The 10th-floor lab had been spacious for just the three of us, but the even larger 31st floor was almost too much. So this time, I set up bookshelves in the spare space to create a library. It was a collection of all the papers and books that I, Doloren, and Giuseppe wanted to read.
Now the lab smelled of paper, not parchment.
‘This is what a proper theory lab should be like.’
Feeling refreshed, I faced the envelopes before me. Laura had already filtered out the ones that didn't even look like answers, leaving 700 sheets. All of these were answers to the method of least squares problem.
I wondered if a genius on the level of Gauss existed in this world...
Rustle, rustle.
I checked people's answers one by one. Most of them were disqualified without even needing to be read in full. Still, I had to stay focused, not knowing when a real answer might pop up...
"This is just astounding. Wow...!"
My surroundings were distracting. It was because Doloren was being loud. She had foregone her own spacious desk and was lying on a cot, reading the paper I gave her.
What Doloren was reading was [The Optimality Theorem of the Method of Least Squares], my paper which contained the solution to the recent question.
"To think you can create the best solution even with errors..."
I could understand being surprised or moved by it. I was the same way as an undergraduate.
"Wow..."
But her frequent exclamations were a bit distracting. Moreover, Doloren wasn't the only one causing a disturbance today. At the desk right in front of me, Giuseppe and Gedrick were racking their brains.
"Professor Gedrick, do you understand this?"
Senior Giuseppe asked with a troubled look. Gedrick crossed his arms and furrowed his brow.
"Hmm... I can grasp it intuitively, but explaining it logically would be difficult. The formulas are too complex."
It was understandable. Matrix notation didn't exist in this world yet. So, having to write out the multivariable normal equations one by one, the formulas stretched on for dozens of pages.
It was as inefficient as proving planetary orbits using only geometry, without calculus. But to devise matrices from scratch myself... I wasn't a major in that field, and I was afraid I'd get the basic definitions wrong.
So I didn't touch it. The mages of this world would probably create it on their own later. ... I'd rather not be asked why I touched statistics when it wasn't my major either.
I didn't want to know this stuff either. I somehow gathered my focus and turned my gaze back to the answer sheets. Please, just let there be one.
I wasn't even hoping for a perfect answer. I just hoped for someone who had at least attempted something starting with the idea of 'minimizing the sum of squared errors.' After flipping through for a while, I discovered an interesting answer.
"Ooh..."
It was an answer that approached the problem by minimizing the sum of absolute values. A very excellent attempt.
‘But this creates non-differentiable points, so it's a fail.’
If you don't like it, you should have had an engineer grade it, not a theoretical physicist.
Next!
As I flipped through the answer sheets, I suddenly stopped.
‘This handwriting looks familiar...?’
[Answerer: Ludwig Everglow] ‘Count Ludwig?’
He was the most important person among the answer sheets I'd seen today. Moreover, the answer itself was perfect.
"Ooh..."
I admired it, but as I looked at the solving process, something was strange.
‘There's no... process?’
To be precise, there was a single function that not even Gauss's mother would know how it was derived, and next to it was a table with only the experimental data that fit the correct function cherry-picked.
‘Don't tell me... he guessed the correct function by intuition first, and then worked backward to select the measurements that fit it?’
To reach the result by omitting the cause... Is such a bizarre feat possible if you only dig into fire magic for generations? The world of magic is truly astounding...
"Well... it's amazing, but..."
Unfortunately, it wasn't the answer I was looking for. I'm sorry, Count Ludwig, but... you fail. By the time I finished grading, the sun had already set.
I looked around and saw that Doloren and Professor Gedrick had already left for the day. Only Giuseppe remained, wrestling with my paper. ... I really hope things go well for my Senior.
"Hoo..."
As I put down my pen, Senior Giuseppe looked up.
"Are you done?"
"Yes."
"Were there any correct answers this time? I doubt there were..."
I replied with a smile.
"No. There were."
"Oh?"
"Two of them, in fact."
"There were two?! Who on earth are they?"
Senior Giuseppe's eyes went wide. Who else would it be? The same people as always.
I held up the two answer sheets with the familiar names written on them.
The next morning, familiar faces gathered in the lab.
"Congratulations. You two are the only ones with the correct answer."
I clapped for Anna and Sir Valerius. I had secretly hoped to discover new talent... but the old guard is the best guard, as they say. Sir Valerius was smiling brightly as he skimmed through my paper.
"This time, at least I didn't get the definition wrong from the start. I like it."
