Deep within the Stringlight Research Institute, there lay a special quiet chamber. It had no windows; its walls and ceiling were entirely covered with sound-absorbing material, while the floor was made of cold black metal plates. The only source of light came from a holographic projection device suspended in the center of the room, currently displaying an immensely complex, slowly shifting multi-dimensional geometric structure. This was Yue'er's "thought furnace"—a place where her inner visions of the mathematical universe could be externalized. She had been in seclusion here for three whole weeks, and apart from the most basic physiological needs, her entire consciousness was immersed in that grand riddle that had haunted the pinnacle of human intellect for over half a century—the P versus NP problem.
Within the framework of her "information-geometric field theory," this classic computer science problem was transformed into a geometric image in a high-dimensional spacetime: all problems that can be quickly solved in polynomial time (the P class) constituted a relatively "flat," easily navigable "plain"; while all problems whose solutions can be verified in polynomial time (the NP class) formed an extremely complex structure, filled with countless "folds" and "labyrinths"—a "folded surface." Whether P equals NP essentially asks whether this "folded surface" can be effectively "ironed flat" in some way, making it as easy to traverse as the "plain."
Over the past few weeks, she had been attempting to understand the precise geometric structure of this "folded surface." It was too complex, its dimensionality unimaginably high, and the manner of its folding seemed to follow some chaotic, unpredictable pattern. Traditional differential geometry tools proved inadequate here, unable to capture its intrinsic, defining characteristics. She felt like an explorer lost in an endless maze, where every seemingly promising path eventually led to deeper confusion.
Fatigue washed over her mental barriers in waves. Highly concentrated thinking consumed enormous mental energy, even inducing mild hallucinations at times—those writhing geometric lines in the holographic projection would sometimes seem to transform into living tendrils, reaching out to entangle her. Several times she almost gave up, ready to admit that her "geometrization" path might have hit a dead end.
Yet, at a critical point where consciousness was about to be completely overwhelmed by exhaustion, an extremely faint yet exceptionally clear "spark" flashed deep within her mind. This inspiration did not come directly from her contemplation of the "folded surface" itself, but rather from a long-buried memory—years ago, when she was still a student, fascinated by the elegance of algebraic geometry, she had deeply studied Riemann surfaces and their concept of "genus."
Genus. A classical invariant measuring the topological complexity of a surface. For a closed surface, genus intuitively corresponds to the number of "holes" it has. A sphere has no holes, genus 0; a torus has one hole, genus 1; and so on. Genus is a global property; it does not concern the specific curvature of the surface, only its overall connectivity structure. It is flexible, capable of penetrating local, subtle geometric deformations and pointing directly to the topological essence.
This ancient concept, like a flash of lightning in the dark, instantly illuminated a blind spot in her thinking.
Why must one remain fixated on understanding the despair-inducing local geometric details of the "folded surface"? Why not attempt to find a similar topological invariant belonging to this high-dimensional "folded surface"? A global indicator capable of piercing through its countless local folds and twists, directly measuring its overall "complexity"?
This idea jolted her awake from the brink of collapse. Almost trembling, she began constructing a new model on the holographic projection. She abstracted the infinitely complex high-dimensional "folded surface" representing NP-class problems, temporarily viewing it as a pure geometric object.
Then, she began trying to define a new kind of "genus."
This was by no means easy. Traditional genus definitions rely on classical properties of two-dimensional surfaces and cannot be directly generalized to such high-dimensional and peculiarly structured "folded surfaces." She needed to find an intrinsic definition, one not dependent on specific embeddings, purely based on the inherent properties of the geometric object itself.
She recalled some concepts she had introduced while constructing her "information-geometric field theory," particularly ideas about "information curvature" and "computational path homology." Could the "obstacle" degree of information transmission during computation be linked to some generalized concept of "holes"? Could the core computational steps required to verify a solution be regarded as some kind of "non-trivial cycles"?
Time lost all meaning in such intense focus. She mobilized all her lifelong learning, fusing tools from algebraic topology, differential geometry, complex analysis, and even quantum information theory, engaging in bold analogies and creative grafting. The structure on the holographic projection changed and rearranged frantically with her thoughts; countless formulas and symbols seemed to come alive, flowing and colliding.
Failure. Adjustment. Failure again. Readjustment.
She forgot hunger, forgot drowsiness, forgot herself. Her entire existence seemed to have transformed into a pure thought process pursuing the ultimate answer.
Finally, after an uncountable number of attempts and rejections, an exquisite, self-consistent mathematical definition clearly emerged from the chaotic flow of thought, like a lotus rising from water.
She named it—Complexity Genus.
This "Complexity Genus" no longer relied on the intuitive concept of "holes," but was defined by analyzing the rank of a certain generalized homology group formed by all possible "computation paths" on the "folded surface." It captured how many essentially different, mutually non-transformable "verification-path entanglement patterns" existed within this geometric structure representing NP problems. The larger its value, the more complex the inherent "verification logic" of the problem, and the more difficult it is to be reduced to an efficient solving algorithm.
It was like a powerful probe, capable of directly piercing through the dazzling local folds of the "folded surface," reaching its topological core, and reading its intrinsic "complexity genes."
Eagerly, she applied this new definition to several classic NP-complete problems, such as the Boolean satisfiability problem (SAT) and the traveling salesman problem (TSP). By constructing their corresponding "folded surface" models and calculating their "Complexity Genus," the results were exhilarating—these known hard problems all exhibited Complexity Genus significantly greater than zero!
This strongly hinted that P ≠ NP.
