  Singularities arise in diverse disciplines and play a key role in both
exploring fundamental laws of physics and making highly-sensitive sensors.
Higher-order (>3) singularities, with further improved performance, however,
usually require exquisite tuning of multiple (>3) coupled degrees of freedom or
nonlinear control, thus severely limiting their applications in practice. Here
we propose theoretically and confirm using mechanics experiments that, cubic
singularities can be realized in a coupled binary system without any
nonlinearity, only by observing the phase tomography of the driven response. By
steering the cubic phase-tomographic singularities in an
electrostatically-tunable micromechanical system, enhanced cubic-root response
to frequency perturbation and voltage-controlled nonreciprocity are
demonstrated. Our work opens up a new phase-tomographic method for
interacted-system research and sheds new light on building and engineering
advanced singular devices with simple and well-controllable elements, with a
wide range of applications including precision metrology, portable
nonreciprocal devices, and on-chip mechanical computing.
