Chapter 140: The Ghost Knocks

The summer in the capital was more sweltering than in previous years.

The air circulation system inside the tower was turned to its maximum setting, but it still couldn’t disperse that lingering sense of restlessness.

This agitation wasn’t caused by the weather.

Ever since the publication of the two formulas by Rayleigh and Wien, the air throughout the entire magical community seemed permeated with the scent of scorching ink and the crackle of static electricity.

Every day, new papers were published proposing various modification methods.

Some attempted to modify the attributes of heat, such as its viscosity or vortex characteristics.

Others simply proposed that perhaps the thermometers were malfunctioning at high temperatures.

The more they failed to find an answer, the more people tended to drift toward extremes.

Yesterday, during an academic exchange, two high-level mages from the Thermodynamic school actually engaged in a duel.

The cause was merely because one claimed the other’s cited constant was inaccurate at the first decimal place.

The Tower of Truth had to urgently issue a ban, prohibiting the use of offensive magic above the Third Circle during academic discussions.

Lia sat on a high stool in the laboratory, holding a vacuum tube she had just finished calibrating.

“These people have gone mad.”

Adèle pushed the door open from the outside, a handful of sweat-soaked flyers in her hand.

She slapped the flyers onto the table.

“Look at this. Someone is saying the anomaly in blackbody radiation is because microscopic particles possess ‘free will’ and don’t want to be bound by formulas.”

Lia set down the vacuum tube and scanned the flyer.

It featured drawings of several molecules with smiling faces, seemingly mocking Rayleigh’s formula.

“And this one.” Adèle pulled out another. “It says the gods have set up a forbidden zone in high-temperature regions that mortal wisdom cannot reach.”

“Nonsense.”

Klein stood at the other side of the experimental bench, calibrating a spectrometer.

He didn’t even look up, his voice steady.

“The groans of the incompetent.”

“It truly is incompetent fury.”

Adèle pulled over a chair and sat down, grabbing the cold water pitcher on the table and taking a heavy gulp. “But Mentor, right now, no one can truly solve this problem.

The experimental data is right there; no matter how one calculates, the theoretical values simply won’t match the observations.”

“Either the low frequencies don’t match, or the high frequencies explode.”

Adèle let out a sigh.

“I’m starting to suspect that the edifice of physics was truly built lopsided.”

Lia didn’t speak.

She picked up a lens cloth and slowly wiped the glass.

The edifice wasn’t lopsided.

There was just something buried beneath the foundation.

Current physics was built upon the assumption that nature was continuous.

Energy was like flowing water, like time—that was common sense.

It was a concept everyone had accepted since birth.

No one would doubt common sense.

Just as no one would doubt that the sun rises in the east and sets in the west.

“Has the new issue of the journal arrived?”

“It has, it’s downstairs.”

Adèle pointed at the floor.

“But I suggest you don’t read it; it’s all just shouting matches.”

“The current journal editorial department likely only accepts those types of vitriolic articles, as if whoever shouts the loudest is the one holding the truth.”

“Go bring it up.”

Klein spoke.

Adèle stood up resignedly and trudged downstairs with heavy steps.

Before long, she returned carrying a thick stack of kraft-paper-covered journals.

Physics Reports.

This was a publication newly created after the establishment of the Physics Department, and it had now become the most authoritative academic battlefield on the entire continent.

Adèle tossed the journals onto the table.

“Here. The latest issue.”

Lia reached out and took the top one.

The cover was simple, with no fancy magical images—only a line showing the date in black.

She flipped open the table of contents.

Her finger slid across those sensational titles.

On the Fatal Flaws of Rayleigh’s Formula

Wien’s Formula: A Product of Empiricism?

Do We Need New Mathematical Tools?

Lia’s finger stopped on the last line.

The title was long and very plain—it even appeared somewhat dull.

On the Improvement of the Law of Energy Distribution in the Normal Spectrum.

Author: Max Planck.

An unfamiliar name.

Not very famous, likely an old lecturer at some ducal academy or the head of a remote laboratory.

But in Lia’s eyes, it was a name like a clap of thunder.

Lia flipped to the corresponding page.

The article wasn’t long.

There was no fierce rhetoric, nor were there claims of discovering some world-shaking universal truth.

Between the lines, there was instead a sense of sincerity and helplessness.

