= Solution
The preceding constructions lead to the <Iwasawa theorem on local cyclotomic units>. Its significance is that an explicitly calculated <local unit> quotient is governed by a <p-adic L-function>; <class field theory> then suggests that the same analytic object should govern global arithmetic. The latter prediction is the <Iwasawa main conjecture>, rather than a consequence of the local calculation alone.
Fix compatible <roots of unity> $\zeta_{p^n}$ and put $K_n=\mathbb Q_p(\zeta_{p^n})$, $\pi_n=\zeta_{p^n}-1$. Let $V_\infty$ be the norm limit of the local <principal units>. Take its even part under <complex conjugation> to obtain $U_\infty^1$, the local principal-unit limit in the real tower. Let $C_\infty^1$ be the closed <subgroup> generated by norm-compatible real <cyclotomic units>, with their finite-order residue factors removed. Write
$$
G=\mathbb Z_p^\times/\{\pm1\},\qquad \Lambda_G=\mathbb Z_p[[G]],\qquad I(G)=\ker(\Lambda_G\longrightarrow\mathbb Z_p).
$$
Here the last map is augmentation. Normalize the <p-adic zeta pseudomeasure> $\zeta_p$ by
$$
\int_Gx^k\,d\zeta_p=(1-p^{k-1})\zeta(1-k)=-(1-p^{k-1})\frac{B_k}{k}\qquad(k\geq2\text{ even}).
$$
The expression $x^k$ is well defined on $G$ for even $k$. A pseudomeasure lies in the total fraction <ring> of the <Iwasawa algebra>; the pole on its trivial finite-character component is removed by the <augmentation ideal>, so $I(G)\zeta_p\subset\Lambda_G$. In this normalization the local theorem is
$$
\boxed{U_\infty^1/C_\infty^1\simeq\Lambda_G/(I(G)\zeta_p).}
$$
The following steps outline its proof and explain where the earlier questions enter.
First encode <local units> by <Coleman power series>. Evaluation at $\pi_n$ is surjective from $R$ to $\mathcal O_{K_n}$, and the product defining the <Coleman norm operator> gives
$$
(Nf)(\pi_{n-1})=N_{K_n/K_{n-1}}(f(\pi_n))\qquad(n\geq2).
$$
Thus norm-fixed series give <norm-compatible sequences of local units>. Conversely, lift $u_m$ from such a sequence to $f^{(m)}\in R^\times$ and take $g_m=N^{m-r_m}f^{(m)}$, where $r_m=\lfloor m/2\rfloor$. This has $g_m(\pi_{r_m})=u_{r_m}$. The estimates in Question 2 give $Ng_m/g_m\in1+p^{m-r_m+1}R$. Consequently for fixed $n\leq r_m$, the series $N^{r_m-n}g_m$ differs from $g_m$ modulo $p^{m-r_m+1}$ and its value at $\pi_n$ is $u_n$. The <compactness> of $R$ in the coarser of the <topologies on integral formal power series> supplies a convergent subsequence. Continuity of the <Coleman norm operator> and of evaluation shows that its limit is norm-fixed and takes every value $u_n$. It is a <unit> because the original <principal units> have residue $1$. Uniqueness follows from the <Weierstrass preparation theorem>: a nonzero series in $R$ has only finitely many zeros in the open unit disk, whereas the distinct $\pi_n$ give infinitely many interpolation points. This constructs the <Coleman power series> $f_u$.
Next turn its multiplicative information into an integral <p-adic measure>. For a norm-fixed <unit> $f$, use the <Frobenius-corrected logarithm>
$$
\mathcal L(f)=\frac1p\log\left(\frac{f^p}{\phi(f)}\right).
$$
The ratio belongs to $1+pR$ by reduction modulo $p$, and the logarithm divided by $p$ is integral because $p$ is odd. Equivalently $\mathcal L(f)=(1-\phi/p)\log f$, interpreting the constant coefficient with the <p-adic logarithm> on <units>. The averaging identity for the logarithm is understood on the open $p$-adic unit disk, where the translated logarithmic series converge; the corrected result itself lies in $R$. The logarithmic product formula and Question 1 give
$$
S\log f=\frac1p\log(Nf),\qquad S\mathcal L(f)=\frac1p\log(Nf/f)=0.
$$
Under the <Amice transform>, the root-of-unity average projects a <p-adic measure> onto its restriction to $p\mathbb Z_p$: the averaged factor $(1+T)^x$ survives exactly for $x\in p\mathbb Z_p$. Consequently $S\mathcal L(f)=0$ says that $\lambda(\mathcal L(f))$ is supported on $\mathbb Z_p^\times$. This gives a <Galois group>-equivariant homomorphism from local <principal units> to the <Iwasawa algebra> of $\mathbb Z_p^\times$.
