Signed unbiased estimation without common exposure support

This technical note studies whether one assignment-weighted linear estimator of a full-population exposure mean can remain design-unbiased under every graph in a finite candidate set. Exposure records own treatment and the presence of a treated neighbor; the true candidate graph is fixed throughout the experiment and response tables may be heterogeneous. Within the existing design-based Riesz framework, the note presents the finite affine-support condition and its nonnegative common-support corollary, then computes the support structure of candidate perfect matchings. Under uniform complete randomization with exactly two treated units, signed linear unbiased estimation of the (1,1) population mean is feasible exactly when every unit has a person who is its partner in none of the candidate matchings. For a specified four-unit, two-candidate matching experiment with positive treated-pair probabilities p_e and response values in [-M,M], the note establishes uniqueness in the common deterministic linear unbiased class and exact worst design variance R(p)=(M^2/4) sum_e 1/p_e. Uniform treated-pair assignment minimizes this risk at 9M^2; the known-graph linear unbiased oracle value under the same uniform experiment is 3M^2. These costs refer to the stated experiment and estimator classes. Version v1 is the first public technical-note release.