Because, if P equaled NP, it would mean that all NP-problem "folded surfaces" could be effectively "ironed flat" into the P-class "plain." And the "Complexity Genus" of the "plain," under her definition, should be zero. Now, these classic NP-complete problems displayed non-zero Complexity Genus—just as a surface with multiple holes can never be continuously deformed into a sphere without holes, they were topologically different!
This was not yet the final, rigorous proof. She still needed to prove that this Complexity Genus was non-zero for all NP-complete problems, and to establish its strict relationship with other computational complexity classes. There remained a vast technical chasm to cross.
But this was undoubtedly the clearest, most powerful, and most mathematically beautiful path discovered so far! She had found an unprecedented key that might unlock the mystery of P versus NP! This was not a vague conjecture, but a concrete, computable judgment tool built upon a solid geometric framework!
Immense joy and an almost debilitating sense of relief swept over her. Supporting her near-collapsing body, she shut down the holographic projection. The dizzying geometric structure vanished, and the quiet chamber fell into pure darkness and silence. Yet within her heart, it was as if ten thousand stars had simultaneously exploded, radiating brilliant light.
She needed to share. Not with the entire world, but with the only person who could understand the extreme hardship and supreme joy of such exploration.
Almost instinctively, staggering, she walked out of the quiet chamber, passed through empty, deserted corridors, and arrived at the rooftop terrace atop the institute. The cool night air slightly cleared her burning mind. Leaning against the railing, gazing at the sleeping institute below and the city lights in the distance, she took a few deep breaths, then dialed Mozi's private communication channel.
The video connected almost instantly. Mozi seemed still at the headquarters of Stringlight Fund, with the iconic, pulsating data nebula screen in the background. His face bore a trace of weariness, but upon seeing Yue'er appear on the screen, his expression immediately shifted to concern.
"Yue'er? You've emerged from seclusion? You look..." He paused, as if searching for the right words, "...very different."
Yue'er's hair was somewhat disheveled, her complexion somewhat pale from the long seclusion, but her eyes—Mozi had never seen her eyes so bright. They burned with a light of such purity that it surpassed any star they had ever gazed upon together. It was the kind of awe-inspiring brilliance that bursts forth when intelligence breaks through its limits and glimpses a corner of truth—a light so intense it even penetrated the screen, making Mozi's heart tremble.
"Mozi," Yue'er's voice trembled slightly with excitement, yet carried an undeniable clarity, "I... I may have found it."
She didn't directly say what she had found, but Mozi understood instantly. Only the problem she had devoted half her life to could make her this distraught.
"P and NP?" Mozi's voice unconsciously lowered, as if afraid to disturb something.
"Yes." Yue'er nodded vigorously. She tried to organize her words, to explain her breakthrough to this lover who was not a mathematician. "In my geometric framework, NP problems are like infinitely folded surfaces... I... I've defined a new quantity, called 'Complexity Genus'..." Quickly, almost incoherently, she described the concepts of "folded surfaces" and "topological invariants," attempting to convey the core insight.
Mozi listened intently, his brow slightly furrowed. He could not fully grasp those exquisite mathematical details, could not construct that high-dimensional "folded surface" in his mind, and could not thoroughly comprehend the precise definition of "Complexity Genus." Those terms were like a foreign language to him.
But he understood her excitement, saw the light in her eyes that outshone the stars. He heard the certainty in her words—not a dogmatic assertion, but a conviction born from rigorous logic and deep insight, almost intuitive in its certainty. He saw an explorer, after a long, lonely journey, finally catching sight of that decisive dawn on the distant horizon—a moment of ecstasy and awe.
That was enough.
He didn't need to fully comprehend the complex mathematics. He understood her, understood the value of this pursuit itself, understood the significance of this breakthrough for her, for the Stringlight Research Institute, and even for the boundaries of human cognition.
When Yue'er finally stopped, looking at him somewhat anxiously, seeming to fear her overly abstract descriptions might be incomprehensible, Mozi's face broke into an exceptionally warm and affirming smile.
"I don't quite understand what that 'genus' specifically is," he admitted frankly, his gaze firmly locked onto those sparkling eyes on the screen, "but I can see the light it has ignited in your eyes."
He paused, his tone becoming utterly solemn, as if stating the most important fact: "Yue'er, that is the face of truth."
These words, simple yet striking at the core.
Yue'er was stunned; then a great warmth, mingled with understanding, moved her to tears. She didn't need to prove anything to anyone, didn't need to explain to the entire world. At this moment, there was someone who, even without fully understanding her language, could read the light in her eyes and recognize it as a reflection of the "truth" they both sought.
Her eyes grew slightly warm. She nodded gently, all her unspoken words condensed into that simple gesture.
"You need rest." Mozi's voice softened. "The road ahead is still long, but now you have seen the direction. That is more important than anything."
The communication ended. Yue'er remained alone on the rooftop terrace, yet the extreme sense of isolation from her seclusion had dissipated. She looked up at the starry sky. The universe remained silent and vast, but now in her eyes, it seemed filled with some comprehensible, internal rhythm.
The dawn of P versus NP had appeared. Though the road ahead remained long, the complete edifice of proof still needed to be built brick by brick, and unforeseen difficulties and external skepticism might still be encountered, that fundamental obstacle seemed to have found a lever point.
She tightened her grip on the railing, her fingertips feeling the cold metal. Yet within her heart, the flame that had just been ignited—the one named "Complexity Genus"—burned brightly. This flame would illuminate every step she took from now on, until the day that puzzle that had troubled humanity for over half a century was finally solved.
The sky before dawn, the east already revealing a faint light.