What it says about democracy
The dilution you can settle by subtraction
Add up the population of each of the enacted plan’s five districts and compare it to the average. That average, the county’s population divided by five, is the ideal population of 32,727, and a district’s deviation is how far above or below it the district sits, as a percentage of the ideal. District 3 is the largest at 33,090 people, which is 363 over the ideal, or +1.11%. District 4 is the smallest at 32,550, or -0.54%. The distance between those two, 1.65 points, is the plan’s maximum deviation, and it is the number a court asks for.
Total spread 1.65 points; the shaded band is the 10 points a legislature drawing state or local districts is presumed to be within.
Figure 1. The enacted plan against equal population. One bar per Board of Education district, running left or right of the ideal of 32,727 people. Bars from a zero line fit this question, because the question is distance from a single number in two directions. The shaded band is the 10-point maximum deviation a state or local plan is presumed to be within.
Reynolds v. Sims, 377 U.S. 533 (1964) required that legislative districts hold substantially equal populations,4 and Brown v. Thomson, 462 U.S. 835, 842 (1983) collected the rule that a state or local plan under a 10% maximum deviation is a minor one and does not by itself make a case.5 This plan spans 1.65 points, a sixth of that allowance. Read as a price, a vote in the smallest district is worth 1.66% more than a vote in the largest, because one seat is shared among 32,550 people there and 33,090 here. That is the entire malapportionment inquiry, done in one subtraction.
If it is that easy, what was the rule for? Until 1963 Georgia nominated statewide officers under the county unit system, in which counties cast two to six unit votes each regardless of size.6 Gray v. Sanders, 372 U.S. 368, 381 (1963) struck it down, and gave the rule its name: political equality “can mean only one thing — one person, one vote.”7
| Quantity | Value |
|---|---|
| People per unit vote, largest county | 177,785 |
| People per unit vote, smallest county | 780 |
| Ratio between them | 228 to 1 |
| Counties that together hold a unit-vote majority | 103 of 159 |
| Share of Georgians living in them | 15.3% |
A vote in the smallest county was worth 228 votes in the largest, and 15.3% of Georgians could outvote the rest. The equal-population rule worked so completely that it stopped being where the argument is.
The dilution you cannot settle by subtraction
The argument moved to the other kind. Section 2 of the Voting Rights Act, 52 U.S.C. § 10301(b), makes out a violation where a group has “less opportunity than other members of the electorate to participate in the political process and to elect representatives of their choice.”8 Thornburg v. Gingles, 478 U.S. 30, 50–51 (1986) turned that into three things a plaintiff must show first, now called the Gingles preconditions.9 The group must be large enough and geographically compact enough to be a majority in one district; it must vote cohesively, meaning largely for the same candidates; and the majority must vote as a bloc against it, so that the group’s candidates usually lose.
Preconditions two and three are claims about elections that happened. Precondition one is a claim about a district that did not, and it is the only one a census file can answer.
Its arithmetic half is quickly done. Each of five districts holds 24,424 voting-age people, so a bare majority takes 12,212 Black adults, and the county has 39,605. “Sufficiently large” is satisfied 3.24 times over. The other half, which Gingles called “geographically compact,” is a fact about where those adults live.
All 3,269 census blocks, shaded by the Black share of their voting-age population; the 875 hatched blocks contain nobody at all. Red lines are the five enacted districts. The county’s index of dissimilarity is 39.9.
Figure 2. Where the Black adults are. All 3,269 census blocks, shaded by the Black share of their voting-age population, with the five enacted districts in red. The 875 hatched blocks contain nobody at all, which is the absence of the quantity rather than a low value of it.
There is no single Black neighbourhood to enclose. The index of dissimilarity, meaning the share of Black adults who would have to move blocks for the two groups to be spread identically, is 39.9.10 Only 306 of 3,269 blocks are majority-Black among adults, holding 36.4% of the county’s Black adults, so the other 63.6% live where they are outnumbered.
So build the district and find out. Start at the block with the most
Black adults and annex neighbours until the district holds 32,727
people. At each step the mapmaker looks at the
reach nearest available blocks on the
frontier, the blocks touching the district so far, and takes whichever
leaves the Black share of adults highest. At reach = 1 one
candidate is ever considered, so race never enters the rule for growing
the district; at reach = ∞ every frontier block is a
candidate.
Each point is one setting of the dial. Hover for the district it produced.
