50: Chapter 50: The Warning of 0.01 Millimeters

On the twentieth day of practicing his fitter skills, Jiang Lin reopened Electrodynamics.

On this Wasteland timeline, if he counted the years of his previous four reincarnations, he had already forcefully traversed undergraduate Electrodynamics once.

Maxwells Equations were no longer unfamiliar; electric fields, magnetic fields, displacement currents, electromagnetic waves, boundary conditions, and energy flows—these terms were no longer just bolded words in textbooks.

He had derived them on his handwriting tablet during countless drafty nights, keeping up with the thought processes of most standard textbooks and solving most of the basic after-class problems.

Some classic standard-answer conclusions, such as the power formula for dipole radiation, he could even rebuild in his mind with his eyes closed.

Yet this did not mean this knowledge truly belonged to him.

He was well aware of the huge gulf between being able to derive something and having it truly ingrained in one's mind.

In that Knowledge Skeleton Calibration Table at the very end of the fourth Wasteland, the self-assessment he wrote for Electrodynamics was exceptionally lucid.

[Undergraduate entrance passed.]

[Field imagery has formed.]

[Complex boundary value problems, anomalous derivations of radiation, relativistic covariant forms: require retaking.]

He still remembered the specific context when he wrote these three lines.

Back then, he was no longer young.

Years of toil in the Wasteland had given him a lumbar disc herniation, and whenever it rained, his back would ache as if pierced by needles.

His eyes had grown dim, and when reading books, he had to wear that pair of reading glasses whose frames were wrapped in several layers of tape.

On that old e-ink screen reader that had accompanied him through decades, a large unfixable gray dead zone had appeared in the bottom right corner; he no longer dared to casually open large PDF files with complex layouts, fearing that a single lag would completely ruin this old antique.

Therefore, that table was never a victory list flaunting progress, but rather a heavy ledger of debts.

It was a bill that an aging Jiang Lin had borrowed from time yet was powerless to repay.

Now, the fifth Wasteland had just begun, and his body had returned to eighteen years old.

Yet his memories were not refreshed, and those debts had not vanished either.

Sitting before a stone table, Jiang Lin propped up his phone and played the pre-downloaded Electrodynamics course videos.

The folder name had not been changed.

[Electrodynamics_Fields and Boundaries_Chapter 1]

The professor in the video was an old man with graying hair, his handwriting was clear and forceful, he never beat around the bush when speaking, and throughout the entire lecture, there were no forced witty remarks inserted just to liven up the classroom atmosphere.

Jiang Lin very much liked this minimalist style that did not waste time.

Unsurprisingly, the first class still started from the most basic electrostatic fields.

[Review One: Gauss's Theorem.]

[Requirement: Do not ask yourself if you know how to apply formulas to plug-and-chug questions. Ask yourself whether, completely without relying on textbook prompts, you can rebuild its physical imagery, scope of application, and bottom-most proof skeleton in your mind.]

The standard statement of Gauss's theorem was very clear: the electric flux through any closed surface is equal to the total charge enclosed within the surface divided by ε₀.

In high school Physics, teachers taught them to use it to handle spherically symmetric charge distributions.

In university General Physics, it was used to handle infinitely long charged straight lines, infinitely large uniformly charged planes, and uniformly charged spheres. The routine was none other than finding a highly symmetric Gaussian surface, pulling the electric field strength E out from the integral sign, and then calculating an area.

In Electrodynamics, it transformed into the first equation of Maxwells Equations, becoming a foundational part within a grander, magnificent structure.

But what Jiang Lin was asking himself right now was by no means these problem-solving routines.

What he asked was: By what right did it grow to look like this, and why did the universe allow such a theorem to exist?

He picked up his pen and drew a tiny black dot in the center of his scratch paper, representing an isolated point charge q.

Then, taking this point as the center, he drew two concentric circles of different sizes, representing two spherical surfaces.

According to Coulomb's law, the electric field strength decayed inversely with the square of the distance.

Meanwhile, the surface area of a sphere increased with the square of the radius.

This was an extremely exquisite coincidence, or rather, a geometric necessity of three-dimensional space.

One field strength became smaller, one area became larger; when the two were multiplied during the flux calculation, the r^2 representing distance happened to be perfectly canceled out.

The flux, which was the total effect passing through the spherical surface, had nothing to do with the radius r.

He wrote down this bottom-most logic.

Then, around that point charge, he drew a closed surface with an extremely irregular shape, like a crumpled potato skin, enclosing the point charge within.

On this potato skin, the surface was no longer pretty, the normal vectors no longer uniformly pointed radially, and the angle between the local electric field vector E and the area element vector dS varied wildly, so complex that direct analytical integration was utterly impossible.

Yet Gauss's theorem ruthlessly proclaimed that as long as the surface was closed and the point charge was inside, no matter how distorted the surface was, the total flux remained that same constant.

He wanted to write down the five characters 'conservation of electric field lines' on paper to explain this phenomenon.

Just as the pen tip touched the paper, it stopped.

