Higher-order interactions—simultaneous couplings among three or more dynamicalunits—produce collective phenomena such as explosive synchronization, multistability,and hysteresis that are absent in pairwise-coupled networks. We prove that these phe-nomena arise from a formal mathematical correspondence between higher-order networkdynamics and classical Landau theory of phase transitions. Specifically, we show thatthe Ott-Antonsen reduced equilibrium equation for the higher-order Kuramoto model isstructurally identical to the critical point condition of a Landau free energy functional, withthe effective order of each interaction term mapping to the degree of the correspondingLandau expansion term. On this basis, we establish two main results. First, an interactionof effective order 𝑝 generates a term of degree 2𝑝 in the Landau free energy (Theorem 1).Second, higher-order interactions of effective order 𝑝 ≥ 2 are necessary and sufficient to ac-tivate subcritical (discontinuous, first-order) bifurcations; pairwise-only coupling is confinedto supercritical (continuous, second-order) transitions (Theorem 2). We derive a generalformula for the maximum number of coexisting stable states as a function of interactionorder and show that the correspondence is preserved through the entire phase reductionchain from Hopf bifurcation to the macroscopic order parameter. These results reframe thediverse phenomenology of higher-order network dynamics as the progressive activation oflatent bifurcation modes within the Landau-Stuart framework, with implications for social,biological, and ecological systems where polyadic interactions dominate.