Methodology

All geometry is measured in the shapefile's current coordinate reference system; the scorer does not reproject it. For district d, Mosaic reconstructs its area and perimeter from precinct areas, exterior edges, and shared boundaries, then calculates:

PP_d = clip(4π × area_d / perimeter_d², 0, 1)
PP penalty = 100 × (1 − mean_d(PP_d))

Approximate Reock selects each precinct's extreme point in 16 evenly spaced directions. For each district it finds the farthest pair among the retained directional extremes and treats half that distance as a radius:

radius_d = max pairwise distance among 16 directional extremes / 2
Reock_d = min(area_d / (π × radius_d²), 1)   when radius_d > 0; otherwise 0
Reock penalty = 100 × (1 − mean_d(Reock_d))

This is deterministic but is not a minimum-enclosing-circle calculation and does not verify that the implied circle encloses the district.

Combined Compactness applies a scorecard to the two mean ratings and averages the resulting penalties. In clipped mode, a rating x maps linearly from penalty 100 at lo to 0 at hi. The bands are PP [0.10, 0.50] and Reock [0.25, 0.50].

B_clipped(x; lo, hi) = 100 − 100 × (clip(x,lo,hi) − lo)/(hi − lo)
Compactness penalty = 0.5 × B(PP) + 0.5 × B(Reock)

Default “unclipped” mode is still bounded. It follows the same line until PP 0.44 or Reock 0.4625, then eases from penalty 15 to penalty 0 at raw rating 0.65, using exponents 3.5 and 5.0 respectively. The chart shows 100 − penalty.

Let M_cd be the population shared by county c and district d, P_c = Σ_d M_cd, Q_d = Σ_c M_cd, and P = Σ_c P_c. For each county, wholly contained districts are combined before taking one square root; partial pieces receive separate square roots:

S_c = Σ_partial d √(M_cd/P_c) + √(Σ_whole d M_cd/P_c)
raw_county = Σ_c (P_c/P) × S_c

The district direction is symmetric: exchange counties and districts to obtain raw_district. The ordinary no-split reference is 1 in either direction. Default mode uses:

penalty = 0.5 × 45 × max(0, raw_county − 1)
        + 0.5 × 45 × max(0, raw_district − 1)

This default penalty is not capped at 100. The optional clipped scorecard maps each direction between a problem-size reference and 1.33 × reference, averages the two 0–100 results, then multiplies by 0.20; its actual range is therefore 0–20. County-Edge Bias is a proposal preference, not part of this score.

Classic Splitting has two independent weighted components, not one combined objective. If touches_c is the number of districts containing any population from county c:

allowance_c = ceil(P_c / max(I,1))
ExcessCount = Σ_c max(0, touches_c − allowance_c)
Excess penalty = 10 × ExcessCount

For the single-county component, with population tolerance t:

min district population = max(I × (1−t), 1)
population-only bound = Σ_c floor(P_c / min district population)
actual clean = number of districts touching exactly one county
Single-county penalty = 10 × (population-only bound − actual clean)

The displayed “maximum feasible” line is only a population arithmetic bound. It does not prove contiguity or simultaneous drawability. The chart shows raw ExcessCount and actual clean districts, not the two optimizer penalties.

For each district, Mosaic measures the absolute fractional difference from ideal population, subtracts the optional safe-harbor band, squares the remaining excess, and sums across districts.

The chart shows maximum and mean absolute deviation in percent, not this penalty.

Alignment is a one-sided reference-district cohesion measure. Let M_ad be the selected weight from reference district a that lies in proposed district d, G_a = Σ_d M_ad, and f_ad = M_ad/G_a. Then:

cohesion_a = Σ_d f_ad²
penalty = 100 × Σ_a G_a × (1 − cohesion_a) / Σ_a G_a

The weight basis is general population by default, or the selected party's votes when party focus is available. An optional restriction includes only reference districts where that party's two-party share is strictly above the threshold (default 0.535). The chart shows 100 − penalty and the minimum reference-district cohesion.

Because the measure is one-sided, merging several intact reference districts into one proposed district is not itself penalized.

