Spring in Princeton arrived slowly; tender buds on branches tentatively unfurled in the crisp wind, sunlight filtering through the study window with a lingering chill. Yue'er sat at her desk, gaze fixed on the screen where academic updates and email lists continuously refreshed, her fingers unconsciously rubbing the warm ceramic of her coffee cup. Two weeks had passed since she posted her preprint on arXiv, titled "On the Geometrization of the P/NP Problem and Preliminary Connections with the Langlands Program—A Perspective Based on Information‑Geometry Field Theory." These two weeks felt like a silent tsunami within her—starting from the relief and emptiness after pressing the submit button, swiftly filled, impacted, even shaken by the enormous, surging waves of external feedback.
The preprint, that double‑edged sword of academia, with its unparalleled dissemination speed, had delivered her highly disruptive conception—like a depth charge dropped into the mathematics community's tranquil lake—instantly to every relevant mathematician and theoretical computer scientist worldwide. Ripples, no, giant waves, were spreading outward at astonishing speed.
The first few days brought an almost suffocating silence. Yue'er knew this wasn't indifference, but the calm before the storm—peers holding their breath, dissecting her dozens‑page paper word by word, filled with novel concepts and complex "generalized lifting functor" constructions. She tried to immerse herself in other readings or organize earlier research notes, but her attention kept drifting toward that webpage refreshing citations and comments. Every notification chime made her heart tighten slightly.
Finally, the first wave of feedback emerged. Private emails began crowding her inbox; scholars she'd never met, even a few seniors whose names appeared in textbooks—admired since her student days—sent inquiries and discussion emails. Among them were sincere admiration and astonishment.
"Breathtaking imagination, Dr. Yue'er! The boldness of the bridge you've built connecting P/NP and Langlands is astonishing."
"The introduction of the generalized lifting functor offers a completely new perspective for addressing certain long‑standing duality problems; regardless of the final conclusion, this itself is a significant contribution."
"Your use of information geometry is highly inspiring; perhaps it opens a new window for understanding the internal structure of complex systems."
These affirmations, like winter sunlight, briefly dispersed the cold in her heart, making her feel her solitary exploration wasn't worthless. Her "child," at least in some eyes, was clever and full of potential.
Yet, following closely, came more massive, sharper **skepticism and criticism**. The mathematics community, a collective taking logical rigor and absolute proof as its highest standard, is inherently **conservative and cautious** toward any theory attempting to challenge existing paradigms—especially "radical" theories involving core problems like P/NP. This conservatism doesn't stem from stubbornness or malice, but from the supreme defense of mathematical certainty. A wrong, not‑strictly‑proven "breakthrough" could cause far more harm than a hundred mediocre but correct results.
Skeptical voices first concentrated on rigorous scrutiny of technical details:
"There seems to be a gap in the proof of Lemma 3.5 in Section 3.2; when applying Serre duality, is the assumption about base‑space connectivity for the fiber bundle sufficiently strong? A stronger condition might be needed here; otherwise, the subsequent lifting construction might fail."
"Regarding compatibility between the generalized lifting functor and classical Langlands correspondence, the argument you give in Section 4.1 resembles a heuristic explanation rather than a strict mathematical proof. Clearer categorical characterization and verification of natural transformations are needed here."
"You claim your framework can circumvent certain known complexity‑theory obstacles, but the paper doesn't provide convincing, detailed comparisons with existing complexity‑class hierarchies (like the polynomial hierarchy). Is this an evasion?"
These were precise, technical doubts from peer experts—like surgical knives dissecting possibly fragile connections in her theoretical framework. Each such email, each publicly discussed point on academic forums, sank Yue'er's heart further. She had to expend enormous mental energy repeatedly checking her derivations, thinking how to respond, or admitting oversights and considering repair strategies. This process drained her spirit and was fraught with uncertainty.
What pressured her even more were those **fundamental doubts** arising from different mathematical‑philosophy schools and thinking habits:
"Does reducing P/NP—a core problem of computational complexity—entirely to a highly abstract geometric/topological property oversimplify? Can the dynamics and resource consumption of computational processes truly be completely captured by static geometric structures?"
"The Langlands program itself still holds many mysteries; forcibly bundling it with the P/NP problem might introduce unnecessary complexity, even potentially compounding the difficulties of both problems."
"This paper resembles more a grand philosophical vision and a declaration of future research plans than a solid, verifiable mathematical achievement. Its core 'unification' assertion remains at the conjecture stage."
These doubts touched the foundation and motivation of her theory. They forced her to step outside technical details, to re‑examine and defend the rationality and value of her entire research path. This wasn't merely an intellectual contest, but a test of faith.
The mathematics community's typical process toward breakthrough theories—especially grand visions like hers attempting to connect two seemingly distant domains—goes exactly like this: first shock and curiosity, then overwhelming strict scrutiny and skepticism from all angles. Only after enduring this purgatorial "peer‑review" prelude, withstanding all reasonable criticism and providing convincing responses or corrections, might the theory slowly, partially gain acceptance. Many once‑earth‑shattering theories, like non‑Euclidean geometry and Gödel's incompleteness theorems, experienced similar, even more intense, storms of skepticism in their early stages.
