Chapter 293 - 269: Teacher, It’s Very Physical
"That’s not the step."
"The mathematical inversion approach... it’s a dead end."
The room fell silent.
If Li Dong had said this to him ten minutes ago—
Wu Kai would have, at most, offered a polite smile out of respect, while silently making a mental note: ’This kid still needs a lot more polishing.’
After all, it was a statement with nothing to back it up.
But things were different now.
Now, Li Dong had pushed past the step that had stumped over a dozen research groups worldwide for three years, and he’d done it in a single morning.
Not only had he pushed past it, but he had also casually filled several more pages with further derivations.
So when Li Dong said, "The inversion approach is a dead end,"
it was no longer the impulsive remark of a young man running on intuition.
It was the words of someone already standing atop the wall, looking back down at those still ramming their heads against it.
...
Wu Kai forced himself to calm down before he spoke.
"Then tell me."
"Why do you think it won’t work?"
Li Dong didn’t beat around the bush.
He organized his thoughts and began to explain.
"Professor."
"The inversion part is, in essence, an inverse problem of an ill-posed operator."
"Where is it ill-posed?"
"Its kernel is incomplete. In the process of inverting the tunneling spectrum back to the Ligand Field tensor, any noise in the input gets amplified by several orders of magnitude by the operator’s small singular values."
"All the mathematical tools we have now..."
"Gikhmanov regularization, spectral truncation, the Backus-Gilbert method... they’re all essentially just suppressing that singular value spectrum."
"The methods of suppression are different, but the underlying Logic is the same..."
"We force-fit a prior into it to tell the operator what the solution ’should’ look like."
Wu Kai nodded.
This was a topic he had become very familiar with over the past two years of supervising applied mathematics students.
Li Dong continued.
"The problem is, all these priors are *mathematical* priors."
"What do I mean by mathematical priors?"
"It means, ’I want the solution to be smooth,’ ’I want the solution to be sparse,’ ’I want the solution to have the minimum norm.’"
"These ’wants’ have no physical connection to what the electron cloud around that single iron atom actually looks like."
"We’re using a mathematical preference to constrain a physical solution."
"The Logic from both sides is fundamentally incompatible."
"That’s why the peak for the third coordination shell is always blurry."
Wu Kai’s brow furrowed slightly.
Li Dong added.
"And another thing..."
"The inversion problem itself is about taking the result of a process and trying to deduce the process."
"Have you thought about what this machine is actually measuring?"
"It’s measuring the current generated by a complete physical process: X-ray excitation, electron transition, unoccupied orbital extension, and tunneling to the probe tip."
"It is, by its very nature, the *result* of a process."
"Yet we’re working backward, trying to infer the process from the result."
"In doing so, the physical details of any intermediate step get compressed into a single effective operator, and then we try to brute-force solve that operator."
"Isn’t that the same as taking a piece of living, breathing physics and butchering it into a set of cold, lifeless operators?"
"And then asking those operators what the original physics looked like?"
"There’s an inherent wall in the middle of all this."
"A wall not built by physics, but by ourselves."
Hearing this, Wu Kai’s frown deepened.
He said nothing.
Li Dong took a deep breath.
"Professor, you asked if I have a solution."
"I have a direction."
"I’m a little uncertain about it."
"But I’d like to try and explain."
Wu Kai looked at him.
"Go on."
Li Dong briefly outlined the new approach he had sketched out on his draft paper earlier that morning.
"We don’t do inversion."
"We do forward modeling."
"We rewrite the inverse problem of that set of measurement operators into a forward model parameterized by physical quantities."
"The parameters are no longer the abstract components of the Ligand Field tensor."
"The parameters are..."
"The dipole transition matrix element, the symmetry of the unoccupied orbitals, the orientation of the orbitals relative to the probe tip, and the WKB decay constant κ for the vacuum gap."
"Then, we openly bring in the X-ray polarization vector as an input boundary condition for the model."
"The polarization vector determines at the source which unoccupied orbital gets excited..."
"This is something physics already does; we don’t need to let the basis functions fight it out."
"And as for that blurry peak of the third coordination shell..."
"We don’t try to single it out."
"We let it emerge on its own."
"Each coordination shell has its own independent WKB decay constant, κ, corresponding to its tunneling barrier."
"The further out the shell, the smaller its κ, and the longer the tail that extends out from under the probe tip."
"In other words, the third shell already has its own unique tail in the tunneling current’s decay curve."
"We don’t need to single this tail out."
"It will reveal itself naturally as the scanning distance is increased step by step."
"It relies on the Gamov factor."
When Wu Kai heard the two words "Gamov factor," his expression flickered.
Li Dong concluded.
"Professor, to summarize briefly."
"We don’t push against the two walls."
"Because between those two walls, there was always a door."
The distant BEEP BEEP of the equipment on the lab bench outside could be heard.
No one in the office spoke.
Wu Kai’s hand, which had been resting on the table, slowly clenched into a fist.
His first reaction was disbelief. ’Impossible.’
𝒇𝒓𝒆𝒆𝙬𝒆𝒃𝓷𝒐𝓿𝙚𝙡.𝒄𝓸𝒎
’How could that be?’
’No one has proposed a new approach in this field for ages.’
’Because other paths were dead ends, everyone eventually converged on mathematical inversion.’
’This was the path that remained after everyone had filtered out all other options, again and again.’
’How could someone suddenly pop up and say, "between those two walls, there was always a door"?’
’How could that door be physics?’
Wu Kai couldn’t help but speak up.
"Li Dong."
"Let’s not even talk about whether your mathematical physics holds up."
"Let me bring up a more fundamental issue."
"You’re talking about forward modeling."
"That means you need to parameterize the entire X-ERT process from start to finish with physical quantities, right from the source."
"But there’s a part in there..."
"That short stretch of vacuum tunneling above the probe tip, plus the interaction between the Ligand Field and the Au(111) substrate..."
"No one has ever actually measured the shape of its effective potential barrier."
"That κ in your Gamov factor, what is its value?"
"You don’t know, I don’t know, nobody knows."
"The very reason we use inversion is because we can’t get around this part."
"We can’t directly measure that potential barrier."
"That’s why we’re willing to admit that even though inversion is ill-posed..."
"...at least every term in it is an ’experimental value.’ As ill-posed as it is, it’s still supported by data."
Wu Kai paused for a moment.
"Your ’physical parameterization’..."
"...it’s empty from the very beginning, right at the source."
"What are you going to fill it with?"
...
The more Wu Kai spoke, the more he felt he had hit the nail on the head.
The core conflict of this entire problem was right there.
It wasn’t that people didn’t want to do forward modeling; it was that the source couldn’t be measured directly.
That’s why they had taken a step back and settled for inversion.
So, he finally shook his head.
His voice was soft, but firm.
"Li Dong, your approach is beautiful in its overall structure."
"But according to the Logic of chemistry..."
"It’s too speculative."
"I’ll be blunt with you."
"This whole approach... it’s just not chemistry!"
The room fell silent again.
Li Dong looked down at his pile of scratch paper.
On the topmost sheet was the section he had scribbled over in red ink.
Beside that red scribble were the five words he had written.
"Let the atoms speak for themselves."
He raised his head and looked at Wu Kai.
His gaze was calm, without a hint of defiance.
"Professor, I understand what you mean."
"On the path of chemistry, it seems there’s no way forward."
"But..."
Li Dong said, with equal conviction.
"Professor."
"It *is* physics!"