Predictive Geometry of Hidden Trajectories in Transformers

Timur Mudarisov, Mikhail Burtsev, Tatiana Petrova, Radu State

Decoder-only transformers are trained only through a terminal next-token prediction loss, yet this loss constrains every intermediate hidden state through the fixed downstream computation. We formalize this constraint by studying layerwise loss-to-go functions: the terminal loss obtained by continuing a candidate hidden state through the remaining transformer blocks. Around successful validation trajectories, we show that the local second-order geometry of these functions is governed, up to low-loss residual terms, by a pullback Fisher operator on hidden-state space. Its spectrum identifies output-sensitive directions and approximately prediction-null directions, yielding a local observable subspace of the residual stream. For causal transformers, the same geometry induces a tokenwise curvature score: a Fisher-weighted sensitivity of the target logits to perturbations of each token's hidden state. This score vanishes outside the causal ancestor set of the target and is controlled by downstream Jacobian couplings, making it a loss-aware alternative to attention magnitude. We estimate these quantities using matrix-free Jacobian-vector and vector-Jacobian products and evaluate them…