Stratospheric aerosol injection has emerged as a candidate climate intervention strategy to partially offset global surface warming. Previous work has demonstrated that lower stratospheric heating drives modifications to the stratospheric thermal profile, circulation, and water vapor, with downstream consequences for surface climate; however, the impact of stratospheric heating has not previously been characterized across a multi-model ensemble. This work presents first results from the Stratospheric Heating Model Intercomparison Project (SHeatMIP), in which an idealized 0.3 K/day tropical lower-stratospheric heating tendency is imposed across five climate models (CESM, GFDL, GISS, MIROC, UKESM). All models show robust increases in lower-stratospheric temperature, water vapor, and polar night jet strength in both hemispheres, with corresponding surface shifts in the subtropical and eddy-driven jets and a polar cap pressure response resembling a positive North Atlantic Oscillation phase. Despite qualitative agreement, inter-model spread is substantial, with differences of 1 K in the cold point temperature adjustment and 0.38 K in the global mean surface temperature response. The surface temperature spread strongly co-varies with the stratospheric water vapor response (R² = 0.82), implicating water vapor as a key source of surface warming uncertainty. The forced polar vortex response shows strong co-variability with the climatological polar night jet strength (R²=0.98), and projects onto the surface as a polar cap pressure anomaly. The results show that inter-model differences in the response to stratospheric heating may be traceable to the climatological mean state, offering a pathway toward observationally-constrained evaluation of model suitability for SAI research.