What a Zero Readout Certifies Zero as the failure locus of retained distinction

rithmetic supplies the symbol 0. It does not, by itself, say what a measurement returning that symbol hasestablished. This paper answers that question for a setting whose primitive is the retained difference—the assertionthat two things are told apart under finite access—and whose readings are finite and rational. In such a setting zerois anomalous: it arrives in the same register as every other value, yet reports that the primitive did not fire. Weshow that a zero readout is not an absence but a classification, and prove which one. For the retained-informationoperator on a positively weighted comparison graph, the zero locus coincides, in both directions, with the locusat which no difference is retained across the declared structure; equivalently, with the fields constant on eachconnected component. The characterisation is machine-checked over the rationals without classical axioms. Twoclarifications carry much of the philosophical weight. The claim is not a redefinition of arithmetic zero, whichremains an ordinary rational and an ordinary constructor of number; we mark the arithmetic value and its fibrewith different symbols throughout. And the class is relative to the declared structure of comparison rather thanabsolute, as an example shows in which the operator returns zero on a field that is neither empty nor pointwisezero, and a second in which it returns zero on a field that is not constant. We locate the position against Frege,structuralism, Spencer-Brown’s unmarked state, pseudometric quotients, operational equivalence, and the standardkernel result for graph Laplacians, stating in each case what is borrowed and what is not.