THE FOLD / GLITCH / OFF BY ONE / THE ALABAMA
THE ALABAMA
add a seat to the house — a state loses one
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Alabama paradox. Under a natural method for dividing seats among states in proportion to population, adding a seat to the legislature can make a state lose one. It is not a rounding slip — it is a real flaw in Hamilton’s method (largest-remainder apportionment).
Each state gets a fair-share quota = population/total × house size, rounded down; the leftover seats go to the states with the biggest fractional remainders. The trap: growing the house rescales every quota and remainder at once, and a state can have its leftover seat snatched by two faster-rising rivals. It is named for the 1880 U.S. census, where Alabama would have received 8 seats in a 299-member House but only 7 in a 300-member House. The discovery, and its cousins, eventually drove Congress to abandon the method — and Balinski & Young later proved no apportionment method can be free of every such paradox.
LIT verified live: for populations 6, 6, 2, Hamilton’s method gives seats (4,4,2) in a 10-seat house but (5,5,1) in an 11-seat house — state C drops from 2 to 1 while the house grew (window.__alabama). FIG no framing; the paradox is exact arithmetic under the stated method.
Each state gets a fair-share quota = population/total × house size, rounded down; the leftover seats go to the states with the biggest fractional remainders. The trap: growing the house rescales every quota and remainder at once, and a state can have its leftover seat snatched by two faster-rising rivals. It is named for the 1880 U.S. census, where Alabama would have received 8 seats in a 299-member House but only 7 in a 300-member House. The discovery, and its cousins, eventually drove Congress to abandon the method — and Balinski & Young later proved no apportionment method can be free of every such paradox.
LIT verified live: for populations 6, 6, 2, Hamilton’s method gives seats (4,4,2) in a 10-seat house but (5,5,1) in an 11-seat house — state C drops from 2 to 1 while the house grew (window.__alabama). FIG no framing; the paradox is exact arithmetic under the stated method.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in OFF BY ONE — the glitch domain of the count that moves the wrong way. The Alabama paradox is the purest off-by-one in politics: add exactly one seat and a state loses exactly one, the total up while a part goes down. AVAN (AI) built the instrument: the quota/remainder apportioner, the house-size slider, the monotonicity-impossibility inverse.
The weave: David names the seat (add one, lose one); I make growing the house strip a seat from a state and show why no method escapes it — the seat drop in 1D, the live apportionment in 2D, the fairness-vs-monotonicity inverse in 3D. The sphere is the seam. Credit: noticed after the 1880 census (C. W. Seaton); Alexander Hamilton’s method; the impossibility theorem by Michel Balinski & H. Peyton Young (1982).
The weave: David names the seat (add one, lose one); I make growing the house strip a seat from a state and show why no method escapes it — the seat drop in 1D, the live apportionment in 2D, the fairness-vs-monotonicity inverse in 3D. The sphere is the seam. Credit: noticed after the 1880 census (C. W. Seaton); Alexander Hamilton’s method; the impossibility theorem by Michel Balinski & H. Peyton Young (1982).
3 ONE DIMENSION
The seat counts at house size 10 and 11, side by side. Two states climb from 4 to 5; the third falls from 2 to 1 — even though there is now one more seat to hand out. The total rose; a part sank.
4 TWO DIMENSIONS · INTERACTIVE
Populations and their apportionment. Slide the house size and watch the seats update by quota-then-remainder. Cross the threshold and a state visibly loses a seat as the house grows — the quotas and remainders shown so you can see the leftover seat change hands.
5 THREE DIMENSIONS + AVAN’S INVERSE
The seat allocation turning — the green forward result: Hamilton’s method, fair by quota, handing every state close to its exact share.
AVAN’s addition (the inverse-companion): the magenta is the property everyone assumes and the method quietly breaks — monotonicity: more total seats should mean no state ever loses one. Reverse the reasoning and the fault appears — the inverse expectation, ‘growing the whole weakly grows each part,’ is false here, because the leftover seats are handed out by a ranking that the very act of adding a seat reshuffles. And the deep inverse is Balinski & Young’s theorem: the method you actually want — one that stays within each state’s quota and never suffers the Alabama or population paradoxes — does not exist. You may have fairness-to-quota or monotonicity, never both. So the magenta ideal is provably unreachable: every apportionment rule betrays some intuition somewhere. Green is Hamilton’s fair-but-fickle split; magenta is the paradox-free method that cannot be built; and the gap between them is a small, exact, permanent flaw in the arithmetic of representation.
LIT Genuine Alabama paradox (noticed after the 1880 US census by C. W. Seaton; Hamilton's method; impossibility theorem by Balinski & Young 1982). Verified live: for populations 6, 6, 2 Hamilton's method gives seats (4,4,2) in a 10-seat house but (5,5,1) in an 11-seat house — state C drops from 2 seats to 1 while the house grew, with correct totals (window.__alabama.alabamaParadox && sumsCorrect). The paradox is exact arithmetic under the stated apportionment method.
FIG No framing: the seat allocation and the non-monotone drop (house grows, a state loses a seat) are exact and computed in-browser by Hamilton's method. The AVAN inverse is the genuine impossibility content — Balinski & Young proved no apportionment method can both stay within quota and avoid the Alabama/population paradoxes, so the monotone-and-fair ideal is provably unreachable, stated as the established theorem.
FIG No framing: the seat allocation and the non-monotone drop (house grows, a state loses a seat) are exact and computed in-browser by Hamilton's method. The AVAN inverse is the genuine impossibility content — Balinski & Young proved no apportionment method can both stay within quota and avoid the Alabama/population paradoxes, so the monotone-and-fair ideal is provably unreachable, stated as the established theorem.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of OFF BY ONE · David Lee Wise (ROOT0), with AVAN