Projective Trace Residues in a Palatini Trace Sector: A Local Conditional Classification

Projective freedom is a standard feature of the trace sector in Palatini and metric-affine gravity. This note studies a narrower question: after a quotient-adapted trace reduction, does a projectively invariant trace residue remain inside a selected local operator class? In four dimensions the adopted law is the last-index projective transformation \(\delta\Gamma^{\lambda}{}_{\mu\nu}=-\delta^{\lambda}{} {\nu}\xi_{\mu}\), with compensator shift \(\delta A_{\mu}=+3\xi_{\mu}\). The induced raw trace shifts are \(\delta(T_{\mu},Q_{\mu},\widetilde Q_{\mu},A_{\mu})=(3,-8,-2,3)\xi_{\mu}\). The three horizontal one-forms \(q_{\mu}=Q_{\mu}+\tfrac{8}{3}A_{\mu}\), \(u_{\mu}=\widetilde Q_{\mu}+\tfrac{2}{3}A_{\mu}\), and \(t_{\mu}=T_{\mu}-A_{\mu}\) are coordinates on the quotient of the extended trace-plus-compensator space \((T,Q,\widetilde Q,A)\), together with the vertical coordinate \(V_{\mu}=A_{\mu}\). They are not coordinates on the intrinsic quotient of the uncompensated triple \((T,Q,\widetilde Q)\) alone. The one-form \(A_{\mu}\) is a nondynamical spurionic reference along the projective fibre. The analysis uses only the compensated residue \(t_{\mu}\), in the restricted pure-trace density \(f_{1}(\phi)R(\Gamma)+f_{2}(\phi)\,t_{\mu}t^{\mu}+f_{3}(\phi)\,\bar\nabla_{\mu}t^{\mu}+U(\phi)\). On a connected local region where \(f_{1}\neq 0\), elimination of \(q_{\mu}\) and \(u_{\mu}\) from the Palatini block is a Schur decoupling: \(S_{t}=2f_{2}\), \(J_{t,\mathrm{eff}}=-\partial f_{3}\), and therefore \(\Delta_{2}=\Delta_{3}=0\), \(\widehat f_{2}=f_{2}\), and \(\widehat f_{3}=f_{3}\). No \(f_{1}\)-dependent \(t^{2}\) term or \((\partial f_{1})\cdot t\) source is induced. Strict removal means that the resulting residue expression vanishes identically, without solving for \(t_{\mu}\) and without imposing an additional scalar constraint. On the relevant connected local branch this holds if and only if \(f_{2}(\phi)\equiv 0\) and \(f_{3}'(\phi)\equiv 0\). The statement is local, conditional, and confined to this compensated operator class.