If it were in the early days of the fourth Wasteland, he certainly would have loved this statement.

Electric field lines started from positive charges, shooting into endless space like a hedgehog's spikes, passing through the closed surface surrounding it; they would neither vanish nor appear out of thin air in a vacuum.

Therefore, as long as the source was enclosed, the number of lines passing outward would definitely be constant. This was called flux conservation.

Pretty, intuitive, useful, capable of forming images in the mind.

But not hard enough.

Jiang Lin frowned slightly, heavily circled the five characters for conservation of electric field lines with his pen, drew a big cross over them, and then wrote a marginal note.

[Warning: This is imagery, not a proof. Electric field lines have never been physically existing entities; they are merely auxiliary tools invented by Faraday to understand fields. You must absolutely not use metaphors and imagery to replace rigorous mathematical logic.]

[Core logic of point charges: inverse-square law field + solid angle geometry = constant flux for any enclosing surface.]

[Any closed surface: must be understood using solid angles. No matter how tilted the surface is, the total solid angle it subtends at the point charge is always 4π. Alternatively, via Gauss's divergence theorem, surface integrals are directly transformed into volume integrals.]

[Multi-charge generalization: relies on the principle of linear superposition. Fields generated by external charges entering the surface must also exit, making the net flux contribution zero; only internal charges make a substantial contribution to the total flux.]

After finishing these three paragraphs, he felt that the skeleton regarding Gauss's theorem in his mind finally clicked into place, fitting together seamlessly.

Because this time he did not rely on any illusory metaphors of electric field lines, but instead steadied his footing with pure geometry and calculus.

In the video, the professor was deriving it with chalk on the blackboard.

The professor's train of thought was exactly the same as what he had just deduced on paper.

Starting from the point charge and spherical symmetry, introducing the concept of solid angles to generalize to general closed surfaces, then utilizing the superposition principle, and finally giving the extremely symmetric integral and differential forms.

While listening, Jiang Lin gently checked off the items beside his scratch paper with his pen.

Where the logical chains matched, he put a checkmark. Where the derivation details were completely identical, he put another checkmark.

When the professor lectured on the mathematical Gauss's divergence theorem and transformed the macroscopic integral form into the microscopic differential form, Jiang Lin did not rush to continue listening and pressed pause.

In the lower half of his scratch paper, he wrote down two rows of formulas side by side.

On the left was the macroscopic, overall integral form, describing the overall conservation behavior on a massive closed surface.

On the right was the microscopic, local differential form, describing the absolute causal relationship between the divergence degree of the electric field at a certain infinitesimally small geometric point in space and the local charge density.

The whole and the local, the macroscopic and the microscopic.

As he looked at it, he suddenly thought of a matter that seemed completely unrelated to fitting work and Electrodynamics, yet was closely connected in its underlying logic.

Back then, in order to mount the bench vise, he had spent a full three days building the workbench.

During that period, he discovered that the dirt ground itself had a tilt of several millimeters.

At that time, the problem seemed extremely simple: the ground was just uneven, causing the table to sit unevenly. Shimming a few pieces of scrap iron to adjust the level would settle it.

But later, when he filed workpiece No. 001 on that table and filed out that directional error where the right was lower and the left was higher, he profoundly understood that an uneven table was never an isolated Physics fact.

The extremely minute tilt of the ground, the thickness error of fractions of a millimeter in the table leg shims, the deflection in the clamping direction of the vise, the unconscious shift of his body's center of gravity during filing, the two-degree inward rotation generated by his right wrist to compensate for the stroke...

All these microscopic, local variables would eventually converge and superimpose in an incredible coupling manner, like a hundred rivers flowing into the sea, finally manifesting on the end face of that small piece of steel as a macroscopic, indisputable overall cutting error.

A minute local action would ultimately manifest on the overall surface profile.

When you disassembled an overall macroscopic error backward using a try square and feeler gauges, it could precisely fall back onto a certain local force-application variable.

This of course did not equal Gauss's theorem.

To forcibly equate the error analysis of fitting work with the electric field divergence, or to rigidly shoehorn surface profile errors into Gauss's theorem, was chaotic analogy and acting like a Physics rogue.

But Jiang Lin realized that this mode of thinking that switched perspectives back and forth between the overall accumulated quantity and the local source quantity, this awareness of piercing through appearances to see the essence, was the exact same rigorous mental training.

Physics and engineering intersected on this crude workbench in the Wasteland.

He picked up his pen and wrote his insights below the two lines of Maxwells Equations.

[The integral form looks at overall conservation boundaries; the differential form looks at local source generators.]

[The same reasoning applies to handling engineering manufacturing problems: one must strictly distinguish between macroscopic overall error manifestations (such as surface profile tilt) and microscopic local error sources (such as wrist exertion angle).]

[But warning: Do not forcibly apply formulas. The fitter system is not a linearly superimposed electromagnetic field; it is a nonlinear chaotic system full of friction and deformation.]

After finishing writing, he looked at it twice and re-traced do not forcibly apply formulas.

