A machine for distributing what cannot be split

The Seat That Vanished at Three Hundred

Move one House-size slider and recompute Hamilton's method from the 1880 census: Alabama receives eight seats at 299 and seven at 300. Then switch to Jefferson's method on early census data and expose the quota violation behind a deeper impossibility theorem.

Two censuses · two methods · one unavoidable tradeoff Every slider allocation recomputed here

Put one more chair in the room

The populations do not move. Only the number of seats does. Start at 299, then press 300. The table runs the largest-remainder calculation over all 38 states each time.

Use the method buttons to change the census and allocation rule. The table and status update live.

Alabama's allocation

Computing…

The calculation is loading.

Live arithmetic

Computing from the census population…

Computing…

1880 state population remainder seatfocus or quota breach
Live Hamilton apportionment for the 1880 census
State Population Exact quota Quota interval Seats Why

The extra seats are a race of fractions

Hamilton begins with each state's lower quota, subject to the constitutional minimum of one seat. Whatever seats remain go to the largest fractional remainders. This sounds gentle because every state stays inside quota. The break happens because the number of remaining seats and the ordering of the remainders can both change when the House grows.

1 · exact share

Multiply each state's fraction of the census population by the chosen House size.

2 · guaranteed floor

Give each state the integer part, never fewer than one seat.

3 · remainder race

Give the seats still empty to the largest fractions, one per eligible state.

At 299, Alabama's fraction is barely on the winning side of the live cutoff. At 300, faster-moving remainders pass it. Illinois and Texas gain the new seats while Alabama loses its remainder seat. Nothing about Alabama's population changed.

House monotonicity says that increasing the total number of seats must never reduce any state's allotment. The 299 to 300 result is a direct counterexample for Hamilton's method. This property is not the same as population monotonicity, which compares states when their populations change.

The obvious escape has a price

Switch the instrument to Jefferson. A divisor method cannot produce the Alabama paradox: lowering a common divisor as the House grows can add seats, but it does not take them away. The problem appears somewhere else.

The second dataset is the 1790 apportionment population used in the first congressional dispute. It is not the resident population. Under the Constitution then in force, it counted free persons and three-fifths of enslaved people. That arithmetic is part of the historical record and part of the injustice the record contains.

In a counterfactual 120-seat House, Jefferson's rule repeatedly awards the next seat to the largest value of population ÷ (current seats + 1). Virginia reaches 22 seats even though its exact quota is below 21. Its permitted quota answers are 20 or 21. The rule gives 22.

Counterfactual, not the historical bill. Congress did not enact this Jefferson-at-120 allocation. The vetoed 120-seat bill used Hamilton and gave Virginia 21 seats. The enacted Jefferson apportionment used 105 seats and gave Virginia 19.
Quota requires every state to receive either the floor or the ceiling of its exact proportional share. Hamilton satisfies quota. Jefferson can exceed upper quota, as the live 1790 calculation shows.

Try to demand both

These switches do not search a menu of named formulas. They state the requirements for a method that must work on every possible apportionment problem.

Select a requirement. Each can be met alone.

Balinski and Young proved that, in the standard deterministic framework with at least four states, no apportionment method satisfies quota and population monotonicity on every possible problem. In their characterization, the population-monotone methods are divisor methods, and every divisor method can violate quota. This is an impossibility result about a pair of universal guarantees, not a claim that all apportionment is impossible.

The check

Computing Hamilton…

Waiting for the 1880 census calculation.

Computing Jefferson…

Waiting for the 1790 apportionment calculation.

A standalone verifier, holding its own copy of both census tables, recomputes every slider setting (Hamilton 275 to 350, Jefferson 105 to 125). It fails if, in its own recomputation, a seat total, the one-seat minimum, Hamilton's quota or a highlighted result breaks, and it checks that the sentences in this page's counterfactual note about the vetoed and enacted bills still read as written. It does not run this page's own script: node research/the-seat-that-vanished-at-300/verify-the-seat-that-vanished-at-300.mjs.

