Modulation of the energy landscape by external perturbations governs various thermally-activated phenomena, described by the Arrhenius law. Thermal fluctuation of nanoscale magnetic tunnel junctions with spin-transfer torque (STT) shows promise for unconventional computing, whereas its rigorous representation, based on the Néel-Arrhenius law, has been controversial. In particular, the exponents for thermally-activated switching rate therein, have been inaccessible with conventional thermally-stable nanomagnets with decade-long retention time. Here we approach the Néel-Arrhenius law with STT utilising superparamagnetic tunnel junctions that have high sensitivity to external perturbations and determine the exponents through several independent measurements including homodyne-detected ferromagnetic resonance, nanosecond STT switching, and random telegraph noise. Furthermore, we show that the results are comprehensively described by a concept of local bifurcation observed in various physical systems. The findings demonstrate the capability of superparamagnetic tunnel junction as a useful tester for statistical physics as well as sophisticated engineering of probabilistic computing hardware with a rigorous mathematical foundation.