A deterministic alternative to gerrymandering.

Partisan gerrymandering is structurally entrenched and getting worse. Rucho v. Common Cause (2019) foreclosed federal review of partisan maps. Alexander v. NAACP (2024) and Louisiana v. Callais (April 2026) substantially narrowed Section 2 of the Voting Rights Act, leaving race-conscious districting in a tighter constitutional window than at any point since the VRA's adoption. With federal partisan review off the table and federal racial review constrained, several states have begun redrawing maps mid-decade in response to electoral results rather than census revision. Redistricting is shifting from a once-per-decade event into a continuous partisan exercise.

DualBalance Districting starts from a different premise:

If a human draws the lines, the lines can be drawn to advantage the human's party. The only structurally honest response is to remove the human from the line-drawing.

The algorithm is a pure function of the state's geometry, census populations, and apportioned district count. No tuning knobs, no random seed, no iteration count. Same input → same output, byte-identical, every time. The map updates only with the decennial census.

The U.S. Constitution divides representation along two axes: the House apportions seats by population (Art. I §2); the Senate by sovereign state regardless of population (Art. I §3). Madison's defense in Federalist 54–58 reads this as a principled refusal to collapse representation onto a single dimension.

We don't claim the Senate represents land area; it represents states. What we take from the bicameral compromise is a weaker intuition: within a chamber, the choice to make districts equal in only one extensive quantity — population — is a choice, not a necessity. A second extensive quantity, geographic area, is available and computable from the same data. A district that spans both metropolitan and rural territory forces each representative to answer to constituents across the state's density range, rather than to a single dense metro or a single rural hinterland.

Real states are not uniform in population density: a typical state varies by two to three orders of magnitude between its densest urban tract and its sparsest rural one. Equal population, equal area, contiguity, and compact shape are not simultaneously achievable in general. Some trade-off is unavoidable. The question is which trade-off to make, how transparently to make it, and whether the resulting procedure can be defended as impartial.

The mathematical-gerrymandering literature has produced a large catalog of metrics: Polsby-Popper and Reock compactness, the Efficiency Gap, mean-median, declination, ensemble outlier methods from Duke and the MGGG Redistricting Lab. All of these are forensic: they take an enacted plan as input and ask whether its properties are consistent with maps that some reference process — random redistricting, ensemble sampling — would have produced absent partisan intent. They are differential, comparing the enacted plan against a counterfactual distribution to infer the line-drawer's motive.

A generative metric is a different object. It states, directly, what a district should look like, and is used as the objective the line-drawing procedure pursues, rather than as evidence about that procedure. Population balance is the only generative metric U.S. redistricting law currently uses; Wesberry and Reynolds require each district to hold roughly 1/N of the relevant population. Every other criterion enters the law as a forensic instrument.

This matters because forensic metrics presuppose a line-drawer to investigate. Removing the line-drawer — replacing it with a deterministic generator — preserves their descriptive role but evacuates their inferential content. There is no intent to infer. We need a generative criterion.

The DualBalance Score (DBS) is that criterion:

$$ \mathrm{DBS} = \frac{1}{1 + 0.5,\overline{\mathrm{pop_dev}} + 0.5,\overline{\mathrm{area_dev}}} $$

where pop_dev_d = |Pop(d) − P*| / P* and area_dev_d = |Area(d) − A*| / A*, averaged over districts. DBS reaches 1.0 for a perfectly balanced plan and approaches 0 as deviations grow. It is the objective the algorithm minimizes against, not a test it must pass after the fact.

Iowa's Legislative Services Agency operates under a written procedure that has been in continuous use since 1980: county-aggregating, deterministic, no judicial back-and-forth in recent cycles. A natural question is whether the same procedure would work elsewhere.

Empirically, the algorithm's behavior is sensitive to the input geography. Iowa is unusually homogeneous: 99 counties, no county exceeding the per-district population cap, no metropolitan area large enough to dominate state politics. We re-implemented the procedure (we call it Cascade in this repo to avoid confusion with the agency) and ran it on six states:

The county-integrity priority that produces clean maps in Iowa -- where no county is too large -- produces unconstitutional maps in any state with a county large enough to swallow a district. The Iowa procedure is exemplary for the geographic conditions in which it was designed; on these results, it is not a transplantable template.

A deterministic procedure that hopes to scale to the full set of states needs a different structural commitment.

DualBalance Districting is a three-stage deterministic pipeline. Each stage is a pure function of its inputs; the composition is therefore a pure function of (state geometry, census populations, N, Karcher tolerance T).

The seed step is DualBalance. Three sub-steps, no iteration:

1. Radial seed placement. Compute the population-weighted centroid of the units. Place N seeds on a small circle around that centroid (radius = 0.1% of the bounding-box diagonal) at equally-spaced angles 2π·d/N. Seed 0 points due east; seeds advance counter-clockwise.
2. Capacitated first-fit assignment. Sort all (unit, district) pairs by normalized Euclidean distance ascending; assign each unit to its first district with remaining population capacity P* = total_population / N. Population balance is enforced as a hard cap, not a soft penalty.
3. Contiguity repair. For any district whose dual-graph subgraph is disconnected, dissolve the smaller components into adjacent districts by lowest-cost transfer.

