arXiv:2608.01497 2026 Architecture 1 ideas extracted · analyzed Aug 31, 2026
The paper gives an explicit formal-power-series construction of lower-triangular Sprugnoli operators whose square is an aerated Appell operator, including exact involutions. The transferable asset is not the particular combinatorial arrays, but the ability to parameterize nontrivial sequence-mixing linear maps with an analytically guaranteed inverse equal to the map itself. Such maps can serve as reversible token or channel mixers inside neural networks, enabling constant-memory backpropagation and stable exact undo operations without numerically inverting a learned matrix. The most practical experiment is to insert a truncated involutive Sprugnoli mixer into a small Transformer and compare it with a standard learned triangular mixer and reversible residual baselines at matched parameters and FLOPs.
Replace a learned sequence-mixing matrix with a structured lower-triangular Sprugnoli operator whose square is exactly the identity. Applying the same operator in reverse reconstructs activations exactly, so it can be used as a reversible Transformer mixer or reversible channel permutation while retaining nontrivial long-range mixing.