The Contingency of Persistence: Construction, Null, and Autocorrelation Sensitivity in Topological Inference on Relational Data

A persistence diagram, and any scalar derived from it, is routinely treated as a property of the data it summarizes. That treatment is unsafe. A persistence-derived summary is not a property of a relational system alone. It is a property of a tuple: the system, the construction by which a filtered complex is built from it, the null model against which structure or excess is judged, and the spatial or graph autocorrelation of the filtration values. This article demonstrates three dependency axes on real substrates spanning spatial and non-spatial, physical and social domains. On a groundwater depletion surface, four defensible pipeline constructions move the analysis from no signal to strong signal, and on a citation graph, four filtration constructions move total zero-dimensional persistence across a factor of thirty. On a fixed citation graph, a persistence-derived reserve metric changes sign between a degree-preserving null and an Erdos-Renyi null. A within-substrate control that permutes the spatial positions of one surface while holding its value distribution fixed removes a negative reserve reading entirely, isolating autocorrelation as the driver rather than any intrinsic property of the system. This article proposes a sensitivity index that makes the contingency reportable, gives an algorithmic protocol for constructing the admissible family over which it is computed, and states plainly which of its properties are demonstrated and which are conjectural. The stability theorem for persistence diagrams bounds the effect of perturbing a filtration function. It does not bound the effect of choosing a construction, of choosing a null, or of ignoring autocorrelation. That gap is the subject of this article.