76: Chapter 76 Critical

After this round of reckoning ended, Jiang Lin actually didn't immediately start writing to prove the Narrow Theorem.

Because before that, he still had to personally push G-index V0.1 to a dead end.

Push it until it bled its last drop of blood, push it until it was smashed to pieces on the cliff edge of logic and Physics.

Only after confirming that all wide avenues were dead ends could he safely walk into that narrow alley that could only accommodate one person.

He began feeding V0.1 extreme conditions.

With the first slash, what he did was literature extrapolation, reduced-order scanning, and low-resolution trend testing.

The results quickly exposed the first crack.

In the extremely high Lundquist number interval, the Plasmoid channel and turbulence channel were no longer always serial, and turbulence might take over the current sheet structure early.

V0.1 did not collapse from calculations.

It was the Physics boundary that pushed it out of the applicable zone.

The second slash, the extremely low β interval.

The plasma thermal pressure was negligible in front of the strong magnetic pressure.

Jiang Lin introduced a guiding field that was not quite strong.

Just this seemingly harmless extra component, in a low β environment, directly twisted the structure of the dissipation layer into a pretzel.

The originally symmetric tearing structure was broken, the ghost of the Hall effect began to wander in the background, and the two-fluid effect forced V0.1 to abandon the single-fluid MHD assumption.

The system of equations collapsed on the spot under the coercion of singular perturbation.

The third slash, asymmetric reconnection like the magnetopause.

The real Physics scene at the junction of the solar wind and Earth's magnetosphere.

Here, the density, temperature, and magnetic field strength of the plasma were completely asymmetrical on both sides of the interface.

Jiang Lin looked coldly at the simulation results; under this scenario, even the current sheet length L itself lost its unique natural definition.

How should you define something whose boundaries are constantly twisting and whose thickness varies drastically in space?

Without the characteristic length, all the dimensionless parameters relied upon by V0.1 became a joke.

Then came the fourth slash, the fifth slash...

For a whole year, Jiang Lin was like a ruthless executioner, cutting off V0.1's ambition piece by piece.

Two years later, he opened the folder named [Narrow_Theorem_Draft], and created a new README document.

At the very top of the document, he typed out seven tentative application boundaries.

The Narrow Theorem was really not a pretty topic he actively picked out.

Those directions grand enough to be written into a grand narrative were too big, too dirty, and too dependent on boundary conditions, making them unsuitable to become the first rigorous proof.

The Narrow Theorem was the last crack of a door remaining after counterexamples blocked all other paths.

Only this path: two-dimensional incompressible resistive MHD, anti-parallel magnetic fields, long thin current sheets with finite aspect ratios, Plasmoid tearing, Sobolev spaces, energy methods.

Only this path was narrow enough to be proven.

And only by proving this narrow path could he lay the first true foundation pile for that crumbling theory in this Wasteland World.

In December of the thirty-fourth year, the temperature outside the Stone House dropped to minus twenty degrees Celsius.

The Moss was all frozen into grayish-brown hard shells.

Jiang Lin began writing the first line of proof.

Not sub-proposition one; that was too big.

He started from a smaller place called the lemma.

[Lemma 0.1]

In the simplified anti-parallel magnetic field model of two-dimensional incompressible resistive MHD, given an initial value that is sufficiently smooth and satisfies the boundary conditions, the magnetic flux function ψ possesses an H¹ weak solution within a finite time window.

Conditional uniqueness can be obtained under additional regularity conditions.

This was the foundation of the foundation.

If even this lemma could not stand, all subsequent derivations regarding the tearing threshold and G-index would be wastepaper.

Without a weak solution, there wasn't even the qualification to discuss instability.

For the first proof, Jiang Lin chose the most orthodox path.

He used Galerkin approximation to construct approximate solutions.

Projecting infinite-dimensional partial differential equations into finite-dimensional spaces, he cut continuous Physics processes into discrete modes like cutting potatoes.

The nib of his pen glided rapidly across the paper, formulas pouring out line by line.

Short sentences, hurried derivations.

Velocity field u, magnetic field B.

Energy equation.

The dissipation term provided control of the critical gradient norm.

Through energy estimates, he proved that the approximate solution family was bounded in H¹.

At least at this norm level, it did not qualify to blow up in a finite time.

Next, through the weak compactness theorem of the Banach space, he extracted a subsequence from this bounded approximate solution family.

Then letting the dimension tend to infinity, through a limit process, he obtained the weak solution of the original equation.

Finally, utilizing higher-order additional regularity and the Gronwall inequality, he hard-fought to smash out conditional uniqueness.

Twelve pages of drafts, densely packed.

After finishing the final stroke, Jiang Lin shook his sore wrist, picked up his mottled thermos cup to take a sip of Moss tea, and then spread these twelve pages of paper out on the desktop to review from the beginning.

Flipping to page seven, a problem appeared.

The constant dependence in the energy estimates was too fine.

When scaling the nonlinear convection term, to suppress that annoying boundary term, he borrowed a Poincaré inequality.

The constant C here inevitably carried the information of the initial energy.

It was not that it could not depend on the initial value.

