The statistical essence of the Transformer architecture has long remained elusive: Is it a universal approximator, or a neural network version of known computational algorithms? Through rigorous algebraic proof, we show that the latter better describes Transformer's basic nature: Ordinary Least Squares (OLS) is a special case of the single-layer Linear Transformer. Using the spectral decomposition of the empirical covariance matrix, we construct a specific parameter setting where the attention mechanism's forward pass becomes mathematically equivalent to the OLS closed-form projection. This means attention can solve the problem in one forward pass, not by iterating. Building upon this prototypical case, we further uncover a decoupled slow and fast memory mechanism within Transformers. Finally, the evolution from our established linear prototype to standard Transformers is discussed. This progression facilitates the transition of the Hopfield energy function from linear to exponential memory capacity, thereby establishing a clear continuity between modern deep architectures and classical statistical inference.

Deep learning-based methods have recently been established as fast and accurate surrogate simulators for optical multilayer thin film structures. However, existing methods only work for limited types of structures with different material arrangements, preventing their applications towards diverse and universal structures. Here, we propose the Opto-Layer (OL) Transformer to act as a universal surrogate simulator for enormous types of structures. Combined with the technique of structure serialization, our model can predict accurate reflection and transmission spectra for up to $10^{25}$ different multilayer structures, while still achieving a six-fold degradation in simulation time compared to physical solvers. Further investigation reveals that the general learning ability comes from the fact that our model first learns the physical embeddings and then uses the self-attention mechanism to capture the hidden relationship of light-matter interaction between each layer.