The Poetry of Formulas Spun by GPT Episode 68: [The Derivative Formula of the Inverse Sine Function] - From a Closed Wave, a Slope Emerges

You've reached the end of the wave, haven't you?
From here on, it looks like there's no turning back.

Yet, inside the closed curve,
the slope is quietly rising.
Cold starlight
thinly illuminates your hesitation.

The voices that drifted away,
and the time that was interrupted,
are connected deep within life.

Existence may not be an answer,
but a small tremor left behind in the universe.

The derivative formula of the inverse sine function is an expression that represents how the original angle changes from a value connected by a sine. It can be said to be a relationship where you trace the slope behind it while watching the height of the wave.

$$
\frac{d}{dx}\arcsin x=\frac{1}{\sqrt{1-x^2}}
$$

The change is relatively gentle near the center, but as the input approaches 1 or -1 at both ends, the denominator becomes smaller and the slope becomes steeper. The closer you get to the edge of the closed range, the more a slight difference appears as a large change.

When a limit approaches, don't you find that your heart is shaken more than usual by smaller events?

On a tiring day, a casual word might sting deeply, or conversely, a small kindness might feel unexpectedly warm. That is not weakness, but perhaps because the slope of where you are standing has become steep. The derivative formula of the inverse sine function quietly teaches us that even with the same step, the magnitude of change differs depending on where you are standing.

Even in a closed place, a slope still remains.

If your heart was shaken greatly today, perhaps you don't need to rush to blame yourself. Near an invisible limit, even small sounds and silences can resonate more strongly than usual. Still, inside you, there is a thin light that has continued since yesterday. Even if you don't decide where to go right now, just quietly confirm that the light has not gone out.

[The Derivative Formula of the Inverse Sine Function] - From a Closed Wave, a Slope Emerges

The Room Inside the Wave

On the morning seven days before closure, the bay was covered in a thin silver film. The boundary between the sea and the sky unraveled in the mist, and only the breakwater extending offshore floated like an arc drawn with a wet pencil. Inside it, the waves were breathing. Without swelling greatly or becoming completely still, they carried a low sound to the shore, as if trying to remember something and then forgetting it.

Toru Sakaki boiled water at 6:40 AM. While the water began to tremble finely at the bottom of the electric kettle, he wiped the dust off the observation panel, checked the communication status, and opened the previous night's records. It was a routine that hadn't changed for over ten years. Even the number of times he moved the cloth from right to left was almost the same.

The metal desk by the window was cold. When he placed his fingers on it, the heat of his skin was absorbed beyond the thin plate. There were two chairs in the room, and he always used only the one in front. A paper box was stacked on the other chair, but he never tried to move that box to the floor.

There was a strange rise in the nighttime wave height graph. The line, which was as gentle as if sleeping in the center, suddenly changed its angle as it approached the edge. The difference in the numerical value was small. It was within a range that would be fine to treat as equipment error. Still, he zoomed in on the screen.

The blue curve, the moment it touched an invisible wall, stretched upward as if trying to escape from something.

Toru took his hand off the mouse. In the distance, a seabird cried once. The sound traveled a long way through the fog and lost its strength before the window glass. He did not know where it had come from, but the mere fact that the sound had been uttered left a thin scar on the air in the room.

The observatory window was slightly curved inward. When Toru's reflection in the glass overlapped with the breakwater beyond it, it looked as if he himself were sealed inside a large arc. Beads of condensation slid from top to bottom, their paths intersecting with the reflections on the sea surface.

The morning's work was to separate waste from archived materials. After the closure, observations would be taken over by an offshore automatic buoy. There would be no need for anyone to wipe the windows every morning or record tide levels in a paper logbook. In the back of the shelves, the past years had piled up as the weight of moisture-laden paper.

When he opened the bottom drawer, an old recorder appeared.

