199: Chapter 199 Where exactly did the Third Domain lose?

At 8:47 AM in the fourth month of the twelfth year of this cycle.

On the third day after the A11 engineering package was archived, the cumulative runtime across multiple machines had surpassed 100,051 hours.

Jiang Lin packed the wiring harness samples, interface scan records, and three process passports from the left side of his workbench into a turnover box, and the vacated space was soon filled again by a failed draft from eleven years ago.

The draft was only twenty-seven pages long, and the file name was still [cubic_field_failed_03] left behind by that doctoral student back then.

It had followed him from that discrete geometry classroom at Tsinghua University to the Wasteland, changing four hard drives, three file systems, and six proof engineering versions, while its page count had surged from twenty-seven to 4,962.

In the upper right corner of the screen, a parameter search that had lasted for ninety-seven days had just ended.

[Quintic Field Candidate: K5-1847]

[Finite-Scale Fitting Index: 1.00631]

[After Widening the Window: Continuous Fallback]

[Fixed Positive Exponent Certificate: Failed]

[Recommendation: Expand Discriminant Range and Continue Search]

Continuing the search was projected to occupy the main computing array for 381 days and would also crowd out two sets of storage defect repair tasks.

K5-1847 was the best-looking loser in eleven years; in the first half, its curve looked even more like a path leading to a counterexample than Jiang Lin's earliest conceived cubic field construction.

As long as the horizontal axis was pushed further to the right, it, like the previous 1,846 candidates, would slowly curve back toward that grey line approaching zero.

Jiang Lin terminated the next round of search.

The task slots released by the main computing array were quickly filled by the verification queue of OUTER CACHE-07, and the progress bar continued to grow from the point of interruption.

He dragged that fallback curve to the left screen and pulled up the cubic field original draft from eleven years ago.

The three accounts left to him by his classroom teacher back then were still lined up on the first page of the file.

[How many points can remain within the window]

[How many unit modulus difference vectors can be found]

[How many edges can each difference vector repeatedly produce]

At the very bottom was a judgment written by Jiang Lin when he was eighteen years old.

[The degree might not be enough.]

Eleven years had passed, and this statement was still correct, yet it was correct only on the most superficial layer.

In the second year of this cycle, when Jiang Lin first completely recalculated the failed cubic field draft, it took him four months to recreate that fallback curve identically.

Algebraic numbers only showed a pair of complex coordinates in the plane, but their other conjugate embeddings would not disappear because of this.

A cubic field with complex embeddings would leave two types of coordinates for the same algebraic number; a pair of conjugate complex coordinates could be projected into the plane, while the remaining real coordinates still determined whether these points could be packed into a finite window.

The most enticing set of calculations in the original draft used highly truncated samples, restricting only the complex embeddings on the plane.

Hundreds of thousands of sample points were crammed into the disk, and the unit-distance difference vectors also increased rapidly, making the image on the screen pretty enough for one to pre-assign an ideal title.

Switching to a full-embedding window and adding the third real coordinate back into the constraints, Jiang Lin found that the disk remained crowded, yet those difference vectors stretched out of the window one by one on the hidden coordinates.

Continuing to enlarge the window could pull them back in, but the number of ordinary points grew even faster, and the unit edges averaged per point dropped again.

He had changed the shape of the window, stretching the disk into a long ellipse, and had also tried letting the hidden coordinates use independent scales.

Every time the window yielded a position to a certain type of difference vector, another ledger of points rushed in from the same gap.

Cubic fields did surpass ordinary lattices several times on a finite scale, except that the margin of victory each time continued to thin out as the scale expanded.

Jiang Lin recorded this layer of thinness with a number.

[δ_eff = log (number of unit edges) / log (number of points) - 1]

The Erdős conjecture allowed δ_eff to slowly approach zero.

The counterexample Jiang Lin was looking for had to stay above zero even when the number of points grew infinitely.

The δ_eff of the cubic fields slipped from the percentage place to the permillage place, and then into even smaller decimal places; batches of parameters were replaced one after another, but the direction remained unchanged.

That failed draft had already given a result, yet it failed to answer a more important matter.

Did it fail because this particular cubic field was chosen too poorly, or would all cubic fields fail?

Jiang Lin handed the problem over to MPS-Proof, demanding it to search for counterexamples that could pierce through this judgment.

The proof engineering immediately split into three attack routes: changing the discriminant, changing the denominator prime factors, and abandoning the symmetric window.

In the first two years, it altogether pierced through seventeen editions of Jiang Lin's obstruction lemmas.

One version underestimated the number of unit-modulus algebraic numbers, which was instantly surpassed by a combination of split prime ideals.

Another version treated every ideal combination as an available difference vector; in reality, one first had to find equivalents within the class group, and then use their ratio to generate algebraic numbers, causing the count to be divided by a layer of the class number.

