Four cases spread over two thousand years, compared on one page, by one person. Here is what that costs, stated plainly, because a lesson that hides its seams is doing the same thing the orders in it do.
The coordinates are editorial. Every dot in stage four is my reading of a scholarly consensus rendered as a number so it can be plotted, and reasonable historians would move several of them. They are meant to support the claim about the shape and the ranking of the lines, not the second decimal place. The exit scores in stage five are the same kind of object.
The cases are not equivalent objects. Rome was a territorial empire that annexed and administered directly. Britain ran a mixed system of settler colonies, direct rule, indirect rule through princely states, and informal commercial dominance over places it never governed. America built a system of formally sovereign allies, treaty organisations, bases and financial infrastructure, and mostly did not annex. The claimed Chinese order is, at present, finance, contracting and infrastructure with one overseas military base. Putting them in one row is useful for asking the same eight questions and misleading if you read the row as a sequence of the same thing.
There is no clean handoff. The four overlap, decay unevenly, and depend on regional intermediaries who had their own agendas throughout. Britain was building the naval order while the Spanish and Ottoman systems were still going. American primacy took decades to consolidate and rested on institutions built while Britain was still solvent. Anything that looks like a relay race here is an artefact of the diagram.
The mortality figures are contested and I have kept them out of the instrument for that reason. Estimates for the famine deaths under British administration in India, for the Jewish revolts under Rome, and for several twentieth century interventions vary by multiples between serious scholars, and a lesson that plotted them would be presenting a live argument as a settled measurement.
Finally, the fourth name. A term only belongs in this series if a single governing centre actually held a preponderance of force over a wide field for a sustained period, and organised law, trade and security across it. Client kingdoms fail that test, and so do trading diasporas, alliance blocs and religious communities, however often the word pax gets attached to their names by people who want the association more than the evidence. Pax Sinica is here because a governing centre is visibly assembling that apparatus right now, on the record, in public, and because you can watch the legitimating story being drafted before it hardens. That is a rare thing to be able to observe, and it is the best reason to run the comparison at all.