Temporal Sheaf Diffusion: Memory under Changing Geometry

When relational geometry changes, should temporal memory preserve past comparisons or reinterpret history using current ones? For a prescribed evolving sheaf, we characterize the exact finite-polynomial historical projection through signal moments and a positive-definite Gram state, deriving continuous dynamics, exact held-history updates, and a compressed representation using $2N-1$ operator moments for $N$ temporal modes. A three-node graph example separates historical regularization, current-geometry filtering, and compression of filtered inputs, showing why time-varying pointwise filtering can leave the polynomial space. Synthetic reconstruction shows that the appropriate geometry depends on the meaning of the observed changes, with little benefit from the coupled solve over filter-then-compress, while a compact Temporal Sheaf Diffusion predictor is competitive on temporal knowledge graphs.