On the top floor of the Mathematical Sciences Department at XianGuang Research Institute, Yue'er's office resembled a universe gradually quieting after a creation‑level storm. On three writable walls, once densely covered with symbols and formulas that evolved and collided like nebulae, most areas were now cleared, leaving only a central region displaying an exceptionally concise, elegant, and tightly structured derivation process. These symbols were no longer restless explorers; they had transformed into solid, solemn cornerstones indispensable to a grand edifice. The air carried a blend of electronic‑device heat‑dissipation, old‑paper ink fragrance, and some indescribable afterglow of intense concentration—silent, yet seeming to resonate with countless silent chords of thought.
Yue'er sat at her desk. On the ultra‑thin graphene display before her, a clean document interface showed. The document's title read: "On the Geometric Separation of P and NP: A New Proof Based on Rigidity Obstacles in High‑Dimensional Manifolds." This was not the ultimate answer she had initially envisioned that would completely solve the P vs. NP problem. Rather, it was the most solid frontier fortress she could reach on the current cognitive boundary—after months of near‑seclusion‑style frenzied thinking, deduction, and negation‑of‑negation, following her inspiration from Xiuxiu's computational lithography and the proposal of the "Geometrization of P vs. NP" framework. This was the **final draft** of her paper on the "PNP‑geometrization" framework.
Her gaze calmly swept over the core section of the document, which stated and proved a crucial theorem. This theorem was the first substantive, weighty breakthrough she achieved after transforming the abstract problem of computational complexity into the study of high‑dimensional geometric structures:
**"Theorem 4.7 (Main Result):** Under the specific geometric representation constructed in this paper (mapping computational problems to points on a compact Riemannian manifold and encoding computational‑complexity information in a 'complexity‑topological invariant' of that manifold within a generalized function space), the statement **P ≠ NP** is equivalent to the following geometric assertion: **There does not exist a global, smooth (C^∞) diffeomorphism that preserves the 'complexity‑topological invariant' and can globally 'smooth‑unfold' the submanifold formed by the point set representing NP‑class problems—which possesses high 'intrinsic foldability'—into the relatively 'flat' submanifold corresponding to the point set representing P‑class problems."**
This highly condensed mathematical language contained the essence of all Yue'er's thinking over these months. She explained it to herself in a more intuitive way: In her geometrized world, NP‑class problems were like an extremely crumpled, folded sheet of paper (high‑dimensional manifold) that had been kneaded countless times into an incredibly complex internal structure. P‑class problems, in contrast, resembled a relatively flat sheet of paper, easy to unfold and navigate. Her theorem stated that **P ≠ NP if and only if there exists no "magic iron" (smooth diffeomorphism) that could completely flatten that NP sheet into the flat P sheet without destroying the intrinsic "fold texture" (complexity‑topological invariant) that represented its complexity.**
In other words, the "difficulty" of NP problems was not because we weren't smart enough to find an algorithm, but because the corresponding geometric structure itself inherently, essentially **resisted** being "smoothed out"; its complex folded structure was intrinsic, ineradicable. This ineradicability was guaranteed by a solid **geometric‑topological obstacle**. She had found and characterized that obstacle—the "complexity‑topological invariant."
This was not merely a clever mathematical analogy; it was a **rigorous mathematical proof** that transformed a millennium‑old problem of computer science into a concrete, profound mathematical problem attackable by modern geometry and topology tools. It did not directly, frontally prove P ≠ NP (that might require more powerful tools or entirely new mathematical paradigms). But it provided an extremely strong and novel **existential support** for the conjecture "P ≠ NP." It pointed the direction, built bridges, and anchored the possibility of proving P ≠ NP onto solid geometric ground.
This paper was her most powerful, most elegant **counter‑reply** to Professor Bo‑so‑lay's questioning letter. Bo‑so‑lay had challenged that the "rigidity" assumptions of her theoretical framework were overly idealized. Yet she, by introducing deeper geometric structures (like a manifold's intrinsic curvature, topological invariants) and more flexible "soft" handling (e.g., allowing a certain degree of structural complexity but proving it cannot be smoothed away), not only responded to the challenge but also pushed the entire research to a higher plane. This was no longer a fragile sandcastle; it was a fortress built with geometric boulders.
Yue'er's fingers hovered above the virtual keyboard. The "submit" button on the screen emitted a faint, certain glow. She knew that once pressed, this paper—condensing her effort, wisdom, and bearing "XianGuang"'s reputation in the theoretical field—would traverse virtual networks to reach *Annals of Mathematics*, one of mathematics' supreme sanctuaries, to undergo the harshest, most merciless scrutiny from top global peers. This would undoubtedly place both herself and her theory at the very center of an academic storm.
Yet in her heart there was no hesitation, nor stirring excitement. After completing the final check, confirming each lemma, each derivation step meshed like precision gears, what she felt was an unprecedented, near‑absolute **void and calm**.
