= Solution
Write $F_n=K_n^+$, $F_\infty=\bigcup_nF_n$, $G=\operatorname{Gal}(F_\infty/\mathbb Q)=\Delta^+\times\Gamma$, where $|\Delta^+|=(p-1)/2$ and $\Gamma\simeq\mathbb Z_p$. Set $A=\mathbb Z_p[[G]]$. Choosing a topological generator $\gamma$ of $\Gamma$ identifies its <Iwasawa algebra> with $\Lambda=\mathbb Z_p[[S]]$, $S=\gamma-1$; this group variable is distinct from the local-unit interpolation variable $T$. The <Teichmüller character> decomposes $A$ into a product of copies of $\Lambda$, one for each character of $\Delta^+$, equivalently each even Teichmüller character of $\operatorname{Gal}(K_0/\mathbb Q)$. All characteristic-ideal statements below are componentwise in this product.
Let $U_n^1$ be the local <principal units> at the unique prime above $p$. Let $E_n^1$ and $C_n^1$ be respectively the p-adic closures of <global units> and <cyclotomic units> inside $U_n^1$, and take their <inverse limits> under the <field norms>. Define $X_\infty$ as the <Galois group> of the maximal abelian pro-p extension of $F_\infty$ unramified away from $p$, and $Y_\infty$ as that of the maximal everywhere unramified extension. This is the real tower: its prime-ramified module is torsion, by the <Leopoldt theorem for abelian number fields>. It must not be confused with the positive-rank prime-ramified module of the full complex cyclotomic tower. The <unramified Iwasawa torsion theorem> gives torsion for $Y_\infty$.
A convenient formulation of the <Iwasawa main conjecture> uses the <p-adic zeta pseudomeasure> $\zeta_p$ on $G$. With the sign convention
$$
\int x^k\,d\zeta_p=(1-p^{k-1})\zeta(1-k)
=-(1-p^{k-1})\frac{B_k}{k}\qquad(k\ge2\text{ even}),
$$
its product with the <augmentation ideal> $I(G)$ is integral and principal. The pole on the trivial-character component is cancelled by that ideal; simply writing an integral quotient by $\zeta_p$ would be incorrect. The main assertion in this convention is
$$
\boxed{\operatorname{char}_A(X_\infty)=I(G)\zeta_p.}
$$
We explain the local calculation, the remaining global obstruction, and how the <cyclotomic Euler system> removes it.
The local calculation is the <Iwasawa theorem on local cyclotomic units>:
$$
U_\infty^1/C_\infty^1\simeq A/(I(G)\zeta_p).
$$
One can see why this theorem has exactly this analytic term. For a norm-fixed <Coleman power series> $f$, its corrected logarithm is
$$
\mathcal L(f)=\frac1p\log\frac{f(T)^p}{f((1+T)^p-1)}.
$$
The ratio lies in $1+pR$, so the expression is integral. The norm identity puts its <Amice transform> in the kernel of the <Coleman trace operator>, hence gives a measure on $\mathbb Z_p^\times$. Its kth moment is $(1-p^{k-1})\delta_k(f)$. The full local <exact sequence> has p-power <roots of unity> at its kernel and cokernel. Passing to the real, or even, part removes both, since complex conjugation acts as minus one on their Tate module and $p$ is odd. The corrected logarithm therefore identifies $U_\infty^1$ with $A$.
Question 2 now shows that the image of $c(a,b)$ is $([b]-[a])\zeta_p$: its even moments are $(1-p^{k-1})(a^k-b^k)B_k/k$. A choice of integer $e$ generating $\mathbb Z_p^\times$ topologically gives a norm-compatible generator $c(e,1)$, after multiplication by its constant <Teichmuller lift> to make it principal. That constant has zero logarithmic derivatives. Its conjugates generate the closed cyclotomic-unit module, and $[e]-[1]$ generates $I(G)$. This yields the displayed local quotient and its <characteristic ideal>. The generation statement is an important ingredient; the mere calculation of a few moments would not prove the theorem.
The <class-field unit sequence> gives the global comparison
$$
0\longrightarrow B\longrightarrow H\longrightarrow X_\infty\longrightarrow Y_\infty\longrightarrow0,
\qquad B=E_\infty^1/C_\infty^1,\quad H=U_\infty^1/C_\infty^1.
$$
Multiplicativity of <characteristic ideals> in <exact sequences> gives
$$
\operatorname{char}(B)\operatorname{char}(X_\infty)
=\operatorname{char}(H)\operatorname{char}(Y_\infty).
$$
Thus the local theorem proves the main conjecture precisely when one proves \b[$\operatorname{char}(B)=\operatorname{char}(Y_\infty)$]. These terms measure the global-unit and ideal-class obstructions. Discarding them would silently impose a much stronger arithmetic hypothesis.
