In the top‑floor office of XianGuang Research Institute's Mathematical Sciences Department, Yue'er's office resembled a crucible of thought. On three writable walls, symbols and formulas seemed to possess life, constantly evolving, colliding, and recombining. On one side was the more robust, "flexibility"‑ and "robustness"‑encompassing Langlands‑program research framework that had withstood Professor B‑so‑lay's critique. Symbols representing L‑functions, automorphic forms, and Galois representations resembled stable constellations, emitting light of certainty in the mathematical cosmos. On the other side lay a newly opened battlefield—explorations providing mathematical support for Xiuxiu's computational lithography project, especially inverse lithography technology (ILT). Here were mathematical tools more closely aligned with physical reality: partial‑differential‑equation numerical solutions, large‑scale nonlinear optimization, uncertainty quantification… less elegant and perfect than pure‑mathematical theory, yet brimming with raw power and vigorous vitality conversing with the real world.
Yue'er immersed herself in this mental jungle woven from abstraction and concreteness. Becoming Chief Science Advisor to Xiuxiu's team was not merely providing intellectual support; it felt more like a window was flung open, revealing a brand‑new mathematical landscape. She'd long been accustomed to wandering in a realm of pure logic and infinite abstraction, where rules were set by definitions, truth established by proof—all clear and self‑consistent. Yet the problem ILT confronted pulled her into a domain filled with noise, approximation, and complex constraints. Here, perfect analytical solutions often didn't exist; what was pursued was the best approximation to physical reality under limited computational resources and time. This shift from "existence and uniqueness" to "optimality and robustness" thinking had a subtle yet profound influence on her.
She repeatedly pondered ILT's core paradox and wisdom: to obtain a "perfect" target pattern on silicon, you must design a seemingly "distorted" and "ugly" pre‑distortion pattern on the mask. This pre‑distortion pattern is the inverse‑mapping result of the physical process (optical proximity effect). Success hinges on possessing deep enough understanding and computational power of the physical process's mathematical model, thereby precisely predicting how "distortion" will be "distorted," ultimately miraculously returning to "perfection."
This process resonated wonderfully in her mind with that ultimate problem haunting theoretical computer science and mathematics for over half a century—the **P vs. NP problem**.
The P vs. NP problem, simple yet profoundly, asks: Are those problems whose solutions can be verified quickly (NP class) also solvable quickly (P class)? In other words, are all problems that are "easy to check" also "easy to solve"?
Yue'er's earlier research attempted to connect the PNP problem with other profound mathematical domains through the grand bridge of the Langlands program, seeking a unified perspective. She'd made some progress, constructed connections, but that core mystery—what is the essential distinction between P and NP—still resembled a snow‑capped peak shrouded in mist, visible yet unreachable.
Now, under the strong impact of ILT's "pre‑distortion" thinking, an unprecedented, breakthrough conception, like lightning tearing through night sky, suddenly illuminated a chaotic region in her mind!
What if… what if computational problems themselves underwent a fundamental **geometrization**?
This idea made her heartbeat accelerate abruptly, blood seemingly rushing toward her brain. She rose abruptly from her desk, almost lunging toward the still‑largely‑blank writing wall, grabbing an electronic pen, fingers trembling slightly from excitement.
She began outlining, transforming the vague yet powerful imagery in her mind into concrete mathematical language.
**"Suppose,"** she wrote a huge title in the wall's center: **"Geometric Characterization of Complexity Classes."**
"Every computational problem," she murmured softly, swiftly sketching a high‑dimensional‑space diagram, "can be mapped to a point in some high‑dimensional space. This point's 'coordinates' encode all the problem's information."
"Then, all 'easy'‑to‑solve problems—the P‑class problems—what kind of point set would they form in this high‑dimensional space? And all 'easy‑to‑verify but hard‑to‑solve' problems—the NP‑class problems—what kind of point set would they form?"
ILT's inspiration played a crucial role here. ILT, to get the target pattern, needs to find a complex, distorted mask pattern. This resembled a process of "encoding" and "decoding." A complex, seemingly patternless mask pattern (hard to directly "solve" for its corresponding target pattern?), after undergoing the physical process (a deterministic, quickly‑simulable "verification" process?), yields a clear target pattern (easy‑to‑"verify" correct solution?).
