Source
Robinson (2017), LIT-1658, §4, pp. 22–25 of arXiv 1603.01446v3; read in NOTE-1221.
What was actually shown
The setting. A presheaf S of pseudometric spaces on a topology T over a set X of entities: a space of observations S(U) for each open U, with continuous restrictions S(U ⊆ V): S(V) → S(U) that compose (Axioms 1–5, pp. 8–16). An assignment picks one observation per open set. Assignments are compared by D(a, b) = sup_{U ∈ T} d_U(a(U), b(U)) (Definition 18, p. 23). A global section is an element of S(X).
The radius. An assignment s is ε-approximate if d_V(s(V), S(V ⊆ U)s(U)) ≤ ε for all V ⊆ U, and its consistency radius is the least such ε (Definition 20, p. 24).
Zero radius is consistency. Every global section induces a 0-approximate section by restriction; for metric spaces this is a homeomorphism onto the 0-approximate sections (Proposition 21, p. 24). With only pseudometrics it is not: Remark 22 gives two global sections and four 0-approximate sections (pp. 24–25).
The bound. If the restrictions are K-Lipschitz and a has radius ε, then D(a, s) ≥ ε/(1 + K) for every global section s (Proposition 23, p. 25). The proof is the triangle inequality: d_V(a(V), S(V ⊆ U)a(U)) ≤ d_V(a(V), s(V)) + K d_U(s(U), a(U)) ≤ (1 + K)D(a, s), since s(V) is the restriction of s(U). The printed last line writes d_V for d_U; the conclusion is unaffected. Data fusion, the search for the global section nearest an assignment (Problem 19, p. 23), therefore distorts the data by at least ε/(1 + K): “The consistency radius of the original data places a lower bound on the amount of distortion incurred by the fusion process” (p. 3).
None of this uses Axiom 6, the sheaf condition.
What this does not say
- That the nearest global section is unique. The paper’s Proposition 25 (p. 29) claims a unique solution for sheaves of Banach spaces on finite topologies; its proof only shows that D is a max-norm metric, under which nearest points to a subspace need not be unique. NOTE-1221 gives a two-point counterexample.
- That the radius locates anything. It is a single number. In the paper’s simulation the biased case had a smaller radius than the unbiased one; the bias was found from the pattern of the discrepancies (THEORY-1092).
- That a small radius means correct data. Data that agree under wrong restriction maps, or that share a common error, have a small radius. The radius measures agreement under the given maps, not accuracy.
- That fusion improves accuracy. It gives consistency, with a minimum cost; whether the consistent state is closer to the truth is an empirical question the paper answers once, in simulation.
- How to weigh different units. The supremum runs over spaces whose distances may be in km, degrees or hours; the result holds for whatever pseudometrics are chosen, and the choice decides what the radius means.