200: Chapter 200 - One-Quarter Relationship Short

Year 13, Month 2 of the current cycle.

Four days after the Proof Engine turned [Prove Tower Existence] red, Jiang Lin terminated the task of enumerating the finite quotient of the 28th layer.

That task had already consumed 176 hours of the main computing array, and the generated group order had grown so large that the terminal needed to open a separate file to save it completely.

The 27th layer was still growing, and the 28th layer would likely grow as well, but the 300th layer and the 3000th layer could always abruptly stop at places yet unreached.

No matter how deep the finite layer enumeration went, it could only prove how high a tower had already been built.

What Jiang Lin needed was for it to never be capped again.

The released computing power was taken over by the OUTER CACHE-07 verification queue, and the dense tower layers on the left side of the screen instantly froze, leaving only the red box at the very top glowing.

[Target: Construct a family of CM fields with the degree approaching infinity, a controlled root discriminant, and a fixed rational prime maintaining complete splitting]

[Current Gap: Infinity]

Jiang Lin changed [Field Family] to [Infinite Tower], and added a group below it.

[G = Galois group of the maximal pro-p extension allowing ramification at finitely many places and complete splitting of designated primes]

Whether the tower existed or not was thus compressed into a much harsher question.

Was this group a finite group or an infinite group?

The previous version of the construction selected 12 rational primes and required them to split completely at every layer.

Doing so was very generous to the planar point set; with each additional completely splitting prime, a large batch of paired prime ideals would emerge in the higher number fields, and the unit-modulus difference vectors would multiply accordingly.

Yet the Galois group had to pay for this generosity item by item.

Designating a prime to split completely was equivalent to killing its Frobenius element; putting it into the group presentation meant adding another relation.

The directions brought by the 12 primes had not yet entered the plane, but the 12 new relations had already squeezed out the margin of the Golod–Shafarevich inequality.

Jiang Lin deleted one prime, and the growth base of the unit directions dropped accordingly; continuing to delete until the tower could be proven infinite caused the exponent on the geometric side to drop back below zero.

That table had been pulled between the two requirements for more than seven months, and the best version was still three relations short.

The suggestions given by the Proof Engine were always dutiful.

[Expand the set of ramified primes T, increase the generator rank]

The more ramified primes there were, the more independent generators the group could acquire, and the cost of the root discriminant would rise along with all the tower layers.

Stuffing more small primes into T could indeed win back the relation margin, but after returning to point set counting, the discriminant would take this profit back from another column.

Jiang Lin stared at the parameter table that had already changed color schemes 16 times, and split the unit direction column back to the initial ideal combination.

Above each completely splitting prime, he had always taken only two choices.

Taking the conjugate prime ideals P and P̄ once each, the number of combinations was two.

This was a setting carried over from the failed draft of the cubic number fields, and also the default value of all search programs over the past eleven years.

Jiang Lin changed the upper limit of the exponent from one to two.

The choices instantly changed from two to three: P², PP̄, P̄².

Changing the upper limit to three, the choices became four.

The column of numbers in the terminal expanded downward sequentially; for each increase of k by one, there was one more local ideal combination, yet the Frobenius relation responsible for forcing the split remained only one.

The denominator cost would rise from p² to p to the 2k-th power, and the lattice where the point set resided would thus become denser.

This cost would suppress the final obtained fixed exponent to be very small, but would not force it to equal zero.

Jiang Lin changed the optimization objective from [Maximize δ as much as possible] to [Strictly prove δ > 0], and then set the number of completely splitting primes to one.

The 12 Frobenius relations disappeared from the conflict graph one by one, leaving only one in the end.

When the symbolic search produced the first feasible region, k had already entered an 18-digit number, and the candidate δ would shrink to more than thirty decimal places behind the decimal point.

Such an exponent would be of no use on any actually drawable point set.

The Erdős conjecture discussed n approaching infinity; as long as a fixed positive number refused to shrink back to zero along with n, the 38th digit and the 1st digit possessed the same veto power.

Jiang Lin saved this version of the parameters and began searching for a tower capable of bearing that single Frobenius relation.

Five odd primes allowing ramification entered the proof first.

The corresponding maximal elementary 2-extension could only provide four independent generators; with the Shafarevich relation bound plus the splitting condition of the designated prime, the upper bound of the number of relations was five.

The Golod–Shafarevich threshold was four squared divided by four, which was exactly equal to four.

Five could not exceed four, and this path ended here.

Jiang Lin added the sixth ramified prime allowed.

The number of generators changed from four to five, the upper bound of the number of relations changed from five to six, and the threshold that the finite group had to pay jumped from four to twenty-five divided by four.

[Generator rank d(G) = 5]

[Upper bound of relation rank r(G) ≤ 6]

[d(G)² / 4 = 6.25]

For the first time, a margin of a quarter of a relation appeared in the parameter table.