That was unrealistic.

Any PDE energy estimate would inherently depend on the initial energy; this was common Physics sense.

How much energy the system had at the very beginning determined the upper limit of subsequent evolution.

The problem was that this constant C depended on the specific shape details of each sample.

If in the subsequent proof, what he wanted to discuss was the initial value of an entire family of long, thin current sheets, searching for the critical condition that triggered Plasmoid, then this constant would become case-by-case.

If the initial value was slightly distorted or the current sheet became slightly thicker, constant C would change.

Then the instability threshold finally derived would also drift along.

Calculated as this number today, tomorrow changing the initial value shape would calculate another number.

What kind of unified theory was this?

The Narrow Theorem would lose the significance of unified guidance.

"No, we can't do this."

For the second time, he adjusted his goal.

The constant must be clean.

It could depend on the uniformly controlled upper limit of the initial value, system parameters, boundary conditions, and the geometric hierarchy of the current sheet.

But it must absolutely not depend on the detailed shape of a specific sample.

Once this requirement was added, the weight of the proof instantly doubled.

Jiang Lin felt that he was not writing mathematics, but rather clearing mines.

For every inequality scaling, he had to look back to trace the source of the constant.

For every Sobolev embedding, he had to put a parenthesis beside it, noting the geometric assumptions of the region.

Writing to the eighth page, he got stuck again.

This time the problem lay in the active layer.

To make the estimate more precise, he had originally restricted the integration region to the vicinity of the active layer of the current sheet.

Because that was where the Physics reaction was most intense, with the largest gradient.

However, as time t passed, the position and shape of the active layer would change.

The magnetic field was reconnecting, the plasma was jetting, and the active layer was constantly twisting and deforming like a living creature.

If the region was changing, the Sobolev embedding constant would change along with the geometry of the active layer.

Once the constant became an unknown function of time t, the subsequent unified estimates would completely spiral out of control, and the Gronwall inequality would have nowhere to start.

Jiang Lin threw his pen down and rubbed his temples.

By the dusk of the third day, the sun was about to set.

The dim, yellowish light passed through the narrow window of the Stone House and hit the paper filled with formulas.

Jiang Lin stared at those symbols, a thought suddenly flashing through his mind.

Why must he follow the active layer?

Since the active layer was moving, he would just find something stationary to cover it.

He didn't need to make estimates on the active layer; he could instead use a fixed external domain Ω_ext.

Wrapping all active regions where the current sheet could possibly evolve into this fixed external domain that was large enough and whose boundary was smooth enough.

Then, using extension operators, all functions within the active layer were smoothly extended to the entire fixed external domain.

In this fixed smooth domain, he would track the unified constant.

This was not a trick he originated.

In his memory, back when he was reading Brezis's functional analysis exercise collection, there had been similar boundary extension techniques in it.

However, he had never seen it in any literature he had read when transplanting it to this highly nonlinear simplified MHD problem to suppress the boundary constants of the moving current sheet.

"Give it a try."

He pulled out a fresh sheet of paper and started writing again.

By the eleventh page, the logical chain closed.

Under the protection of the extension operator, all the inequalities obediently retreated within the limits.

Re-evaluating.

This time, all the constants C relied solely on the uniform initial value upper bound, the geometric characteristics of the fixed external domain \Omega_{\text{ext}}, the system parameters, and the macroscopic boundary conditions.

He successfully drove out that ghost-like sample shape dependency.

Lemma 0.1, temporarily passed.

In March of Year 35, Jiang Lin transcribed this pile of heavily modified and unrecognizable drafts into a formal electronic manuscript.

A full twenty-three pages of LaTeX code.

Compile, generate PDF.

Clean and rigorous mathematical formulas appeared on the screen.

Title: [Lemma 0.1: Existence and Conditional Uniqueness of Finite-Time Weak Solutions]

Status: [Completed.]

Date: [March 7, Year 35.]

Jiang Lin looked at this document, his heart beating a little fast. In the README of the [Narrow_Theorem_Draft] folder, he solemnly typed out a few lines:

This is the first brick of a Great Wall.

There are many more to come.

One by one.

In late March, after a brief rest, Jiang Lin began tackling Lemma 0.2.

[Lemma 0.2]

The weak solution constructed in Lemma 0.1 can be upgraded to a local strong solution under additional conditions.

Going from a weak solution to a strong solution sounds like just a step away, but mathematically it is crossing a chasm.

A weak solution only requires the function to have first-order derivatives in H¹, allowing a certain degree of physical roughness.

However, to discuss the real spallation phenomenon within the narrow current sheet, finer control is necessary, requiring at least H² or even higher-order norms.

It must be guaranteed that the function is not only continuous, but also that its second-order derivatives exist in the sense of square integrability.

Boundary regularity needs to be handled.

It is necessary to ensure that those highly nonlinear convection terms and Lorentz force terms do not snowball and blow up the estimation constants when taking higher-order derivatives.

This time, the progress was excruciatingly slow.

After writing five pages, he got stuck.

When calculating integration by parts, he missed a normal derivative term on the boundary.

Adding this term back meant rescaling the previous inequalities all over again.