When you do not understand a complex real system, never use clever-sounding metaphors to cover up your ignorance.

Comparing an electric field to water flow and voltage to water pressure—such metaphors were very useful when popularizing science to fool children. But when one truly needed to calculate electromagnetic radiation energy or solve boundary reflection problems, the water flow metaphor would instantly collapse and lead you astray.

A metaphor was merely a bridge that brought you closer to the truth; it was never the other shore itself. To reach the other shore, you could only rely on ruthless mathematics and true measurement.

...

On the twenty-second day, the calm and smooth review came to an abrupt halt.

In the advancement of his theories, Jiang Lin encountered the first theoretical node in the fifth Wasteland that truly got him stuck and made him feel physically uncomfortable.

Vector potential, A.

In magnetostatics, such a quantity was introduced such that the magnetic induction intensity B equaled the curl of A.

This formula could be said to be an old acquaintance.

During the years of the fourth Wasteland, he had derived it step by step.

In calculating magnetic field exercises for current-carrying straight wires or circular coils, he had skillfully used the vector potential form of the Biot-Savart law to simplify the integration process.

Even when lecturing on the section of electromagnetic field gauge transformations, he had stopped before this concept and pondered for a very long time.

But strangely enough, every single time, as long as his thoughts touched the letter A, an extremely uncomfortable sense of foreignness would surge from the bottom of his heart.

The reason was also very simple.

The magnetic field B was real.

It was something that solidly existed in the Physics world.

It could be measured.

Holding a compass, the magnetic needle would deflect; this was B at work.

Placing a current-carrying coil inside, the coil would experience the torque of the Ampere force and rotate; this was also B at work.

Shooting a charged particle in with velocity v, the particle would experience the Lorentz force F = q(v × B), and its trajectory would bend.

As long as there were instruments, B could give definitive readings in the laboratory.

Then why did this group of pioneers engaged in Physics have to introduce, at the bottom-most layer of theory, a vector potential A that fundamentally could not be measured and was extremely non-unique?

If something was not unique, it meant it carried subjective arbitrariness.

Yet a ghostly variable carrying subjective arbitrariness was grandly written into the equations describing the core laws of the universe.

For the Jiang Lin of his early years who firmly believed that Physics meant measuring reality, this was simply challenging the bottom line of his faith.

At night, Jiang Lin wrote in his logbook.

[Review Two: Ghostly Variable—Vector Potential A.]

[Known mathematical structure: In a local region, or within a simply connected space satisfying corresponding topological conditions, because the magnetic field is source-free ∇ · B = 0, a vector field A can be introduced such that B = ∇ × A.]

[Core question: If the value of A is fundamentally not unique, does it count as a meaningful physical quantity at all? Or is it merely an algebraic trick played by mathematicians for the convenience of solving partial differential equations.]

He in the late period of the fourth Wasteland, in order to cope with his inner doubts, had actually memorized the definitions of gauge degrees of freedom in the textbook until he could recite them fluently.

As long as you added the gradient of any scalar function X to the existing A, it became a new A'.

A' = A + ∇X

Then when you went to find its curl, because the curl of any scalar gradient is always identically zero, the calculated magnetic field B remained that same original B, without the slightest deviation.

This was in Physics called gauge freedom.

Even in time-varying electromagnetic fields, the electric potential φ and vector potential A could hold hands together to perform joint gauge transformations.

As long as they changed according to this rule, the real electric field E and real magnetic field B calculated at the end could maintain absolute physical invariance.

These words, these formulas, were all correct.

Flawless in logic.

But Jiang Lin knew very well that being able to recite these tongue-twister-like definitions and passively write down gauge transformation formulas on an exam paper by heart did not mean you understood them in the depths of your soul.

He still felt deeply uneasy about using uncertain ghosts to describe a deterministic Real World.

That arbitrarily selectable scalar function X was like mocking his Physics intuition built upon hard matter.

Irritated, Jiang Lin put down his pen, pushed away the stone stool, stood up, and turned to walk over to the workbench.

Clamped tightly inside the jaws of the bench vise was practice piece No. 006, which he hadn't completely finished filing yesterday afternoon.

It was a typical piece of Q235 low-carbon steel, cut into small blocks from a long bar using a manual saw. Slight burrs left over from sawing could still be felt along its edges.

He picked up the flat file from the tool rack.

Instead of striking a pose and starting to exert force as usual, he simply placed the toothed surface of the file gently and flatly against the rough surface of the workpiece.

Applying slight pressure with his fingers to hold down the wooden handle, he closed his eyes and tried to feel that tiny bit of tactile sensation where metal met metal.

In this corner smelling of iron filings, the world of a bench worker was actually filled with countless invisible and unmeasurable things.

You could see the stout lead screw of the vise, but you couldn't see how large the stress distribution clamped on both sides of the steel piece was at that moment, nor whether it was within the elastic limit or had already caused minor plastic deformation.

You could see the off-white surface of this steel piece, but you couldn't see what kind of intricate internal stresses remained inside its crystal lattice due to the previous rolling and sawing.

