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 $\\mathcal{H}=\\mathcal{H}_+\\oplus\\mathcal{H}_-$ carry a unitary reflection $\\Theta$ exchanging the two halves, let $C=C^*$ commute with $\\Theta$, and assume Osterwalder--Schrader (OS) positivity on $\\mathcal{H}_+$. For a reflection-symmetric orthogonal projection $\\Pi$ that also preserves $\\mathcal{H}_+$, we characterize exactly when the projection-sandwich residual $$\\widetilde{C} = C - \\Pi C \\Pi$$ remains OS-positive: this holds if and only if the positive-half projection $\\Pi_+$ commutes with the OS Gram operator $W=P_+\\Theta C P_+$. 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 $\\mathrm{id}$, 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 $4\\times4$ rational example gives a standalone exact failure certificate with positive original covariance and a negative residual OS direction. A larger audited $8\\times6$ 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.