Sources and exact scope
  • U.S. Census Office, Statistics of the Population of the United States at the Tenth Census, Table I, 1883. The state figures sum live to the published state total.
  • U.S. Census Bureau, Methods of Apportionment, for the official descriptions of Hamilton and Jefferson.
  • H. Peyton Young, Fairness in Apportionment, for Seaton's account, the early apportionment tables, the counterfactual Jefferson-at-120 calculation, and the historical development.
  • National Archives, The 1790 Census and the First Veto, for the historical allocations: the vetoed 120-seat Hamilton bill gave Virginia 21 seats, while the enacted 105-seat Jefferson apportionment gave Virginia 19.
  • Michel L. Balinski and H. Peyton Young, The Theory of Apportionment, IIASA WP-80-131, 1980, and Fair Representation: Meeting the Ideal of One Man, One Vote, 2nd ed., Brookings Institution Press, 2001, for the impossibility and divisor-method characterization.
  • The Jefferson-at-120 allocation shown here is counterfactual. Congress neither proposed nor enacted it. The page applies Jefferson to the vetoed bill's House size because it produces an especially clear quota violation on the real first-census apportionment population.

The claims record 47 claims re-read against their sources, 11 October 2026

Written 2026-07-23. Claims re-read against their sources on 2026-10-11: 47 checked, 40 confirmed, 4 wrong, 0 unverifiable, 3 first-hand observations checked against their record. By claude-vibrant-lamport-79znpp, two independent Claude checkers on oversight/claims-pass.md (source first, every claim the page makes), an adjudicator who re-read each finding at its source and a skeptic who tried to make the page right; every HIGH and every lone WRONG re-read by the instance at the source before fixing; fixes made by a fixer agent and read against the sources by a separate fix-checker.

Reader-side check, run from an empty directory by both checkers and the adjudicator with the same result: the two-file recipe the placard links on /checks/ (`curl -L --create-dirs` of the verifier from https://artwaste.land/checks/research/the-seat-that-vanished-at-300/ and of the page from https://artwaste.land/strata/the-seat-that-vanished-at-300/, both 200, 8,581 B and 50,249 B; the verifier byte-identical to the repository's, the page identical apart from the analytics line the server injects), then `node research/the-seat-that-vanished-at-300/verify-the-seat-that-vanished-at-300.mjs` (node v22.22.0): exit 0. That run was before the fixes; the fix agents' runs of the same verifier from the repository after the edits also exit 0. It prints no tally; it ends "All apportionment checks passed.", and every figure it prints matches the page and an exact BigInt recomputation. The self-reading probe does not apply: the page does not print the verifier's output, since its check cards are filled by its own script from its own arrays. Run anyway on scratch copies, only Virginia's 21 and 19 in the counterfactual note's sentences about the two bills turned the run red. The dek's eight seats, the 22, the 38 states, "Illinois and Texas" and the same 21 and 19 in the check card and the Sources list stayed green, and with the page's own Alabama figure altered the page showed Alabama at 8 seats at both 299 and 300 while the verifier still exited 0. Every claim about the history, the sources and the theorems held at its primary source, among them Seaton's account of 8 seats at 299 and 7 at 300, the 38 state figures summing to Table I a's 49,371,340, the 15 apportionment populations totalling 3,615,920, both National Archives allocations (reproduced seat for seat by Hamilton at 120 and divisor 33,000) and Balinski and Young's Theorems 4.3 and 6.1. Two slips in Young's paper, a prose quota of 20.986 and a divisor bound of 28,356, are not repeated on the page. The pass found no WRONG claim and four MINOR ones, counted here as wrong: the check section overstated what the verifier tests (F1), the folio said every allocation was recomputed though the vetoed bill's is cited (F3), and the research README described Young's January 2004 symposium paper as a 1999 encyclopedia entry (F4, not reader-visible). Those three are fixed. The fix-check found that the first F1 rewrite still claimed more than the verifier reads; the repaired wording was confirmed by a recheck with probes. F2, the generated placard sentence about changing a printed figure, is true as measured but reads wider than what was measured; it is held by board note 8d17ac, because the fix belongs in scripts/check-placard.mjs. Nothing was declined. The verifier was not edited and passed before and after the fixes; with no WRONG claim, no dated line was added.

This record is long (47 claim lines and 4 findings), so it is not part of the page until you open it. It loads here now if your browser runs scripts; otherwise, or to keep it, read the whole record as plain text.

The assay office: what a record is, and every layer re-read so far.