could speak from his corner and move with him instead of around him.

After Daniel Hayes finished that first point, the office fell quiet.

Chris Fly pressed his lips together and said nothing.

If Kevin Shaw had been the one making the criticism, he would have pushed back immediately. But this was a student. What exactly was he supposed to do, start swatting at a doctoral candidate in front of everyone? The only dignified move was to smile and let it pass.

Brandon Brooks also pursed his lips, though for a different reason. The truth was that he, too, suspected something might be off in the precursor study. But the earlier data had already been checked multiple times. Even if they doubted it, what practical step remained? Review it again? Recalculate everything again? The most likely result was still the same conclusion.

But Daniel Hayes was not finished.

He hesitated for a brief moment, then continued.

"Over the last two days, I've been working on a mathematics problem connected to the third-order Kawahara-type equation," he said. "It's one of the equations commonly used in electromagnetic experimental analysis."

"I found a way to simplify the computational burden enough for broad computer-based coverage. In principle, that should allow the precursor-study data to be recalculated and reanalyzed with much more complete numerical support."

The room went dead still.

Every gaze in the office fixed on him, and the looks were strange enough to border on surreal. Even Kevin Shaw stared at him differently now. He had expected Daniel Hayes to offer a sensible procedural opinion, maybe even a useful one. He had not expected this.

Two days?

A result already?

And not just any result, but one tied to the third-order Kawahara-type equation used in electromagnetic data analysis?

For a moment, the silence in the office carried more weight than any outburst could have.