# Connectedness Atlas

> A mathematical research programme and interactive companion by Bernat Espigulé, Universitat de Girona, for four-dimensional connectedness loci of planar binary similarities and stable complex-tree contact families.

Connectedness Loci for Pairs of Planar Similarities and its research companion, Contact Families, Sections and Landmarks, are in preparation. Both titles are provisional. The manuscript PDFs await author review and release approval. The BMD poster is the exact submitted edition of 27 September 2026. The catalogue records these statuses and the actual authors of related works.

## Explore

- [Programme and three-orientation explorer](https://complextrees.com/4D/): DD, DO and OO maps, exact families and linked specimens.
- [Live six-locator atlas](https://complextrees.com/4D/atlas/): full-angle DD survey and movable specimens.
- [DD laboratory](https://complextrees.com/4D/lab/): original 43,173-ray restricted-sector survey and diagnostics.
- [Mathematical guide](https://complextrees.com/4D/docs/): definitions, results and evidence.

## Research and structured resources

- [Manuscript announcements and citations](https://complextrees.com/4D/publication/).
- [Submitted BMD poster](https://complextrees.com/4D/publication/poster.pdf).
- [LaTeX bibitems](https://complextrees.com/4D/publication/bibitems.tex).
- [Technical agent guide](https://complextrees.com/4D/docs/AGENT_GUIDE.md).
- [Programme registry](https://complextrees.com/4D/data/programme.json).
- [Publication catalogue](https://complextrees.com/4D/data/publications.json).
- [Research evidence ledger](https://complextrees.com/4D/data/research-ledger.json).
- [Dataset catalogue](https://complextrees.com/4D/data/datasets.json).
- [Publication checksums](https://complextrees.com/4D/data/publication-assets.json).
- [Parameter schema](https://complextrees.com/4D/schemas/parameter.schema.json).
- [Exact interval example](https://complextrees.com/4D/examples/real-contact.json).
- [BibTeX](https://complextrees.com/4D/citations.bib).
- [CITATION.cff](https://complextrees.com/4D/CITATION.cff).
- [Source repository](https://github.com/espigule/espigule.github.io/tree/master/4D).

## Essential conventions

Canonical maps: f−(z)=−1+λ−Cε−(z), f+(z)=1+λ+Cε+(z), where C0 is identity and C1 conjugation. The translations ±1 are generally not fixed points. λ−=(2w/ρ)exp(i(α−γ)), λ+=(2(1−w)/ρ)exp(−i(α+γ)); 0<w<1 and ρ>2max(w,1−w). Angles are radians. Plane orientation and endpoint traversal are separate fields.

Connectedness is equivalent to 2∈Dp, with Dp=λ−Cε−(Kp)−λ+Cε+(Kp). A legacy diagnostic Δ=Dp−2 uses target 0. Retain this convention with each figure.

Finite-address hues are computational fingerprints, not topological invariants. The four-family manuscript projection instead colours by weight. Finite survivors and resource caps are unresolved outcomes. Complete rational separation certificates and validated stability cells have distinct evidence records. The toroidal opening is imposed by the projection; it does not classify the topology of a connectedness locus.