Bounds, Decompositions and Null Behaviour of KRATOS: A Mathematical Specification of a Recognition-Comparability Diagnostic

KRATOS is a group-structured bibliometric diagnostic that compares the distribution of documents (participation) with the distribution of citation weight (recognition) over a fixed, finite universe of analytical groups. This note gives its complete mathematical specification and derives the properties that govern its interpretation. Beyond bounds and equality conditions for each component, we show that the recognition-alignment score depends on citation data only through ratios of group mean citation rates to the corpus mean; that the composition ceiling of the participation--recognition factor is an affine function of the total variation distance to the uniform reference; and that the factor itself satisfies Fr'echet-type bounds in terms of this ceiling and a participation-weighted recognition score. We establish an exact logarithmic decomposition of the composite index, an ordering between the primary and a reciprocal-symmetric recognition score, invariance properties, and closed-form first and second moments of the recognition ratios under global and stratified permutation nulls. These results explain why composite orderings can be sensitive to small groups, heavy-tailed citation distributions and metadata reassignment. All results are verified numerically with an accompanying script. The specification concerns measurement structure only; it does not define a measure of epistemic change or justice.