Eruption Column Height as a Test of Plume Rise Scaling Against Observed Plinian Events

This paper addresses a single, sharply posed question: whether the quarter-power plume scaling predicts the column heights of well-observed Plinian eruptions from their independently estimated mass eruption rates. The approach is deliberately narrow. Published values are taken as given, combined through explicit physical relations, and the arithmetic is carried out in a deposited calculation script so that every number reported below can be reproduced exactly. The headline outcomes are that predicted height Pinatubo1991 km = 45.65; ratio pred to obs Pinatubo1991 = 1.141; mean ratio pred to obs = 0.8582. Beyond those figures, the paper sets out how each result responds to the uncertainties in its inputs, which input dominates the error budget in each case, and what observation would most improve the constraint. The intention throughout is that a reader should be able to disagree productively: every assumption is stated in one place, every relation is written down, and substituting different published values requires changing a single line of the calculation rather than reconstructing an argument from prose. Where the result agrees with earlier estimates obtained by different routes, that agreement is noted; where it differs, the source of the difference is traced to a specific input or a specific modelling choice rather than left as an unexplained discrepancy between literatures.