A stunning geometric scene materialized in her mind:
**"Perhaps… the point set corresponding to P‑class problems constitutes a relatively 'smooth,' 'flat,' 'regular' surface or manifold within this high‑dimensional space!"** She drew a smooth, gently undulating surface on the wall, labeling it "P." "In this region, 'finding a solution' (locating a specific point on the surface, or determining whether a given point lies on the surface) is easy, because the surface's structure is simple, navigation straightforward. Like searching for a specific location on a flat prairie—you can follow clear paths to quickly arrive."
"And NP‑class problems…" Her pen shifted, beside the smooth surface began depicting an extremely complex, countless‑folded, furrowed, self‑intersecting, fractal‑structured, highly distorted surface, labeling it "NP." "…their corresponding point set constitutes a 'highly folded,' topologically extremely complex surface! On this surface, 'finding a solution' (locating a specific point on this complex folded surface) becomes extremely difficult, because the surface's own structure is like an infinitely complex maze, paths entangled, lacking obvious patterns."
"But!" Yue'er's tone grew excited, eyes shining with light like discovering new land. "**Verifying** whether a point lies on this complex NP surface might be relatively easy! Because once you give me a candidate 'point' (a claimed solution), I can use a fast computational process (e.g., checking whether it satisfies the verification algorithm defined in the NP‑problem definition) to confirm whether this point indeed lies on that highly folded surface! It's like: even if I can't easily find treasure in a maze, if you bring the treasure to me, I can easily confirm whether it's genuine based on the treasure map (verification algorithm)!"
This geometric metaphor intuitively transformed computational‑complexity differences into differences in **spatial‑structural complexity**! P class is flat, traversable prairie; NP class is folded, distorted maze. The difficulty of solving stems from navigating the maze itself; the ease of verification stems from directly checking whether a given position lies on the maze map (i.e., on the folded surface).
She further deepened this conception:
"Then, the question of whether P equals NP transforms into a profound geometric‑topological question: **In this high‑dimensional 'complexity space,' can that 'smooth‑flat' P surface and that 'highly‑folded' NP surface be transformed into each other via some continuous transformation preserving certain key geometric or topological properties (like homotopy equivalence, or some finer 'complexity‑preserving' transformation)?**"
"If P equals NP, that means, however folded‑distorted the NP surface appears, it can essentially be 'ironed flat' into that smooth P surface—there exists a global, efficient method to turn the maze into a plain. If P does not equal NP, that means such 'ironing‑flat' transformation is impossible—the NP surface's complex folded structure is intrinsic, inherent, cannot be efficiently eliminated; the maze is a maze—its complexity is a real barrier."
This geometric perspective provided her with a brand‑new, extremely powerful tool for studying the P vs. NP problem. She could employ differential geometry to study local properties like curvature and connection of "P‑surface" and "NP‑surface"; employ algebraic topology to study global invariants like homology groups and homotopy groups; even employ her familiar Langlands program to explore possible symmetries and dualities behind these geometric structures.
This was no longer merely a game of logic and language; it placed the problem onto a geometric stage filled with intuition and imagination. She seemed to see those abstract computational problems transformed into concrete, microscopic space‑time structures, and the millennium‑old mystery of P vs. NP transformed into probing the essence of these structures.
Just as she reveled in this ecstasy of inspiration, eagerly wanting to further organize and formalize surging thoughts, her office door was gently knocked. She didn't even turn, subconsciously responding, "Come in."
The door opened, revealing Mozi and Xiuxiu. They seemed to have just finished a joint discussion on High NA project progress, passing by to check on her. Mozi still wore well‑tailored dark suit, bearing calm aura; Xiuxiu wore practical workwear, face carrying slight fatigue from technical challenges, yet eyes bright.
When they saw her standing before the writing wall, back trembling slightly from excitement, the wall covered with fresh, vibrant geometric sketches and keywords about "P," "NP," "geometrization," "folded surface"—both immediately realized something extraordinary was happening here.
"Yue'er?" Xiuxiu tentatively called, tone curious.
