How Linear Is a Transformer Feed-Forward Block? Per-Block Linear Recoverability Is Learned, Not Architectural

We present a systematic empirical study of transformer compression through over 40 experiments on GPT-2 (124M parameters) and Mistral 7B (7.24B parameters). Our analysis covers spectral compression, block-level function replacement, rotation-based quantization, activation geometry, and adaptive early exit. We identify five structural properties relevant to compression. (1) Variance is not importance: high-variance activation directions are approximately 96 percent uncorrelated with predictive directions (measured via CCA), and projecting onto these subspaces preserves over 90 percent of variance while degrading perplexity. (2) Block linearity is conditional: transformer blocks are approximately linear (R^2 ~ 0.95 on GPT-2, 0.93 on Mistral block 31) only under the correct upstream distribution; modifying earlier blocks induces distribution shift that degrades downstream approximations. (3) The reconstruction wall: approaches that factor weights into quantized components amplify errors through cross-terms, making direct quantization strictly superior. (4) Linearity increases with depth: Mistral 7B exhibits a progression from R^2 = 0.17 (block 0) to R^2 = 0.93 (block 31), indicating a division between nonlinear feature construction and linear refinement. (5) Approximately 30 percent of tokens are computationally easy, confirmed via exit heads and KL divergence sensitivity. We demonstrate that single-block linear replacement achieves 34x compression with a 1.71 perplexity increase on the final block of Mistral 7B, while multi-block replacement fails due to residual error accumulation and distribution shift. These findings suggest fundamental limits to static post-training compression and motivate adaptive, per-token computation as a more effective direction.

We show that smooth two-layer feed-forward networks (FFNs) expose an additional structural model extraction channel under a chosen-input raw-output oracle at the FFN branch; consider transformer FFN branches with GELU or SiLU activations under chosen-input raw-output access, without access to parameters, gradients, or internal activations; exploit a second-order leakage channel in which projected input Hessians form different mixtures of the same hidden symmetric rank-one factors induced by the FFN input weights. We formalize resulting Hessian collection as a partially symmetric decomposition to establish conditions for local identifiability and stability to exploit vector-output stencil reuse to reduce the structural query cost by a factor of 16. On independently trained CIFAR-10 vision transformers, only 16 projected Hessians, corresponding to 8193 black-box queries, recover the hidden FFN directions with average absolute cosine alignment above 0.91, with 86.7.1 % of GELU and 91.1 % of SiLU directions exceeding 0.90 alignment. Recovery remains high across independently trained models, repeated extraction runs, and all transformer blocks. The recovered structure supports functional extraction too. Keeping the recovered directions fixed and fitting only the remaining FFN parameters yields high-fidelity substitutes with more than 92 % top-1 agreement, while test accuracy remains within 1.24% and 0.57% of the GELU and SiLU targets. Output rounding and Gaussian noise substantially reduce recovery under a fixed attack configuration, but adapting the finite-difference step restores average alignment to 0.9188 and 0.9081. This is an end-to-end path from black-box second-order observations to hidden FFN-structure recovery and functional replacement. Under the stated oracle model, smooth FFN curvature exposes internal parameter geometry that behavioral fidelity alone cannot reveal.

Attention-only dynamical theories model Transformer residual directions as particles aggregating on a sphere. We extend this framework by incorporating the feed-forward network (FFN) term as a local steering field acting on each token state. The resulting theory predicts that the tangential component of the FFN field is necessary for motion in residual-direction space, that critical residual directions correspond to nonlinear projective equilibria, and that a commutator defect determines when a finite attention--FFN block can be accurately approximated by a parallel, additive flow. Across GPT-2, Pythia, Mistral, and Llama models, the extended theory improves one-step angular prediction relative to an attention-only baseline, with the contribution of the FFN increasing from GPT-2 to Llama-3-8B. Intervention experiments show that retaining only the tangential FFN component preserves most model quality, whereas retaining only the radial component causes performance to collapse. The tangential component also preserves output diversity under aggregation pressure. As a practical application, layers with small commutator defects can be approximately parallelized with only a modest increase in loss, whereas layers with large defects degrade rapidly. These findings support the interpretation of FFN layers as directional steering fields that shape Transformer residual geometry and govern the feasibility of block-level interventions.