# Unit-distance H2-to-H5

## Setting

You are solving a mathematical proof problem. Assume the stated starting result and establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.

## Problem

You may assume H2 below without proof. Prove H5 below. No other checkpoint result is granted.

**Given: H2**

There exist an absolute constant \(C>0\) and an integer \(\ell_0\ge2\) such that, for every integer \(\ell\ge\ell_0\), putting
\[
t=\left\lfloor\frac{(\ell-1)^2}{100}\right\rfloor,
\]
there exist a totally real cyclic cubic field \(F/\mathbb Q\), distinct rational primes \(q_1,\ldots,q_t\), and a tower
\[
F=F_0\subset F_1\subset F_2\subset\cdots
\]
satisfying all the following properties.

1. Every \(q_b\) is congruent to \(1\pmod4\) and splits completely in every \(F_j\), for every \(j\ge0\).

2. Every \(F_j/F\) is finite Galois and everywhere unramified, and its Galois group is a \(3\)-group. Every \(F_j\) is totally real.

3. Writing \(f_j=[F_j:\mathbb Q]\), one has
\[
f_j\longrightarrow\infty,
\qquad
\operatorname{rd}(F_j)=\operatorname{rd}(F),
\qquad
\log\operatorname{rd}(F)\le C\ell\log\ell.
\]

For each fixed \(\ell\), the primes \(q_1,\ldots,q_t\) are chosen once and are independent of \(j\).

Here \(\operatorname{rd}(L)=|\operatorname{Disc}(L)|^{1/[L:\mathbb Q]}\) is the root discriminant of a number field \(L\), and all logarithms are natural.

**Target: H5**

For a finite set \(P\subset\mathbb R^2\), let
\[
\nu(P)=\#\bigl\{\{x,y\}\subset P:x\ne y,\ \|x-y\|_2=1\bigr\},
\]
where \(\|\cdot\|_2\) is the Euclidean norm. For every positive integer \(n\), define
\[
\nu(n)=\max\{\nu(P):P\subset\mathbb R^2,\ |P|=n\}.
\]
There exist a fixed real number \(\delta>0\) and a sequence of finite planar sets \((P_j)_{j\ge1}\) such that, writing \(n_j=|P_j|\),
\[
n_j\longrightarrow\infty,
\qquad
\nu(P_j)\ge n_j^{1+\delta}\quad\text{for every }j\ge1.
\]
Consequently, infinitely many distinct positive integers \(n\) satisfy
\[
\nu(n)\ge n^{1+\delta}.
\]
The same \(\delta\) applies to every set in the sequence.

Write the full proof to answer.md in the workspace root. Write the statements and hypotheses of any external results you cite to cited_results.md; if none are used, write 'No external results cited.' Identify any gap you cannot close.