The flow of dense emulsions underlies applications from food and pharmaceutical processing to bioengineering, yet their rheology remains difficult to interpret under conventional volume-imposed conditions. Here, we change the control variable to osmotic-pressure using a recently developed instrument—the Capillarytron. This approach reveals a unified rheological structure where the osmotic pressure Π , by controlling droplet deformation, sets a pressure-dependent jamming volume fraction. When expressed in terms of the distance to this jamming point, all rheological data—spanning both Π - and ϕ -imposed measurements—collapse onto a single power-law divergence, akin to granular suspensions. The resulting constitutive relations provide a predictive, parameter-free description of emulsion rheology across Newtonian, yielding, and shear-thinning regimes. Together with recent results on soft spheres, our findings point to a unifying paradigm: Soft amorphous materials—from soft spheres to emulsions and likely foams—obey the same hard rules as granular suspensions, with softness entering through a pressure-dependent jamming point. This framework rationalizes Herschel–Bulkley rheology, assigns its parameters microscopic meaning, and opens perspectives on rigidity transitions in soft, deformable systems, including biological tissues.