You could see the rows of sharp teeth on the file, but the moment your hand exerted force to push it out—the microscopic cutting angle when the teeth truly cut into the metal grain boundaries, the instantaneous high temperature generated locally, the slippage and tearing of crystal grains...

These were all microscopic processes invisible to the naked eye and absolutely unmeasurable by ordinary measuring tools.

He couldn't directly see the variables of these processes.

He could only deduce those invisible causes through the final macroscopic results, much like blind men touching an elephant.

From whether the iron powder falling into the chip box was curled or pulverized, from whether the sound made when pushing the file was a crisp hiss or a grating crunch, from the magnitude of the impedance felt by his wrist, and finally, from a faint strip of light leaking under the merciless 90-degree square.

Through these observable macroscopic consequences, he was convinced that those invisible stresses and deformations truly existed.

Then, what about A?

Since A was not unique, the same objective magnetic field B could be produced by tens of thousands of different A's.

Then A could not directly correspond to some unique physical entity.

If a quantity could change arbitrarily with the gauge function X of your subjective choice, then it itself lost the qualification to be an independently measurable physical quantity.

Having figured this out, Jiang Lin turned around, hurried back to the stone table, grabbed his pen, and wrote rapidly on the paper.

"Breakthrough strategy: Do not treat A as matter."

"A is not a directly measurable field quantity with substantial energy like the electric field strength E or magnetic induction intensity B. Do not try to find an instrument in the Real World to read out the value of A; that is futile."

After writing these two lines, he felt that simply denying what it wasn't did not explain why it existed, which wasn't thorough enough.

So he continued to add below.

"However, just because $\vec{A}$ cannot be directly measured, one must never arrogantly say that it is merely a mathematical trick for solving equations."

"If the underlying laws of the physical world truly allow the existence of gauge degrees of freedom, then A might not be a variable describing material entities; it might be an advanced language describing the underlying connective structure or interaction topology of the universe. It is more fundamental than B; it is just that, limited by our three-dimensional macroscopic measurement methods, we can only see its curl projection (namely B)."

Writing up to this point, Jiang Lin's hand trembled slightly.

This was an extremely bold inference, and he didn't dare be sure whether it was correct.

So behind this grand speculation, he added a set of parentheses with extreme caution.

"(Note: This inference is tentative. Before learning the Aharonov-Bohm effect in Quantum Mechanics, do not take this sentence as absolute truth. Remain skeptical and keep moving forward.)"

That night, he opened the dog-eared Electrodynamics textbook again and reread the derivation of the vector potential section word by word.

With the psychological preparation from the evening, those mathematical derivations became incomparably smooth.

Because $\nabla \cdot B = 0$, mathematically, A can inevitably be introduced such that $B = \nabla \times A$.

Flipping further ahead, to simplify the complex wave equations derived from Maxwells Equations, predecessors cleverly utilized the gauge degrees of freedom of A.

Choosing the Coulomb gauge ($\nabla \cdot A = 0$), the electrostatic equations became exceptionally concise.

Choosing the Lorenz gauge ($\nabla \cdot A + \mu_0 \varepsilon_0 * \partial \varphi / \partial t = 0$), the equations for the electric potential and vector potential were completely decoupled, turning into a perfectly symmetrical d'Alembert wave equation, so beautiful it made one want to cry.

As for the derivation, Jiang Lin could walk through it step-by-step with his own pen without a single error.

Yet when the lead type in the book reappeared with the words "gauge invariance means physical essence remains unchanged," the pen in his hand still stopped mid-air.

He wrote down today's final reflection on the margin of his logbook.

"Vector potential A summary: I can recite its mathematical definition, I can calculate its substitution into specific models, and I can write down the partial differential equations of gauge transformations meticulously."

"However, how on earth does a ghost quantity with non-unique values participate in constructing physical reality at the deepest level? This sense of unease still exists and has not been truly understood."

"Handling method: Do not bang your head against the wall here, do not forcibly pursue sudden enlightenment here."

This mental method of not forcing sudden enlightenment was the only life-saving trick he figured out back then in the fourth Wasteland when he was driven nearly crazy by counter-intuitive Quantum Mechanics.

Some cognitive barriers could not be broken just by staying put and pondering deeply.

The reason it stuck you was that the road ahead of you had not yet been finished, and more advanced mathematical tools of the world behind had not yet opened up to you.

Forcibly digging this pit in place with existing low-dimensional tools would only make the hole deeper and deeper, eventually burying oneself alive inside, without necessarily digging through the tunnel of truth.

The best way was to tag this problem with a bright red label, put it into your backpack, carry it on your back, endure the discomfort, and keep walking forward.

...

If theoretical breakthroughs required time to brew, then the progress of craftsmanship in reality often engaged in some extremely tiny moments.

On the thirty-third day, the physical memory of a bench worker and the theoretical intuition of Physics appeared for the first time on Jiang Lin in that kind of piecemeal yet scalp-tingling sense of engagement.