It seems he was still bothered by not being able to complete calculus because of the Method of Continuity. Anna, on the other hand, was glaring at my paper with sunken eyes. It looked like she'd stayed up several nights over my problem.
"... To think that data selection, which I've always done by intuition, has to be proven through such an arduous process. It's almost bizarre."
It seems Anna had also approached it with intuition at first, like Count Ludwig. But in the end, she found a logical solution. Just like my sister.
I explained the future plans to the two of them.
"As you can see, the formulas in this paper are too messy for ordinary mages to understand."
Nod, nod.
They both agreed. I pointed at the paper and said.
"So, I'd like to ask the two of you to take on the role of commentators for this paper. On the condition that you are listed as co-authors, of course."
"Same as with calculus."
Sir Valerius chuckled.
"I'm counting on you."
"Fine. But this time, disciple of Doloren, you're the first author. Don't even think about running away like last time."
"Of course."
There was no reason to step back anymore. My reputation was already sky-high. I was only publishing it jointly because my conscience pricked me at the thought of taking all the credit for a new discovery myself.
I turned to Anna and asked.
"You're okay with that, right, little sister?"
"I am, but... I have something to ask you, brother."
Anna hesitated before speaking.
"If I graduate early, can you accept me as a researcher for the School of Magical Surveying right away?"
"... Early graduation?"
Anna explained.
"It's a special track only for those who have published a paper with a certain number of citations."
A student like that can apparently graduate from the academy early, regardless of their grades, if they publish one more 'high-quality' paper to the Magic Tower. And Anna had already written a paper with [Principia].
"So you plan to graduate with this paper?"
"Yes. I've met all the citation requirements with [Principia]. All that's left is for the people of the Magic Tower to deem this paper 'high-quality'."
I could guarantee it would be. If she wanted to become a graduate student that quickly, I had no choice but to accept...
"Are you sure you want to graduate so soon?"
Anna stared at my face and began to speak.
"... From the beginning, I was too naive to be called an academy student. After [Principia], I arrogantly believed I could predict anything."
"But I've gotten everything I wanted from academy life."
Anna took her academy student ID from her pocket and placed it on the desk.
"Little sister. If you do that, you won't be a student anymore."
Pfft. Anna laughed.
It was a laugh that held a genuine maturity, different from when she was just pretending to be an adult.
"Even if I leave the academy, everyone will still be there."
Was what Anna needed not an epiphany, but an advisor? I didn't know who she had met, but I hoped she would see it through to the end. I said to Anna.
"That is the right answer, mage!"
"?"
Take it all!
Soon after, the correct answerers were announced to the public. It wasn't separately featured in the Upperhills Newspaper, but... According to rumors, Isaac Upperhills had sent handwritten letters to the challengers who had unfortunately failed.
"Holy crap!! I got a reply! From Sir Isaac!"
"... What? Why'd only you get one?!"
"Well, it's probably because I wrote a better answer than you!"
All the mages who received Isaac's letter were delighted... but there was one person who couldn't be purely happy. Count Ludwig Everglow picked up Isaac's letter with a trembling hand.
[Greetings, Your Excellency the Count] [In the realm of flame, it would be no exaggeration to say that Your Excellency's magical intuition reaches the heavens...]
[However, since such magical intuition cannot be generalized...]
Squeeze.
Ludwig closed his eyes tightly.
"So that's how it is..."
It had been a strike with all his might. But it seemed it wasn't enough to surpass the heavens. A moment later, Ludwig opened his eyes and read through the model answer.
"To minimize the sum of squared errors..."
A refined solution. To be able to approach the truth without relying on instinct.
"Hahahaha..."
Although it did nothing to resolve the unpredictable attacks from his wife, Count Ludwig...
"Uhahaha."
He just laughed. Afterward, the magical society buzzed with the news of Valerius and Anna's joint correct answer. ... However, unlike this magical society, the general public was enthusiastic about different news.
[Upperhills Newspaper] [Dome Stadium Completed! -A New Symbol of the Capital's East- . In an interview with this paper, an official from the Magic Tower’s Administrative Office stated, "From now on, all thesis Demonstration Defenses (verification duels) conducted by the capital's Magic Tower and the Magic Academy will be held here at the Dome Stadium"...
The first to take this glorious stage will be the early graduates of the capital's Magic Academy this semester. Among them, it is known, is 'The Mage Who Reads the Stars,' Anna Upperhills. Furthermore, the Magic Tower official revealed that their verification will be handled by 'the Standard of the Empire,' Isaac Upperhills.]