Scarlet really needed to do something about her hunger problems, but she wasn’t sure what she could do.

She couldn’t just turn her stomach off using magic, because all of her mana usage burned energy like you wouldn’t believe. In just a single day, Scarlet burned more calories than a normal person does in a month. Trying to go a normal length of time without eating was impossible. She’d starve to death if she went a single day without food. She needed a more effective way of getting energy, like a super energy drink or something.

...She wasn’t going to take off her mana regeneration clothes. So you can just forget that idea.

That was what was really going through her mind as she explained the finer details of her proof to the professor, realizing she had just introduced two new branches of mathematics to the world as a part of her proof, simply because she had been too hungry to care.

“I see, so once the countability of the collection has been proven you can use this… induction method. How useful. To think that problems could be approached in such a way.” The Professor mumbled to himself as Scarlet drew symbols in the air with mana, lacking a chalkboard.

Scarlet felt extremely awkward as the lesson continued and she taught the man what to her was basic math, but to him was space age logic from the future. Honestly, how have these people managed to get so far without proper mathematics? Mathematical induction wasn’t that young, was it? Oh god. These people didn’t have any sort of calculus either or anything beyond basic algebra and geometry. Did they even know what a matrix was?

It was no wonder all of their magic circles sucked, their engineers all had the equivalent of a high school education, and not even from a good high school. She supposed she should feel glad that the professor at least knew what a bijection was, even if he only knew it for finite sets and had to be explained to what it would mean for functions on infinite sets.

Scarlet herself had dove pretty deep into high level algebra and differential equations in her previous life in order to learn how to make an image recognition program for a bot to assist with one of the more unfair minigames she was struggling with. The fact that it was easier to learn how to make such a bot from scratch than it would have been to beat the minigame fairly just went to show how impossible it was.

...Everyone was watching her giving the professor a lesson, weren’t they?

A quick glance around the cafeteria showed her that yes, everyone was watching. Just great. Well, flying under the radar was always going to be a lost cause anyways. Before she ever arrived, she was known as that boy Mehal found in a cave somewhere.

A bell told in the distance and Scarlet stopped. “Professor, while I would like nothing more than to continue, I have to complete my placement exams.” Scarlet said, hoping to get out of this.

“It’s fine. I will inform the headmaster that you passed them all. Just continue. The exam for magic isn’t important. If you can get a magic based Class, then you will have no trouble with it.” The man said, waving her on.

“But… I need to demonstrate my practical combat skills, or else they won’t allow me to enter the Labyrinth.” Scarlet said.

“Enter the Labyrinth? Why in the devil would you want to do that?” The older man said, blinking in confusion. “Leave the Labyrinth for the Soldier Track Students. Magic Track Students have more important things to do. The magics that we create are far more valuable than any singular sword could ever be, continuing to protect the Kingdom for generations, even after our passing.”

...What?

It was true that individuals who managed to get a magic class were uncommon. Only a few hundred of them in the entire Kingdom, and the results that even a low Tier Class could achieve far outstripped their competitors. Even a Tier 1 Mage would be considered a valuable asset. Too valuable to be risking inside of a Labyrinth.

The enchantments and rituals that they could do made them worth a hundred swordsmen of a higher Tier.

Class Changes wouldn’t be achieved without hunting, but they weren’t common enough to be banking on, and it was usually the mage’s higher general skills in magical engineering that were valued, rather than their Class Skills or high stats.

...They wouldn’t ban her from the Labyrinth, would they?

“I am sorry sir, but I fully intend to enter the Labyrinth. As such, I really must be taking those exams.” Scarlet said, allowing the symbols she had been writing in the air to fade away.

“What!? No! I won’t allow it!” The old man bellowed as Scarlet grabbed her tray and made to buss her plates. “A youth as talented as yourself is too important to let die inside of the Labyrinth.”

“What I chose to do is none of your business. I came here so that I could fight alongside my young master. Not so that I could become some old man’s teacher.” Scarlet snapped, annoyed with the old man.

“Sca… Bran, that old man is Professor Hamlie. He’s um… One of the world’s foremost experts in mana field research.” Nico nervously stammered as he looked between the annoyed girl and the old man who looked like he was about to cry.

“Don’t care. I’m going.” Scarlet said flatly. 

He might have been an expert when it came to mana field research, but true to character, Scarlet just didn’t care. Not when the matter of growing levels was being threatened.

It wasn’t like he was better than Aqua was, and even the former lieutenant of the Order of the White Raven was basically just Scarlet’s intern at this point, charged with the tasks that Scarlet herself didn’t want to do. Like purchasing materials.

Besides, his knowledge on mana fields was pathetic compared to hers, since he didn’t have vector calculus to work with.

“Wait. Please. Can’t you just finish explaining the proof first?”

“It would take me another 10 hours to explain to you all the background you're missing.”

“I’ve got time!”

