  The Wilson-Fisher criticality provides a paradigm for a large class of phase
transitions in nature (e.g., helium, ferromagnets). In the three dimension,
Wilson-Fisher critical points are not exactly solvable due to the
strongly-correlated feature, so one has to resort to non-perturbative tools
such as numerical simulations. Here, we design a microscopic model of
Heisenberg magnet bilayer and study the underlying Wilson-Fisher
$\mathrm{O}(3)$ transition through the lens of fuzzy sphere regularization. We
uncover a wealth of crucial information which directly reveals the emergent
conformal symmetry regarding this fixed point. In specific, we accurately
calculate and analyze the energy spectra at the transition, and explicitly
identify the existence of a conserved Noether current, a stress tensor and
relevant primary fields. Most importantly, the primaries and their descendants
form a fingerprint conformal tower structure, pointing to an almost perfect
state-operator correspondence. Furthermore, by examining the leading rank-4
symmetric tensor operator, we demonstrate the cubic perturbation is relevant,
implying the critical $\mathrm{O}(3)$ model is unstable to cubic anisotropy, in
agreement with the renormalization group and bootstrap calculations. The
successful dissection of conformal content of the Wilson-Fisher universality
class extends the horizon of the fuzzy sphere method and paves the way for
exploring higher dimensional conformal field theories.
