Reflection Positivity under Projection-Type Coarse Graining: Rigidity, Null Escape, and Exact Finite-Scale Obstructions

Reflection positivity is a positivity condition on a reflected half-space, and it is not inherited automatically by covariance pieces produced by coarse graining. We isolate a finite-dimensional obstruction for a common projection-type construction. Let H=H₊₋ carry a unitary reflection Θ exchanging the two halves, let C=C^* commute with Θ, and assume Osterwalder--Schrader (OS) positivity on H₊. For a reflection-symmetric orthogonal projection Π that also preserves H₊, we characterize exactly when the projection-sandwich residual C = C - Π C Π remains OS-positive: this holds if and only if the positive-half projection Π₊ commutes with the OS Gram operator W=P₊Θ C P₊. Equivalently, the mixed positive/negative-time commutator block vanishes. This gives a direct algebraic test for projection-type coarse graining. We then record three consequences. First, in the scalar infinite-lattice setting a finite-range, fine-translation-invariant orthogonal projection is necessarily $0$ or id, so nontrivial local projections must break fine translation invariance or leave this class. Second, a null-escape mechanism shows that a projection which sends an OS-null vector to a strictly positive OS-energy direction forces the residual to violate reflection positivity. For the massive free covariance with bond reflection we identify the OS null cone explicitly and obtain its dimension. Third, a complete 4×4 rational example gives a standalone exact failure certificate with positive original covariance and a negative residual OS direction. A larger audited 8×6 block-harmonic computation is retained only as supplementary provenance because the closure archive does not contain the complete projector table required for independent recomputation. The obstruction is deliberately narrow: it applies to projection-sandwich residuals and does not exclude non-projective Markov kernels, conditional expectations, completely positive maps, nonlinear coarse graining, or continuum constructions.