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Take a dipole made up of a set of nodes (i.e. resistors) \$R_1,R_2\ldots,R_n\$ that are connected by vertices (i.e. wires) in any given way. Rayleigh's monotonicity law states that the effective resistance of the dipole, \$R_\mathrm{eq}\$, is an increasing function of each of its variables \$R_1,...,R_n\$; that is to say that increasing any \$R_i\$ will amount to an increase in the effective resistance \$R_\mathrm{eq}\$.

A consequence of this principle is that adding an extra vertex (i.e. a length of wire) within the dipole will always reduce the effective resistance. Indeed, we may consider that the absence of a length of wire is equivalent to the presence of a resistor with infinite resistance. So adding a vertex is equivalent to reducing its resistance from \$\infty \ \Omega\$ to \$0 \ \Omega\$. Hence we may apply Rayleigh's monotonicity principle.

I was wondering if this can be generalised to the norms of complex impedances. That is to say, does adding a vertex within the dipole always reduce the norm of the impedance of the dipole?

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Trying your claim on the simple counter-example below

schematic

simulate this circuit – Schematic created using CircuitLab

proves itself untrue.

Unfortunately that's enough to wipe out the generalization you were after

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