It was a silver machine that fit in the palm of his hand. The paint on the corners was peeling, and faint fingerprints remained around the stop button. On the side, there was a line carved with something like a needle. It looked like a semicircle. But the scratch was shallow, and it blended into the rust along the way.

Toru turned the recorder over. Inside the battery cover, the character "M" was written in small letters.

It was the character for Mio.

The air seemed to stop in the back of his throat, but he gave a single cough and forced it out of his body. He pressed the play button on the recorder. Nothing happened. It took him a few seconds to realize the battery was dead.

Mio had visited this observatory in the spring five years ago. She had appeared without notice, saying she was nearby for acoustic design work. She opened the window slightly and pointed the recorder toward the sea. When he asked what she was recording, she placed her index finger to her lips instead of answering.

That memory ended there.

Toru placed the recorder next to the paper logbook. He could replace the batteries. There were spares left in the storage room. However, he stood up and carried a box for disposal from another shelf. The sound of the bottom of the box scraping against the floor was closer than the waves and further than his own breathing.

After midday, the fog thinned. The tip of the breakwater and the sea beyond it began to come into view. The bay was not completely closed. Between the ends of the arc, there was a gap wide enough for a single ship to pass through. The tide entered and exited through there.

As he was about to wash a white mug that was no longer in use, Toru stopped his hand in front of the faucet. A dry ring of salt remained on the bottom. It was the kind of thing that would disappear immediately if he ran water over it. He did not turn on the water and returned the cup to its original place.

In the evening, a beam of light fell from a rift in the clouds. It cut diagonally across the sea surface and severed the arc of the breakwater midway. Only a portion of the gray waves turned a pale orange, and its brightness changed shape every few seconds.

Toru watched the light grow thinner without touching the recorder. The light did not reach the room. It simply showed, without telling anyone, that an angle still remained inside the closed bay outside the window.

The sound of approaching the limit

When there were three days left until the closure, a low-pressure system had approached the offshore area. From the morning, the wind blew as if searching for the corners of the building, making a thin whistling sound in the gaps of the window frames. A flat surface of water still remained in the center of the bay, but at both ends of the breakwater, the waves were beginning to break into white foam.

Toru broke the seal on the spare batteries.

When I opened the lid of the recorder, a dry smell of metal wafted from within. I inserted new batteries and pressed the play button. A small lamp lit up red, and the rotating mechanism began to move faintly.

The first thing I heard was the wind.

It was not a low, steady wind, but a wind too close, like fingers stroking the surface of the recorder. Beyond that, there were footsteps. A few steps on a hard floor, then a stop. Clothes rustling. A bird chirping in the distance. Something touching a desk. Further away, waves breaking.

There was no voice of Mio.

Toru turned up the volume. The grains of noise grew larger, and even the gaps that had been inaudible began to take shape. I felt as if words were buried there. He rewound the same section over and over. Between the wind and the footsteps, between the bird's cry and the rustle of clothes, there was only the presence of someone about to open their mouth.

But no matter how much I listened, it never became words.

He tried to infer the missing parts from the sounds before and after, just as he would when filling in missing observational data. Mio was by the window. She was looking at the sea. She had tried to say something. Perhaps she had called a name. Perhaps she had smiled. Perhaps, before stopping the recording, she had been searching for words to leave for someone.

If I connected the inferences, a story was formed.

However, the moment the lines were connected smoothly, that story was no longer Mio's, but the shape he needed it to be.

Toru stopped the playback.

In the afternoon, I found Kuze's notebook behind the document shelf. The cover was warped by the sea breeze, and the glue on the spine had peeled off. Beside the observational values, several short notes unrelated to the weather were left behind.

The more closed a space is, the more sounds that cannot be heard from the outside are amplified.

Only that one line had a different pen pressure. Below it was the name Mio, written small. I didn't know if Kuze had written down words he heard from her, or if she had written it herself.