That was definitely not the bullshit engagement written in fantasy novels where one picked up a file and felt like they had divine assistance just because they understood some advanced Electrodynamics formulas.

That violated the conservation of mass and the laws of thermodynamics.

That was just a very tiny operational moment.

That morning, the sky was overcast and windless.

Before starting work on a new practice piece No. 007, he habitually dealt with the edge burrs of piece No. 006 left on the vise yesterday first.

Holding a fine-toothed flat file with a standard posture and a steady center of gravity,

as the file smoothly pushed across one side of piece No. 006's surface, Jiang Lin's wrist keenly captured an unharmonious tactile feel.

It was uneven.

Throughout the entire pushing process, the first third was very smooth, but when it reached a certain specific two-centimeter-wide area in the middle, his wrist felt a noticeably increased resistance.

It was like cutting into a slightly harder piece of bone.

After pushing past those two centimeters and moving further ahead, the resistance instantly became a little lighter again, returning to smoothness.

This difference in resistance was extremely subtle. If he were still a beginner practicing for the first time, he would likely have dismissed it as an illusion.

But the current Jiang Lin stopped and was not in a hurry to push a second time.

He removed the file, straightened his back, and readjusted his breathing.

Then, he put the file back to its starting point and pushed it again from the same direction at an extremely slow speed, with the exact same angle and the exact same downward pressure.

The change in resistance still existed in that specific area, arriving as expected.

He narrowed his eyes, changed to a cross-intersecting angle—that is, from a direction perpendicular to the side—and pushed again.

This time, that obvious sudden change in resistance weakened and became very vague.

He unscrewed the crank of the vise, took down workpiece No. 006, and wiped away the remaining iron chips and oil stains on the surface with a clean cotton cloth.

First, he felt back and forth across that area with the pads of his fingers.

He couldn't feel anything out of the ordinary.

Then he lifted the workpiece into mid-air, letting the light hit the metal surface at a very small grazing angle.

Aside from a few intersecting file marks, no obvious depressions or bumps could be seen.

So he grabbed the square to inspect it; that edge did not show any obvious warping or deformation.

Finally, he even took out a vernier caliper to precisely measure the thickness of that section.

The readings were consistent, and no macroscopic dimensional thickness changes could be found to explain the sudden change in tactile feel.

The dimensions hadn't changed, and the surface wasn't bulging.

Then what was going on?

Jiang Lin returned to the stone table, opened his logbook on benchwork, and began recording this abnormal phenomenon.

"Piece No. 006 local phenomenon: During surface push-filing, a noticeable difference in local resistance appeared in the middle section. After multiple repeated tests in the same direction, the phenomenon was stably reproduced; after changing the pushing and cutting direction, the perception of the resistance difference weakened."

"Possible cause investigation:"

"1. Uneven surface roughness: The initial file grain direction in a specific area may interfere with the current pushing direction, affecting the dynamic cutting friction."

"2. Local work hardening: During previous sawing or rough filing, the stress applied locally in this area was too large, causing the crystal grains there to undergo severe slip deformation, producing a work-hardening effect that increased the local hardness of the material."

"3. Clamping deformation release: The hidden micro-elastic deformation generated when the vise was clamped was released after filing away the surface material, causing the local stress field to change."

"4. Material defects: This cheap piece of Q235 steel itself had uneven carbon content or tiny inclusions internally."

"5. File teeth clogged with chips or locally embedded chips: Low-carbon steel chips may embed into the file teeth or the workpiece surface, causing friction to suddenly turn gritty during a certain section of push-filing. Re-testing is required after cleaning the file teeth."

After writing these 5 items, he calmly made an objective judgment below.

"Evaluation conclusion: Under the current rudimentary conditions on the Wasteland where there isn't even a metallurgical microscope, it is fundamentally impossible to distinguish from physical testing which exact cause led to the resistance change. Do not jump to conclusions."

"Core takeaway: My tactile sense can already distinguish the differences in cutting feedback across different areas on the same piece of material. This is a qualitative leap. Remember: being able to sense the existence of a difference is a huge step forward, but this by no means equates to automatically knowing the physical source of the difference. Maintain awe."

He closed the benchwork logbook.

That night, wrapped in a tattered quilt by the light, Jiang Lin continued to review the Electrodynamics textbook.

The progress had advanced to the boundary conditions of the medium interface.

This section was full of tedious formulas and derivations.

Listed strikingly on the textbook: At the interface between two different media, the tangential component of the electric field strength must be continuous: $E_{1t} = E_{2t}$.

The normal component of the electric displacement vector is continuous in the absence of free surface charge; if there is a free surface charge $\sigma_f$, a jump occurs: $D_{1n} - D_{2n} = \sigma_f$.

Similarly, the normal component of the magnetic induction intensity is always continuous: $B_{1n} = B_{2n}$.

Meanwhile, the tangential component of the magnetic field strength H is controlled by the free surface current density $K_f$. If the normal vector is taken pointing from medium 1 to medium 2, a common writing is: $\hat{n} \times (H_2 - H_1) = K_f$.

