# Unit-distance H3-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 H3 below without proof. Prove H5 below. No other checkpoint result is granted.

**Given: H3**

There exists an absolute constant \(C\ge1\) such that, for every totally real number field \(F\) of degree \(f=[F:\mathbb Q]\), the field
\[
K=F(\mathrm i),\qquad \mathrm i^2=-1,
\]
satisfies
\[
\operatorname{rd}(K)\le2\operatorname{rd}(F),
\qquad
h(K)\le\bigl(C\operatorname{rd}(F)\bigr)^{2f}.
\]

Here \(h(K)\) is the ideal class number of the ring of integers of \(K\). The constant \(C\) is independent of \(F\) and \(f\).

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.