PT-Projected Projective Palatini Gravity: Two-Derivative Operator Basis, Admissible Equivalences, and a Local IR Residual

We study the local infrared content of four-dimensional Palatini gravity in the projective equivalence class, with the observable sector defined by a scalar PT projection. Restricting to a strictly local, curvature-linear two-derivative truncation, we (i) give an explicit and complete basis for all PT-even, projectively admissible bulk scalars in the trace/scalar channel and (ii) define the admissible equivalence relations that preserve the posture, including IR closure of the matter sector and tensorsector locking diagnostics used as auditable admissibility tests. A key structural consequence of full one-form projective invariance, \( \Gamma^\lambda{}_{\mu\nu}\to \Gamma^\lambda{}_{\mu\nu}+\delta^\lambda_{\mu}\xi_\nu \), is the appearance of a projectively invariant residue one-form \( \mathcal{T}_\mu \equiv T_\mu-A_\mu \), where \( A_\mu \) is a compensator transforming as \( A_\mu\to A_\mu+3\xi_\mu \). We then prove a conditional local no-go: within the closed two-derivative operator class and modulo admissible equivalences, there exists no reformulation that removes \( \mathcal{T}_\mu \) from the bulk dynamics in the observable trace/scalar channel while simultaneously (a) preserving IR closure of the minimal matter-coupling posture and (b) preserving the tensor-sector locking diagnostics (in particular luminality). Any attempted bulk removal is necessarily exhausted by a small set of controlled failure modes, including collapse to a trivial residue-free branch, departure from the admissible operator class / IR non-closure, or locking failure. On admissible domains one may restrict to longitudinal representatives, where a scalar \( \epsilon \) parameterizes the physical longitudinal content of \( \mathcal{T}_\mu \); \( \epsilon \) is not a compensator and cannot be eliminated as a pure gauge artifact. We summarize the exclusion logic in a compact diagnostic table and provide a minimal counterexample on standard backgrounds, thereby making the local IR residual (“IR island”) operationally auditable.