Which components remain continuous and which components undergo cliff-like jumps do not depend at all on the elegant form of Maxwells Equations in a vacuum.

It depends on whether external free charges and free currents exist on the interface, and even more on how the inherent permittivity $\varepsilon$ and permeability $\mu$ of the media on both sides of the interface suddenly change.

The universality of physical equations themselves did not disappear because of the existence of media.

Yet it was precisely this extremely realistic interface that forcefully made the perfect calculus equations grow boundary conditions that had to be handled on a case-by-case basis.

Reading up to here, Jiang Lin's mind suddenly went buzz with a hum, instinctively associating it with the piece of steel No. 006 that he had repeatedly explored on the workbench during the day.

Those hardness changes and residual stress zones on the surface of the material were certainly not any kind of medium interface in strict electromagnetic definitions.

If one mechanically applied physical concepts so indiscriminately, they would be scolded out of the classroom by professional professors.

Formulas could not be applied blindly.

However, the word "interface" or "boundary" hit like a pebble, heavily plunging into the deep waters of his thinking.

He suddenly realized that the so-called interface in the real physical world was never that thin geometrically meaningful straight line drawn in textbook illustrations.

An interface was that gray zone where physical states underwent drastic changes.

In Electrodynamics, it manifested as the twists, turns, and jumps that the electric field vector E and magnetic field vector H had to undergo to satisfy material properties.

In the world of a bench worker holding a file, it might manifest as that trace of stiffness in the pushing tactile feel, as the transition of the file's friction sound from deep to sharp, as the falling iron chips suddenly turning from a curled state into finely powdered shapes, or even as a slight drift in the vernier caliper readings at the 0.01-millimeter magnitude.

Extending a bit further, in that patch of farmland behind the house trying to feed him, how did an interface manifest?

It manifested as digging thirty centimeters down from the surface, where the soil color suddenly changed from a dried yellowish-brown to a slightly moist dark black.

It manifested as that dividing line where the water-retaining capacity of the water just poured yesterday plummeted by today.

It manifested as the degree of difficulty when the plant roots, which originally plunged downwards smoothly, had to grow in a curved manner after encountering a hardened soil layer.

The word "interface" must absolutely not be used indiscriminately across different complex systems.

Yet its existence, at a fundamental philosophical level, forced humanity to ask the same ruthless question.

Where down to the width of a single hair did the physical states in this system begin to mutate?

After crossing this invisible line of mutation, which physical quantities still maintained continuity and loyalty?

And which physical quantities underwent drastic leaps due to the obstacles of reality?

What on earth was the deepest root cause of such a leap?

Jiang Lin felt his breathing grow somewhat hurried, grabbed his pen, and heavily wrote a sentence at the very bottom of the draft paper filled with formulas for electromagnetic boundary conditions.

["So-called boundaries are never merely decorative lines in mathematics. A boundary is a rough hand forcefully thrust into the equations of a perfect theory by the Real World just to demonstrate its own existence."]

After finishing this sentence, he slightly frowned.

This sentence was written far too much like those pretentious philosophical summaries.

Survival in the Wasteland did not need this kind of flowery, pretentious literary sickness.

Without hesitation, he picked up a red pen and added a heavily realistic warning right after the sentence, as if setting rules for himself.

["This insight is kept solely for the purpose of broadening the mind. In the future, when encountering any specific engineering or survival problems in the Wasteland, the analysis must strictly fall back on fundamental Physics equations, material properties, or real measurement data. It is absolutely forbidden to write only this kind of seemingly pretty nonsense. Theory is used to guide the direction, and hands are used to verify the truth."]

Forty-six days.

This fitting and turning training, which was so dry it made one want to vomit, finally ushered in an extremely minor yet crucially important turning point on this gray afternoon.

That day, Jiang Lin was dealing with the No. 007 practice piece on the bench vise.

The initial state of this piece of stock was very poor, and the saw cut was severely slanted.

But in the process of handling it, Jiang Lin suddenly had a somewhat unexpected discovery.

He found that he no longer needed to stare constantly like a taskmaster at that damn wrist of his right hand.

Thinking back to the first dozens of days practicing 001 and 002, it had simply been a nightmare.

Every time he pushed the wooden handle forward, his brain had to act like a radar, allocating the vast majority of its computing power to monitor every joint.

It could not turn inward, the shoulders had to be extended, the center of gravity could not be pressed entirely on the file, the right hand could not rush to exert force, and the left hand absolutely could not just be placed in front for show; it had to provide a stable and balanced downward pressure.

The result often was that the more frantically his brain reminded him, the stiffer and more disobedient his body's muscles became.

The stiffer he was, the more the surface profile error filed out resembled a dance of demons, utterly devoid of any regularity.

But today, the situation had changed.

Unconsciously, the time had arrived at the evening of the forty-seventh day.

Standing in front of the workbench, Jiang Lin had been filing continuously for nearly three hours.

