LithoSAGE is a spatially adaptive, geometrically equivariant neural operator designed for fast yet accurate lithography simulation. The method learns a continuous spectral weighting using a sum of Gaussian functions. This representation lets the model keep a single set of spectral parameters for any window size, automatically adapting the number of retained frequency modes to the physical width. Because the cutoff in physical frequency is fixed, the operator remains rotation equivariant while the mode count scales with window size. On the ViaSet mask‑to‑aerial image task, LithoSAGE uses only 0.089 million parameters—about 1.9 % of the 4.745 million parameters employed by the classic Fourier neural operator (FNO). Across both ViaSet and MetalSet datasets, LithoSAGE shows the lowest root‑mean‑square error (RMSE) and relative L2 error, and its print‑pattern intersection‑over‑union (IOU) and pixel accuracy (PA) stay above 0.99. The operator’s translation equivariance holds under periodic boundary conditions. When the input is shifted by sub‑pixel amounts that fall between grid points, residual prediction errors arise, indicating that the property is limited to exact grid translations. These results illustrate that a carefully designed spectral–geometric approach can produce a compact surrogate for lithographic simulation while achieving or surpassing the accuracy of larger, less adaptive architectures.