= Solution
Let $F_n=\mathbb Q(\mu_{p^{n+1}})$ and $F_0=\mathbb Q(\mu_p)$. The <cyclotomic character> identifies
$$
G=\operatorname{Gal}(F_\infty/\mathbb Q)
=\Delta\times\Gamma,\qquad
\Delta\cong(\mathbb Z/p\mathbb Z)^\times,\quad
\Gamma\cong1+p\mathbb Z_p.
$$
Choose $\gamma$ with cyclotomic value $u=1+p$, and put
$$
\Lambda=\mathbb Z_p[[\Gamma]]=\mathbb Z_p[[T]],\qquad T=\gamma-1.
$$
The <Iwasawa algebra> of $G$ is $\Lambda[\Delta]$. Conjugation, independent of lift as in the preceding solution, makes $X$ a <compact Galois module> over this ring. It is the <p-ramified Iwasawa module>, not the unramified class-group module.
Because $p-1$ is a unit in $\mathbb Z_p$, the <Teichmüller character> $\omega$ gives orthogonal <character idempotents>
$$
e_\chi=\frac1{p-1}\sum_{\delta\in\Delta}\chi(\delta)^{-1}\delta,
\qquad \chi=\omega^i,\quad0\leq i\leq p-2.
$$
They sum to one, so $X=\bigoplus_\chi X_\chi$, where $X_\chi=e_\chi X$. Each is a <module> over the one-variable ring $\Lambda$. Complex conjugation is the element $-1$ of $\Delta$; characters with $\chi(-1)=1$ are even, and those with $\chi(-1)=-1$ are odd.
The basic \b[Iwasawa-module rank theorem] in this tower gives
$$
\boxed{
\operatorname{rank}_\Lambda X_\chi=
\begin{cases}
0,&\chi\text{ even},\\
1,&\chi\text{ odd}.
\end{cases}}
$$
In particular,
$$
\boxed{\operatorname{rank}_\Lambda X=\frac{p-1}{2},\qquad
X^+\text{ is torsion},\qquad
\operatorname{rank}_\Lambda X^-=\frac{p-1}{2}.}
$$
Thus it would be incorrect to describe all of $X$ as a <torsion module>.
Here is the <class-field unit sequence> explaining these ranks. Let $U_\infty$ be the inverse limit, under local norms, of the pro-$p$ completions of the unit groups at the unique prime over $p$ in $F_n$. Let $\overline E_\infty$ be the inverse limit of the closures of the global units in these local unit groups, and let
$$
Y=\varprojlim_n\operatorname{Cl}(F_n)(p)
$$
under ideal norms. The <Artin reciprocity> maps yield
$$
0\longrightarrow\overline E_\infty
\longrightarrow U_\infty
\longrightarrow X
\longrightarrow Y
\longrightarrow0.
$$
The final module is the <unramified Iwasawa module> and is finitely generated torsion, as proved in the other essay. Local-unit theory gives $U_\infty$ <Iwasawa-module rank> $p-1$, with rank one in each $\Delta$-character. The <Leopoldt theorem for abelian number fields> gives $\overline E_\infty$ rank $(p-1)/2$, with rank one in each even character and zero in each odd character. In odd characters the possible norm-compatible roots of unity contribute a rank-zero term, not a free summand. Subtracting ranks in the exact sequence gives the displayed result and also finite generation.
For any character, the <Iwasawa module structure theorem> describes $X_\chi$ up to <pseudo-isomorphism>:
$$
X_\chi\sim
\Lambda^{r_\chi}
\oplus\bigoplus_j\Lambda/(p^{a_j})
\oplus\bigoplus_k\Lambda/(f_k(T)^{b_k}),
$$
where the $f_k$ are irreducible <distinguished polynomials> and the kernel and cokernel of a <pseudo-isomorphism> are finite. For a torsion component, its <characteristic ideal> is generated by
$$
p^{\sum_j a_j}\prod_k f_k(T)^{b_k}.
$$
The <Iwasawa invariants> are $\mu=\sum a_j$ and $\lambda=\sum b_k\deg f_k$. A characteristic ideal describes the elementary divisors only in aggregate; it is not an assertion that the module is cyclic or that an odd component is actually free.