Districts are equally weighted. The default directional bound is B=0.20. The chart shows the signed raw fraction rather than a percent or penalty.

Static mode uses strict sᵢ>0.5 for a Democratic win. Democratic wasted votes are (sᵢ−0.5)Tᵢ when Democrats win and sᵢTᵢ otherwise; Republican wasted votes are the complementary losing or surplus votes. The raw gap is Σ(wasted D − wasted R)/ΣTᵢ.

Default Robust mode uses a closed-form expectation:

σEG = √(0.03² + σdistrict²)
qᵢ = Φ((sᵢ−0.5)/σEG)
raw = Σ Tᵢ × ((2sᵢ−0.5) − qᵢ) / ΣTᵢ

Robust EG always uses 0.03 for its statewide swing term; the shared Swing setting does not change it. The default directional bound is B=0.35. The chart shows the signed raw fraction.

Let S(v)=meanᵢ Φ((sᵢ+(v−V)−0.5)/σcombined) be the smoothed Democratic seat-share curve under uniform swing.

Partisan Bias raw = 0.5 − S(0.5)
Bias default bound B = 0.25

Positive Bias favors Republicans under Mosaic's sign convention. Partisan Gini evaluates |S(v) − (1−S(1−v))| on the grid 0.35, 0.37, …, 0.65, integrates it by the trapezoid rule, and uses:

The Gini chart shows this good-low penalty, not the raw area and not a reversed rating.

Default mode compares expected Democratic seat share E_S=mean(pᵢ) with statewide vote share V. It discounts a winner's bonus within |V−0.5| and scales the remaining magnitude against 0.42; the within-bonus multiplier is 0.25.

Inversion risk is the quadrature-integrated probability that the statewide popular-vote loser wins a strict chamber majority. The popular-vote winner switch is smoothed with a normal CDF of scale 0.005. Tied chambers count for neither party.

The chart shows good-high 100−penalty and bad-high 100×inversion risk on the same panel.

kernelᵢ = 1 − 4(pᵢ−0.5)²
cDf = mean(kernelᵢ)

Clipped mode maps cDf=0 to rating 0 and cDf≥0.75 to rating 100. Default mode follows that line through cDf=0.5625 (penalty 25), then eases to penalty 0 only at cDf=1 with exponent 7/3. The chart shows 100−penalty.

Expected Democratic seats E_D = Σᵢ pᵢ
Favor Dem penalty = 100 × (k−E_D)/k
Favor Rep penalty = 100 × E_D/k

Chance of Majority uses a strict majority. For even-sized chambers, a tie belongs to neither party, so Democratic and Republican majority chances need not sum to 1.

Hinge uses the selected party's probability of reaching at least the selected threshold. Both probability objectives use penalty=100×(1−probability)^1.5; charts show 100×probability.

For group g in district d, aggregate selected-universe total U_d and group count G_gd. Defaults are midpoint m=0.44, steepness τ=0.05, and solid reference s=0.55.

share_gd = G_gd / max(U_d,1)
P_gd = 1 / (1 + exp(−(share_gd−m)/τ))
ref_g = max(sigmoid((s−m)/τ), 10⁻⁹)
T_g = (statewide G_g / statewide U) × k
target_g = round(T_g)

Let x=P_gd/ref_g. Hard district credit is min(x,1). Default soft credit is (x⁻⁸+1)⁻¹⁄⁸. Mosaic sums only the largest target_g credits. Displayed Seats and per-group Rating always use hard credit; the optimizer uses soft credit when “Unclipped” is on.

Smart Targets supplies a greedy reference credit D_g and applicability signal. Hard credit is displayed as Opportunity count. An applicable group's rating is 100×min(hard credit/D_g,1). The overall penalty is 100×(1−weighted success), weighting groups by unrounded T_g. It is not the simple average of the displayed group ratings.

Smart Targets is the optional geographic reference used by Electoral Opportunity. Before scoring a plan, it greedily assembles nearby high-share precincts into hypothetical district-sized bundles. The resulting credits determine whether a group is included and what reference its opportunity rating uses.

