Torsion-Free Weyl Connections in Dimension n >= 3: Explicit Ricci Decomposition, Anisotropy, and Global Energy Functionals

We present a complete and explicit derivation of the Ricci tensor associatedwith a torsion-free Weyl connection in dimension n >= 3, using fixed geometricconventions and line-by-line curvature expansions. Adopting the Ricci contraction Ric_{mu nu} = R^lambda_{ lambda mu nu}, we showthat the Weyl--Ricci tensor decomposes canonically into three independentcontributions: a Levi-Civita part, a purely antisymmetric linear term given byn (dQ)_{mu nu}, and a purely symmetric quadratic term proportional to(n-2)(Q_mu Q_nu - |Q|^2 g_{mu nu}). This structure highlights dimension two as a sharp rigidity threshold.In n = 2, the Levi-Civita Ricci tensor is necessarily pure trace, and thequadratic Weyl contribution vanishes identically due to the (n-2) prefactor,forcing a complete collapse of the symmetric anisotropic sector.For n >= 3, this collapse no longer occurs: genuine local anisotropy survivesand is encoded in the symmetric part of the Weyl--Ricci tensor. We further derive the associated scalar curvature shift,R^nabla = R^LC - (n-1)(n-2)|Q|^2,and introduce canonical global energy functionals measuring non-metricity andantisymmetric curvature, namely the L2 norms of Q and dQ. All results are purely geometric and rely solely on the defining properties oftorsion-free Weyl connections. No field equations, variational principles, orgauge choices are assumed. The presentation is fully self-contained, with alltensorial identities derived explicitly and conventions fixed once and for all.