Chapter 342 - 289: Li Criterion

Chapter 342: Chapter 289: Li Criterion

Third line: When this coefficient falls within a specific open interval, [the iteration is anchored]!

Falling outside the interval, [the iteration converges normally].

Three lines of equations, cleanly and completely explaining the "Ghost Wall," an unresolved problem that has plagued the field for thirty years, ever since 1993.

Las Vegener half-rose from his chair.

’This... Is this fucking it? The answer?’

’It’s that simple?’

He looked back up at the screen.

At the end of the Comment, Li Dong had only left a short note.

"This criterion provides the necessary and sufficient conditions for ’when a pseudo-convergence anchor will appear.’"

"When the conditions of the criterion are met, the visual representation of the iteration is almost indistinguishable from genuine convergence, unless someone knows the true solution beforehand."

"The author advises colleagues in this field that, going forward, any iterative scheme involving cyclic weights should have this criterion explicitly verified in the paper to avoid unnecessary controversy."

It was only a recommendation.

It didn’t name anyone.

And yet, this recommendation was posted right below Engelhardt’s paper.

’So... should Engelhardt’s paper be checked against Li Dong’s criterion?’

Without a doubt, it had to be checked.

And the result... would naturally be that the paper was invalid!

Las Vegener put down his pen and cursed under his breath.

’Fuck.’

...

Li Dong’s Comment had been on arXiv for less than twenty-four hours.

The entire field exploded.

By nine o’clock the next morning.

The first person to publicly share it was Jack Sutton of Durham University.

He was one of the top researchers of the next generation in the field of inverse problems.

Sutton posted a link to the PDF on his X account.

He included only a single sentence.

"I’ve derived it to the twelfth step. Every step is correct. The Ghost Wall has been cracked open."

In less than an hour, it had over a thousand shares.

Next up was Bram De Vries from Amsterdam.

He posted a photo of a handwritten draft.

It was another equivalent form of the Li Criterion, which he had derived himself after pulling an all-nighter.

His accompanying caption was even more brutal than Sutton’s.

"If Li is right, a significant number of papers from the last thirty years will need to be re-evaluated against this criterion."

Then came Princeton, ETH, Stanford, ICTP...

Everyone posted their own handwritten calculations.

By the next afternoon,

the number of shares had surpassed twenty thousand.

This was the first time a Comment in the mathematics field had racked up twenty thousand shares in less than twenty-four hours.

But for all the excitement in the field, controversy was brewing.

A few senior figures, people who had built their careers in the Tikhonov field for thirty years, began to publicly question the Comment.

The first to step forward was Eric Lindgren from Copenhagen.

He posted a long article on LinkedIn.

His meaning was quite clear.

First, Li’s criterion provides an "open interval."

Something like an open interval is very fragile in engineering practice.

Slightly alter the discretization method for the boundary conditions, and the coefficient will jump out of the interval.

Therefore, while this criterion might be correct "mathematically," it doesn’t necessarily have any true guiding significance "in an engineering context."

Second, if "pseudo-convergence anchors" truly exist, they are something that should have been discovered back in the 1980s.

A twenty-year-old kid doing pure math just now managed to crack it open?

Based on his experience, Lindgren found it hard to believe.

Third, Lindgren suggested that everyone calm down and wait for Engelhardt’s group to respond before drawing any conclusions.

After Lindgren posted his piece, the field immediately split into two camps.

Supporting Lindgren was a group of researchers in their forties and fifties, people who had been firmly established in the Tikhonov field for years.

Their logic was simple:

’Something I couldn’t figure out in twenty years, you cracked open at age twenty?’

’A criterion I couldn’t find in twenty years, and you just conjured it out of thin air?’

’By what right?’

’I don’t believe it.’

Supporting Li Dong were mostly young people in their early thirties, along with a few senior researchers who had painstakingly pored over every line of Chapter Six of *Spectral Method for Nonstandard Eigenvalue Problems*.

The debate raged from arXiv to X, to MathOverflow, and across several professional mailing lists in the inverse problems field.

They argued for three days.

Neither side could convince the other.

「On the fourth day.」

「Princeton Institute for Advanced Study.」

「Peter Sarnak’s Office.」

Inside, the nearly seventy-year-old professor was sipping a coffee and slowly reading Li Dong’s Comment on his iPad.

When he reached the last page, he chuckled.

’This kid... he’s not even in applied mathematics, but he’s already causing such a huge stir.’

’This business with cyclic Tikhonov... I suspect it won’t be called the Ghost Wall anymore. From now on, it’ll probably be the Li Criterion.’

...

The shockwaves from Li Dong’s Comment continued to spread.

「The morning of the fifth day.」

「London. IOP Publishing Group Headquarters.」

「Office of the Deputy Editor-in-Chief, *Inverse Problems*.」

The Deputy Editor-in-Chief, Maria Torres, had a grim expression on her face.

Two documents lay on the table before her.

One was the galley proof of Engelhardt’s paper.

The other was a printout of Li Dong’s Comment.

Sitting nearby were several members of the journal’s editorial board.

The two independent external reviewers hired by the journal were also present.

They had been sitting in this room for two hours.

They had gone over Li Dong’s criterion, line by line.

Every line was correct.

"I suggest..."

"We launch an internal review."

"All the papers we’ve published in the last ten years that used that little trick."

"We run them all through the Li Criterion."