The Imaginary Number Submersion refers to the act of using a specific and special magical mystic code to prove one's existence within an imaginary space where location, time, and distance do not exist. Usually, this requires the use of a special mystic code known as an imaginary number pointer. In the Chaldea timeline, Chaldea obtained such a mystic code, the Moon of the Surface, and installed it on their combat vehicle for imaginary number submersion.
The reason a vehicle is required is that imaginary number submersion needs a completely sealed container, isolated from the outside world. This prevents external observers from perceiving the interior, allowing for a special kind of transformation between the real and the imaginary for the sake of existence proof. Aoko also uses a similar method to achieve her objective.
Moreover, she has an even better option.
"This is..." Pepe seemed to realize something, his eyes widening as he looked at the orb in Aoko's hand and asked, "...is that from one of the Twenty-Seven Dead Apostle Ancestors?"
Aoko nonchalantly reached out her right hand and grasped at the air, causing a sphere roughly the size of an apple to materialize out of thin air. Aoko did not respond to Pepe's question, and Artoria beside her showed no curiosity or surprise, for this orb was precisely the Principle Blood Ring she had collected.
As the crystal-clear orb began to glow faintly, the scenery before Pepe's eyes suddenly blurred. He squinted against the strong light, but after a brief moment, he found himself standing in an unfamiliar place.
"This is...!" Pepe looked in astonishment at the magnificent, beautiful scenery before him. Rather than nature's craftsmanship, he was more willing to believe that this was a fantasy world meticulously crafted by a masterful artist.
He discovered that he was standing on a hill in the center of a town. In front of him was a beautiful city with a strong European style, and beyond the city lay forests, rivers, and mountains of the natural world.
The whole environment was harmonious and wonderful. The only unnatural thing was that, in this fairy tale-like town, aside from themselves, there was not a single resident.
Looking back, Pepe saw a majestic, beautiful, and mysterious castle. As a magus, he could sense the powerful magic flowing through every corner of the castle—this was not an ordinary building!
"Alright, alright, stop daydreaming," Aoko waved her hand in front of Pepe's eyes and said, "This world is our ark for submerging into the imaginary number world. As a sealed container completely isolated from the outside world and under my control, I can use it to reach that abandoned piece of history—the place where Hinako was transferred."
"…A failed history?" Pepe pondered and asked his question, but Aoko did not answer. Instead, she activated the Soaring Wind Realm, bringing the three of them flying toward the royal garden at the top of the castle. This was because the core node of the artificial leyline she had set up earlier was here, and she wanted to use this spot to drive the entire Principle Blood Ring to submerge into the imaginary number space. It was the most reasonable location.
"This is a beautiful place!" Pepe showed off his excellent aesthetic sense as a top fashion designer, instantly spotting the fine details of the castle. "Living here, you can't help but imagine yourself as a princess in a fairy tale!"
Aoko glanced at him without replying, but she did intend to win this Shugendo magus over. As long as he didn't become a tool of the Alien God, he was a rather capable talent, worth her effort to persuade.
Although his orientation and habits were a bit problematic, Aoko didn't dislike this big sister of a man—especially since the original story guaranteed his reliability.
"In the real-number space, we look for a specific real-number coordinate..." Aoko explained to herself while taking out a mystic code that looked like it had a conical spinning top floating at its tip. "...Then, using this, we add extra imaginary number elements to that specific real-number coordinate, achieving the transformation between the real and the imaginary."
"As for the specific coordinates, I already got them from that fairy's mind," Aoko carefully adjusted the mystic code in her hand. "The general location is still the British Isles, but the worldline is different. Artoria, you understand what I mean, right?"
"...Ah, I get it," Artoria nodded. "In other words, we're heading to another doomed Britain?"
"Those guys at Atlas introduced little Shion to Touko, so I can place a secret order to the Atlas Institute through Touko," Aoko said. "The price is to create a materialized soul for the Atlas Institute as the core of a perpetual magical furnace—not a cheap deal…"
"Wait, you said this mystic code is from the Atlas Institute?" Pepe asked in surprise, covering his mouth. "Could it be one of those seven...?"
"It's not," Aoko shot him a look and cut him off. "But it is one of their heirlooms, slightly modified as a commissioned product for me. In a sense, I snatched it from Marisbury."
Pepe didn't ask further, because massive magic circuits on Aoko's body were already lighting up. The enormous magical energy pressed down on him like a storm, making it hard to breathe. The Imaginary Number Compass: The Moon of the Surface had started to function, and the Moon Map at the center of the disc was unfolding. If someone were observing them from outside at this moment, they would see that the Principle Blood Ring orb floating in the air had disappeared from the real-number world, having completed its submersion.
"Submersion successful," Aoko said. "We are now inside the Imaginary Number Sea. Next, we need to search for that pruned Lostbelt according to the coordinates."
...Lostbelt?
Pepe listened to this term in confusion. He vaguely remembered Marisbury mentioning it in Chaldea once, but he didn't know exactly what it meant, so he could only make some guesses.
Since this was her first time piloting an Imaginary Number Ocean submersion, Aoko wasn't very skilled yet, so it took longer than expected.
Artoria, meanwhile, skillfully summoned an automaton tending the garden to bring her some snacks, while Pepe sat on a nearby bench, watching Aoko float in midair as she carefully adjusted the Moon of the Surface.
The entire submersion process lasted about an hour and a half. Aoko finally breathed a sigh of relief—her skills were decent. She had completed her first imaginary number submersion by manually operating the Moon of the Surface, without using a computer at all.
"Alright," Aoko clapped her hands to get their attention. Artoria hurriedly stuffed the last piece of cake into her mouth, quickly chewing and wiping her mouth clean with a napkin. "We've arrived at our destination and are about to surface."