 Solution

The relevant notion is a root-of-unity Euler system, which contains auxiliary-prime data as well as the p-power tower. Fix a finite set $S$ of primes containing two but not $p$, and let $W_S$ consist of all roots of unity whose orders are prime to every member of $S$. It is a map $\Phi :W_S\to {\overline{\mathbb{Q}}}^{\times }$ satisfying these three axioms:

The first axiom puts $\Phi (\zeta )$ in the maximal real subfield of $\mathbb{Q}(\zeta )$. For $\zeta ={\zeta }_n$, the second axiom is exactly norm compatibility from $K_n^{+}$ to $K_{n-1}^{+}$ for $n\ge 1$. Auxiliary roots with prime orders different from $p$ are essential: mere norm compatibility of a sequence in the tower is not a verification of all three axioms.

The real cyclotomic unit group at a p-power conductor is generated by $-1$ and the symmetric cyclotomic units $c_n(a,b)$ with $p\nmid ab$. Galois conjugates are included, since a conjugate of $c_n(a,b)$ is $c_n(ua,ub)$ for an integer $u$ prime to $p$. Thus a prescribed unit can be expressed aswhere $d_a(z)=z^{-a/2}-z^{a/2}$, with half exponents interpreted in the odd-order root group. Choose $S$ to contain two and every prime divisor of the $a_j$. For $z\ne 1$ in $W_S$ putEach factor at $z\ne 1$ is nonzero because raising $z$ to $a_j$ preserves its order. At $z=1$ each $d_{a_j}$ has a simple zero with first coefficient $-a_j$; the total exponent zero cancels the zeros and signs, giving the stated removable value. Consequently $\Phi ({\zeta }_n)=\epsilon$.

We verify the axioms individually. First, $d_a(z^{-1})=-d_a(z)$; the product's sign is $(-1{)}^{\sum e_j}=1$, so $\Phi$ is even. The unique half-root operation commutes with every field automorphism, all exponents are integers and $\eta$ is rational. Hence $\Phi (\sigma z)=\sigma \Phi (z)$, including at $z=1$.

Second, $q\notin S$ is odd and does not divide any $a_j$. For a variable $x$, choose half powers consistently and useHere $\rho \mapsto {\rho }^a$ permutes ${\mu }_q$, and the product of its half powers is one. Raising these identities to $e_j$ and multiplying gives the distribution identity, since ${\eta }^q=\eta$. If $x=1$ or $x^q=1$, take the removable values after cancellation; the rational-function identity remains valid there. Thus the distribution axiom holds also at the special roots where the uncancelled factors vanish.

Third, fix $q\notin S$, a root $z$ of order prime to $q$, and a prime $\mathfrak{q}$ above $q$. Every $\rho \in {\mu }_q$ reduces to one modulo $\mathfrak{q}$. If $z\ne 1$, its reduction retains its exact order: prime-to-q roots have distinct reductions because their order polynomial is separable in characteristic $q$. Therefore $1-z^{a_j}$ is a local unit, and all the factors and inverse factors are regular at that reduction. Reduction of the expression for $\Phi (\rho z)$ then gives $\Phi (z)$, since the half roots reduce compatibly. This proves the congruence and local integrality for nontrivial $z$.

At $z=1$, write $\rho =1+T$. The cancelled expressionis a unit formal power series over ${\mathbb{Z}}_q$, because every $a_j$ and two are q-adic units. Its constant term is $\eta {\prod }_j{a}_j^{e_j}$. Evaluating at $T=\rho -1$ proves the remaining congruence modulo every prime over $q$. This also covers $\rho =1$. Hence every real cyclotomic unit at any layer is a value of an Cyclotomic Euler system satisfying all three axioms.

For completeness, at a nontrivial root $z$ every ratio $d_a(z)/d_b(z)$ is a global unit. After removing a root-of-unity factor, both it and its inverse are geometric sums because $a,b$ are invertible modulo the order of $z$. Thus the values used at nontrivial roots really are units, not just nonzero algebraic numbers. Values at one may be rational S-units, which is compatible with the definition.

The auxiliary norm factor can also be read off explicitly. If $z$ has order $m>2$ and $q\nmid m$, then for $\rho \ne 1$ of order $q$,Here the arithmetic Frobenius sends $z$ to $z^q$. This convention makes the sign of the Euler factor explicit, while the case of a repeated prime $p$ gives the exact tower norm relation already proved in Question 2.