Chapter 23: The Calculus of Magic

The barrier atop the highest spire vanished.

Adèle perceived it instantly.

She set down the alchemy crucible she held, then ascended the stairs with swift steps, halting at the threshold of Klein’s study.

The door opened from within.

Klein stood behind it, his already pale complexion seeming even more pronounced after two months of secluded study, his black hair meticulously neat.

No trace of magical fluctuation emanated from him; his presence was as still and profound as a deep well.

“Master, you have concluded your research,” Adèle said, bowing respectfully.

Klein nodded, then walked behind his desk and took a seat.

“What has transpired in the tower these past two months?”

“Everything is in order.

Martin’s period of penance concludes in ten days, and experimental materials have been fully replenished according to the quarterly inventory,” Adèle reported with clear, concise articulation.

She paused for a moment.

“And . . . Lia, during our discussions in the alchemy lab, put forth a hypothesis concerning the composition of matter.”

Klein remained still, listening in silence.

“She posits that the myriad materials in the world might all be composed of dozens of ‘fundamental materials’ that cannot be further broken down,” Adèle recounted.

“She had christened these fundamental materials ‘elements’.”

The study was quiet.

Adèle continued, “She believes these elements can be discovered through decomposition.

I’ve already begun attempting to decompose cinnabar and malachite, following her reasoning, and I’ve already achieved some preliminary results.”

“Ancient texts?” Klein inquired.

“Lia did not mention any,” Adèle replied.

“She said it was merely a hypothesis.”

Klein did not press further.

“You may leave.”

“Yes, Master.”

***

After Adèle departed, Klein remained seated in his study for a considerable time.

Before him lay two stacks of parchment: one, Lia’s somewhat hastily penned initial draft of *Principles of Calculus*, and the other, his own meticulously rewritten and organized version, nearly double the thickness.

He rose, picked up the thicker stack of manuscript, and exited the study.

Inside Lia’s room.

She was sprawled across her desk, idly prodding a parchment, covered in whimsical drawings of small turtles, with the blunt end of a quill.

Two months had passed.

Klein hadn’t emerged.

Edgar remained unheard from.

Adèle was hopelessly engrossed in decomposing minerals.

She felt as though mushrooms might soon sprout from her.

The door swung open.

Lia looked up, seeing Klein enter.

He appeared no different than he had two months prior.

“Master,” Lia said, straightening her posture.

Klein approached her, placing the thick stack of parchment he held onto her desk.

“I have finished reading your manuscript.”

“Oh,” Lia responded.

‘As long as my head wasn’t blown off, I suppose.’

‘Casualties were an inevitable part of academic discourse, but it was certainly preferable if her boss wasn’t among them.’

“This is my organized version,” Klein said, gesturing to the new stack of manuscript on the desk.

Lia reached out and took it.

It felt heavy.

She opened the first page.

His script, meticulously neat, appeared almost printed.

The title remained unchanged: *Principles of Calculus*.

However, a subtitle now accompanied it: “A Universal Computational Method for Describing Change and Accumulation and Its Application in Magical Model Construction.”

She continued to flip through the pages.

Chapter One: Limits.

Lia’s original definition had been rephrased by Klein, rendered in language more concise and more aligned with the habits of scholars in this world.

And at the end of the chapter, an entirely new section had been added.

“Appendix One: Application Calculus of the Extremum Method in the ‘Spell Delayed Release’ Model.”

Beneath it was an example problem.

“Example: Construct a ‘Force Field – Push’ spell such that, upon casting, the thrust does not instantly reach its maximum, but rather increases smoothly to its peak over 0.5 seconds.

Determine the instantaneous rate of change of the thrust at t=0.1 seconds, and optimize the mana output curve based on this.”

Following this were dense pages of calculation, employing Lia’s proposed derivative formulas, ultimately yielding a precise mana output function.

Lia’s mouth fell open.

Chapter Two: Derivatives.

Klein had added “Application of Derivatives in Finding the Optimal Explosion Range for the ‘Element – Fireball’ Spell.”

Chapter Three: Integrals.

Klein had included “Application of Integral Methods in Calculating the Total Pressure Borne by Non-Standard Curved Surfaces in a ‘Force Field – Shield’.”

He had even independently derived several new fundamental formulas, such as the chain rule for composite functions, and annotated it: “Chain Rule for Composite Functions I.”

Lia turned page after page, her expression slowly shifting from initial surprise to one of increasing complexity.

Klein had not merely understood it.

He had fully mastered it.

Furthermore, in just two short months, he had extrapolated from it, transforming this purely mathematical tool into a weapon capable of optimizing spells and solving practical problems.

He had even thoughtfully revised some of Lia’s phrasing.

“I’ve altered some of the expressions,” Klein stated.

“The original text in your manuscript was too obscure.”

Lia flipped to the modified page.

Her epsilon-delta language definition had been completely deleted by Klein, replaced instead with a lengthy passage describing the concept using terms like “infinitely approaching” and “arbitrarily small range.”

‘Obscure?’

‘That was the very bedrock of modern mathematics, the crown jewel of logic, the crystallization of human intellect!’

‘And he called this obscure?’

‘And these application problems, using derivatives to calculate a fireball’s explosion?’

‘How could he even conceive of that!’

‘This was meant for calculating planetary trajectories, and he’s using it for fireballs?’

‘This was like using an ox-cleaver to slaughter a chicken!’

“This set of tools . . .” Klein looked at Lia.

“It is not intended to explain the world.

It is a pure computational method.”

“Yes,” Lia said, closing the manuscript and placing it on the desk.

“A tool.”

“A tool for calculating planetary motion,” Klein stated directly, revealing its ultimate purpose.

Lia’s heart seized for a beat.

“Yes,” she admitted.

Klein gazed at the two manuscripts on the table.

“It seems we lack methods, not theories.”

He pushed Lia’s original manuscript back to her.

“Keep this initial draft safe.”

Then, he pointed to his own organized version.

“This one, under the name of the Magic Association, I will have copied and distributed as an advanced textbook to force field and incantation mages of at least Six Rings for study, and I will recommend that they must achieve complete mastery of it.”

Lia was stunned.

To announce it so soon?

“People will die,” she said instinctively.

“Mediocre talents were destined for elimination,” Klein replied, his voice devoid of any emotional inflection.

“The path to truth demanded its paving stones.”

‘Master Valerius’s coffin lid must surely be rattling wildly.’

With these words, Klein turned and left the room.

The door closed softly.

Lia stood alone before the desk, gazing at the two copies of *Principles of Calculus*, so starkly different in thickness.

One was a pure theory from another world.

The other was that same theory, now imbued with fangs and claws by this world’s foremost genius.