When Is Softmax Attention a Gradient? Exact Characterization and Conditioning

Energy and potential representations provide structural explanations for the fixed points and dynamics of tied or specially structured attention maps. Fixed-memory softmax attention can be a gradient even when its keys and effective values are untied, but this depends on the additive geometry of the keys. For $F(q)=U^{\top }\mathrm{softmax}(\beta Kq)$, we give a finite necessary-and-sufficient criterion for a global potential by grouping equal key-pair sums. Generic memories force a common scaling and translation of the keys; binary Cartesian memories admit exactly independent coordinate scalings. We characterize constant-metric convex representations and derive two-sided finite-query bounds on the distance to the conservative value space from measured Jacobian asymmetry, with explicit conditioning and observation-error dependence. A near-Cartesian family shows why conditioning is essential: curl tends to zero while both the parameter distance and the best field approximation within the same-key conservative class remain bounded away from zero. Deterministic numerical checks validate the exact characterization and finite-query certificates, and illustrate the predicted loss of conditioning near pair-sum collisions. Finally, an analytic bifurcation construction shows that arbitrarily small incompatible perturbations can produce oscillation near a reference map's stability boundary. These results distinguish exact compatibility, reliable approximate certification, and dynamical stability.