Yue'er abruptly turned. Her face flushed with excitement; that special light in a mathematician's eyes, immersed deep in thought‑world, hadn't yet faded, yet mixed with a burning eagerness to share discovery. Seeing Mozi and Xiuxiu standing side‑by‑side at the doorway, she was momentarily stunned, then an indescribable sense of happiness and completeness surged in her. At the moment her life's perhaps most important theoretical breakthrough descended, the two people she most wanted to share it with were right beside her!
"Mozi! Xiuxiu!" Her voice was somewhat hoarse from excitement. She hurried toward them, almost dancing while pointing at wall sketches. "I've thought of it! I may have found a brand‑new direction! A direction to **geometrize** the P vs. NP problem!"
She forced herself to calm, using clearest language possible to expound to these two partners—not mathematics professionals yet possessing top‑tier understanding—the conception just born. She started from ILT's "pre‑distortion" inspiration, explaining mapping computational problems to high‑dimensional space, the startling metaphor of "P‑class smooth surface" and "NP‑class folded surface," the enormous potential of transforming a millennium‑old problem into a geometric‑topological question.
She spoke somewhat hurriedly; some places even seemed incoherent due to thought leaps. Yet the huge joy and passion originating from intellectual breakthrough were unreservedly transmitted to Mozi and Xiuxiu.
Xiuxiu's jaw dropped. Though those profound geometric‑topological concepts were somewhat beyond her grasp, the vivid metaphor of "smooth surface" and "folded maze," and the almost‑burning fire in Yue'er's eyes, made her clearly sense that a remarkable idea was born. This excitement seemed purer, closer to wisdom's source, than when she'd successfully lit the EUV source.
Mozi listened quietly. His profound gaze swept over the unfamiliar geometric figures and mathematical symbols on the wall, finally settling on Yue'er's face, unusually bright from excitement. He perhaps couldn't fully understand all mathematical details of this geometrization framework, but he could keenly capture the powerful strength within—that strength of insight into complex‑system essence and transforming it into researchable models. How similar in spirit this was to his effort of distilling market chaos into computable models! Moreover, he could sense what this breakthrough meant to Yue'er personally, and to "XianGuang"'s mission of pursuing ultimate truth.
When Yue'er finally stopped, slightly breathless, looking at them with eyes filled with expectation and shared joy, the lab fell into brief, respectful silence.
Then Xiuxiu was first to jump up, hugging Yue'er tightly, shouting: "Sister Yue'er! So awesome! Though I don't fully understand, it feels so amazing! Is this that 'boundary'‑generated inspiration you mentioned? So magical!"
Mozi's face revealed an exceedingly rare, unguarded smile of admiration and delight. He stepped forward, not as excited as Xiuxiu, extending his hand, gently gripping Yue'er's arm—movement steady yet firm.
"Yue'er," his voice low yet full of affirmation, "this sounds… like a perspective that truly could change the game. Congratulations."
No more words needed. At this moment, theoretical leap and emotional resonance perfectly merged. In this office stacked with books and formulas, mathematician, engineer, and capital architect were closely linked because of a newly born conception that could shake the foundations of theoretical computer science.
Yue'er looked at Mozi and Xiuxiu before her, at the unreserved support and joy in their eyes. Her heart filled with immense warmth and strength. She knew this geometrization road would inevitably be long and arduous, filled with unknown challenges. Yet this inspiration's birth itself was a milestone victory. More importantly, on this lonely journey exploring truth, she wasn't traveling alone.
"Thank you… thank you both." Yue'er's voice somewhat choked, yet her smile was brilliantly radiant. "It's just a beginning, but… this feels so good."
Mozi and Xiuxiu exchanged smiles. Xiuxiu excitedly proposed: "We must celebrate! Right now! I know a new dessert shop opened outside the institute—their chocolate lava cake is simply divine!"
Mozi smilingly nodded agreement.
At this moment, theoretical starlight and friendship's warmth together illuminated this office at world wisdom's forefront. The geometrization of P vs. NP—this newly sprouted seedling of thought, at its birth, bathed in capital support, engineering inspiration, and deep‑affection nourishment. Its future is immensely anticipated.