The first global step is the <Euler-system divisibility for real cyclotomic class modules>: a generator $f_Y$ of $\operatorname{char}(Y_\infty)$ divides a generator $f_B$ of $\operatorname{char}(B)$ on every character component. Here is the mechanism. Choose auxiliary primes $q$ splitting completely in a finite layer and satisfying $q\equiv1\pmod{p^m}$. For a generator $\sigma_q$ of its auxiliary cyclic group, put
$$
D_q=\sum_{i=1}^{q-2}i\sigma_q^i,\qquad
(\sigma_q-1)D_q=(q-1)-N_q.
$$
Apply products of these operators to the <unit> <Cyclotomic Euler system> of Question 3. The norm axiom and this group-ring identity make the derived elements invariant modulo $p^m$-th powers, so Kummer descent gives classes in $F_n^\times/(F_n^\times)^{p^m}$. The congruence axiom identifies the valuation at each newly introduced prime with a residue symbol of the preceding class; valuations away from the chosen auxiliary primes vanish modulo $p^m$. The <Chebotarev density theorem> supplies primes in prescribed ideal classes with the required residue-symbol behavior. Adding these primes successively forces the elementary divisors of the class group to divide the available cyclotomic-unit index. Taking the norm limit yields $f_Y\mid f_B$. Finite control defects do not survive localization at height-one primes; using two coprime annihilators of such a finite defect also shows that no hidden p-power factor is left. This is the substantive global argument furnished by the <Cyclotomic Euler system>, not a consequence of Iwasawa's local theorem alone.
For the reverse comparison use the <cyclotomic unit index formula> and norm descent, keeping invariants as well as coinvariants. At the base <field> put $A_0$ for its p-primary <ideal class group>, and let $N_\infty(E_0^1)$ be the <universal norm of units in a Zp-extension>. Class-field and <unit> descent, using total ramification and the principal prime over $p$, give
$$
Y_{\infty,\Gamma}\simeq A_0,\qquad
B_\Gamma\simeq N_\infty(E_0^1)/C_0^1,\qquad
\#(E_0^1/N_\infty(E_0^1))=\#Y_\infty^\Gamma.
$$
These are the required norm-defect control statements. The <cyclotomic unit index formula>, with the p-adic closure justified by Leopoldt, gives $\#A_0=\#(E_0^1/C_0^1)$. Also $B^\Gamma=0$: its coinvariants are finite by the displayed control, so its invariants are finite by the <Iwasawa module structure theorem>, whereas $B$ is a submodule of the cyclic local quotient $H$, which has no nonzero finite submodule. One must retain $Y_\infty^\Gamma$; dropping it would incorrectly identify all <global units> with universal norms.
Define the finite <Euler characteristic of a one-variable Iwasawa module> by $\chi_\Gamma(M)=\#M_\Gamma/\#M^\Gamma$. The preceding identities now give an exact cancellation:
$$
\chi_\Gamma(Y_\infty)
=\frac{\#(E_0^1/C_0^1)}{\#(E_0^1/N_\infty(E_0^1))}
=\#(N_\infty(E_0^1)/C_0^1)
=\chi_\Gamma(B).
$$
For a torsion $\Lambda$-module with finite invariant and coinvariant groups, the structure theorem gives $\chi_\Gamma(M)=|f_M(0)|_p^{-1}$. Write $f_{B,\chi}=f_{Y,\chi}h_\chi$ using the Euler-system divisibility. After forgetting $\Delta^+$, the <characteristic series> are the products of their character-component series. Equality of these <Euler characteristic of a one-variable Iwasawa module> values gives
$$
\sum_\chi v_p(h_\chi(0))=0.
$$
Every summand is nonnegative, since $h_\chi\in\mathbb Z_p[[S]]$. Each is therefore zero. A <formal power series> with p-adic-unit constant term is a <unit>, so every $h_\chi$ is a <unit>. Hence $\operatorname{char}(B)=\operatorname{char}(Y_\infty)$, and cancellation in the class-field <exact sequence> completes the boxed main conjecture.
The familiar reflected odd class-group formulation follows from <Kummer reflection in Iwasawa theory>, with the inversion of the group variable and the <Tate twist> both included. For example, for nontrivial even $\chi$, if $f_\chi((1+p)^s-1)=L_p(\chi,s)$, then the p-ramified series is $g_\chi(S)=f_\chi((1+p)(1+S)^{-1}-1)$, and $g_\chi((1+p)^{1-s}-1)=L_p(\chi,s)$. The proof above includes the exceptional trivial component through augmentation regularization. It assumes neither a vanishing real class group nor Vandiver's conjecture: the Euler-system divisibility and the norm-defect cancellation replace that extra hypothesis.
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