Numerator, Denominator or Source Ratio A three-term decomposition of retreats in governance projection: a public analysis protocol for existing data

Abstract A political centre projects taxation, military force and administrative orders over its territory, and the reach of that projection both expands and retreats. A retreat of the screening length can come from a fall in transport capacity (the numerator) or from a shortening of the time for which central orders stay effective locally (the denominator); and where the centre’s reach is bounded by its frontier with a competitor, coverage can also shrink because the ratio of the two sources’ strengths changes (the source ratio). This paper registers a public analysis protocol for existing historical data designed to separate these three. It gives a three-term decomposition of the two-source boundary and its algebraic consequence, source-ratio weakening: the numerator and denominator terms are both weighted by the logarithm of the source ratio, so that when the centre and its rival are evenly matched, no change in the screening length moves the coverage boundary appreciably. Classification proceeds in two steps: the first, on the screening-length layer, uses the two-mechanism rule of Numerator or Denominator v3.0; the second, on the coverage layer, apportions the boundary shift between a screening term and a source-ratio term. Two provisions specific to this domain are registered: the main analysis excludes narrative texts compiled by successor regimes, and dissipation produced by the strategic evasion of the governed is flagged rather than modelled. The principal result produced here is a negative simulation: at a standard error of 0.10, as the source ratio between centre and rival falls from 3 to 2, 1.5 and 1.2, recall for the screening-dominated type on the coverage layer falls from 0.56 to 0.28, 0.07 and 0.01, while recall for the source-ratio-dominated type stays near 0.65. The protocol therefore identifies the decline of single-source polities best, and can only describe phases in which several powers contend. The data drawn on were public when this was written, so every test is post hoc and provides exploratory evidence only.