Before evening, the rain began. The drops hitting the window were sparse at first, but as time passed, they lost their intervals and became a single, thick sound. The pipes groaned inside the walls, and the metal fittings on the ceiling vibrated in small increments. Every time the warning light on the observation panel flashed, the shadows in the room shifted slightly.

The center of the wave height graph was still gentle. However, at the observation point corresponding to the end of the breakwater, the curve was rising sharply. A change in input, however slight, appeared on the screen as a large angle.

Toru looked out the window. The water rising and falling in the center of the bay looked calm from a distance. But the waves approaching the gap were pushed into the narrow space, suddenly rising high and breaking. Even though it was the same sea, the magnitude of the change differed depending on where one stood.

The mobile phone on the desk vibrated.

Kuze's name was displayed on the screen. Toru stared at it for a few seconds before answering the call.

"Is it rough over there?"

"Yeah."

The machine is normal.

It is normal.

That was the extent of the conversation. Kuze said nothing about the lockdown, and he seemed to know nothing about the recorder. After the call ended, Toru's face was reflected in the black screen.

Another incoming call screen overlapped in his memory.

On an afternoon five years ago, an alarm signaling abnormal values had been sounding. He chased the numbers in front of the observation panel and turned his mobile phone face down. Mio's name had appeared three times. He couldn't answer now. He would call her back later. That was his judgment. It was not the wrong judgment. At least, not at that moment.

When he did call back, it was not Mio who answered the phone.

The wind shook the observatory. The legs of the metal desk touched the floor, making a short sound. Toru realized his hands were gripping the edge of the desk tightly from the pain in his fingers.

Being rational was not the same as not losing anything. Even if one piled up the right choices, it did not guarantee that the world would remain in the right shape. That contradiction did not fade with the passing years; like a wave approaching the shore, it suddenly increased in height with a trivial sound.

At 9:13 PM, the power went out.

The lights on the observation panel went out all at once, and the driving sound of the machinery ceased. Immediately after, the emergency light near the floor lit up white. The light illuminated only the legs of the chair and the edge of a paper that had fallen under the desk.

In the room where the mechanical noise had vanished, the building groaned like a different living creature. The walls creaked, water moved deep within the pipes, and the window glass vibrated low under the wind pressure. What he had thought was silence until now was filled with countless small sounds.

Toru played the recorder in the darkness.

Wind. Footsteps. The rustle of clothes. Birds. Waves.

And breathing.

Mio's breathing was shallow, stopped once in the middle, and then continued. Toru realized he was matching his own breath to it. The air of the past and the air of the present overlapped inside the small machine. Even though their lungs were in different times, for a few seconds, they moved at the same interval.

Near the end of the recording, a sound rang out three times.

Tap.

Tap.

Tap.

It sounded like a knock on a door, the sound of a finger touching the casing, or perhaps the sound of something falling in a distant place.

Toru did not stop the playback. He waited for the answer to continue.

However, after the third sound, there was a long silence. Only the faint sound of magnetic tape rubbing flowed, and eventually, the recorder stopped automatically.

In the low light of the emergency lamp, only Toru's breathing remained. It was not that he understood something. It was as if the building, creaking somewhere in the darkness, was telling him that even the things he could not hear had not vanished.

A Slope Without a Name Yet

On the morning after the storm had passed, the bay held the memories of the night in its body. Seaweed was tangled on the breakwater, and branches and shells were scattered on the wet concrete. Rusted metal fragments shone faintly at the bottom of puddles, and a thin iridescence remained inside the broken shells.

The sky was still dark. Only at the eastern edge was a layer slightly brighter than blue being born.

On the final day of the closure, Toru boiled water at the same time as always. He wiped the observation panel from right to left, checked the communication status, and saved the final record. In the paper logbook, he wrote the date and time, and below that, he wrote only 'Observation Ended'.

The letters were smaller than he had thought.

