Structural Obstructions in Two-Dimensional Torsion-Free Weyl Geometry

We identify five exact structural obstructions governing torsion-free Weyl geometry in dimension two.These obstructions are purely algebraic or geometric in nature and hold independently of gauge choices, perturbative schemes, or dynamical assumptions. Quadratic anisotropy obstruction.The quadratic contribution of the Weyl form Q to the symmetric Ricci tensor carries an explicit prefactor (n−2). In dimension two, this factor vanishes identically, forcing the complete disappearance of all anisotropic quadratic Ricci components. Pure-trace symmetric Ricci obstruction.As a consequence, the symmetric Ricci tensor of any torsion-free Weyl connection in dimension two is necessarily of pure trace type. No local symmetric tensorial degrees of freedom can be supported in this sector. Local dynamics obstruction.Any field equation whose principal part is constructed from the symmetric Ricci tensor collapses to a scalar constraint. Hence, no local anisotropic dynamics can arise from symmetric Ricci data in two-dimensional Weyl geometry. Conformal triviality obstruction for exact structures.If the Weyl form is exact, the Weyl connection reduces globally to the Levi-Civita connection of a conformally rescaled metric. Exact Weyl structures therefore carry no local non-metric content. Perturbative blindness obstruction.Perturbative expansions around the Levi-Civita connection fail to detect these constraints: the loss of anisotropic degrees of freedom is non-perturbative and algebraic, and cannot be recovered order by order. The only remaining local invariant beyond trace data is the antisymmetric curvature scalar associated with dQ; all further distinctions between Weyl structures in dimension two are global or cohomological.These five obstructions show that two-dimensional torsion-free Weyl geometry is not a simplified version of higher-dimensional behavior, but a rigid and singular regime in which entire classes of commonly assumed constructions are mathematically excluded.