Late autumn in Princeton grew colder. Yet the light in Yue'er's office seemed frozen by the chill, staying lit till dawn for several nights in a row. Ever since that walk under the stars with Mozi, she seemed to have truly received an "apple"—a fresh inspiration from modular‑form theory that allowed her to cleverly circumvent the lemma obstacle that had stalled her for so long.
It was a kind of ecstasy, like trudging through a dark tunnel for ages and finally glimpsing a glimmer of exit light. With remarkable efficiency and concentration, she unfolded this inspiration, advancing along the newly opened path, completing the proof of the key lemma in one breath, then seamlessly embedding it into the grand framework of "geometrization of computational complexity." The logical chain seemed interlocked; mathematical symbols flowed like tamed spirits under her pen, ultimately pointing to an exhilarating conclusion—under the specific algebraic‑geometric model she defined, P‑class problems indeed corresponded to a class of varieties with "simple" geometric structure, while a typical NP‑complete problem corresponded to a variety exhibiting high "complexity" and "singularity."
For several days she remained in an excited creative state. She began drafting the paper, putting this breakthrough into words. Each definition weighed, each theorem stated, each proof step refined, brought her a solemn joy of constructing a truth‑edifice. She seemed to have touched the cornerstone of the bridge connecting two seemingly separate worlds—discrete computation and continuous geometry.
Yet the goddess of mathematics is severe; she never grants laurels easily.
Just as Yue'er repeatedly reviewed the completed draft, preparing to share it with a few trusted peers for preliminary feedback, an utterly ordinary, almost‑overlooked transitional step caught her attention. It was an assertion about the "well‑definedness of a map," so natural in context that she had almost subconsciously skipped it, considering it self‑evident.
But this time, some intuition—perhaps the instinctive alertness to potential contradictions nurtured by long rigorous training—made her halt at this seemingly flat, harmless spot. She set down her pen, frowned slightly, re‑examined the assertion.
She tried constructing an extreme, even somewhat bizarre **counterexample** in her mind. Not to overthrow the whole theory, but as a stress test, a necessary means to examine the logical chain's resilience.
At first everything seemed normal. But when she substituted this specially constructed example into subsequent derivations, tracking its behavior under the entire framework, a tiny, almost imperceptible crack appeared. This counterexample didn't directly lead to contradiction, but it challenged the "universality" on which a later crucial inference relied. This map, in the special scenario she constructed, exhibited a kind of ambiguity, a "vagueness" not fully covered by existing definitions.
Cold sweat instantly beaded her temples.
This wasn't a glaring error, not a yawning chasm. It was more like discovering an extremely fine internal stress line inside a seemingly perfect crystal—almost fused with the crystal structure. It didn't shatter the crystal immediately, but it meant this crystal wasn't as flawless as imagined; inside lay a latent weak point potentially destabilizing the whole structure.
**The rigor of mathematical proof** lies precisely here. It demands the logical chain be diamond‑hard, no link resting on "approximately," "possibly," or "under normal circumstances." It must withstand the interrogation of any counterexample, however extreme, however bizarre. One counterexample suffices to overturn a seemingly perfect conjecture (e.g., Euler's conjecture being disproved); an un‑excluded potential ambiguity can make the entire proof edifice tremble.
Yue'er felt a dizzying vertigo. She forced herself calm, checking that step over and over, trying to mend the crack or prove the counterexample invalid within her main framework. She tried various approaches—adjusting definitions, adding constraints—but each attempt either ruined the framework's elegance and generality or introduced new, more intractable problems.
That night she lay completely sleepless. All earlier sense of achievement and excitement vanished, replaced by icy fear and profound self‑doubt. The crack indeed existed; tiny yet fatal. The grand framework she had poured months of effort into building likely had an irreparable flaw at its foundation.
Had she been too hasty? Blinded by initial victory, overlooking the utmost caution and trepidation needed in mathematical exploration? That seemingly inspired "apple"—was it a tempting but poisonous forbidden fruit?
Immense frustration nearly engulfed her. She felt like a traveler crossing a desert, finally seeing an oasis mirage, only to touch it and find it just heat‑distorted light. The psychological plunge from peak into abyss was harder to bear than remaining in darkness all along.
The next day she couldn't do any effective work. The draft was locked in a drawer, not daring to glance. She wandered campus aimlessly; autumn leaves drifted onto her shoulders unnoticed. Colleagues greeted; she responded mechanically, mind wholly absorbed in that inescapable mathematical crack.
Night fell again; loneliness and doubt flooded her. Sitting in her dark office, lights off, she felt abandoned on a rational wasteland, surrounded by endless, cold‑logic‑composed badlands.