“…In view of the current limitations of Rayleigh’s formula and Wien’s formula, and the mutual exclusivity of their physical meanings across their respective frequency bands, I have attempted to find a mathematical interpolation function…”

“…This is not based on a derivation from any physical principle, but merely a mathematical attempt…”

Lia looked at the text.

Just like in her past life.

This person was very sincere.

He admitted he didn’t understand the principle.

He was merely playing with numbers.

He placed Rayleigh’s formula and Wien’s formula on two sides of a scale, attempting to find a balance point and forcibly fuse these two clashing things together.

Lia’s gaze moved downward.

In the middle of the paper, a line of equations was written.

That was the formula Planck had forcibly pieced together through mathematical interpolation.

This was the embryonic form of that ghost.

In this world, it was the first time it had appeared in this form.

“What is it?”

Klein sensed that Lia’s gaze had lingered for too long.

He set down the tool in his hand and walked over.

Lia pushed the journal to the center.

“Look at this.”

Klein looked down.

His gaze fell upon the formula.

A few seconds of silence followed.

“The mathematical structure is very strange,” Klein commented.

“The exponential form in the denominator minus one. Is this to approximate Wien’s formula at high temperatures and low wavelengths, while approximating Rayleigh’s formula at low temperatures and high wavelengths?”

“It should be.”

Lia grabbed a sheet of scratch paper and dipped her pen in ink.

“Let’s verify it.”

She copied the formula onto the paper.

“Assume the wavelength λ tends toward infinity.” Lia’s pen tip slid across the paper. “According to the Taylor expansion, theeindex term can be approximated as 1 plus the exponent…”

Klein watched her derivation.

“The denominator becomes…” he said. “The numerator and denominator cancel out.”

Lia wrote down the final result.

“It returns to the form of Rayleigh’s formula. Proportional to temperature T and inversely proportional to the fourth power of the wavelength.”

“It matches perfectly.”

“As the wavelength tends toward infinity, the energy tends toward zero. There is no ‘Ultraviolet Catastrophe’.”

“Now look at the short-wave direction.”

Lia started a new line.

“When  λ tends toward 0, the exponential term becomes extremely large, and the minus 1 can be ignored.”

Her pen moved rapidly.

“The formula degenerates into the form of Wien’s formula.”

Lia set down her pen.

Two lines of derivation were left on the paper.

Perfect.

This formula was like a bridge, perfectly connecting the two broken ends.

It retained Rayleigh’s accuracy in long waves while preserving Wien’s convergence in short waves.

There were no infinities or gaps.

A smooth and elegant curve that spanned the entire spectrum.

Adèle leaned in, looking at the equations on the paper.

“This… this solves it?”

She widened her eyes, her face full of disbelief.

“Just like that? Just piecing two formulas together?”

“It’s not that simple.”

Klein picked up the journal, staring at Planck’s name.

“To think of using this mathematical form to construct an interpolation—this person has a very solid mathematical foundation.”

“But…”

Adèle pointed at the formula.

“What does this mean? I mean, its physical significance.”

She looked at Lia, then back at Klein.

“Rayleigh’s formula was derived from the Equipartition Theorem. Wien’s formula was derived based on a thermodynamic analogy. What about this formula? What is it based on?”

Lia leaned back in her chair.

“It’s based on fudging numbers.”

“Huh?”

“The author said so himself.”

Lia pointed to the passage in the paper. [In order to fit the experimental data, a mathematical function was constructed. He doesn’t know why it looks like this; he only knows that by writing it this way, the calculated numbers are correct.]

Adèle opened her mouth.

“What is this? A blind cat running into a dead mouse?”

“If this dead mouse is the truth, then this cat is the greatest cat in the world.”

Klein closed the journal.

“To the lab.”

“What for?”

“Verification.” Klein picked up his staff. “We are going to run this formula using our data.”

***

Three hours later.

The laboratory was dead quiet.

Only the slight hum of the mana recorder could be heard.

Dozens of sheets of parchment filled with data floated in the air.

Klein stood before a massive blackboard, holding the final set of comparison data.

A curve was drawn on the blackboard.

That was the theoretical curve of Planck’s formula.

Upon this curve, hundreds of red dots were densely marked.

These were the blackbody radiation experimental data measured at different temperatures and with different materials over the past few months.

Every single red dot.

Every single one.

They all fell precisely on that curve.

The error did not exceed one-thousandth.

This level of fit was practically hair-raising.