Yue'er felt an unprecedented weariness and… **wavering**. During the day, she forced herself to read and analyze each doubt, attempt to draft replies, or re‑calculate on her scratch paper. But late at night, alone facing the study's silent walls, those sharp questions and negative evaluations would surge into her heart, echoing repeatedly.
"Have I truly gone in the wrong direction?"
"Is this 'lifting' structure essentially an un‑rigorizable castle in the air?"
"Is linking P/NP with Langlands merely a mathematician's romantic fantasy, not a viable path?"
Confidence, like a sandcastle, began cracking under the continuous washing of skeptical waves. She started doubting her intuition, wondering whether that exhilarating inspiration about "connectedness" was just a self‑suggested illusion. The feeling was like bringing a carefully nurtured child before people, only to receive countless criticisms and remarks like "this child looks a bit strange"—making her, as the "mother," heart‑aching and bewildered.
Just as she reached her emotional lowest, nearly drowning in self‑doubt one deep night, two requests simultaneously lit up on the study's encrypted communication line. Mozi and Xiuxiu.
She hesitated, then connected. Images of the three appeared on screen: Mozi's background showed Shanghai's predawn nightscape; Xiuxiu seemed still in the lab's rest area, a trace of weariness on her face.
"Yue'er," Mozi spoke first, his voice transmitting through the waves with a steady strength. "I've seen some discussions about your paper on mathematics forums." He didn't use words like "controversy" or "skepticism," but Yue'er understood.
Xiuxiu followed immediately: "Yue'er‑jie, though I can't grasp those mathematical details, I can feel… you've been under great pressure lately. Don't take those comments too much to heart."
Yue'er looked at their concerned expressions on screen; the tough shell she'd been maintaining finally cracked. She lowered her head, voice slightly choked: "I… I don't know. Some of the questions they raise… are very sharp. I've been checking; some places… maybe I really didn't consider enough. I've even started doubting whether the whole direction is…"
"Yue'er." Mozi interrupted her, his tone unusually firm and unquestionable. "Remember my model? Every major iteration performed amazingly in back‑testing, but once deployed live, it always encounters new, unforeseen market behaviors. People always question whether the model's logic has fundamental flaws. Skepticism is the norm in exploring unknown territory—even a necessary whetstone."
He paused, then continued: "Your work is opening up a mathematical frontier no one has set foot in. If everyone could immediately understand it without any dissent, that would actually indicate limited value. True breakthroughs often come with massive controversy and uncertainty."
Xiuxiu nodded vigorously, chiming in: "Right, Yue'er‑jie! Think about our EUV light source—how many failures did we go through from concept to first ignition? How many people doubted it as 'impossible'? Even now, with the prototype tape‑out successful, there are still various voices internationally—some calling it a fluke, some questioning data authenticity, some waiting to see if we'll stumble next. But what matters is we know what we've done, we know how we've walked each step, we know our direction is right!"
She looked at Yue'er, eyes clear and sincere: "Your mathematics is even more abstract, more frontier than our lithography machine. Facing skepticism is completely normal. But you can't negate yourself because of it! That 'connectedness' idea you proposed gave me such great inspiration! I trust your intuition; I believe that 'bridge' you see must exist!"
Mozi said gently: "Yue'er, skepticism isn't scary. What's scary is letting skeptical voices drown out your inner voice. What you need to do isn't doubting the path you've chosen, but picking up sharper weapons to answer those doubts, to perfect your theory. This itself is part of the proving process—a battle harder than writing the first draft, but also more valuable."
Hearing their words, Yue'er felt a warmth rise from her heart, slowly dispelling the lingering chill and self‑doubt. They didn't understand those complex mathematical details, but they understood her, understood the hardship of exploration, the value of persistence. Their support wasn't blind encouragement but the deepest comprehension and firmest trust based on their own experience of breaking through thorns in their respective fields.
She raised her head, wiping the moisture at the corner of her eye, taking a deep breath. On screen, Mozi's gaze was steady as bedrock; Xiuxiu's eyes blazed like flame.
"Thank you," her voice regained calm, even gaining a resilience not there before. "I understand. Skepticism is inevitable; I can't retreat. I'll analyze those questions one by one—those I can answer, I'll answer to the best of my ability; those where gaps exist, I'll find ways to mend them; even if ultimately some parts need to be rebuilt from scratch, that… is also part of moving forward."
Her gaze grew clear and resolute again: "This 'bridge'—I won't give up."
Seeing the rekindled fighting spirit in her eyes, Mozi and Xiuxiu both showed relieved smiles. They knew that wise, tenacious Yue'er had returned.
The call ended; the study returned to quiet. But Yue'er's state of mind was completely different. She reopened those doubt‑filled emails and forum posts; her gaze no longer flickered evasively but carried a calm, almost combative scrutiny. The conjecture's ripples had spread, bringing storm, but also opportunities for clarification and tempering. And she wasn't sailing alone in the storm. She had her anchor, her light. The road ahead meant responding, revising, more arduous climbing. But now, she feared nothing.