Professor Hamlie followed her all the way out the practice fields and continued to pester Scarlet as she stood in line for magic examination with Charles. 

They went to the magic line first, mostly because it was the shorter of the two lines, as even among Nobles, those capable of freely controlling their mana to the point of being able to use magic at the age of 10 were uncommon. Out of the 400 new students from across the Kingdom, only around 30 of them already had some form of magic.

Scarlet eventually gave in and just continued her lecture while standing in line, drawing stares from the proctors, who knew damn well who Professor Hamlie was, and were confused to see him receiving lessons from a child.

“Wait… Please. I can’t keep up… I have questions.” Professor Hamlie whimpered as Scarlet had started to use her [High Speed Invocation] skill to try to speed the process along. Sadly, it didn’t seem like he could understand 1.75x speed. “So many things I have never heard before. Child, where did you learn it all?”

“Ah… Learn it?” Scarlet said, swallowing a little. 

“Yes. Who was your teacher?” Professor Hamlie asked, directly this time.

...What was she supposed to say? She couldn’t exactly say she was self-taught, could she? That would make her seem like more than she already appeared to look like in the man’s eyes.

Should she say Aqua was her teacher? No, Aqua won’t be able to answer any follow up questions that the man might ask. Typhus? She might be expected of heresy just by association. ...Wouldn’t necessarily be untrue. William had a better understanding of the theory than Aqua at this point, but that might lead them to the fact that she was just Scarlet Tailor dressed as a boy.

“My teacher was someone by the name Rebecca Garza.” Scarlet said, tossing out her alias. Seemed like a safe thing to do. She had a few publications under that name, and it wasn’t like anyone would be able to track her down in order to ask her whether or not it was true. But the man’s reaction to that was larger than she was expecting.

“R...R...Rebecca Garza!? You mean the famed Hermit Alchemist!?” The old man shouted.

“Um… Hermit?” Scarlet replied awkwardly. She hadn’t exactly been keeping up with her own reputation, so his response came as a surprise.

When they said Hermit… that made her feel like they were calling her an old hag.

How rude!?

Apparently Professor Hamlie had been following her publications for the last 4 years, ever since one she did on the properties of different carbon structures and the creation and uses of carbon nanotubes.

...Also, it seemed like publishing around 150 papers in 7 years was considered a lot in this world. ...Was that a lot?

Honestly, she only kept doing it because William kept pestering her too.

“Please tell me, where can I find the master? There are so many things that I wish to ask her.” Professor Hamlie asked eagerly.

“I’m… not sure.” Scarlet said hesitantly. Not having expected things to go in this direction.

“I see. I suppose the Hermit Alchemist is a drifter then, always on the move. What a shame. I truly wished to meet her in person.” Professor Hamlie said with a look of complete and utter disappointment.

Scarlet didn’t trust herself to open her mouth. She could already see Charles biting his lip, trying to stop himself from laughing, with Leon rolling around on his side, bellowing loudly.

‘It isn’t that funny.’ Scarlet thought bitterly, wondering if she should teach the lion a lesson, when her ears perked up.

“A… Amazing. Mana velocity is 125.2 meters per second, with a mana purity of 93.0% and a density of 10.1 units per cm2.” The proctor at the station said.

He was operating a ring which would measure the speed and quality of the mana people pushed through it, which was the first of three tests that students were expected to complete.

And the proctor was right, those results were impressive. Impressive enough that Scarlet wondered why she hadn’t sensed the person who had expelled the mana doing it.

Though when she glanced forward, that started to make a bit more sense.

The child in front of the ring looked like he was only 7 years old, with a delicate frame and short blond hair that didn’t cover his slightly pointed ears.

An elf, or rather, a half elf.

Elves had bodies that could more naturally take in and use external mana, resulting in them being much more adept with [Mana Control]. Though because the mana flowed through them so unabated, they didn’t leave a strong fingerprint on the mana as it moved through their bodies.

Because of the lack of such a fingerprint, they weren’t able to manifest a Bond through the standard Awakening Ceremony. Even a half elf, like the boy here, didn’t have one. Something that felt as conspicuous as if he was missing an arm or a leg. Scarlet had had to tamper with things in order to allow Racheal to have a Bond of her own, even though she was ¾ human, using the blood of a full human relative, Charles, in order to muddy the water and allow something to grow. 

Scarlet suspected that this was the reason why Racheal was so... fond of her cousin after only a few meetings. Because her Bond had a bit of Charles in it.

“As expected of Prince Isaka.” An attendant boy said enthusiastically, though the small boy only hummed, looking away disinterestedly towards the part of the field where the weapon combat tests were going on, his fingers strumming the pummel of one of a pair of short blades at his waist.

While his elvish blood made him a natural when it came to controlling mana, it was clear where his interests were.

“Prince? Is he another one of the King’s grandchildren?” Scarlet whispered to Charles.

