Chapter Ninety-Four: Appraisal of Conduct

Although the actual conference was still a month away, Wu Tong had only ten days to prepare her report, which meant the schedule was rather tight. As soon as she returned home, she found herself about to break her promise; apologizing to her family, Wu Tong immediately reached for her laptop and began browsing through materials. She didn’t bother going to the study, instead choosing to work right beside her family, so that whenever she looked up, she could see them, and so they could see her as they came and went.

Once immersed in her studies, she was able to focus completely, undisturbed by her surroundings—she had little demand for an ideal environment. Even a noisy commotion wouldn’t have mattered, let alone the relative tranquility of her home. Since she had already set her goal, she had to make good on her word. Empty promises were not her way; she believed that what was spoken must be done.

The Polignac Conjecture posits that for every natural number K, there exist infinitely many pairs of primes (p, p+2K), where K is a positive integer. The special case when K=1 is the Twin Prime Conjecture. But how could this form be extended to arbitrarily large natural numbers?

If one could prove this conjecture, it would put another period to part of Hilbert’s eighth problem. Yet all this rested upon the premise: proving it.

Continuing along the train of thought she had developed while researching the Twin Prime Conjecture, Wu Tong wanted to first attempt to extend and refine the bridge she had built for that problem, to see if it might become the tool she needed to tackle the Polignac Conjecture.

To do a good job, one must first sharpen their tools. With a fine instrument, solving the problem would follow naturally.

Within the realm of prime numbers, Wu Tong’s reserves of knowledge were vast. She had studied every resource available, including the myriad of innovative and unconventional ideas published on arXiv.

She had pored over countless papers and treatises on the subject, yet the true “dragon-slaying blade” was still out of reach. The pearls of insight she had gathered remained scattered; without a thread to string them together, they could never become a necklace that would dazzle the world.

Where, then, was the proper path forward?

Once more, a storm of mathematics raged in Wu Tong’s mind. She scoured references, made endless deductions, her pen serving as the extension of her thoughts, sketching out calculations on draft paper as she systematically excluded or confirmed the factors she needed.

The broad direction—of creating a tool—was correct. The bridge she’d constructed, which could successfully prove the Twin Prime Conjecture, pointed the right way. However, relying solely on this bridge was not enough to extend the case of K=1 to arbitrarily large natural numbers. It needed new enhancements, to transform it into a grand bridge that could traverse and unify all, a bridge of ultimate harmony—such was her aspiration.

After three days of relentless, multidirectional deduction, Wu Tong turned her attention to group theory. She recalled Mr. Hilbert’s proof of the infinitude of primes, which leapt out from her mental storm and offered her a glimmer of hope toward a final proof.

Suddenly, what had been a hopelessly tangled skein now revealed its beginning and end, becoming clear and concise. Wu Tong smiled quietly. Her pen glided smoothly across the page, line after line of crisp equations flowing forth.

Given...
C(g): GG, Xgxg
...

Thoughts welled up like a spring, sparks of inspiration igniting a blaze that swept across the plains. A brand-new proof tool was born. Using the principle of infinite topological extension, supplemented by sieve methods, and employing group theory, she now approached the problem of the infinitude of primes from an overarching perspective.

She named it the Infinite Group Proof Method. Mathematical symbols and numerals interwove beneath her pen, paving a broad road toward the truth.

She spent half a day organizing the core content and concrete steps of the Infinite Group Proof Method.

With her tool refined and preparations complete, Wu Tong was ready for battle. She applied her method to her current goal: the Polignac Conjecture.

The problem, once shrouded in a thick veil, was finally unveiled, revealing its long-hidden true nature.

...a·c ≡ b·c (mod m) thus ac – bc ≡ 0 (mod m), which gives (a–b)·c ≡ 0 (mod m).
...(m, c) = 1, that is, m and c are coprime, so c can be canceled, yielding a–b ≡ 0 (mod m), thus a ≡ b (mod m)...
...

For every natural number K, there exist infinitely many pairs of primes (p, p+2K)—this proposition was proven true.

As the final word "established" was written, it was as if celestial music descended to earth. Another conjecture that had long troubled mankind was now transformed into a theorem.

She had succeeded in extending K to all arbitrarily large natural numbers!

Flexing her slightly stiff fingers, Wu Tong leaned back to relax, closed her eyes to rest, and let out a long, contented breath. At last, she had not boasted in vain; she had conquered the “minor” goal of the Polignac Conjecture!

Within her mind’s eye, the Enlightenment Stele shone with a radiant glow, suffusing and expanding, bringing nourishment to Wu Tong’s nearly exhausted brain, like sweet rain falling from the heavens, replenishing what her two days of intense mental effort had consumed.

The aura surrounding the stele grew even richer, no longer as thin and ethereal as it had once been. Through Wu Tong’s ceaseless study and near-epiphanic advances, as she verified major conjectures and absorbed the power of knowledge, the stele’s light had thickened to a luster as rich as milk. This was a sign of its gradually restored power.

A gentle clarity flowed through her heart—the Enlightenment Stele had regained another faculty: absolute deduction.

Anything Wu Tong laid eyes upon could now be subjected to in-depth analytical deduction, with the stele’s assistance. Of course, this ability had only just begun to recover; for now, it could only simulate data or the like, and still depended on Wu Tong’s own knowledge reserves.

But to Wu Tong, this was already a delightful surprise. Absolute direction and absolute deduction—a seemingly perfect pair.

After a brief rest, Wu Tong launched into a new round of work. The proof was finished, but it existed only as a draft. She still needed to organize it into a complete report.

But this didn’t require much extra effort. After a day’s work, in the early hours of the tenth day—her actual birthday, July 29th—she finished compiling both the report and a draft of her paper, uploading them directly to the website provided by the ICM.

Another night passed, the moon setting and the sun rising. After a deep rest and a bout of morning exercise, Wu Tong went downstairs to enjoy the sumptuous breakfast her family had prepared.

Golden, crispy fried dough sticks—plump and chewy, refreshing little pickled vegetables, salted duck eggs split in half, the yolks glistening with red oil—these were her grandmother’s specialty.

Palm-sized, flaky sesame buns stuffed with layers of thinly sliced braised beef, and a bowl of eight-treasure porridge, glistening with rice oil. Just the sight of it was enough to fill one with joy; to taste it was a truly heavenly delight.

Wu Tong smiled and gave her grandmother and auntie a thumbs-up in praise. These past few days, she had been working tirelessly, day and night, and her family had gone to great lengths to accommodate and care for her.

“Tongtong’s research report is finished!” As soon as Xu Meifeng saw the radiant spirit on Wu Tong’s face—so different from the serious intensity of previous days—she couldn’t help but ask, beaming.

“It’s done. There’s nothing left to do now but go there next month and deliver my presentation!” Wu Tong nodded in confirmation.

(End of this chapter)