Characteristic-ideal bridge from local units to the main conjecture ($\mathrm{char}(E/C)\mathrm{char}(X)=\mathrm{char}(U/C)\mathrm{char}(Y)$)

In the real cyclotomic tower, the class-field unit sequence gives $0\to E/C\to U/C\to X\to Y\to 0$, where $X$ is the p-ramified Iwasawa module and $Y$ the unramified one. The modules are torsion over each one-variable character component. Splitting the four-term sequence into two short exact sequences proves the displayed identity by multiplicativity of characteristic ideals. The Iwasawa theorem on local cyclotomic units computes $\mathrm{char}(U/C)$ analytically. Thus the main conjecture for $X$ is equivalent to $\mathrm{char}(E/C)=\mathrm{char}(Y)$; local interpolation alone does not prove this remaining global equality.