The Entropic Bound for Transformers: Why Static Rank Fails and Attention-Native Rank Recovers

How much matrix rank is required to preserve every bounded value output of normalized softmax attention? We study the unrestricted maximum-row-$\ell_1$ approximation rank $r_\varepsilon(A)$, exactly the least rank achieving uniform error over all bounded vector-valued values. Row softmax exposes the intrinsic interaction $C=P_m(\log A)P_N$, whereas invertible $Q/K$ gauges leave $A$ fixed while changing the Euclidean geometry of a chosen query/key factorization. We replace that coordinate-dependent description by a projective residual $q(C-T)$ and an attained factor-radius size $κ(T)$. For every rank-$r$ retained interaction with $τ(T)<\varepsilon$, we prove $r_\varepsilon(A)\le \min\left\{ N,\; C_r\left( 1+\frac{κ(T)} {(\varepsilon-τ(T))^2} \right)^{r/2} \right\},$ with the same unknown dimension constant as the underlying weighted Gibbs-row cover. The profile is gauge invariant, termwise no worse than native retained-subspace bounds at the same declared dimension, and has a worst-case sharp $r/2$ size exponent at fixed $r$ and $\varepsilon$. We then measure $r_\varepsilon(A)$ directly on learned attention using 9,978 certified brackets across BERT, GPT-2, Qwen2.5, and two ViT checkpoints; where certificates do not close, the optimum remains interval-valued. A pre-specified 2,302-cell held-out study further shows that the historical native-coordinate geometry block contains coarse, mostly head-level information but no detectable incremental information beyond a strong calibrated baseline. The new intrinsic descriptor is not evaluated in that study. Together, the theory and measurements distinguish an operator-intrinsic complexity control from a stronger empirical explanation that the learned-head evidence does not support.

A widely cited result by Dong et al. (2021) showed that Transformers built from self-attention alone, without skip connections or feed-forward layers, suffer from rapid rank collapse: all token representations converge to a single direction. The proposed remedy was the MLP. We show that this picture, while correct in the regime studied by Dong, is incomplete in ways that matter for architectural understanding. Three results are established. First, layer normalisation is precisely affine-rank-neutral: it preserves the affine rank of the token representation set exactly. The widespread claim that LN "plays no role" is imprecise; the correct statement is sharper. Second, residual connections generically obstruct rank collapse in real Transformers such as BERT-base, in a measure-theoretic sense, without contribution from the MLP. The MLP's irreplaceable function is different: generating feature directions outside the linear span of the original token embeddings, which no stack of attention layers can produce. Third, a phenomenon distinct from rank collapse is identified: head-channel non-identifiability. After multi-head attention sums per-head outputs through the output projection, individual contributions cannot be canonically attributed to a specific head; n(H-1)d_k degrees of freedom per layer remain ambiguous when recovering a single head from the mixed signal. The MLP cannot remedy this because it acts on the post-summation signal. A constructive partial remedy is proposed: a position-gated output projection (PG-OP) at parameter overhead below 1.6% of the standard output projection. The four collapse phenomena identified in the literature -- rank collapse in depth, in width, head-channel non-identifiability, and entropy collapse -- are unified under a symmetry-breaking framework, each corresponding to a distinct symmetry of the Transformer's forward pass.

All Transformer-based large language models compute attention via the Euclidean inner product, an architectural choice that Dong et al. (2021) proved causes representational rank to decay doubly exponentially with depth in pure self-attention stacks. We develop a theoretical framework that targets this structural limitation at the mathematical level by replacing the flat Euclidean metric with learned per-token Riemannian metrics. Our contributions are threefold. (1) We prove that Riemannian attention scores with heterogeneous per-token metrics are non-Gram---they cannot be factorized as QK^T with factorization dimension O(d). We are explicit that this is a structural observation, not a proof of rank preservation. (2) We establish that low-rank metric factors render all geometric operations tractable: geodesic distance in O(dr) per token and metric inversion in O(dr^2) via the Woodbury identity---both far below the O(d^3) cost of a general matrix---making Riemannian attention feasible at billion-parameter scale with negligible overhead. (3) We present the Fiber Bundle Transformer, a complete architecture specification in which each token position carries its own Riemannian metric, attention is geodesic distance computation, feed-forward updates use metric-preconditioned steps, and the connection carries explicit curvature and torsion proxies. We derive formal predictions about correctly implemented geometric architectures and identify the central open problem: proving or disproving that heterogeneous Riemannian metrics prevent the rank collapse that row-stochastic attention matrices otherwise cause. This paper presents theoretical analysis and architectural design; empirical validation is the subject of future work.