Mosaic ranks distance between representative points inside precincts. It does not use adjacency, shared boundary, road access, or travel time. Longitude/latitude-like coordinates are flattened by multiplying x by cos(mean latitude); projected coordinates are used as supplied.

ideal = ΣU/k
rounds = max(round(statewide group share × k), 1)
queried neighbors = min(N, max(32, 4 × ideal/mean(U)))
candidate centers = min(N, 600, max(100, 1200 // rounds))

1. Choose the highest-share available precincts as candidate centers.
2. For each center, take the nearest available precincts up to twice ideal population; skip a pool holding less than 0.999 × ideal.
3. Within the pool, add precincts in descending group-share order until group count reaches solid × ideal or total count reaches ideal.
4. Keep the highest-credit bundle, remove its precincts, and repeat through the proportional target.

candidate denominator = max(bundle total, ideal)
candidate share = bundle group count / candidate denominator
candidate credit = min(sigmoid((share−midpoint)/steepness) / solid reference, 1)

applicable when first credit ≥ 0.10
reference credit = sum of greedy credits through the proportional target

When Smart Targets is off or coordinates are unavailable, Mosaic instead bundles precincts from a statewide descending group-share ranking. That fallback may combine distant places.

Limit: neither method enforces contiguity, the run's population tolerance, compactness, county boundaries, complete-plan construction, or legal constraints. Smart Targets is a heuristic comparison reference, not proof that a valid district can be drawn and not a maximum number of drawable districts.

For each real adjacency edge e=(u,v), group weight is the product of the endpoint group shares. Virtual edges receive weight zero.

q_gi = G_gi/max(U_i,1)
w_ge = q_gu × q_gv
A_g = Σ_real edges w_ge
f_g = Σ_cut edges w_ge / A_g
E(k,N) = 0.6972 × (k−1)^0.5150 / N^0.4091
r_g = f_g / E(k,N)

Applicable groups are pooled before calibration with mass √A_g and a weighted root-mean-square of r_g. The calibration is a monotone four-piece map anchored at penalties 5, 55, and 95; its district-count anchors are:

x = max(k−1,10⁻⁹)
R5 = max(1 − 1.1362 × x^(-0.2447), 0.02)
R95 = min(1 + 13.9170 × x^(-0.6368), 20)
Rs = clamp(1, R5+0.05, R95−0.05)

The optimizer uses the calibrated pooled penalty. Each group chart shows 100−calibrated group penalty, so the overall line cannot be reconstructed by averaging group lines.

Mosaic identifies connected group-concentration cores at share thresholds 0.50, 0.35, and 0.20, using only real adjacency. Their multipliers are 1, 0.5, and 0.25. Cores with general population below 20,000 are discarded; qualifying nested cores remain separate records.

P_cd = general population of core c placed in district d
effective pieces N_eff,c = core_pop_c² / Σ_d P_cd²
forced pieces m_c = max(1, ceil(core_pop_c/ideal_pop − 10⁻⁹))
r_c = N_eff,c / m_c
core weight = √core_pop_c × layer multiplier
r_pool = weighted root-mean-square of r_c

The same four-piece calibration shape is used, with district-count anchors:

x = max(k−1,10⁻⁹)
R5 = clamp(1 + 0.000074 × x^1.784031, 0.02, 3)
Rs = max(1 + 0.259077 × x^0.267266, R5+0.05)
R95 = min(max(3.923603 − 3.232672 × x^(-0.688527), Rs+0.05), 20)

The overall calibrated penalty is optimized; group charts show 100−group penalty. This calibration is provisional.

Neighborhood Severance and Community Dispersion use the same map F(r) after calculating metric-specific anchors R5 < Rs < R95:

r ≤ 0:    F = 0
r ≤ R5:   q = min(18R5/max(R95−R5,10⁻⁹),12)
           F = 5(r/R5)^q
r ≤ Rs:   F = 5 + 50(r−R5)/(Rs−R5)
r ≤ R95:  F = 55 + 40(r−Rs)/(R95−Rs)
r > R95:  tail = (R95−Rs)/18
           F = 100 − 5tail/(tail+r−R95)

Therefore F(R5)=5, F(Rs)=55, and F(R95)=95; the upper tail approaches but never reaches 100.