Writing three more pages, he got stuck again.

The trouble this time was even greater.

It was that the embedding theorem was used too hastily.

During the proof, in order to control the infinity norm of a nonlinear term, he casually used a Sobolev embedding in a two-dimensional space.

H² → C⁰

Logically there was no problem.

Two-dimensional H² functions are indeed continuous.

But he ignored the constants.

He ignored boundary regularity and the dependence of the constants on the domain geometry.

In the fixed smooth domain of the previous Lemma 0.1, this embedding constant K was an extremely well-behaved fixed value.

But now, to capture the critical thickness of spallation, his estimates inevitably had to sneak back to the vicinity of that increasingly thin moving layer.

Once the domain began to thin and the aspect ratio began to stretch, the embedding constant K would explode out of control like a wild horse breaking its reins.

Modify.

Redesign the weight function.

Write again.

Try using anisotropic Sobolev spaces to separately control the derivatives in both the length and width directions.

Time ticked away bit by bit.

By the end of May in Year 35, the draft paper for Lemma 0.2 had piled up to seventeen pages, yet the conclusion remained nowhere in sight.

It was not that there was completely no progress.

Every day, he could push forward by a line or half a page.

But every day, new minor problems were also exposed.

Fixing this minor loophole meant the entire preceding structure had to be fine-tuned accordingly.

In Jiang Lin's eyes, the entire proof process had turned into an increasingly bloated and heavy mechanical device.

The further along he went and the deeper the derivation became, every mathematical bolt in the front began to bear weight exceeding its design.

At the slightest disturbance, the entire proof system would emit a dangerous creak.

This was probably the normal state of rigorous proofs in modern mathematics.

Behind those glamorous theorems written in textbooks were all these dirty and tiring jobs of repeatedly plastering and patching like a bricklayer.

Entering June, progress became even slower.

Lemma 0.2 was barely pushed to the twenty-fourth page.

However, his attitude toward this draft had changed from initial anticipation to deep dissatisfaction.

Even disgust.

Some steps logically did hold up.

Through various exquisite mathematical techniques and forcibly pieced-together interpolation inequalities, the conclusion could barely be connected.

However, it was wrong.

It didn't look like something that grew naturally out of the physical imagery.

It was too rigid.

It was like scaffolding he had forcibly erected in a mire just to cross a river by force.

The scaffolding might be usable, and it might let him bluff his way through this paper.

But Jiang Lin did not trust this kind of proof.

A truly solid mathematical proof should not be like this.

It should be like a well-polished piece of crystal, allowing people to see clearly the physical structure behind it through mathematical symbols.

Rather than just seeing complex techniques and dazzling constant controls.

If mathematics and Physics disconnected here, then this theorem would be vulnerable in the Real World.

For a whole week in June of Year 35, Jiang Lin stopped.

He stopped writing, stopped reading papers, and stopped running those hair-pulling verification scripts.

Every day, he only did the most basic survival maintenance.

Waking up, eating potatoes, putting on a protective suit to go outside and inspect the bearings of wind turbine no. 2.

Walking two kilometers away to change the data card of the observation point.

Then returning to the Stone House.

Sitting in front of that scratched desk, staring at the twenty-fourth page of Lemma 0.2.

He was waiting.

Waiting for those rigid formulas to speak.

Waiting for the draft to tell him where its truly stuck soul lay.

On the afternoon of the sixth day, the sunlight slanted across the desktop, dust dancing in the column of light.

Jiang Lin looked at the anisotropic embedding constant circled in red, and suddenly understood.

What trapped him was not at all that a certain inequality scaling was insufficiently fine, nor that a certain embedding constant had failed to find the optimal bound.

It was the physical imagery.

It was that his own brain lacked a piece of the puzzle.

He still didn't know what should ultimately determine the lower bound of the secondary sheet thickness.

He knew that spallation would definitely occur; this was determined by first principles.

He also knew that after cascading splitting, the secondary sheets could not become infinitely thin, because there were no true singularities in the physical world.

He also knew that some microscopic scale would inevitably step forward and forcibly stall this cascading process.

But what was the specific image of that scale?

He wasn't clear.

Was it the intrinsic scale of magnetic diffusion?

Was it the equilibrium point of energy dissipation and convection?

Was it the reflection feedback of the boundary conditions?

Was it the kinetic bottleneck of the outflow channel?

Or some deeper geometric topological restriction he hadn't yet realized?

He only vaguely knew that a bottom line existed.

But the two words "vague" could not be written into a theorem.

Without a clear and intuitive physical imagery as a guide, mathematical derivation could only be walking a tightrope over a cliff blindfolded, relying purely on brute-force pushing.

Conclusions derived through brute-force, even if they temporarily fooled formal checks, could not fool oneself.

On the last day of June in Year 35.

Jiang Lin did something that had never been on any schedule for decades.

Shouldering a carrying frame, he went on a spontaneous hike.

No research goals, no task lists, no pre-set conclusions.

He just felt that if he continued to be trapped in this Stone House, trapped in those cold LaTeX codes and divergent partial differential equations, his thinking would dry up just like this Wasteland.