**5. A Counter Example**

In previous papers related to the infinite-state representation, we have already demonstrated that the so-called initial conditions of the Caputo derivative are unable to correctly express the dynamics of FDE/FDS free responses. However, attracted by the apparent simplicity of the Caputo initial conditions, most fractional calculus researchers ignore the more complex (in fact not too complex) infinite-state approach.

Consequently, this paper intends to prove the fundamental errors of the usual Caputo derivative approach using an elementary theoretical counter example. Then, with the frequency-distributed integrator allowing the theoretical computation of the true free response, we prove the necessity to use frequency-distributed initial conditions to solve any FDE/FDS initial-condition problem.
