98: Chapter 98 Dead End

The desk lamp in the study was still on, casting Chen Youhua's shadow onto the wall covered in scratch paper.

Early morning sunlight squeezed through the gaps in the curtains, adding a hint of warmth to the cold equations.

He was currently plunged into a state of immense confusion and a faint sense of self-doubt.

The excitement and sense of breakthrough brought by the system's prompts yesterday were still fresh in his memory.

Following the new path of Local Perturbation Thresholds and Measure Concentration, he felt unstoppable as he wrote down numerous derivations.

He even faintly felt that he might have touched upon a glimmer of hope for solving the existence and smoothness problem of the 3D Navier-Stokes Equation.

That sense of achievement from pushing the frontier of mathematics forward by a small step had brought him to the office early today to follow up on his success.

However, when he tried to press his advantage today, continuing deep along yesterday's line of thought in hopes of taking down this Millennium Prize Problem that had plagued the mathematical world for a century in one fell swoop, things went wrong.

He had encountered a new bottleneck.

Some key coupling terms became exceptionally complex, and conventional mathematical tools seemed difficult to handle effectively.

This was normal; after all, it was a world-class problem.

But strangely, no matter how much he racked his brains, trying every possible transformation and estimation, even deliberately forcing himself into a mental dead end...

...the system silent in the depths of his consciousness remained completely unresponsive, as if it had fallen into a total hibernation.

"What's going on?"

Chen Youhua stopped writing and rubbed his throbbing temples.

Had the system failed?

He tried to temporarily move his thoughts away from the Navier-Stokes Equation and toward another slightly simpler mathematical problem he had considered before.

A conjecture regarding the Spectral Distribution of Random Matrices.

Almost the very instant the thought occurred!

"Buzz."

That familiar, slight vibration in his consciousness appeared once more.

[Prompt: Consider introducing a Free Probability Framework, analogous to the concept of independence in classical probability, and focus on the combined application of R-transforms and S-transforms.]

The clear and precise prompt arrived as expected, opening another window of thought for him.

Chen Youhua was completely stunned.

The system wasn't broken; it was still working.

Then why... of all times, when he needed it most for the Navier-Stokes Equation, was it silent?

He irritably flipped through the derivation drafts he had written yesterday, which he had once been quite proud of.

His gaze swept back and forth among those complex symbols, trying to find the key point he had overlooked.

Once, twice... the more he looked, the tighter his brow furrowed.

A faint sense of unease began to grow.

He no longer obsessed over forcing progress, but instead forced himself to calm down and re-examine everything he had done yesterday following the system's prompts, like the most demanding peer reviewer.

Time passed minute by minute; only the rustling of paper being flipped remained in the study.

Suddenly, his finger stopped on a page of scratch paper.

His gaze was fixed intently on a key integral transformation step.

There, to handle the complexity of 3D space, he had subconsciously applied a dimensionality reduction technique that was very effective in 2D proofs.

He had tried to project or constrain the information of the 3D flow field onto a specific 2D surface, using 2D tools to indirectly control 3D behavior.

This operation itself was not mathematically incorrect; in fact, it was an effective approximation in certain specific cases.

"The problem is here..." he whispered to himself, tapping the step firmly with his finger.

He finally understood the reason for the system's silence!

Due to the inherent difficulties brought by 3D nonlinearity, especially Vorticity Stretching, as well as the loss of information and inability to control internal point behavior during the integral dimensionality reduction process, completely and strictly reducing a 3D problem to a 2D one and solving it with 2D methods had become a dead end.

Simply put, trying to compress the information of a 3D continuum with infinite degrees of freedom into a 2D surface would inevitably lead to information loss and blurring.

It's like drawing a map on a piece of paper; it inevitably fails to represent the full picture of 3D terrain.

This is about information loss. It's not that 3D can't be reduced in dimension, but in the process of reduction, can 2D preserve all the information of the 3D?

Assuming a black hole is a singularity, if you throw an object into it, the object's mass certainly won't be lost, but there is currently no means to retrieve the object's information.

Chen Youhua only now realized that the results he obtained through the system yesterday, which he thought were a small step forward for the mathematical world, were actually the end of the road for the 2D existence and smoothness of the Navier-Stokes Equation.

But this endpoint only solved a simplified 3D version where information was lost after dimensionality reduction, rather than the real, complete 3D Navier-Stokes Equation full of chaotic potential!

He thought he was sprinting toward the Millennium Prize Problem, but in fact, he was only solving a neutered model.

Having figured this out, Chen Youhua looked at the thick stack of yesterday's painstaking work on the desk, and a bitter, helpless smile couldn't help but appear at the corner of his mouth.

He let out a long sigh and leaned his body heavily back into the chair.

"So that's how it is..."

he muttered.

"No matter how exquisite 2D tools are, they are ultimately 2D.

3D problems must ultimately be severed with a 3D sword."

The system's silence was not abandonment, but the sternest warning.

This path is a dead end; turn back now.

He reached out, gathered the scratch papers filled with dimensionality reduction derivations, and gently pushed them to the corner of the desk.

Then, he pulled out a fresh sheet of A4 paper and spread it out before him.

His pen tip hovered; his gaze became focused and clear once more, only this time, with less of yesterday's urgency and more calmness.

He no longer tried to find shortcuts.

He had to face the core of 3D chaos directly, searching anew within its turbulent world of infinite degrees of freedom for a mathematical framework that could constrain its fury and prove its constant gentleness.

This required brand new, more powerful mathematical tools that truly belonged to the third dimension.

In the study, the pen tip touched down again, drawing a vector symbol u(x, y, z, t) representing the 3D Velocity Field on the pristine white paper.