It was as if the world didn’t produce this formula—it was simply the way the world was originally, and people just hadn’t discovered it yet.

Adèle slumped in her chair, staring at the blackboard as if looking at a miracle, or perhaps a ghost.

“I’ll be damned…” she muttered to herself.

“It truly matches everything. Even that most difficult 1500-degree high-temperature range matches.”

She turned to look at Klein.

“Mentor, how do you explain this?”

Klein remained silent.

He stared at the formula.

His brow was deeply furrowed.

As an Eighth-Circle Great Mage and a scholar seeking the essence of the world, he instinctively felt discomfort toward this kind of “knowing the how but not the why” phenomenon.

A perfect formula must correspond to an underlying physical mechanism.

Rayleigh’s formula corresponded to the equipartition of energy.

What did this formula correspond to?

That exponential term, that minus one—what exactly did it represent the microscopic particles doing?

“It can’t be explained.”

Klein finally spoke.

“At least, it can’t be explained with existing physical theories.”

He looked toward Lia.

“What do you think?”

Lia sat in the shadows of the corner.

She was twirling a pen in her hand, her gaze fixed on the formula on the blackboard.

The formula was smiling at her.

It was the smile of Quantum Mechanics.

Right now, it was still just a skin—a painted skin forcibly pieced together with mathematics.

Even Planck himself didn’t know what he had done yet.

In order to cater to the data, he was forced to introduce this strange mathematical form.

It was too perfect to be a coincidence.

In order for this mathematical form to hold up physically, he would have to make an assumption that betrayed his ancestors.

He would have to assume that energy was not continuous.

That energy came in discrete portions.

Like buying things with whole coins; you couldn’t pay with half a coin.

This assumption would completely destroy the foundation of classical physics.

“I think…”

Lia stopped twirling her pen.

“This formula is too perfect—too perfect to be a coincidence.”

“Since the math is right, then there must be a mechanism we haven’t discovered yet supporting it in reality.”

Adèle scratched her hair.

“What kind of mechanism? Can energy truly jump one step at a time like a staircase?”

“Wait, isn’t energy supposed to be continuous, like water?”

“How is that possible? Energy is obviously continuous.”

“If energy were in chunks, wouldn’t our mana be jerky when we cast spells?”

Lia glanced at Adèle.

“Who knows.”

Lia withdrew her hand.

“Maybe the world itself exists frame by frame.”

“Stop joking around.”

Adèle waved her hand. “That’s also quite absurd.”

“But truth is often exactly that absurd.”

Klein looked at Adèle.

“Organize the data,” he instructed.

“Publish a bulletin in the name of the Tower of Truth. Confirm the high degree of consistency between Planck’s formula and the experimental data.”

“Yes.”

Adèle jumped up and began clearing the parchment from the table. “This is going to be lively. That Planck fellow is likely going to become famous. Regardless of whether he’s a genius or not, at least he solved the Ultraviolet Catastrophe.”

“He didn’t solve it.”

Lia walked to the window, looking at the blinding sunlight outside.

“He just hid the catastrophe from the surface into the belly of the formula.”

“Now, everyone has to stare at the belly of this formula.”

“Until someone carves out what’s inside.”

Adèle hummed a little tune while organizing the data.

She was happy.

Because she finally didn’t have to face those messy, clashing formulas anymore.

As long as there was a useful tool, she didn’t care about the reason behind it.

Most mages thought this way.

Pragmatism.

Lia looked out the window.

A pigeon flew across the sky.

Its wings flapped against the air, producing a continuous airflow.

But in the microscopic world, it was not continuous.

It was countless quantum leaps.

That middle-aged mage named Planck was likely sitting at his desk right now, tearing his hair out over his own formula.

Lia let out a soft sigh.

the sound was very light, swallowed by the humming of the laboratory instruments.

That ghost.

It had finally knocked on the door.

“Let’s go.”

Lia turned around.

“Let’s go eat. I want the stewed beef from that shop in the west of the city.”

“I’m coming too!”

Adèle raised her hand. “To celebrate that the edifice of physics hasn’t collapsed!”

Lia smiled.

“We just changed the foundation, that’s all.”

Although this process of changing the foundation might cause a few floors to collapse and crush a few stubborn old residents to death.

But those were all necessary prices.

Klein took off his laboratory robes and hung them on the rack.

“Let’s go.”

He opened the door.

Sunlight flooded in.