A group presentation certainly could not accommodate a quarter of a relation.

Either add a complete one and truncate the tower, or leave this quarter empty indefinitely, making finiteness impossible to hold.

Jiang Lin selected six candidates that were the smallest while balancing the real splitting conditions.

[T = {3, 5, 7, 11, 13, 17}]

The maximal totally real multiquadratic subfield generated by them could be written as:

[L_T = Q(√5, √13, √17, √21, √33)]

Next, a rational prime p was still missing.

It needed to split completely in L_T, still split after adding i, and at the same time not fall into T.

The screening program ran through primes within one hundred and stopped at the 101st position.

[101 ≡ 1 mod 4]

[5, 13, 17, 21, 33 are all quadratic residues modulo 101]

The terminal arranged the five groups of square root integers in the same row.

[45² ≡ 5, 35² ≡ 13, 44² ≡ 17, 18² ≡ 21, 2d 101 )]

101 therefore split completely in L_T(i).

Jiang Lin added it to S, requiring its Frobenius element to be identically equal to one throughout the entire tower.

[S = {∞, 101}]

The splitting condition at the infinite real places ensured that the tower layers remained totally real, while 101 contributed that single newly added relation.

The Proof Engine recalculated the generator and relation ranks along group cohomology, reducing the red conflicts from 23 places to 4 places.

The first place questioned whether the six primes in T truly provided five independent quadratic directions, and the maximal multiquadratic subfield gave a Frattini quotient of rank five.

The second place questioned whether forcing 101 to split would simultaneously prune a generator; 101 had already split completely in the elementary quotient, and its Frobenius fell within the Frattini subgroup, increasing relations without decreasing generators.

The third place fell on the real infinite places; the definition of the tower had already included all real positions into the splitting set, and the finite layers would not grow complex embeddings.

The last place was the strict direction of the Golod–Shafarevich inequality.

Jiang Lin unfolded the original theorem, the Shafarevich relation bound, and the current group presentation side by side, unifying the definitions of generators and the number of relations item by item.

Any finite pro-2 group, if presented minimally with five generators, must have a relation rank strictly greater than twenty-five divided by four.

The relation rank was an integer, and the lowest it could take was seven.

The relation rank of the current group was at most six.

Jiang Lin wrote the last line on the proof page.

[6 < 25 / 4]

The red proposition box turned green.

[G_T^S is an infinite group]

[There exist totally real finite subextensions L_j of unbounded degree]

[101 splits completely in each L_j]

The running time in the lower right corner of the screen stopped at 273 days and 11 hours.

Jiang Lin printed out that page of the proof, clipped it into the transparent board at the edge of the workbench, and spent the next three days handling the requalification records of the multi-machine team and the verification gaps of OUTER CACHE-07 as usual.

Every time he returned from the control room, he would re-examine that inequality.

Six was still less than 6.25.

On the morning of the fourth day, he finally allowed the Proof Engine to send [Tower Existence] into the geometric construction as a known condition.

Adding i to each L_j yielded the CM field K_j = L_j(i).

The newly added ramification within the tower was always restricted to the fixed set T, and adding i only brought an additional fixed cost of the prime two; thus, the degree of K_j could grow continuously, while the root discriminant was held down by the same constant.

Jiang Lin denoted this constant as R₀.

[R₀ = 2 × 3 × 5 × 7 × 11 × 13 × 17 = 510510]

The number was large, but fortunately it would no longer grow with the tower layers from then on.

If [L_j : Q] = f_j, 101 would split into f_j pairs of mutually complex-conjugate prime ideals in K_j.

Jiang Lin allocated the total exponent k from each pair P_a and P̄_a; the exponent assigned to P_a could take any integer from zero to k, and the other side automatically got the remaining part.

A pair of prime ideals gave k + 1 choices, and f_j pairs gave (k + 1) to the power of f_j ideals.

They possessed identical norms.

The class group would compress some of these different combinations into the same ideal class, while the pigeonhole principle guaranteed that at least (k + 1) to the power of f_j divided by h(K_j) combinations were squeezed into the same cell.

Taking the ratio of ideals in the same class, the generator of the principal ideal was denoted as a, and then u = a / ā was constructed.

For every complex embedding σ of K_j, there was:

[|σ(u)| = |σ(a) / σ(ā)| = 1]

The conjugate coordinates that caused the cubic number fields to constantly leak out of the window eleven years ago were all tethered to their respective unit circles this time.

These u fell in the ring of integers multiplied by 101 to the power of -2k, and could precisely serve as difference vectors in the same Minkowski lattice.

The class number would still take away a large number of directions, but the upper bound of the root discriminant gave a uniform cost where h(K_j) grew at most exponentially with f_j.