Figure 3. What the dial buys, and what it costs.
Each point is one setting of reach, plotted by the
compactness of the district it produced and the Black share of that
district’s adults. Polsby-Popper is the
area-to-perimeter score: 1 for a circle, near zero for a long ragged
edge. The triangle is the ceiling reached by dropping contiguity
altogether.
The dial buys Black share and pays in compactness, at a brutal rate.
At reach = 1 the district is 41.3% Black among adults with
a Polsby-Popper of 0.324: a respectable shape, nowhere near a majority.
The first setting that clears 50% is reach 40, and its compactness is 9
times worse. At the far end the district reaches 60.0% at 0.008, which
is not a shape so much as a route.
Two points deserve their own story. One setting is out of order: at reach 2 the district drops to 41.1%, below several tighter settings, because a greedy search, one that takes the best-looking next step without looking further ahead, is not guaranteed to improve as its own parameter grows. And the starting block barely matters, since restarting from each of 306 majority-Black blocks moves the answer only 2.1 points.
The honest summary an expert could write from Figure 3 is that a majority-Black district can be built here only by abandoning compactness. That summary is wrong.
The map already exists
The enacted plan has been in force this whole time. Put its District 4 on the same two axes.
| District | Black of adults | Polsby popper | Reock |
|---|---|---|---|
| Algorithm, reach 1 | 41.3% | 0.324 | 0.766 |
| Algorithm, reach 40 | 51.1% | 0.036 | 0.336 |
| Algorithm, reach unlimited | 60.0% | 0.008 | 0.284 |
| Ceiling (contiguity dropped) | 67.1% | 0.006 | 0.056 |
| ENACTED District 4 | 53.7% | 0.266 | 0.514 |
District 4 is 53.7% Black among adults at a Polsby-Popper of 0.266. It is more Black than every contiguous district the algorithm drew below reach 40, and more compact than every one at or above it. 0 of the algorithm’s 15 districts beat it on both measures at once.
The same county three times. The enacted district is both more Black and more compact than the algorithm’s unlimited-reach attempt.
Figure 4. The same county, three times. Left, the
district the algorithm built at reach = 1. Centre, the one
it built with the dial turned all the way up, a route rather than a
shape. Right, the district Georgia enacted, which is both more Black and
more compact than either.
People drew that one, out of the whole county rather than outward from a seed. Exhibiting a map settles the first precondition for good; failing to find one settles nothing, because no search is ever exhaustive. That asymmetry, rather than the arithmetic, is what the fighting in these cases is about.
Majority of what?
Bartlett v. Strickland, 556 U.S. 1, 19–20 (2009) reads the precondition to require a real numerical majority, greater than 50 percent of voting-age population.11 District 4 clears 50% on all three counts. Take the four-district configuration, where it does not.
| Reach | Of population | Of adults | Of registered voters | Majority |
|---|---|---|---|---|
| 1 | 48.2% | 44.3% | 50.4% | on registered voters only |
| 4 | 49.6% | 45.6% | 50.7% | on registered voters only |
| 12 | 50.7% | 46.8% | 51.7% | on registered voters only |
| 24 | 52.2% | 48.3% | 53.2% | on registered voters only |
| 60 | 53.1% | 49.3% | 54.1% | on registered voters only |
| 100 | 53.1% | 49.6% | 53.7% | on registered voters only |
Every one of those districts is majority-Black among registered voters, and not one is majority-Black among voting-age population. The registered share never drops below 50.1% and the voting-age share never rises above 49.6%. Same blocks, same lines, same people, two answers to the question the Court calls dispositive.
Neither number is wrong, and the disagreement is a join. The census counts everybody who lives here; the registration file counts everybody who signed up to vote. The bottom number everything is divided by therefore comes from a different office, on a different date, under a different rule about who counts.
The standard changed four months ago
Louisiana v. Callais, 608 U.S. ___ (2026), decided 29 April
2026, left the three preconditions standing and changed what satisfies
the first.12 A plaintiff now “cannot use race as a
districting criterion” in drawing an illustrative map. That map must
also meet all the State’s legitimate districting objectives “just as
well” as the State’s own (slip op., at 29–30). Every point on Figure 3
except the leftmost was produced by letting race decide which block to
annex. Same county, same census, same code, and a change in doctrine
moves the admissible answer 18.7 points, down to reach = 1
and its 41.3%.