Brazil

The Chamber of Deputies without its constitutional floor and ceiling.

No districting algorithm is needed here and nothing is estimated: the constituency is the state, the states are not going to move, and the electoral authority publishes the votes. The whole question is the arithmetic.

Article 45 of the 1988 Constitution makes Chamber seats proportional to state population and then contradicts itself: no state may have fewer than 8 seats or more than 70. Nine elections from 1990 to 2022 are recomputed with the same votes, the same chamber size and the same within-state rule, changing only that.

São Paulo goes from 70 seats to 112; Roraima, Amapá and Acre from 8 to 2. On the 1991 census the distortion is larger still, 16.6× falling to 1.5×.

There are in fact two distortions, and they are usually discussed as one. The seat table has not moved since 1993, so it is not the Constitution's rule but the rule's 1993 output, left in place for thirty years — Pará should have 21 seats on the 2010 census and has 17. Recomputing the constitutional rule itself on current population disagrees with the real table in eleven to fourteen states at every census. The freeze redistributes unfairness among the middle-sized states; the floor and ceiling create it at the extremes.

And it barely touches the party system

This is the result worth arguing about. Brazil's apportionment is an enormous violation of one-person-one-vote and is nonetheless only a minor source of partisan disproportionality.

The over-represented states are not a partisan bloc, so their gains and losses largely cancel. São Paulo's 42 extra seats move a governing coalition by about four, because São Paulo is not a bloc either: it is less centrist-for-hire than the country but more polarised, running above the national share for both government and opposition, so its extra seats split between the two sides.

What does drive disproportionality is district magnitude. Fourteen of the 27 units elect ten members or fewer, and an electoral quotient in an eight-seat district is 12.5% of the vote — a threshold far above anything Brazil imposes explicitly. Small parties are excluded by arithmetic, not by rule. The floor and ceiling are an equality-of-voters problem, not a proportionality problem, and anyone claiming the cap on São Paulo distorts the party system has the numbers against them.

What it does change is what the chamber can pass

Party composition barely moves, and every president since 1990 clears both the majority and the three-fifths bar under either apportionment. But individual roll calls split regionally even when parties do not, and those are exactly what the apportionment decides. Of 4,318 plenary divisions since 2003, 56 fall within ten votes of the threshold that applied to them; reweighting those, 16 are more likely than not to have gone the other way.

The clearest case is a constitutional amendment on charging tuition at public universities, rejected 304 to 139 in March 2017 — it needed 308 and fell four short. Reweighted, its expected yes count is 314, with a 95% probability of clearing the bar. The mechanism is visible directly: São Paulo gains 41 seats and voted yes at 0.70 per seat, while the states losing seats voted yes at 0.12 to 0.25. This is a real regional split, not an artifact of resampling.

And the one everyone asks about: the amendment lowering the age of criminal responsibility to 16, rejected five short of the bar in June 2015, reweights to a coin flip — 57%. The honest statement is that a chamber apportioned by population would have been on a knife edge, not that it would have passed.

Checked: the 1990 and 1994 seat tables reconcile row by row and column by column against their published totals, and the two independent vote compilations used agree on total valid votes to within half a percent — to 0.000% in 1998. Run with the real apportionment, the reweighting reproduces every tally exactly.