Professor Hibiko took a step towards the big whiteboard in front of the classroom. The faint marks from her earlier writing still glistened under the fluorescent lights of the room. They showed complex staircases and abstract shapes that twisted the mind just trying to grasp them.
This time, instead of putting down her thoughts on the board, she paused. She faced the room filled with students, her expression serious and intense. When she spoke, her voice dropped low, inviting everyone to pay close attention.
"Listen up," she said, her tone firm. "You must get this point clear. No matter how much a lower dimension grows, it can never touch a higher one."
Her words hung in the air, thick and heavy.
"The world we take for granted, our three-dimensional space, can stretch out light-years upon light-years. It can twist, fold, and expand. It can hold entire galaxies, emptiness, civilizations, our dreams, and even destruction. But even though it can grow huge," she said, raising her hand with the marker poised, "it will never, ever reach that fourth dimension."
She shifted to the second box she had drawn before, her movement deliberate.
"Just like how a construct in the fourth dimension can be an endless city with timelines stacked like books on a never-ending shelf. It can grow, split, and simulate reality but can never reach into the fifth dimension. The draw between these levels isn't simply about space. It's not even about time."
With purposeful steps, she walked slowly in front of the class, each step soft against the tiled floor.
"It's about existence itself," she said, looking at them directly.
Her amber eyes seemed to penetrate their uncertainty, imploring them to grasp what she was saying.
"You could fill the entire universe, pack every inch of three-dimensional space, with an unlimited number of three-dimensional entities—planets, realities, countless people—layered one on top of another in a colossal multiverse. And yet," she raised a finger for emphasis, "none of it would leave even the slightest trace on a four-dimensional surface."
The room was quiet. Not from confusion, but from a profound realization.
"Because climbing to a higher dimension isn't just about having more space. It changes what existence means. The rules shift. The basics of reality itself change. And the scale we use to measure it? It becomes irrelevant."
She turned back to the board and finally began to draw again.
A vast field of tiny squares—tightly packed within a larger rectangle. She labeled the small squares as "3D structures."
Next to them, she drew a sphere set apart from the rest.
"This is," she said, "how a single four-dimensional construct appears from below."
Then her voice lowered to a near whisper.
"If you stacked the entire infinite number of three-dimensional slices into this room, it wouldn't cast a shadow in the fourth-dimensional space."
She placed the marker down once again, folding her hands behind her back now.
Her gaze swept over the class, not like a teacher assessing her students, but like someone who had glimpsed deeper truths staring right back at her.
"You've got to understand why," she told them. "Those who exist in higher dimensions can erase our presence without ever needing to interact with us."
A brief silence followed.
"Because to them, we might be less than just dreams."
No further explanation was necessary.
The weight of her words settled heavily in the air:
We may have no way to reach them, but they could ignore our existence entirely.
And yet…
Somewhere within this endless maze of dimensions…
There was something resisting them.
For the first time, the classroom felt less like a safe haven and more like a shelter from the storm.
Professor Hibiko stood next to the board again, the marker uncapped but still, as if waiting for the right moment to move forward.
"Let's shift gears now, and discuss the next chunk of material," she said, her tone taking on a more serious note. "The Axiom Manifold."
After a pause, she began to sketch another diagram. This one wasn't structured like a tower; it resembled a giant web, intricate and linked, much like neurons or roots sprawling beneath a vast forest.
"This is something crucial," she explained. "It's hidden within the underlying layers of dimensions, sitting below even what we call the negative dimension."
She turned back to the class, her expression still unreadable.
"You already know this universe—the material world we inhabit—isn't the whole picture. It's built upon layers. Dimensions. Constructs. But these are merely results."
Her hand swept over the web she had drawn, tapping the center point.
"This is what comes before all of that."
"The Axiom Manifold," she said with gravitas, "serves as the backbone of our reality. It's the foundational layer producing every formal system—be it logical, mathematical, or structural."