Her phone screen lit the darkness—a video‑call request from Mozi.
She hesitated long, finally answered. On screen, Mozi's calm face; background his familiar, screen‑filled trading room.
"Yue'er?" His voice came through the speaker, a touch of concern. "You look… not well."
Yue'er lacked strength to dissemble, nor mood for pleasantries. Driven by a near‑breaking need to confide, she haltingly, even incoherently, told Mozi about discovering the proof crack and the huge self‑doubt it triggered. She described the seemingly trivial yet potentiallyoverthrow ambiguity, that fear of building a truth‑edifice only to find its foundation unstable.
"…I might have been wrong from the start." Her final voice trembled slightly. "All the effort may be built on quicksand. I overestimated myself."
Mozi listened quietly, didn't interrupt, face showing no surprise or disappointment, as if hearing a report about anomalous market volatility. Only after she fell silent did he speak slowly.
"In my world," his voice steady and clear, carrying a strange soothing power, "models also often develop 'cracks.' Perfect back‑test, live trading shows unexplained deviation. Sometimes it's over‑fitting; sometimes it's fundamental change in market structure—we call it 'paradigm shift.' "
He paused, continued. "Finding a crack doesn't mean failure. Quite the opposite—it means you've touched a deeper layer of the problem. The market tells me my model's boundaries through losses; mathematics tells you your theory's boundaries through counterexamples and ambiguities. That itself is progress, a deepening of cognition."
"You say you might be wrong," Mozi's gaze through the screen seemed to peer straight into her wavering heart, "but in mathematics, right or wrong can ultimately be determined through deeper exploration. Unlike the market, where truth often hides permanently under noise. Your problem at least has a definite answer, whether that answer is what you hope for or not."
His tone grew deeper, more certain. "**Your problem is more real than the market.** The market brims with lies, manipulation, group irrationality; its 'truth' is fuzzy, fluid. What you pursue is a definite 'yes' or 'no' in the universe's underlying code. The pursuit itself, and that ultimately found, indisputable answer—its value far surpasses any profit or loss in the market."
"Finding a crack is part of this process, even a crucial part. It forces you back to the start, examining every assumption, polishing every definition. It's painful, but it's the only road to more solid truth."
Mozi's words were like a cool beam piercing the fog and self‑negating vortex in Yue'er's heart. He didn't offer hollow comfort, but used a philosophy of "boundaries" and "determinism" she could understand, redefining her current predicament.
Yes, mathematical rigor lies precisely in its zero tolerance for counterexamples and ambiguity. Finding a crack isn't shameful—it's rigor at work, mathematics' own immune system clearing unreliable conceptions. This itself is part of mathematical exploration, what distinguishes it from many other fields.
Her problem is more real than the market. That phrase deeply moved her. In Mozi's world full of uncertainty and noise, the determinism and rigor she pursued instead became a luxury almost unattainable.
A strange calm began replacing earlier panic. The crack remained; the problem far from solved; the road ahead might mean tearing down and rebuilding, or at least major surgery. But that feeling of being utterly crushed dissipated.
"Thank you, Mozi." She said softly, voice still weary but with a thread of strength. "I think… I know what to do now."
"Go back, face that crack." Mozi's voice held a note of unwavering firmness. "Scrutinize it, analyze it, either mend it or… follow the new direction it indicates, opening a harder but perhaps truer road."
After ending the call, Yue'er sat in darkness a long while. Then she stood up, turned on the desk lamp. Blinding light made her squint, but she didn't shrink back.
She retrieved from the drawer that draft manuscript filled with her heart's blood yet hiding a fatal crack. This time her gaze didn't avoid the place that pained her. She took a red pen, beside that step drew a heavy question mark, then wrote: "**Ambiguity exists here; need to strictify definition or find alternative path.**"
She knew this would be a more arduous, more patience‑ and creativity‑testing battle. Yet now her heart wasn't full of despairing self‑doubt, but kindled a calm, focused resolve similar to Mozi facing market fluctuations—a determination to discover, analyze, and ultimately solve problems.
The mathematical abyss remained awe‑inspiring; the proof crack still starkly visible. But this deep night, a conversation crossing virtual space infused her with a new courage—not blindly optimistic courage, but courage to face error, embrace rigor, rebuild on the ruins of failure. She spread a fresh sheet of draft paper, pushed the cracked draft aside, like a chess player confronting a wrecked board, ready to start a new round of more thoughtful deduction. The long night stretched, but the spark of thought already flickered anew at the crack's edge.