The eleventh version could already handle fixed number fields, but when switched to cubic field families with continuously increasing discriminants, the constants in the estimation formula spiraled out of control along with the discriminant.

Jiang Lin once thought that broader enumeration would allow him to find the ones that slipped through the net; the machine's reward to him was a batch of curves that were increasingly adept at delaying the death penalty.

In the fourth year of this cycle, the candidate range expanded from cubic fields to quartic and quintic fields.

During the day, the force-controlled operation cabin repeatedly opened and closed old valves next door; at night, the proof engineering sent one new number field after another into the full-embedding window.

Some number fields could provide more unit-modulus elements, but their embedding lattices were severely skewed, meaning a window of the same size could not hold enough points.

Some number fields had regular lattices, but their root discriminants were shockingly high; before the points could even be projected into the plane, the covolume had already taken away more than half of the returns.

Another batch of candidates excelled in the first two items, but when it came to turning ideal ratios into actual algebraic numbers, the class-number ledger stripped away a layer of the direction count.

Viewed individually, all three numbers were qualified, but when placed into the same number field, they would dun each other for debts.

Jiang Lin changed the candidate table from sorting by degree to three columns, dividing each column by the number field degree to calculate exactly how much return and how much cost each additional embedding brought.

The categorized ledgers marked out the loss locations of the cubic fields.

Within a fixed number field, expanding the denominator and window could indeed generate more points and yield more unit modulus difference vectors, but the latter only increased at the sluggish speed typical of divisor counts.

The point set had expanded from a warehouse into a city, yet the reusable unit directions had only increased by a few streets; δ_eff would sooner or later be forced back to zero by the scale of the points.

Quartic and quintic fields could trade for better finite constants, but a fixed degree could not change this ledger of growth.

As for continuously changing number fields of the same degree, the discriminant and class number would factor these so-called new directions into the cost.

Once the degree was locked, if the root discriminant was also controlled, there would only be a finite number of number fields left to choose from; if the discriminant was released, the density of the embedding lattice would drop accordingly, and the candidates won back would leak out from the other side of the window.

In the winter of the eighth year, Jiang Lin wrote this judgment into the first edition of the low-degree obstruction lemma.

MPS-Proof pierced through the unified constant within it using an extremely skewed quartic field.

Jiang Lin deleted forty-six pages of proofs and rewrote the lattice parameters into the estimation.

In the spring of the ninth year of this cycle, the new version left a gap in the class number that was non-uniform with respect to the degree.

During the week when the northern branch of OUTER CACHE-07 began long-term coolant supply, he compressed the loss of ideal classes back into the difference vector count, reducing the red conflicts on the proof pages from thirteen to two.

When H-01 experienced a bipedal support failure, the last two places were still hanging on the screen.

After the unmanned operation team completed the recovery of their companion, Jiang Lin guarded the equipment bay for nine hours, casually modifying one of those places into a local estimation that only required a fixed degree.

The remaining place, which involved the window boundary, followed him back to the Outpost and hung on the wall of the assembly room for over two years.

On the day G-Explorer-C withdrew from the front line, Jiang Lin returned from beside the re-inspection platform, connected the full-lifespan samples to the process passports, and sent the final extreme case of the obstruction lemma into the independent verifier.

The verifier ran for nineteen days.

On the third day after the 100,000-hour timer crossed the integer mark, the result stopped before Jiang Lin's eyes.

[Low-Degree Obstruction Lemma: Passed]

[Applicable Framework: Full-Embedding Bounded Window / Bounded Denominator Unit Modulus Difference Vector]

[Arbitrary Fixed Degree Upper Bound d₀: δ_eff approaches 0]

[Continued Enumeration of Cubic, Quartic, and Quintic Fields: Unable to Generate Fixed Positive Exponents]

Jiang Lin crossed out the phrase [The degree might not be enough] on the first page of the original draft and wrote a new line.

[The degree must grow.]

This line of text only cost six characters, but the price paid was eleven years and over 4,000 pages of failed records.

The low-degree obstruction blocked a massive swath of paths and compressed the directions that were still viable into much clearer focus.

If one wanted δ_eff to stay above zero, the scale of the point set could no longer continue to grow by enlarging the denominator within the same number field.

The number field degree had to rise together with the point set; with each additional embedding layer, the number of unit directions also had to increase at a fixed proportion.

Finding a single beautiful high-degree field alone could only give a finite scale; the required object had to be a family of number fields, and every single layer had to be able to pay the same three ledgers.

At the same time, the root discriminant could not expand along with the degree, otherwise the embedding lattice would become increasingly sparse; the small prime ideals providing unit directions also could not lose splitting at higher levels, otherwise the newly added degrees would merely become empty rooms.

