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 associated
with a torsion-free Weyl connection in dimension n >= 3, using fixed geometric
conventions and line-by-line curvature expansions.

Adopting the Ricci contraction Ric_{mu nu} = R^lambda_{ lambda mu nu}, we show
that the Weyl--Ricci tensor decomposes canonically into three independent
contributions: a Levi-Civita part, a purely antisymmetric linear term given by
n (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 the
quadratic 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 survives
and 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 and
antisymmetric curvature, namely the L2 norms of Q and dQ.

All results are purely geometric and rely solely on the defining properties of
torsion-free Weyl connections. No field equations, variational principles, or
gauge choices are assumed. The presentation is fully self-contained, with all
tensorial identities derived explicitly and conventions fixed once and for all.