# Unit-distance H1-to-H4

## 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 H1 below without proof. Prove H4 below. No other checkpoint result is granted.

**Given: H1**

There exist an absolute constant \(C>0\) and an integer \(\ell_0\ge2\) such that, for every integer \(\ell\ge\ell_0\), there exists a totally real cyclic cubic field \(F/\mathbb Q\) with the following properties.

Let \(F^{\mathrm{ur},3}\) be the maximal everywhere-unramified pro-\(3\) extension of \(F\), and put \(G=\operatorname{Gal}(F^{\mathrm{ur},3}/F)\). Then
\[
d(G)\ge\ell-1,
\qquad
\log\operatorname{rd}(F)\le C\ell\log\ell.
\]

Here \(d(G)\) is the minimum number of topological generators of \(G\).

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: H4**

There exist a fixed integer \(m\ge1\), a fixed real number \(\gamma>0\), a sequence of number fields \((K_j)_{j\ge1}\), and finite sets \(U_j\subset K_j\), such that, writing \(d_j=[K_j:\mathbb Q]\), one has
\[
d_j\longrightarrow\infty,
\qquad
U_j\subset m^{-1}\mathcal O_{K_j},
\qquad
|U_j|\ge\exp(\gamma d_j).
\]
For every \(j\ge1\), every \(u\in U_j\), and every field embedding \(\sigma:K_j\hookrightarrow\mathbb C\),
\[
|\sigma(u)|=1.
\]

Here \(\mathcal O_{K_j}\) is the ring of integers of \(K_j\), and \(m^{-1}\mathcal O_{K_j}=\{a/m:a\in\mathcal O_{K_j}\}\). The same constants \(m\) and \(\gamma\) apply to every \(j\).

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.