He turned off the equipment one by one. The communication device's fan stopped, the low hum of the recording terminal vanished, and the transformer against the wall fell silent. Each time a sound decreased, it was not that the room became empty, but that other sounds returned.

The sound of water droplets falling from the roof. The presence of birds fluttering their wings outside the window. Distant waves. The sound of his clothing sleeves touching the desk. The air moving in and out of his own nose.

He took the paper box down from the second chair. A ring of dust remained on the seat. Toru wiped it with a cloth. The chair was empty, but he did not think to have anyone sit there. It was just that what had been piled up was gone, and the chair's original form appeared.

He carried the white mug to the sink. When he turned the faucet, a thin stream of water hit the ring of salt at the bottom. The ring did not disappear immediately, but flowed into the drain while thinning and unraveling. He did not scrub it until it was completely gone. He turned off the water and placed the wet cup upside down on the windowsill.

He played the recorder one last time.

This time, he did not look for a voice.

The wind sounded as wind. Footsteps passed by with the weight of having stepped on the floor. The birds' calls were not a signal to anyone, but a trace that birds had been in an unseen place. The rustle of clothing conveyed that Mio's body had existed there, moved her arms, and touched the air.

Breathing flowed.

Toru did not match his breath to it. He heard Mio's breathing and his own as separate things. There was no need for them to overlap. Even if the past and the present did not take the same shape, they could resonate in the same room at the same time.

The final three sounds rang out.

Tap.

Click.

Click.

He started to think about what it was, then stopped. It didn't mean it would vanish if he didn't assign it meaning. The nameless sound had certainly remained within the recorded time.

Toru stopped the recorder and traced the scratch on its side with his finger. The line he had thought was a semicircle was slightly interrupted near the end. It had just been hidden by rust.

The break was thin, no wider than the tip of a fingernail.

He didn't know if it had been left intentionally or if the hand had simply stopped while carving the scratch. It was too small to call an exit, and too smooth to call a defect.

Toru placed the recorder on top of the disposal bin. After staring at it for a while, he picked it up and put it in the inner pocket of his bag. The small weight settled into place inside the fabric.

When he finally cut the breaker, the observatory was wrapped in complete silence. Yet, even to that silence, the sea reached. Waves broke on the other side of the wall, and a faint vibration passed beneath the floor. Even though the building was closed, the bay did not stop its ebb and flow. The moon's gravity continued to move the sea from the distant sky, and the end of the night was drawing near due to the Earth's rotation.

Toru stepped outside the door.

Cold air touched his cheeks. The temperature was lower than before the storm, but the wind was weak. He inserted the key and turned it slowly. Metal meshed inside, and a short sound rang out.

The door closed.

He put the key in his pocket and began to walk along the path following the bay. The road was not straight; it curved gently in the same direction as the arc of the breakwater. There was a slope beneath his feet he hadn't noticed until yesterday. Even in places that looked flat, there remained an angle that forced his body to lean slightly forward.

Water droplets lined the grass on the side of the road. A small insect was trying to cross a fallen stem. It would advance, slip, and move its legs again. Toru did not stop, but he walked while keeping its movement in the corner of his vision.

In the center of the bay, waves were slowly rising and falling. As he approached the break in the breakwater, the flow of water was narrowed and passed out to the open sea with increased speed. Inside the arc that looked closed, the sea had never once stopped.

The morning sun appeared from the eastern sky.

The light passed through the break in the breakwater and fell diagonally onto the sea surface. Every time the waves moved, the angle changed minutely. The light did not become a single path; it broke and connected, approached and receded.

Toru did not look back.

The closed observatory was behind him, and inside his bag, the recorder touched his body in time with his stride. The lost phone, the unfillable void, and the three sounds all remained just as they were.

Even so, his feet were carried toward the end of the curved road.

The diagonal light stretching from the sea touched the edge of the wet sidewalk, and while changing its angle just like the waves, it reached quietly right up to his shadow.