Jiang Lin set:

[k = ⌈18R₀³ / π⌉ - 1]

This fixed integer was approximately 7.6 times 10 to the 17th power.

It was large enough to make the newly added unit-modulus directions at each layer exceed the joint cost of the class number and the lattice covolume, and since it only depended on R₀, every subsequent layer would use the same k.

As the proof reached here, the degree of the number field finally began to work for Jiang Lin.

Every time f_j doubled, the number of points in the window grew according to a fixed base, and the unit-modulus difference vectors also grew according to another fixed base; the portion by which the latter base crossed the threshold would be completely preserved to the next layer.

Jiang Lin packed the total embedding image of the lattice into a polydisc window of radius R, and then chose an arbitrary complex coordinate from it to project onto the plane.

Field embeddings possessed injectivity, and different lattice points remained different planar points after projection.

Every point in the inner window plus any unit-modulus difference vector would stay within the outer window, and the distance after projection was strictly equal to one.

Boundary losses, lattice density, class numbers, and denominator costs were all put back into the same exponential inequality.

That initial set of 18-digit parameters reappeared on the terminal.

[Upper bound of point count: A^f_j]

[Lower bound of unit distance: A^f_j × B^f_j / C]

[B > 1]

[δ₀ = log B / log A > 0]

The default precision displayed the last line as zero.

Jiang Lin removed floating-point calculations from the proof dependencies, and instead used rational intervals to sandwich π, the logarithmic term, and the upper bound of the root discriminant respectively.

At 256-bit precision, the lower end of the interval was still jumping back and forth near the zero line.

After 512 bits, the positive sign stabilized.

By 2048 bits, the error interval was already nine orders of magnitude narrower than the candidate value.

[δ₀ lower bound 38]

[Interval certification: Passed]

The Proof Engine immediately generated the final proposition.

[There exists an absolute constant δ₀ > 0, and arbitrarily large finite planar point sets P, such that the number of unit distances they determine satisfies]

[U(P) ≥ |P|^(1 + δ₀) / C]

Jiang Lin copied the second line to the last page of the classroom manuscript of Chapter 192, and wrote down the form of the Erdős conjecture below.

[U(n) ≤ n^(1 + o(1))]

The fixed δ₀ would not vanish as n increased, and the two lines of formulas could not hold simultaneously.

The file status changed from [Conditional Counterexample] to [Main Chain Closed / Entering Independent Verification].

From the 14th year to the 16th year of the current cycle, Jiang Lin did not continue to optimize that pitifully small exponent.

A larger δ could make the result look a bit better, but the truth or falsity of the conjecture had already crossed the zero line.

He spent more than two years tearing down the proof.

The number field construction, group theory infinity, class number estimation, unit-modulus element counting, Minkowski lattice point bounds, and two-dimensional projection were divided into six verification packages of non-shared intermediate conclusions.

Every external theorem was traced back to the original offline literature, every [existence] was marked with its selection dependencies, and every floating-point number was replaced with a strict interval that could be re-checked.

The first round of verification found an overcounted root of unity factor in the deduplication of ideal ratios, so the fixed constant C was magnified, while δ₀ remained unchanged.

The second round of verification required proving that the projected points would not coincide, and the injectivity of any field embedding closed off the conflicts.

The third round rewrote the average counting of random translations of the window center into a deterministic existence proposition, and the ratio of the number of points to the number of edges remained within the original boundaries.

The final round of attack directly deleted all finite-layer numerical samples, retaining only symbolic inequalities and existing theorems.

The proof remained closed.

In the third month of the 16th year of the current cycle, at 4:27 AM.

Jiang Lin completed the final backward check from the theorem statement, and the terminal marked all 4,317 dependency nodes green.

[Tower Infinity: Passed]

[Root Discriminant Uniform Upper Bound: Passed]

[Exponential Growth of Unit-Modulus Directions: Passed]

[Planar Projection Counting: Passed]

[Fixed Positive Exponent: Passed]

[Conclusion: Erdős Unit Distance Conjecture Does Not Hold]

On the other side of the workbench lay that 27-page failed draft of cubic fields.

The task metadata on the first page retained the date Jiang Lin wrote in his Tsinghua University dormitory.

[Plan officially launched: November 1, 2022]

The Real World was still two days away from that day.

Jiang Lin had carried this document through more than a dozen winters in the Wasteland.

He divided the final draft into three copies: paper, offline hard drive, and read-only verification disk, and put the version to be brought back to reality into the regression list.

In the manuscript status bar, he filled in [Internal Proof Closed / Pending Real World Peer Verification].

The printer spat out the last page at 5:00 AM.

This page only had the verification signature, file hash, and the shortest line of formulas in the entire proof.

[6 < 25 / 4]