The radial configuration is what carries the dual-balance property: each slice spans both dense (near-center) and sparse (boundary-side) territory. DualBalance alone leaves several percent per-district population deviation — radially well-spread but not yet legally viable. The next two stages close the gap.

Phase 1 — pop tightening. Greedy local search of boundary-unit moves. Each accepted move either reduces the L¹ sum of |pop_dev| or strictly reduces pop_dev_max (max-reducing moves are preferred). When single-unit moves stall in a multi-tied-max local optimum, a length-2 then length-3 augmenting-chain escape — the deterministic analogue of an ejection chain — searches for a transport sequence on the district-adjacency graph that the 1-opt neighborhood cannot express.

Phase 2 — DBS hill-climb. Once Phase 1 converges, picks the boundary move that maximally improves DBS, subject to pop_dev_max not exceeding the value Phase 1 reached.

Phase 1 at VTD scale can stall above Karcher because typical VTDs are too large (avg ~1000–3000 people) to make sub-Karcher pop adjustments. We re-initialize at Census block scale by inheriting each block's district from its containing VTD (a deterministic spatial join), then re-run Phases 1 and 2 at block granularity. Block populations average ~20 people, so the running-max envelope around the Karcher target admits many more candidate moves and Phase 2 has the room it needs to refine area balance.

To make the block-scale stage tractable, contiguity checks are cached. contiguity.py maintains per-district articulation points via Tarjan on CSR adjacency arrays (numba-accelerated when available), reducing each candidate contiguity check from O(V + E) to O(1) and giving a 28× speedup on the per-move cost compared to a networkx baseline.

Six-state validation (MN, IA, MA, NC, WI, TX), block-scale, refinement from VTD-Karcher init at 0.05% tolerance:

(✓ = below 0.05% Karcher line.)

Headline reads:

• 5 of 6 beat enacted on DBS. WI is the one loss.
• 5 of 6 beat enacted on pop_dev_max. Only IA's enacted plan is tighter; we sit exactly at Karcher.
• More partisan-fair than enacted in 4 of 6 (efficiency gap smaller in magnitude on IA, NC, WI, TX; tied on MA, MN).
• More minority-majority districts than enacted on NC (2 vs 1) and TX (22 vs 19).

See out/comparison_block.png, out/comparison_race.png, and out/comparison_partisan.png for the side-by-side plots vs Cascade and enacted.

By design:

• Does not consider political parties
• Does not consider race or demographics
• Does not preserve "communities of interest"
• Does not optimize for competitiveness
• Does not iterate, tune, or anneal

These are all sources of human interpretation — and therefore bias. The procedural neutrality is symmetric: the algorithm cannot be tuned to advantage any group, and equally cannot be tuned to help any of them.

pip install -e ".[dev]"
pytest                                                # 106 tests, ~2s

# Per-state PoC end-to-end (any state in dualbalance.states.STATE_INFO)
python scripts/prep_state_units.py --state MN
dualbalance generate --config configs/mn_vtd.yaml     # DualBalance core, raw
python scripts/compare_state.py --state MN            # vs Cascade, BDistricting, enacted

# Full block-scale refinement pipeline (the headline numbers)
python _test_block_from_vtd.py MN 8                   # VTD-Karcher -> block

# All 50 states in one resumable loop
python -u _run_all_states.py

DualBalance includes a deterministic apportionment step that assigns each state its district count using the Method of Equal Proportions (the U.S. standard since 1941):

priority(s, n) = population(s) / sqrt(n(n+1))

The two-level system:

• National level → population-based apportionment across states
• State level → DualBalance districting within each state

For each run:

• map.geojson — one feature per atomic unit with district_id
• metrics.json — DBS, per-district populations and areas, compactness, partisan and race diagnostics; deterministic and byte-identical for repeated runs

dualbalance score --plan some_plan.geojson --units state_units.geojson --geography vtd

• Population × area cannot both reach exact balance on real US geography. Density varies by ~300× between urban cores and rural counties, so any contiguous-unit partition has a hard floor on area_dev_mean. DBS reaches 1.0 only for uniform-density states.
• Compactness scores will be lower than the hand-drawn norm. Radial slices are deliberately not blob-shaped. Courts have used compactness in race-based gerrymandering cases (Shaw v. Reno and progeny), but those cases turn on intent, not geometry per se. A race-blind, content-neutral generator does not carry the intent that triggers Shaw.
• Section 2 of the VRA has been substantially narrowed by Alexander (2024) and Callais (2026), but is not formally struck down. A race-blind generator may not automatically satisfy Section 2 in jurisdictions where the Gingles preconditions are met. The scoring harness flags this risk without compromising the generator's content-neutrality.

See docs/legal-standards.md for the case law overview and docs/stuck-state-problem.md for the chain-escape problem statement that motivated the augmenting-path Phase 1.

• Deterministic over stochastic
• Transparent over heuristic
• Reproducible over negotiable
• Generative criteria over forensic ones
• No tuning knobs over many configurable parameters