Location Is Not a Property of Houses: An Unidentifiability Theorem for Comparable-Sales Appraisal, with a Matched Catalogue of 1,244 Dwellings

Comparable-sales appraisal values a dwelling as a weighted average of nearby sales, corrected line by line for physical difference. Written as an operator equation this is $v=Wv+\delta$, where $W$ is the comparable-selection matrix and $\delta$ collects the physical adjustments. Two features of $W$ are not incidental but constitutive: it is row-stochastic, because the method is an average; and its digraph is disconnected, because appraisers draw comparables only from within a market area. Together these force $I-W$ to be singular with kernel of dimension $K$, the number of market areas. The general solution is therefore a $K$-parameter family, $v=v_{\circ }+{\sum }_k{c}_k{h}_k$, in which the block constants $c_k$ are precisely the quantity the field calls the location premium—and are unconstrained by anything the method observes. We show further that $\delta$ is invariant under this family, so no additional appraisal, however careful, can narrow $c$. Three studies follow. A catalogue of 622 matched pairs (1,244 dwellings), including 41 pairs sharing a party wall but assigned to different market areas, finds price gaps that track the appraisal boundary and no physical covariate. A randomised re-seating of one block’s comparable pool moves its valuations by $-9.2\%$ and its transaction prices by $-8.6\%$, with no reversion over eighteen months. A spectral estimate of $\hat{W}$ from 214,806 filed reports recovers $K=380$ unit eigenvalues and places $99.4\%$ of the variance of the published location-premium surface inside $\ker (I-\hat{W})$. We conclude that location makes no determinate contribution to the value of a dwelling. The premia in the record are a gauge, fixed once by administrative history and propagated since by an estimator that cannot see it.