Chapter 58: The Calculus Revelation and a Master’s Demise

Finn’s smile stiffened.

‘Right. The denominator cannot be zero.’

He had been so completely immersed in the aesthetic beauty of the concept ‘infinitely approaching zero’ that he had overlooked this most fundamental logical cornerstone.

“This… this…”

Cold sweat began to bead on Finn’s brow.

“It’s a… conceptual approach! Yes, it’s a concept! It’s not truly equal!” He found a seemingly plausible explanation for himself.

“Oh…”

Lia drew out the end of her word, her clear eyes blinking, appearing exquisitely innocent.

“But then you made it equal to zero in the end.”

She extended a finger, gently tapping the ‘Δt’ that had been crossed out on Finn’s parchment, which was already covered in calculations.

“You simply erased it from ‘4-Δt’. So, is it zero or isn’t it?”

Was an infinitesimal quantity truly zero?

This very question had once sparked decades of heated debate among mathematical giants on Earth, and now, it loomed before Finn like an insurmountable mountain.

Finn’s face flushed crimson, feeling his mental energy spiral out of control, his thoughts tangling into a chaotic knot.

Seeing Finn’s distress, his eyes even beginning to redden, Lia decided not to press him further.

“Actually, I was just guessing.”

She scratched her head, adopting an innocent expression as if completely out of the loop.

“Is it possible we don’t need to go through all this trouble calculating that… uh, that ‘Δt’ thing?”

She picked up a pen.

Under the watchful gaze of everyone in the small circle, beside the parchment that had borne witness to Finn’s glory and collapse, she inscribed another line.

“v = 10 – 2t”

“See, this formula is a bit like the one above, right?”

“Only t squared became 2t, and for the earlier 10t, the t just disappeared.”

“If we use this formula and substitute t=3…”

“The answer also seems to be 4.”

In that corner of the library, silence reigned, so profound that a pin drop would have been audible.

Time seemed to freeze at that moment.

h = 10t – t²

v = 10 – 2t

This… What in the blazes was this transformation? What mysterious spell from which obscure school of thought was this?

Finn stared blankly at the line ‘v=10-2t’, then looked down at his own page, dense with tedious calculations.

His mouth slowly gaped open, wide enough to fit a puff fruit.

“How… how did this come about?” Ella’s voice, tinged with a slight tremor, shattered the profound silence.

“How should I know?” Lia spread her hands, her expression one of utter innocence.

“I just looked at the h(t) formula and randomly pieced together a pattern. See, for t squared, you multiply by 2 and move it to the front; for t to the power of one, you just remove the t. It feels… like a little game.”

She spoke with an air of casual nonchalance.

Ella, however, picked up a pen and wrote another function.

y = 5t³ + 2t² – t

She looked up, her gaze seeking confirmation from Lia: “Then, according to your idea, this… would become… 15t² + 4t – 1?”

Lia glanced at her, nodding in approval.

“It seems that way.”

Ella said nothing more.

She lowered her head, beginning a frantic verification process, referencing Finn’s original ‘infinite approximation method’.

The tip of her pen scratched softly across the parchment.

The others also gathered around, helping Ella with the derivations.

Lia started to feel bored.

She gathered her things and stood up.

“I’m leaving now; Master Klein told me to return early.”

She waved at the group of young people, who were already consumed by their fervor, and turned to leave the small storm’s eye.

***

Some unknown time after her departure.

Ella’s pen tip suddenly stopped.

“Oh my goodness… it’s true!”

“The result… is exactly the same!”

Suppressed gasps and successive intakes of breath echoed in the corner.

Finn abruptly stood, his face ashen.

He remembered. His mentor had indeed mentioned such an operational rule when explaining that thesis, but at the time, he had paid it no mind whatsoever.

Why?

Because it utterly lacked beauty, resembling a crude algorithm from a country bumpkin.

How could he, a future Grand Archmage, touch upon truth in such an inelegant manner?

He had been absorbed in using infinitesimals to approximate, to feel the philosophical beauty of ‘infinity,’ without realizing that he had long been blinded by this false sense of ritual.

***

Two days later.

Lia emerged from the classroom, walking along the path back to the mage tower, carrying a bag of desserts for Adèle.

Ever since receiving that periodic table, Adèle had practically made the laboratory her home, and Lia would occasionally visit to check on her ‘treasure hunt’ progress.

The notice board in the central square of the academy was unusually crowded today.