The trivial character component $X_1$ is zero. Indeed, the maximal abelian pro-$p$ extension of $\mathbb Q$ unramified outside $p$ is precisely the rational <cyclotomic Zp-extension>, by the <Kronecker–Weber theorem>. Coprime descent using the <character idempotent> $e_1$, followed by the <abelianization over a Zp-extension> formula, identifies $(X_1)_\Gamma$ with the additional abelian pro-$p$ quotient over $\mathbb Q$. There is no such extension beyond the cyclotomic one, so $(X_1)_\Gamma=0$, and the <Compact Nakayama lemma> gives $X_1=0$.
The central arithmetic description concerns the nontrivial even characters. Let $g_\chi(T)$ be the integral <Kubota-Leopoldt p-adic L-function> power series with convention
$$
g_\chi(u^{\,1-s}-1)=L_p(\chi,s).
$$
For positive integers $k$ of the matching parity, the interpolation is
$$
L_p(\chi,1-k)
=-\left(1-(\chi\omega^{-k})(p)p^{k-1}\right)
\frac{B_{k,\chi\omega^{-k}}}{k},
$$
where the Dirichlet character in the Euler factor is its primitive associate, and $B_{k,\eta}$ is a <generalized Bernoulli number>. The \b[Iwasawa main conjecture], a theorem here, says
$$
\boxed{\operatorname{char}_\Lambda X_\chi=(g_\chi(T))
\quad(\chi\ne1,\ \chi\text{ even}).}
$$
The trivial character is excluded from this displayed analytic normalization: its $p$-adic zeta function has a pole and must be treated separately.
The proof mechanism starts with norm-compatible <cyclotomic units>, for instance ratios
$$
c_n(a)=\frac{1-\zeta_{p^{n+1}}^a}{1-\zeta_{p^{n+1}}}
\quad(p\nmid a),
$$
and their character projections. A <Coleman power series> encodes a norm-compatible local unit by a single power series. Applying its logarithmic derivative to these cyclotomic units produces the <p-adic L-function> above. If $C_\infty$ denotes their closed norm-limit module, this calculation determines
$$
\operatorname{char}_\Lambda(U_{\infty,\chi}/C_{\infty,\chi})=(g_\chi).
$$
The remaining global input is
$$
\operatorname{char}_\Lambda Y_\chi
=\operatorname{char}_\Lambda(\overline E_{\infty,\chi}/C_{\infty,\chi})
$$
for nontrivial even $\chi$. In the <class-field unit sequence> modulo $C_{\infty,\chi}$, multiplicativity of <characteristic ideals> cancels these two terms and gives the boxed formula for $X_\chi$.
There are two established routes to this global input. The <Mazur-Wiles theorem> constructs suitable abelian extensions from Galois representations associated with <modular forms> and the <Eisenstein ideal>; their sizes supply the missing divisibility. The <cyclotomic Euler system> route uses norm relations for cyclotomic units at auxiliary primes and descent to bound the class-group module. The <analytic class number formula> and the cyclotomic-unit index formula then supply the equality of characteristic ideals. These are substantial arithmetic theorems, rather than consequences of the abstract structure theorem alone.
A useful further result is the \b[Ferrero-Washington theorem]: $\mu(Y)=0$ for this cyclotomic tower, more generally for cyclotomic towers of abelian <number fields>. Its proof rules out an identically zero reduction modulo $p$ of the relevant $p$-adic $L$-series, using distribution of $p$-adic digits. Together with the main conjecture, it gives $\mu(X_\chi)=0$ for each nontrivial even $\chi$. Therefore these torsion components are finitely generated over $\mathbb Z_p$ up to finite error.
Finally, <Kummer reflection in Iwasawa theory> relates the even $p$-ramified module to an odd unramified component:
$$
X_\chi\sim Y_{\omega\chi^{-1}}^{\,\iota}(1),
\qquad \iota(\gamma)=\gamma^{-1},
$$
where $(1)$ is the <Tate twist> by the cyclotomic character. The corresponding power-series substitution is
$$
T\longmapsto u(1+T)^{-1}-1.
$$
This explains why the $p$-ramified interpolation involves $u^{1-s}$ while the reflected class-group interpolation involves $u^s$. Both the involution and the twist matter. The rank theorem, the class-field sequence, the structure theorem and the explicit even-character characteristic ideals together give the known <compact Galois module> structure; none justifies replacing every odd component by a free module without further argument.
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