Complex Trees · Algebraic curves and their stable strata

Mirror trees: stable arcs,
Cantor contacts and address spaces

The same infinite addresses connect the historical fan, an algebraic contact curve and a stable set in parameter space. Every exterior branch coefficient satisfies |cⱼ| > 1. A contact equation and a proof of stability remain different objects.

Explore the historical fans Exact junction data

From mirror symmetry to the direct connectedness locus

The binary mirror slice has a = re^iθ and b = re^−iθ, with 0 < r < 1. Both maps preserve orientation. It lies in the direct locus M^ff, where f₀(z) = −1 + az and f₁(z) = 1 + bz. In the 4D chart this is w = ½, α = θ, γ = 0, ρ = 1/r.

A contact equation specifies an identification of infinite addresses. A marked stable family also preserves the complete coding relation on a stated domain. The contact atlas distinguishes the equation, its certified stable subset and the numerical drawing. Families with more than two maps have 2n real multiplier coordinates.

The historical canopy uses Fⱼ(z) = qⱼ(1 + z); its finite branch skeleton is a separate drawing. For a ≠ b, translation by 1 followed by H(z) = ((2 − a − b)z − 2)/(b − a) carries the binary canopy to the anchored maps above. On this page the exterior mirror coordinate is c = 1/conj(q), so (c₀,c₁) = (conj(c),c). Notation and coordinate conventions.

Continuous arity-transition curves 2013 figures · reciprocal charts

Stable set in its algebraic chart

Canopy and the prescribed contact

Show the finite branch skeleton

Position in the direct 4D locus

Fourth coordinate: w = 0.3 … 0.7

Open this binary parameter in 4D →

The exterior mirror plane

From a first-level intersection to all addresses

The double-point hierarchy

Common-prefix levels 3

Cantor endpoint depth 5

Magnify the selected cylinder

Contact address

Common prefix u · L/R

Finite bit prefix

Repeating bit word

Hausdorff dimension · double-coded set

Hausdorff dimension · unique-coded set

Every indicated address is infinite. Parentheses mark its repeating tail; the common prefix is the same in both addresses. Exact multiplicity comes from the theorem, not from a residual or a sampled point cloud.

Reading the three layers

Curve: the address identity is exact on the chosen algebraic family. In the binary case it implies connectedness throughout the strict contraction domain.

Stable subset: a polygon proof or terminating common-return certificate excludes every additional cross-address identification. All four real multiplier coordinates may vary along the two-real-dimensional surface.

Rendering: coordinates and finite tips are evaluated numerically. A small residual is not used as a certificate. Unshaded regions have no negative classification.

c₀ = 1/a, c₁ = 1/b
ρ = 2/(|a| + |b|), w = |a|/(|a| + |b|)
α = (arg a − arg b)/2, γ = −(arg a + arg b)/2
X = (4 + ρ cos α) cos γ
Y = ρ sin α
Z = (4 + ρ cos α) sin γ

This is the same coordinate projection used in the 4D explorer and the linked section cutaways. The central display hole is not a theorem about the connectedness locus.

Every historical template has a validated patch

Template	Arity	θ ≈	R ≈

The defining polynomials and rational τ boxes, not the rounded angle labels, specify the patches. For n > 2 the ambient multiplier domain has real dimension 2n; a two-map picture is not substituted for that system.