Stable sheets in four dimensions
Four zipper families in exterior parameters |c₁|, |c₂| > 1
4,096 certified cells around the island probe, magnified in the c₁ plane. The circular C marker in the atlas locates this patch; its symbol size is independent of the patch size.
Axes show c₁ − c₁(C), in units of 10⁻⁶.
Each sheet approximates the nonlinear image of a certified inner cover with a triangular mesh. Color and the weight filter retain the fourth coordinate w; the three-dimensional view projects it out. Blank space carries no classification. Apparent intersections between projected sheets need not be intersections in four dimensions.
Coordinates of this earlier coefficient-phase projection
This September viewer retains its original projection. The mirror contact atlas and 4D explorer use the manuscript’s (ρ, α, γ) chart. The zipper indices describe traversal choices, separately from map orientation.
θ₁ = arg(a), θ₂ = arg(b).
X = (4 + ρ cos θ₁) cos θ₂
Y = (4 + ρ cos θ₁) sin θ₂
Z = ρ sin θ₁
At fixed w and orientation, this chart is injective on the displayed range 0 < ρ ≤ 2. Direct and mixed systems belong to separate orientation charts. Its central hole comes from the display map. The faint grid is the reference torus ρ = 2. The slider selects a finite band w ± 0.025 around the centres of displayed cells.
Three probes, four families
Select a curve to locate itCurve branches are red and blue where a stable certificate is available. The two neutral C examples are included for comparison; they are not part of the displayed stable sheets. These polygonal drawings have a uniform error bound below 0.0051 in the endpoint coordinate.