An ideal horizontal spring with spring constant \(k\) is attached to a block of mass \(m\) on a frictionless surface. At time \(t = 0\), the block is at displacement \(x_0 > 0\) and moving with velocity \(v_0 > 0\), resulting in simple harmonic motion described by \(x(t) = A\cos(\omega t + \phi)\), where \(\omega = \sqrt{k/m}\). An experimenter observes that scaling both \(x_0\) and \(v_0\) by a positive factor \(\beta\) scales the resulting amplitude \(A\) by \(\beta\) and leaves the phase constant \(\phi\) unchanged, whereas scaling only \(x_0\) by \(\beta\) changes both \(A\) and \(\phi\). Which of the following statements provides the correct physical explanation for these observations?