The integral lifting step in this proof is the <Coleman unit-measure exact sequence>
$$
0\longrightarrow\mathbb Z_p(1)\longrightarrow V_\infty\xrightarrow{\mathrm{Col}}\mathbb Z_p[[\mathbb Z_p^\times]]\xrightarrow{m_1}\mathbb Z_p(1)\longrightarrow0,\qquad m_1(\mu)=\int x\,d\mu.
$$
Here $\mathrm{Col}(u)=\lambda(\mathcal L(f_u))$. The last map is surjective, since $m_1(\delta_1)=1$, and it has the cyclotomic action on its target. Its composition with $\mathrm{Col}$ is zero: $D\phi=p\phi D$ implies $D\mathcal L(f)(0)=0$. The kernel on the unit side consists of norm-compatible $p$-power <roots of unity>. To see the shape of the kernel, write $z=\log(1+T)$; the equation $\mathcal L(f)=0$ becomes $h(pz)=ph(z)$ for $h(z)=\log f(e^z-1)$, so only its linear term can survive. Integral solutions are $f=(1+T)^c$, $c\in\mathbb Z_p$, after excluding finite residue factors by the <principal unit> condition. For the image, use the following explicit <integral lifting of the corrected cyclotomic logarithm>.
Take a <p-adic measure> $\mu$ on the units with first moment zero, and write $g$ for its <Amice transform>. Then $Sg=0$ and $g'(0)=0$. Put $H=\exp(pg)\in1+pR$ and solve $f^p=H\phi(f)$ with $f(0)\in1+p\mathbb Z_p$. The constant equation is solved by $f_0=\exp(pg_0/(p-1))$. In degree one the terms involving $f_1$ cancel, and the remaining condition is $H_1=0$, exactly $g_1=0$; choose any $f_1\in\mathbb Z_p$. For $n\geq2$, after fixing the lower coefficients, the coefficient multiplying the unknown $f_n$ is
$$
p f_0^{p-1}(1-p^{n-1}).
$$
The other terms are divisible by $p$: for the already integral truncated series, $f^p\equiv\phi(f)\pmod p$, while $H\equiv1\pmod p$. Division therefore gives an integral $f_n$, since $f_0$ and $1-p^{n-1}$ are <units>. Recursion constructs $f\in R^\times$ with $\mathcal L(f)=g$. Now $pSg=\log(Nf/f)=0$, and $Nf/f\in1+pR$ by Question 2. The <p-adic logarithm> is injective there, so $Nf=f$. This supplies the desired <Coleman power series> of a <norm-compatible sequence of local units>. The freely chosen linear coefficient accounts precisely for the kernel $(1+T)^c$. It proves the integral <surjectivity> step, including the integrality that rational inversion of $1-\phi/p$ would leave unproved.
<Complex conjugation> acts by $-1$ on both copies of $\mathbb Z_p(1)$. As $p$ is odd, taking the even part is exact and removes both end terms. It follows that \b[$\mathrm{Col}:U_\infty^1\simeq\Lambda_G$]. The integral <exact sequence> is essential here: the support calculation alone would not prove this <isomorphism>.
It remains to calculate the image of the <cyclotomic units>. For $a,b\in\mathbb Z_p^\times$, consider the <symmetric cyclotomic unit> series
$$
f_{a,b}(T)=X^{(b-a)/2}\frac{1-X^a}{1-X^b}.
$$
The quotient has an invertible constant term $a/b$, because both numerator and denominator have a simple zero at $T=0$. The factor $X^{(b-a)/2}$ makes the expression unchanged under $X\mapsto X^{-1}$. The product over the <roots of unity> gives $Nf_{a,b}=f_{a,b}$; one may verify this first for integer $a,b$ prime to $p$ and then extend by continuity. Dividing by the norm-compatible <Teichmuller representative> makes the series correspond to <principal units> and does not change its logarithm. The generating function for the <Bernoulli numbers>, with $z=\log X$, gives
$$
\log f_{a,b}(e^z-1)=\log(a/b)+\sum_{\substack{k\geq2\\k\text{ even}}}\frac{(a^k-b^k)B_k}{k\,k!}z^k.