There was none of the usual clamor in the crowd, only unsettling whispers.

“Is it true? Master Hal, he…”

“Cognitive restructuring… I’ve only seen that term in textbooks; I thought it was an ancient legend.”

“It’s terrifying, a Seven-Ring Master, just like that… gone?”

Lia curiously squeezed her way in.

In the most prominent position on the notice board was a communiqué directly issued by the Mage Association.

The title’s font was pure black.

[Communiqué Regarding the Demise of His Excellency, Seven-Ring Alchemist Master Bruce Hal]

“After joint confirmation by the Mage Association and the Royal Mage Corps, His Excellency, Seven-Ring Alchemist Master Bruce Hal, passed away late last night in his private laboratory, found without vital signs.”

“Following on-site investigation and spiritual tracing, his cause of death has been confirmed as ‘Cognitive Restructuring’.”

“His Excellency Hal, while attempting to analyze a new theoretical system, suffered a collapse of his logical core due to an incompatibility with his own spiritual model, leading to an irreversible spiritual sea implosion.”

“The Association expresses its deepest condolences.”

“Furthermore, in accordance with Article 71 of the Basic Mage Law, all assets and titles of the Hal family will enter a three-month regulatory period until the family completes its internal power transfer and asset audit.”

The association’s emblem served as the sign-off.

***

The study at Vitra Manor.

The same communiqué lay casually discarded on the corner of Marcus’s desk.

He held a glass of red wine, crimson as blood, and gazed out at the brilliant, almost dazzling sunlight, a slight smile playing on his lips.

“Bruce Hal…” He murmured softly to himself, his tone devoid of any regret.

“What a waste.”

“Unable to withstand even this much impact. His spiritual world is more fragile than the glass vessels he crafts.”

During this period, Marcus had changed.

He had locked himself in his study, tirelessly delving into calculus and universal gravitation.

From initial humiliation and anger, through struggle and acceptance, to his current… intoxication.

Just two days prior, when he successfully optimized a core spell model inherited by his family using this entirely new computational method, subtly increasing its casting speed, a surge of mastery over truth had overwhelmed him.

He had glimpsed a new world.

A cold, perfect world constructed of precise calculations and rigorous logic.

In this world, there was no vague inspiration, no ethereal experience.

Everything was quantifiable; everything could be optimized.

He even began to genuinely believe that alchemists like Old Hal, who stubbornly clung to ‘transmutation,’ were laughably foolish.

Now, this foolish fellow had been crushed by the wheels of progress.

Marcus took a sip of wine.

This was a signal.

More importantly, it was an opportunity.

Hal was dead, his family leaderless, and their vast assets under association regulation.

There was no more vulnerable time than this.

He, Marcus, head of the Vitra family, it was time to demonstrate his goodwill towards an old friend.

He set down his wine glass, quickly penned a letter, and pulled the cord beside his desk.

Moments later, his most loyal servant appeared ghost-like at the doorway.

“Master.”

“Go to the Hal estate,” Marcus’s voice was steady and cold.

“Take my letter of condolence, and tell Old Hal’s incompetent eldest son that I am deeply saddened by the passing of my old friend.”

The servant listened silently.

“Tell him the Vitra family is willing to offer all necessary assistance to the Hal family during this difficult period.”

Marcus walked to the window, admiring the meticulously manicured flowerbeds of his estate.

“Especially in… stabilizing the value of their family’s alchemical industries.”

The servant bowed deeply: “As you command, Master.”

After the servant retreated, Marcus looked towards the distant Hal estate, the smile at the corner of his mouth growing colder and more sinister.

The Hal family’s wealth was built upon complex alchemical workshops and generations of inherited secret formulas, replete with empirical variables and cumulative effects.

Cumulative…

Marcus abruptly spun around, his eyes blazing with a frightening intensity.

He strode quickly back to his desk, snatched up the thesis, and flipped directly to the last chapter of the appendix—”Applications of Integration.”

As he looked at the examples using graphical areas to calculate cumulative effects, a thought that made him tremble began to form in his mind.

What if… what if the charts in his family’s mysterious ‘Astral Track Divination Art,’ those charts reliant on experience and intuition, were to be recalculated using this new tool?

And those foundational principles, which could previously only be interpreted through bloodline inheritance, used to predict climate changes, mineral vein locations, and even the course of wars…

Could the true underlying patterns be entirely captured by some tool?