Shotaro narrowed his eyes, interest piqued. He wasn't the only one; even those who seemed disinterested earlier were now shifting forward, captivated by something they couldn't quite understand.
Professor Hibiko carried on.
"There's a concept in contemporary theoretical science, especially proposed by Professor Max Tegmark, known as the Mathematical Universe Hypothesis. This idea suggests that reality—every law of physics, each atom, every interaction between particles—isn't merely described by mathematics; it is mathematics."
She tapped her temple gently.
"Our universe, according to this theory, isn't just governed by equations; it's literally made from them."
"But what Tegmark left out," she said softly, "was the origin of these mathematical structures."
She pointed once more at the web.
"This is the Axiom Manifold. It's not just a catalog of known mathematics; it's where every potential and impossible mathematical reality is born. Everything logic can quantify… and everything it can't."
She started to label branches of the diagram.
Peano Arithmetic. ZFC Set Theory. Category Theory. Hyperset Theory. Acausal Topologies. Non-Computable Frameworks. Theory of Everything. Berkeley Cardinals.
Her voice sharpened with clarity.
"The math we've already unlocked—the structure behind concepts like relativity, quantum mechanics, thermodynamics, and more—that's just skimming the surface, the known portion."
"But the Axiom Manifold contains it all."
She raised her hands, one for each level she described.
"The math we haven't discovered yet." "The math that we may only speculate about without proof." "The math that seems irrational—it's incorrect."
She paused for a moment, allowing the weight of her words to sink in.
"Here, you'll find structures where division doesn't matter. Where infinity has its own uniqueness. Where logic flows through different gateways. Where time isn't measured in seconds, but in proofs.
A gasp rippled through the classroom.
One boy timidly raised a hand, his voice shaking slightly. "But… that means there are laws in play that we don't even have numbers for yet."
Maya Hibiko's faint smile returned, still sharp yet distant.
"That's right."
She stepped closer again, filling the space, causing the room to feel tighter—the walls appearing to stretch as new ideas expanded within them, ideas too big to fit into any one mind.
"And what makes this all the more unsettling," she continued, "is that the Axiom Manifold doesn't care about whether you believe in it or not. It just exists."
She pointed back to the center of the drawing.
"The Axiom Manifold is the code that constructs all existence—serving as the root layer beneath reality. It's where causality, motion, geometry, balance, and reasoning are all formed. Not planned out. Not managed."
"It… emerged. Like the very essence of breath from nothing."
A long silence filled the room.
This wasn't the kind of silence that awaits a test.
No; it was the kind of stillness where you feel something significant is on the horizon, waiting to shatter the calm—not glass, not air, but the very fabric of understanding.
With her back to her diagrams, Professor Maya Hibiko stood tall, the filenames of complex ideas still faintly illuminated under the stark ceiling lights. The sound of her marker cap closing echoed softly in the charged air. She clasped her hands behind her back, her posture as poised as an arrow ready to fly.
Then, a slight quiver.
It came from the far side of the room, where sunlight barely touched, and the walls carried a chill.
"Excuse me, ma'am," a girl spoke, her voice barely above a whisper. "Where did the Axiom Sphere come from?"
The room held a breath. A clock ticked three times—steady yet deliberate.
Professor Hibiko turned gradually, now intent, not shocked. Her amber eyes caught the light again, reflecting a depth that seemed simultaneously wise and ancient.
"Great question," she replied—delivering approval that contained more than just praise. It felt necessary.
She wanted that question. She longed for it.
In her facade of calm, there was a craving—not for being recognized, but to evoke thought. For her, learning wasn't simply structure; it was a kind of faith, and she was willing to carry that belief.
After all her years of education and countless papers that most would never read more than once, it wasn't praise or recognition she sought.
It was this moment—when one brave soul dared to question the edges of reality.
Her heels clicked on the floor as she walked back to the board—deliberate and slow.
The air thickened to meet some unseen force pressing in through the glass windows. A particular scent wafted through, a blend of ozone with the scent of old paper and stale dry-erase markers. Somewhere far below, the pipes rumbled, as if the very building itself clung to secrets long forgotten.