Jiang Lin changed the search conditions to three lines.

[Degree approaches infinity]

[Root discriminant uniformly controlled]

[Fixed small primes provide sufficiently many split prime ideals at each layer]

MPS-Memory retrieved nine categories of candidate number field families from the offline literature library.

Cyclotomic fields were eliminated first; as the degree increased, their root discriminants rose all the way up.

Several composite number fields performed well at lower levels, but upon entering higher degrees, the originally split small primes acquired longer inertia degrees, causing the growth of unit directions to stall accordingly.

Randomly generated high-degree fields were even more straightforward; one column would occasionally jump out with a good score, while the other two columns would typically turn out bad in very characteristic ways.

The screening continued until the end of the twelfth year of this cycle, leaving only one category of structure in the list that could simultaneously accommodate the three lines of conditions.

[CM field tower]

Jiang Lin had already encountered CM fields back in the quartic field stage, but that time he similarly failed due to the fixed degree, and the file was thrown into the elimination directory for six years.

Now he reopened it, and a property from the old construction that was once only enough to save computational volume turned into the interface for the entire new route.

CM fields possessed complex conjugation internally; taking any suitable non-zero algebraic number a and letting u = a / ar{a}, one had u · ar{u} = 1.

This equality would hold true along with every embedding, and the modulus length of u in all complex coordinates was equal to one.

The most troublesome hidden shadows in the cubic fields were tethered to the unit circle here by the same algebraic identity.

Jiang Lin gave up trimming the window individually for each embedding, switching instead to a multi-disk window in Minkowski space.

Each coordinate was a disk of radius R, and all the disks combined together to form a high-dimensional, box-like region.

After the inner radius shrank to R - 1, adding any full-embedding unit modulus difference vector would still leave the result within the outer window.

Jiang Lin then projected one of the complex coordinates onto the plane, and every translatable point within the window formed a unit edge with the point after translation.

The three ledgers that were originally intertwined thus turned into three bases growing exponentially with respect to the degree.

The number of window points had its own growth base, the unit directions had their own growth base, while the root discriminant and class number entered as fixed costs that had to be paid at each layer.

As long as the base of the unit directions surpassed the geometric cost, with each additional layer of degree, the surplus advantage would no longer be diluted by the scale.

Jiang Lin had the computing array verify this ledger on a set of finite tower layers first.

In the selected finite tower layers, a fixed rational prime remained completely split, and Jiang Lin constructed a from the pairwise conjugate prime ideals, then generated unit modulus elements using a / ar{a}.

The ideal classes would consume a portion of those combinations, but the remaining quantity would still increase exponentially with the degree.

The first tower layer only provided a finite-scale advantage.

After the second tower layer expanded the point set, δ_eff dropped slightly, yet stayed above zero.

The third and fourth layers were added in succession; the curve oscillated up and down near the same position and never curved back toward the zero line.

For the first time, a green interval spanning all calculated tiers appeared on the screen.

[Conditional Construction: Passed]

[If the tower can be extended infinitely, there exists a fixed δ₀ > 0]

[Number of Unit Distances: U(n) ≥ n^(1 + δ₀)]

This was not yet a counterexample to the Erdős conjecture, because the word if in the second line of the screen still weighed heavily over the entire construction.

Finite tower layers could only give a finite number of point sets, whereas the conjecture required infinitely many increasingly large point sets.

Jiang Lin needed a CM field tower that could extend infinitely, with the root discriminant controlled at all times, and with the selected small primes maintaining sufficiently good splitting at each layer.

The offline literature library provided several existing entry points: class field towers, controlled ramification, the Golod-Shafarevich inequality, and old methods of adding relations to Frobenius elements.

These terms were adjacent to each other in the index, yet they were still separated from a usable proof by large sections of number theory that Jiang Lin had yet to complete.

In the second month of the thirteenth year of this cycle, the status light of the northern reference position at the Outpost flickered in a low-power cycle.

G-Explorer-C had already exited the danger zone, and the three sets of machines in the testing field were still replacing task pods, with cumulative runtime continuing to tick upward.

Jiang Lin moved [cubic_field_failed_03] into the obstruction theorem directory, retaining that classroom original draft as the first page.

The file status was changed from [Pending Search for Better Number Fields] to [Low-Degree Route Closed].

The 381 days of computing power originally reserved for the enumeration of quartic and quintic fields were all allocated by him to a new task.

[Objective: Construct a CM Field Tower with Degree Approaching Infinity, Controlled Root Discriminant, and Fixed Small Primes Maintaining Splitting]

The proof engineering immediately generated a new conflict graph.

On the entire graph, only the topmost proposition box was glowing red.

[Prove that the tower exists.]