During these three hours of high-intensity physical labor, he had only stopped twice to drink hot water and adjusted his standing foot position once when it felt slightly sore and numb.

Most magically, during these three hours, his brain was in a wondrous state of dissociation.

His brain had initially been replaying the blackboard writing of the professor in the video when deriving the vector potential A.

After filing for an hour, his thoughts leaped back to the medium interface boundary conditions he had pondered bitterly late last night.

Later still, he began to review whether the starting angle when sawing 007 this morning had deviated by a couple of degrees, causing him to waste a whole half hour rectifying the plane with a coarse file afterwards.

For a period in the middle, because the wind blowing in from outside carried a hint of mud and dust, he even briefly lost focus, wondering if the poorly growing crops on the east side of the farmland behind the house were suffering because the soil color leaned yellowish and the boundary line of fertility loss had already spread over.

His thoughts roamed wildly, but his hands, like two precise linkage mechanisms programmed in advance, carried out extremely stable push-pull movements on the bench vise.

When he finally snapped out of this near-flow state, stopped the action in his hands, grabbed a rag to wipe away the iron powder on the surface of 007, and brought the ruthless knife-edge square close to inspect it, he froze.

The gap letting light through was extremely faint.

He pulled out the feeler gauge, his fingers beginning to probe with a sense of disbelief.

After repeated insertion and withdrawal.

He found that for this piece of 007, which was originally severely slanted, the difference in light transmission between the left and right sides had shrunk to a degree almost indistinguishable to the naked eye.

He used the feeler gauge to repeatedly probe that light gap which was so thin it was almost invisible, and changed directions twice to re-apply the square.

Strictly speaking, this was not any standard metrology.

He could not claim that he had truly achieved a flatness of 0.01 millimeters for this surface.

But under the current crude square light-transmission inspection setup, the thickness difference of the feeler gauge that could be inserted on the left and right sides had indeed been suppressed to about 0.01 millimeters.

This was not the final precision.

It was merely a staged estimation of a single-item error.

Yet even so, it was enough to make him stop what he was doing.

Jiang Lin did not overflow with joy.

In the Wasteland, loss of emotional control was fatal, whether it was wild joy or extreme sorrow.

He calmly put away the feeler gauge and returned the square to its original position.

Then, he picked up a clean cotton cloth and wiped the surface of workpiece No. 007 all over again, ensuring that not even a single iron chip interfered.

Next, he picked up the square again and brought it close for inspection.

Changing the direction of the illuminating light, he inspected it once more.

Rotating the workpiece one hundred and eighty degrees, he inspected it again.

Through three consecutive re-measurements, the resistance feedback given by the feeler gauge showed no obvious changes.

The reading was indubitably certain.

0.01 millimeters.

This was the ultimate achievement closest to a flat surface that he had attained in a full forty-seven days, facing this abyss of directionally slanted error, ever since he started touching a file.

Jiang Lin let out a long breath of foul air, turned around to walk to the stone table, opened his logbook, and solemnly wrote.

["Workpiece No. 007 Quality Inspection Report: Under the conditions of square light transmission and feeler gauge rough inspection, the left-right directional error is estimated to be about 0.01 millimeters. Current best historical record."]

["Note: This data does not represent strict flatness, but only represents the single-item comparison result under the current crude testing system. Subsequent re-verification via continuous samples 008 and 009 is required to confirm whether stability is possessed."]

After writing down the achievement, his penmanship shifted, and he began a ruthless dissection.

["Phenomenon Analysis: During today's continuous filing process lasting three hours, my conscious mind did not continuously and deliberately monitor the exertion posture of my wrist and shoulders as it usually did. However, the final mechanical action output instead exhibited unprecedented stability."]

["This may illustrate an extremely critical physiological mechanism turning point: after the first six practice pieces and tens of thousands of repetitive push-pulls containing painful error corrections, the control against wrist inward-turning began to sink from the highly energy-consuming cerebral cortex conscious mind into the underlying action memory patterns of the cerebellum and muscles."]

Writing up to here, Jiang Lin looked at the words sinking into low-level action patterns, feeling that this phrasing seemed somewhat arrogant, as if announcing that he had already become a god-tier master.

He shook his head, decisively crossed out the sentence with his pen, and changed it nearby to more cautious wording.

["Correction: This merely indicates that automated tendrils are beginning to appear in some extremely basic physical exertion control of the body."]

["Extreme Warning: A single accidental meeting of standards absolutely does not equate to the true stability of technology. Contingency is worthless in the Wasteland. There must be sample data support from three or more consecutive pieces to prove that this flesh-and-blood machine has truly memorized this action. Do not pop the champagne in advance."]

Having finished writing these harsh self-warnings, he finally put down his pen, allowed himself to walk over to the workbench, and glanced with a trace of gratification at workpiece No. 007 left on the bench vise.

In fact, purely from its appearance, this piece of steel No. 007 was very ordinary.

It still had the dusty color of low-carbon steel, and if viewed with a magnifying glass, its surface was still covered with small, messy cutting scratches left by the tips of the file teeth.