Maya lifted the marker again but held it hovering just above the surface, lost in thought. Her voice had deepened as she spoke, now sounding less like that of a teacher and more like a voice from history itself.
The hint of dry-erase fluid still permeated the air.
"Now, about the Axiom Manifold," she began. "There's no known starting point for it."
Everyone sat in rapt attention. The typical chatter had vanished; even the faint ticking of the clock now felt forced, all eyes locked on her.
"Some argue that the Axiom Manifold symbolizes God, a logical construct from which every axiom and parameter of possible realities sprouted."
She turned slightly, her marker still in hand, her eyes glimmering with intent.
"Others think it's a remnant, a fossil from something older still. Something that, not just mathematics, but the fundamental fabric of causality itself, emerged from."
Once again, the atmosphere shifted.
It wasn't wind or a breeze but the unsettling weight of untapped knowledge soon bubbling under the surface. That lurking feeling of something monumental approached.
Then, softly, almost reverently, she spoke. "The Seed of Axiom Theory."
As those words slipped out, her body seemed to quiver, just a little, with excitement.
A pause filled the air.
"Shut up, Mr. Mazino," a voice murmured from the back. A moan laced with irritation, quickly drowned out before it could lead to the disruption simmering underneath.
Professor Hibiko remained unfazed, marker still poised in hand.
Where was I? she thought quietly.
Ah, yes. The Seed.
The air was taut again. Silence had descended deeper.
Maya continued. "I believe the Axiom Manifold has a source, a nucleus… a seed that was planted before anything else existed. A spark—not energy, but potential itself."
She sketched a symbol on the board.
It wasn't a mathematical formula. It looked like an ancient script, like an intricate symbol crudely etched into stone by hands long dead.
"The seed blossomed," she murmured, "into every notion of possible—and impossible—mathematics. Each structure branching out and fractalizing like frost creeping across glass."
The air shifted again, and the smell morphed once more—from dry-erase fluids to something ancient like sun-warmed paper, burning copper, and that wet scent of something deep and cloistered beneath wood.
"It didn't merely stop at known mathematics," she elaborated, now pacing the classroom. "It birthed constructs difficult for any minds to touch. Logics that consume themselves. Equations that reveal things you didn't even think to ask yet. A balancing language that sprung forth from the void, called law."
There was movement in the front row, as if something had broken through the layers of understanding.
Drawing closer to the end of her marker, she sealed the tip and stepped away from the board.
Her gaze floated out across the students, her demeanor enigmatic in the dim glow of the projector.
"The seed continues to blossom," she whispered.
And then softly:
"Somewhere beneath us."
Time seemed to hang in the air, and every student suddenly felt it: a shift. A silence drew across the classroom, like something was straining against conventional understanding.
"Ma'am," a boy interrupted, voice steady but tinged with the tension that only those who treaded too far into profound thoughts can feel. "If our world contains an absolute form of infinity… how can it exist within this material sphere?"
Heads turned. It wasn't mockery; it was something deeper.
That question lingered beyond about school or academic grades. It had a pulse of its own.
Professor Hibiko turned slowly, like a pendulum resetting.
"I was waiting for that question," came her response, with a hint of excitement.
Stepping back to the whiteboard, she drew a vertical line, then three horizontal lines that sliced through it, labeling the three pillars: Mathematics, Metaphysics, and Pataphysics.
"This world," she began, each word cutting through the tension, "is built on these three invisible supports."
She ran her finger along, making connecting marks beneath each term.
"Mathematics sets up the framework," she shared. "It's a language of patterns, defined within sets and framed by axioms. It provides the groundwork for everything measurable, from the simple to the truly infinite."
Her gaze caught the boy's eye again, encouraging and steady.
"Modern set theory tells us that infinities are not created equal," she explained. "Some are vast, while some are beyond that, but there is always a larger scale."
She made notation that read:
ℵ₀ < ℵ₁ < ℵ₂ < ... < Ω.
