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We provide a complete first-order analysis of how cross-entropy training reshapes attention scores and value vectors in a transformer attention head. Our core result is an \\emph{advantage-based routing law} for attention scores, \\[ \\frac{\\partial L}{\\partial s_{ij}} = \\alpha_{ij}\\bigl(b_{ij}-\\mathbb{E}_{\\alpha_i}[b]\\bigr), \\qquad b_{ij} := u_i^\\top v_j, \\] coupled with a \\emph{responsibility-weighted update} for values, \\[ \\Delta v_j = -\\eta\\sum_i \\alpha_{ij} u_i, \\] where $u_i$ is the upstream gradient at position $i$ and $\\alpha_{ij}$ are attention weights. These equations induce a positive feedback loop in which routing and content specialize together: queries route more ","title":"Gradient Dynamics of Attention: How Cross-Entropy Sculpts Bayesian Manifolds","url":"https://arxiv.org/abs/2512.22473","vendor":"arxiv_cs_ai"},"summary":"arXiv:2512.22473v5 Announce Type: replace-cross \nAbstract: Transformers empirically perform precise probabilistic reasoning in carefully constructed ``Bayesian wind tunnels'' and in large-scale language models, yet the mechanisms by which gradient-based learning creates the required internal geometry remain opaque. We provide a complete first-order analysis of how cross-entropy training reshapes attention scores and value vectors in a transformer attention head. Our core result is an \\emph{advantage-based routing law} for attention scores, \\[ \\frac{\\partial L}{\\partial s_{ij}} = \\alpha_{ij}\\bigl(b_{ij}-\\mathbb{E}_{\\alpha_i}[b]\\bigr), \\qquad b_{ij} := u_i^\\top v_j, \\] coupled with a \\emph{responsibility-weighted update} for values, \\[ \\Delta v_j = -\\eta\\sum_i \\alpha_{ij} u_i, \\] where $u_i$ is the upstream gradient at position $i$ and $\\alpha_{ij}$ are attention weights. 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