But Jiang Lin knew clearly in his heart that it was already worlds apart from No. 001 lying in the scrap bin at the very beginning.

On the body of No. 001, everywhere was engraved with the ugly marks left by that ignorant Jiang Lin whose muscles were out of control.

The filing marks varied in depth and were extremely messy, the middle of the plane even bulged, and that ridiculous right-low-left-high directional error was like a crooked, ugly line hidden inside the metal that could never be removed.

And this piece No. 007, which still could not be considered perfect and would still fail according to standards, silently proved an extremely important matter to Jiang Lin with its tiny error compressed to 0.01 millimeters:

In this cold universe that followed the laws of Physics, as long as the direction was right, repetition containing feedback was never wasted in vain.

In this mire, walking slowly—even if as slow as a snail crawling—never equated to standing still in place without moving.

The night of the fiftieth day.

Jiang Lin opened a brand new page of his logbook, and at the blank space at the very top, heavily wrote a number with a thick black pen.

[50]

After writing this number, he did not rush to write today's record, but instead flipped this thick logbook back to the first page.

Like an onlooker examining his own past, from the first day he arrived in this fifth Wasteland all the way to the records of the fiftieth day, page by page, he reviewed it all over again.

Inside were complaints about being tired to the point of tasting sour water for plowing farmland, thrilling records of nearly having his hand broken by a concrete slab while building a workbench, theoretical deductions tormented by vector potential A and Gauss's law with pages full of question marks, and surface profile quality inspection reports from 001 to 007 filled with frustration as errors were reined in bit by bit.

In the long years spanning forty years in the fourth Wasteland, Jiang Lin had learned many survival skills, but the most profound and cruel lesson he had learned had nothing to do with skills.

That lesson taught him that time was very long, long enough to rust through steel.

But human life and energy were extremely limited.

No matter how much theory you understood, you only had twenty-four hours a day.

No matter how strong your will was, your lumbar spine and muscles would also go on strike after operating under overload.

The further one went, facing the massive pressure of Wasteland survival and technological reconstruction, the more soberly one needed to know which systematic problems had to be tackled to the bitter end and resolved today, while those questions at the theoretical boundary and technological chasms that could not be bridged for the time being could only be bundled up and carried along as one moved forward.

Having finished reading the records of the first fifty days, Jiang Lin turned to the page with [50] and wrote a staged summary below the number.

["Fiftieth day, milestone review."]

["Evaluation: The initial systematic establishment upon entering the fifth Wasteland is declared preliminarily completed."]

["Operating Status: Currently, the three main lines—camp and farmland maintenance to sustain basic survival, fitting physical action training as an industrial starting point, and Electrodynamics theoretical review as a cognitive foundation—have all established stable, self-operating positive feedback loops. No collapse nodes have appeared."]

After writing these three lines, he turned to the next page.

["Fifty-first day, and subsequent short-term action plans."]

["1. Theoretical Main Line: Electrodynamics cannot stagnate; one must forcefully break through electrostatic and magnetic fields while carrying the discomfort toward gauge field boundaries, advancing to the complete time-varying field form of Maxwells Equations."]

["2. Craft Main Line: Open workpiece No. 008. Target shifted. Since the directional error has been preliminarily controlled on 007, the next focus must turn to monitoring profile errors—namely, the middle protrusion. Ultimate challenge target: compress the middle protrusion arc to 0.01 millimeters."]

["3. Mental Warning: One must never blindly and arrogantly raise the material-blanking precision targets overall just because of a one-time single-item compliance of No. 007. This flesh-and-blood machine is unstable; one must first confirm the stability of that muscle automation through the boring verification of continuous samples such as 008 and 009. If you step too wide, you will definitely stumble."]

Having finished writing the last plan, Jiang Lin closed the logbook, and from the material storage box at the bottom, he skillfully pulled out a new flat piece of low-carbon steel.

His left hand adjusted its position, feeding it into the jaws of the bench vise.

His right hand turned the crank handle. The lead screw rotated, and the jaws closed.

Tightening, applying force.

After feeling resistance, he habitually tapped twice on the top of the steel stock with his knuckle, listening to the sound to recheck whether the clamping produced suspension tremor.

Everything went smoothly.

He removed from the tool rack the medium-tooth flat file that had accompanied him through dozens of days and nights.

Spreading his feet apart, he settled into the optimal configuration center of gravity determined after countless fine-tuning adjustments.

Wrists relaxed, shoulders extended.

["Sha, sha, sha..."]

The low, uniform metal-cutting friction sound rang out rhythmically in the quiet stone room.

Gray, finely divided iron chips drifted downward like light snow in early winter; some fell onto the crude wooden tabletop, and a small portion quietly fell onto the back of his hand, where veins slightly bulged due to gripping.

Within the light and shadow of the lamp, the sound of the file being pushed was extremely uniform, without the slightest bit of stagnation or hesitation.

Accompanied by this friction sound was Jiang Lin's breathing, which was equally uniform, long-drawn, and filled with astonishing patience.