"Greater infinities encompass smaller ones," she explained. "Just like how a volume embeds lines, or how a page contains ink. And at the summit of this hierarchy, some theorize lies Absolute Infinity—not merely a number, but a state of being."
She circled it slowly.
"In abstract mathematics, this exists outside the binary of what's possible or impossible. It does not belong to a system. It is the system. Gödel's incompleteness. Tarski's undefined. The furthest wall where coherence cannot hold."
Her voice pitched slightly lower, still firm.
"But mathematics can only tell you the what. The why is another matter entirely."
Maya extended her hand down to the next level on the board, writing it clearly as her tone deepened.
Metaphysics.
"Metaphysics takes up the questions surrounding why existence matters at all. It isn't about hard rules—it seeks causes and brings up thoughts of being, origin, essence. It is a lens, attempting to understand why structures come into being, investigating why anything is understandable at all."
Each tap of her chalk marked out her words as she continued.
"Why does time flow? Why does identity persist throughout changes? Why does existence show itself in comprehensible equations? Metaphysics searches for significance hidden inside the concerns around how things function."
Then she lowered deeper still.
The final layer she wrote was:
Pataphysics.
"A term I know sounds strange," she began, "but Pataphysics is the shadow of the system itself."
The classroom remained still.
"It explores imaginary resolutions," she said, her voice tightening with gravity. "It's about what lies beyond the norms of metaphysical understanding. Pataphysics doesn't adhere to the rules of mathematics or metaphysics. It stands not above them but lingers afterward."
She stepped back to turn the whole diagram around.
"With Math saying: 'How does this work?' Metaphysics is left asking: 'Why should it work?' while Pataphysics questions: 'Why not something that shouldn't?'"
She turned towards her pupils, intensity brightened in her amber eyes.
"Tegmark believes that all mathematical structures physically manifest. Not just in this reality or parallel ones, but in every system, logical or otherwise—even those at odds with our physics—each holding its spacetime."
She pointed sharply to the apex of the math axis.
"But Tegmark ends there, limiting logic's reach."
Maya drew a slow breath.
"Pataphysics takes it further."
A slow exhale followed her thoughts, allowing the weight of their meaning to settle.
"It creates ideas of imaginary infinities. Structures that couldn't exist even within a higher order of reason. Universes whose frameworks would inherently collapse on their first question but still persist. Not because they're allowed to—rather, because they exist as exceptions."
A moment of hushed understanding washed over the classroom, holding them all captive.
"That's why, in pataphysics, imaginary infinities, even those larger than absolute infinities in set theory, can form," she articulated, "because the question arises—'Why not?'"
That statement collided with the space, bold and daring.
A quiet shudder went through the front row.
A hand rose cautiously—its owner hesitant yet brave.
"Is pataphysics… bigger than Tegmark's Theory of Everything?"
Professor Hibiko turned once more, slowly, deeply contemplating the weight of what was asked.
"In some sense, yes," she replied. "But I don't see it as a contradiction to Tegmark."
She lingered.
"It serves as a recursive extension. A dimension that doesn't replace mathematics or metaphysics but grows within their shadows. It's not hierarchy; it's a loop with gravity that's uneven."
Her speech sharpened again, pulling them back into that familiar intensity.
"And be ready to keep up."
She crossed her arms, chalk still gripped in one hand like a wand of authority.
"This is Toyotaro Miracle High. This isn't just another prep school. You're not here to pass time with memorizing formulas or reenacting high school dramas that would make a bad sitcom."
She stepped closer, heels striking the ground like a countdown.
"You're here because getting by means demonstrating skills at a post-doctoral level. Passing—really passing—means engaging with ideas that very few can comprehend."
No one stirred.
Even the flickering lights seemed to dim, as if caught in her gravitational pull.
Professor Hibiko lowered her eyes, scrutinizing them with the same mixed compassion and understanding you'd find in someone who revered a delicate tool.
"It's a tough road," she admitted quietly.