$$
The symmetric factor cancels the linear term. Applying $1-\phi/p$, and then the moment formula of Question 3, yields
$$
\int x^k\,d\mathrm{Col}(f_{a,b})=(1-p^{k-1})(a^k-b^k)\frac{B_k}{k}=\int x^k\,d\bigl(([b]-[a])\zeta_p\bigr)
$$
for even $k\geq2$. These moments determine an even <p-adic measure> on the units: <polynomials> are dense by the <Mahler theorem>, even projection preserves density, and multiplication by $x^2$ is invertible on the units, so the positive even-degree <polynomials> are dense there as well. Thus
$$
\mathrm{Col}(f_{a,b})=([b]-[a])\zeta_p.
$$
The closed ideal generated by all differences $[b]-[a]$ is $I(G)$. The closed <subgroup> generated by the corresponding <cyclotomic units>, including its <Galois group> translates, therefore has image exactly $I(G)\zeta_p$. Taking the quotient of the <isomorphism> $U_\infty^1\simeq\Lambda_G$ proves the boxed local theorem.
Finally introduce global arithmetic. Let $\overline E_\infty^1$ be the closure of norm-compatible <global units> in the real tower, let $X_\infty^+$ be its <p-ramified Iwasawa module>, and let $Y_\infty^+$ be its <unramified Iwasawa module>. With compatible unit and splitting conventions, <class field theory> gives the <class-field unit sequence>; quotienting its unit terms by $C_\infty^1$ gives
$$
0\longrightarrow\overline E_\infty^1/C_\infty^1\longrightarrow U_\infty^1/C_\infty^1\longrightarrow X_\infty^+\longrightarrow Y_\infty^+\longrightarrow0.
$$
In the cyclotomic tower the prime above $p$ is principal, so the unramified class-field convention also makes it split. On the torsion character components, multiplicativity of <characteristic ideals> implies
$$
\operatorname{char}(U_\infty^1/C_\infty^1)\operatorname{char}(Y_\infty^+)=\operatorname{char}(\overline E_\infty^1/C_\infty^1)\operatorname{char}(X_\infty^+).
$$
The local theorem has already calculated the first factor analytically. Equality of the <global unit> and class-group factors would therefore identify the <characteristic ideal> of $X_\infty^+$ with the same ideal of the <p-adic L-function>. That is the content predicted by the <Iwasawa main conjecture>. More explicitly, for a nontrivial even finite character $\chi$, the character component of $I(G)$ is the whole component and the prediction is
$$
\boxed{\operatorname{char}_{\Lambda_\chi}(X_\chi^+)=(g_\chi),\qquad g_\chi((1+p)^{1-s}-1)=L_p(\chi,s).}
$$
Here $\Lambda_\chi$ is the power-series algebra of the pro-$p$ part with coefficients in the character-value <ring>. The trivial component requires the augmentation correction for the zeta pole. Equivalently the <characteristic ideals> of $\overline E_\infty^1/C_\infty^1$ and $Y_\infty^+$ agree componentwise. A formulation using odd class-group characters must include the appropriate cyclotomic twist and inversion, as in <Kummer reflection in Iwasawa theory>. \b[The motivation is an analytic formula for <local units> together with a global class-field exact sequence; the global equality is the additional arithmetic assertion.]
The related class-number growth theorem illustrates why passage to an <Iwasawa module> is powerful. For the cyclotomic $\mathbb Z_p$-extension, let $p^{e_n}$ be the $p$-part of the <class number> at level $n$. The <unramified Iwasawa torsion theorem>, the <Iwasawa module structure theorem> and the class-group control relations give, for sufficiently large $n$,
$$
e_n=\mu p^n+\lambda n+\nu.
$$
Here $\mu,\lambda$ are the <Iwasawa invariants>. To see the two growth terms, use a topological generator $\gamma$ and, after a fixed sufficiently high level $n_0$, the norm polynomial $\nu_{n,n_0}=(\gamma^{p^n}-1)/(\gamma^{p^{n_0}}-1)$. An elementary summand $\Lambda/(p^a)$ contributes $a(p^n-p^{n_0})$ to the length of its quotient by this polynomial. A summand associated with a <distinguished polynomial> of degree $d$ contributes $dn$ plus a constant: at each root $\alpha$ the valuation of the norm polynomial is $n$ plus a fixed constant for large $n$. If $1+\alpha$ is a root of unity already killed at level $n_0$, its value is instead exactly $p^{n-n_0}$ and gives the same linear contribution. The finite errors in the structure and control maps stabilize and contribute to $\nu$. Thus module structure explains growth, while the <Iwasawa main conjecture> goes further by identifying the <characteristic series of an Iwasawa module> themselves with analytic ones.
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