= Solution
The <idele group> is the <restricted product>
$$
J_K=\prod_v'K_v^*,
$$
with respect to $\mathcal O_v^*$ at the finite places. Thus $a=(a_v)$ is an <idele> if all components are nonzero and $a_v$ is a unit at all but finitely many finite places. Its topology has basic open sets which equal the full unit subgroup outside finitely many places; it is not just the subspace topology of the unrestricted product. The group is locally compact, and the diagonal $K^*$ consists of the <principal ideles>. The <idèle class group> is $C_K=J_K/K^*$.
The finite <valuations> give the <idele-to-ideal homomorphism>
$$
(a_v)\longmapsto\prod_{v<\infty}\mathfrak p_v^{v(a_v)}.
$$
Its kernel is the product of the finite unit groups and the infinite multiplicative groups; <principal ideles> map to <principal fractional ideals>. Thus further quotienting gives the <ideal class group>. The product of normalized local moduli defines $|a|_J$; at complex places the normalized modulus is the square of the ordinary absolute value. The <product formula> says that <principal ideles> have modulus one. These constructions package <ideals>, units and archimedean conditions into one group.
Here is the precise reciprocity equivalence. For a finite <abelian extension> $L/K$, the global law is a continuous <Artin reciprocity map>
$$
\theta_{L/K}:J_K\longrightarrow\operatorname{Gal}(L/K)
$$
which kills $K^*$ and sends a <uniformizer> <idele> at an unramified place to <arithmetic Frobenius>. Its full class-field-theoretic norm statement is $C_K/N_{L/K}C_L\cong\operatorname{Gal}(L/K)$. In the local formulation, the maps
$$
r_v:K_v^*\longrightarrow D_v\subseteq\operatorname{Gal}(L/K)
$$
are continuous, kill local units when unramified, send a local <uniformizer> to the same Frobenius, are compatible with restriction in towers, and obey
$$
\boxed{\prod_v r_v(a)=1\quad\text{for every }a\in K^*.}
$$
Here $D_v$ is the <decomposition group> of any prime above $v$, independent of that choice because the extension is abelian. The usual local norm theorem identifies $K_v^*/N_{L_w/K_v}L_w^*$ with $D_v$. The principal-idele condition is the essential global compatibility; existence of unrelated local norm isomorphisms alone would not establish global reciprocity.
Given the compatible local maps, define
$$
\boxed{\theta((a_v))=\prod_vr_v(a_v).}
$$
Only finitely many factors are nontrivial: there are finitely many ramified places, and an <idele> is a unit almost everywhere. The product is a homomorphism. Continuity follows by choosing local kernel neighbourhoods at those exceptional places and the full unit groups elsewhere. The boxed compatibility kills $K^*$, so the map descends to $C_K$. At every unramified place it has the required Frobenius value. The local norm maps also show that <idele> norms are killed, since the local component of an <idele> norm is $\prod_{w\mid v}N_{L_w/K_v}(b_w)$. Together with the class-field norm-index statement this gives the global norm-kernel formulation.
Conversely, embed $K_v^*$ in $J_K$ with every other component one, and set $r_v=\theta\circ\iota_v$. Each restriction is continuous and has the required unramified normalization. An open kernel of the finite-valued global map contains a basic neighbourhood whose unrestricted finite-place factors are full unit groups outside a finite set. Hence those local restrictions kill units. For an arbitrary <idele>, remove the finitely many exceptional and nonunit components; the remainder lies in that kernel neighbourhood. It follows directly that $\theta$ is exactly the product of its local restrictions, not merely a map agreeing on single-component <ideles>. Its triviality on $K^*$ is exactly the boxed product relation. With the decomposition and norm assertions of the reciprocity theorem, these restrictions are the <local reciprocity maps> into $D_v$ with kernel $N_{L_w/K_v}L_w^*$.
One can explicitly connect this idelic formulation with the classical <ideal> law without assuming an idelic law in advance. A <modulus of a number field> records finitely many nonnegative exponents at finite places and a selected subset of real places. For a modulus $\mathfrak m$, let $U_{\mathfrak m}$ have factors $1+\mathfrak p_v^{m_v}$ at finite places dividing the modulus, full units at other finite places, positive real numbers at specified real places, and full infinite multiplicative groups otherwise. <Weak approximation for number fields> supplies a global $a\in K^*$ making $a^{-1}j$ lie in the specified local factors at the finitely many modulus places. Its <valuations> outside the modulus define an <ideal> prime to $\mathfrak m$. A different such $a$ changes that <ideal> by a ray-principal <ideal>, and every <ideal> prime to $\mathfrak m$ is obtained. Hence
$$
J_K/(K^*U_{\mathfrak m})\cong I_K(\mathfrak m)/P_K(\mathfrak m).
$$
An <ideal> <Artin map> killing $P_K(\mathfrak m)$ therefore extends to $J_K$ via this isomorphism; restricting it gives the local maps just constructed. Conversely, continuous local maps satisfying the product condition have an open product kernel containing some $U_{\mathfrak m}$ and hence make the <ideal> <Artin map> kill $P_K(\mathfrak m)$. This proves the <idelic local-global reciprocity equivalence> at the reciprocity-law level as well as the formal conversion of maps. The <arithmetic Frobenius> normalization and product relation agree with https://www.jmilne.org/math/CourseNotes/CFT.pdf#page=186[Milne's local-global reciprocity formulation].
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