As if all mental energy, all intellectual passion, had been exhausted in those months of solitary struggle with abstract concepts. Now, her brain felt like a sponge completely wrung out—no more chaotic thoughts, no more proving impulses—just a pure, nearly transparent stillness. This wasn't fatigue; it was a deep satisfaction bringing absolute mental rest. She resembled a pilgrim who had just completed a long journey, standing before the temple gate, her heart filled with awe for the journey itself and peace after fulfilling her mission. As for the clamor and judgment beyond the gate, they seemed irrelevant to her present state of mind.
She gently pressed the "submit" button.
The screen showed a "submission successful" prompt. No bells chimed, no confetti flew—just a line of cold system text confirming the voyage of a thought.
She closed the document interface, shut off the display. The office sank into deeper silence. She rose, walked to the window, gazed at the distant city's blurred skyline. The setting sun was sinking, staining the sky a warm, tranquil orange‑red, strangely blending with the void calm within her.
She had completed an important **"proof of existence"** as a mathematician. Her thought, her construction, had escaped the confines of personal thinking and become an independent mathematical object awaiting examination. Whether the external world praised or criticized, it was already there.
In this peculiar calm—relieved of burden yet directionless—a strong impulse suddenly arose. She didn't want to immerse alone in this void; she wanted to see another battlefield, to witness that radically different **"proof of existence"** that transformed abstract theory into material reality.
Without notifying anyone, she quietly left her office, took the dedicated elevator down, deep underground, toward Xiuxiu's ultra‑clean lab.
Passing through layer upon layer of airtight doors, purified in the air‑shower chamber, she arrived at the observation corridor in the lab's core area. Through the huge observation window, she saw the already‑legendary **High NA EUV lithography prototype**. No longer in the tense, uncertain state of testing, it now operated in a low‑power, steady maintenance mode—like a giant who had performed earth‑shaking feats now taking a small rest, its massive form appearing serene and imposing under soft indicator lights.
Xiuxiu stood not far from the observation window, back to her, still wearing the blue cleanroom suit but without the hood, short hair somewhat disheveled. She seemed to have just ended a stretch of high‑intensity work, or perhaps simply habitually kept watch here, quietly gazing at the machine into which she had poured her entire heart. Her back conveyed deep exhaustion, yet carried an indescribable gentleness and satisfaction—like a mother guarding her child.
Yue'er didn't speak, just quietly walked to Xiuxiu's side, stood with her, looking through the observation window at the silent behemoth.
Xiuxiu sensed someone beside her, slightly turned her head, saw Yue'er, and a tired but genuine smile touched her face. She didn't ask why Yue'er had come, didn't exchange pleasantries—just returned her gaze to the machine inside the window.
"It's so quiet," Yue'er said softly, breaking the silence.
"Mm‑hmm," Xiuxiu responded, voice hoarse, "after making so much noise for so long, finally quiet."
Brief silence fell again. Two women—one having just committed the most abstract logical‑geometric conception to paper, submitting it to the world's court of reason; the other having just commanded a grand engineering project that conquered matter, harnessing light, carving theoretical possibility into silicon‑wafer reality.
They stood here: one conquering the **light** of the material world, proving "existence" with engineering might; the other steering the **strings** of the logical cosmos, constructing another form of "existence" with mathematical purity.
No verbal exchange was needed; there was no need to share success‑joy or process‑hardship. In this silence guarded by precision instruments and absolute cleanliness, after each having experienced peak climbing in her own domain and weathering external pressure, a profound, word‑transcending **mutual understanding** flowed quietly between them.
They understood each other's obsession with pursuing their field's utmost limits, the courage to walk alone, the immense sense of achievement and ensuing void calm that came from turning ethereal "possibility" into solid "existence."
Yue'er watched Xiuxiu's focused profile, watched the machine symbolizing the limits of human engineering. The void born from paper‑submission seemed filled by a warm, tangible fullness. Her theory might be abstract, but Xiuxiu's machine was precisely the most resonant echo of abstract theory in the real world.
Xiuxiu, too, could sense the unusually serene aura about Yue'er. She knew that must be the state after completing something extremely important in her own world. She didn't need details; she understood that feeling.
Thus they stood side‑by‑side, in the silent lab corridor, like two lighthouses weathered by different storms yet equally solid—one illuminating the abyss of thought, the other illuminating the boundaries of matter.
A long while later, Xiuxiu spoke softly, voice still hoarse but carrying a trace of barely perceptible reflection:
"Sometimes I think, the stuff we do—one too abstract, the other too concrete." She pointed at the machine inside the window, then at her own head, finally resting her gaze on Yue'er. "But it seems… either one missing, it doesn't work."
Yue'er gave a slight nod, a clear, understanding curve touching her lips.
"Mm," she responded softly, "abstract and concrete, like light and shadow, are one from the start."
Proof of existence, perhaps, never comes in only one form. And at this moment of standing shoulder‑to‑shoulder, two different‑dimensional "existences" achieved perfect harmony and unity in this silent underground space. In their own ways, they proved themselves, and proved that which "XianGuang" pursued—a broader future connecting abstract and concrete, thought and matter—was real and tangible.