“No. Prince Isaka is the King’s youngest. The sixth born, with three elder brothers and two elder sisters.” Charles replied. The age gap in the royal children was great, with his eldest son was 38 while his youngest appeared to be turning 10 this year. He must have been slightly older than Scarlet and Charles, as she had not seen him at the Class selection ceremony. 

Scarlet watched the boy as he continued with his test, showing off a wind barrier spell at the second station before moving onto the third where he healed a mannequin of an artificial wound.

His healing spell wasn’t great, but it would do for small wounds. Though Scarlet took notice of how he was using [Evocation Magic].

Because of how their own bodies handle mana, Elves weren’t talented when it came to charging its properties though [Evocation Magic], but this boy was trying to do it anyways, despite [Conjuration Magic] being the more natural fit. Curious.

“Next please!” The person at the ring said, calling up the next student to take the test. 

“Well, wish me luck.” Charles said before walking forward.

The proctor confirmed Charles’ identity before inviting him to send his mana through the metal ring.

Charles sent his mana through the ring as hard as he could as the proctor took the results. “47.4, 28.9% and 1.3 units per cm2. It’s not bad, for someone who is on the soldier’s track. More than enough to cast basic spells with. Please proceed to the next station. Next!”

Scarlet walked up, with Professor Hamlie still following behind her.

To his credit, the proctor did his best to ignore the man and continue on with business. “Bran Silvana, got it. Please condense you mana as much as possible and fire it through the ring here.”

“Understood.” Scarlet said with a nod before lifting up one finger and pointing it towards the center of the ring.

There was a flash of light and a loud popping sound as Scarlet’s mana shot from her finger like some kind of deathray, passing straight through the middle of the loop and smashing into the wall on the far side of the field. Though the wall remained fine. A large number of eyes turned to see what had just happened.

“Ah…” The proctor simply broke as he looked down at the results. “Mana velocity of 384.2 meters per second, mana purity of 78.9% and a density of 383.1 units per cm2. That’s…”

“Still need to work on my mana purity then.” Scarlet mumbled.

Mana density was related to the scale of a spell’s effect. Mana velocity could affect the drop off rate for spells that are cast at a distance. Meanwhile mana purity affected the intensity of the effect.

“Could I try again?” Scarlet asked as she changed the way the mana flowed through her body in order to further refine her mana.

“N… No. That’s alright. Just go to the next station.” The man said, ushering her on.

“Oh… Alright.” Scarlet said, disappointed, but walking towards the next station.

“Impressive shot.” Charles said with a smile.

“That? It wouldn’t really hurt anything stronger than a level 10 boar.” Scarlet said dismissively. Her finger of death was more like a finger of especially strong winds. “It is no more powerful than a rookie’s arcane bolt spell.”

Her saying this caused Charles to flush in embarrassment as it was his turn to step up to give a display of what kind of magic he could do, and after a minute long chant, he fired off a rookie's arcane bolt spell.

Really, magic wasn’t his specialty. He only learned enough of it to use [Elemental Weapon Arts] and [Mystic Body], as well as some [Enhancing Magic].

“That’s good enough, I suppose.” The proctor at this station said, scribbling down some notes. “Can you use any [Healing Magic]?”

“Not as of yet.” Charles admitted.

“I see. Then you will be placed in a 5th year Practical Magics Class. You don’t need to attend it immediately, but you will not be permitted into the Labyrinth without at least three people who are rated at least at the 7th year. Understood?” 

“Understood.” Charles said with a nod.

So, they would need at least 3 people capable of using healing magic before they would be allowed entrance. 

Scarlet obviously could use it, and Sidney could as well. So they just needed one more. Nico probably had someone who knew it among those who he had gone into the Labyrinth with in his previous years. They could use one of them. Hopefully they would be willing to go as deep as Charles’ party was planning on going.

Since Charles couldn’t use healing magic, he simply stepped to the side without going to the third station, leaving Scarlet to walk up to the second station.

“Whenever you are ready.” The proctor said to Scarlet who nodded in understanding.

She was going to make sure she was fully passed.

Taking in a deep breath, she started her [High Speed Invocation] while releasing orbs of elemental mana from her body to float up into the air. Dozens of orbs, of all different attributes. She produced around a 15 of them every second throughout her entire minute-long cast, after which all one thousand orbs she had produced twisted themselves into several glowing spears and shot towards the training dumbs, hammering them over and over again and kicking up a large amount of dust.

“Is that good enough?” Scarlet asked as she finished her elemental barrage spell.

It had been a spell chosen to show off her high level [High Speed Invocation] and [Dual Evocation] skills, rather than for its destructive power, but it was still impressive.

Both the proctor and Professor Hamlie just gawked in shock at the display for a few moments before saying anything. “Just... go to the next station.”

DyslexicWriter

For those of you who don't know, formalized induction is only around 800 years old. Though some problems had been solved using the method before it.