# Exact families in the binary atlas

The dependency-free `family-registry.js` exposes `AtlasFamilies.FAMILY_REGISTRY`, `evaluateFamily(id, input)`, and `normalizePair(rawMaps)`. It contains 27 entries. Each record provides the mathematical domain, parameter controls, exact formula, source, landmarks, and a connectivity explanation. Formula evaluation uses binary64 arithmetic; theorem-backed family membership describes the stated mathematical family, rather than an independently interval-verified rounded state.

## The 2018 complex-tree families

Source: Bernat Espigulé, *Complex Trees and their families of connected self-similar sets*, Universitat de Barcelona, 28 June 2018, supervised by Xavier Jarque and Núria Fagella. [Repository handle](https://hdl.handle.net/2445/129385).

The original tree `T{a,b}` has maps `F1(x)=1+ax`, `F2(x)=1+bx`. For distinct nonzero strict contractions, its canonical DD multipliers are exactly `(a,b)`. The conjugacy is

```text
h(x) = ((2−a−b)x − 2)/(b−a).
```

Identical letters give a singleton tipset and do not represent the nondegenerate homogeneous DD pair with translations ±1. The map from the original rooted-tree presentation to the canonical chart is singular there.

| Registry ID | Alphabet | Source |
|---|---|---|
| `tfm-quadratic` | `{z, 1+z²}` | Appendix B, Figure 47, p. 60 |
| `tfm-rational` | `{z, z+1/(1+z)}` | Appendix B, Figure 47, p. 60 |
| `tfm-cubic` | `{z, 1+z+z²}` | Appendix B, Figure 47, p. 60 |
| `tfm-fibonacci` | `{z, (z−1±√((z−3)(z+1)))/2}` | Appendix B, Figure 51, pp. 61–62 |
| `tfm-signed` | `{z, −z}` | Section 1, Eq. (1), pp. 2–3 |
| `tfm-symmetric` | `{z, conjugate(z)}` | Section 1, Eq. (2), pp. 4–5 |

The first four families are connected throughout the domain in which both letters are nonzero strict contractions. These conclusions can be checked directly from the following eventually periodic contacts. In the notation below, a prefix is followed by infinite repetition of the parenthesized word.

| Family | Equal addresses | Vanishing factor in their difference |
|---|---|---|
| Quadratic | `1111(2)∞`, `211(2)∞` | `b−1−a²` |
| Rational | `111(2)∞`, `211(2)∞` | `(1+a)b−1−a−a²` |
| Three-term | `111(2)∞`, `21(2)∞` | `b−1−a−a²` |
| Fibonacci-gasket | `12(21)∞`, `22(21)∞` | `b²+(1−a)b+1` |

Every pair starts with different symbols, so each identity proves a first-level contact and hence connectedness. The full rational identities are in the registry's `contactWitness` records and checked as exact integer-polynomial identities by the regression suite. This gives membership, not a complete topology or a zipper parametrization.

The Fibonacci entry's default decimal parameter is near the thesis's algebraic Fibonacci-gasket specimen. Its connectedness follows from the family equation, but the precise Fibonacci-gasket identification is not assigned to those rounded coordinates. The algebraic branch is selected explicitly and the contraction domain is enforced.

The signed and mirror-symmetric slices also contain disconnected parameters. The registry reports the exact real or quarter-turn classifications and the universal ratio bounds; other cases remain unresolved. Real parameters in the mirror-symmetric slice give coincident maps and are excluded from the nondegenerate chart.

## Named zipper families

The 12 entries `zipper-{dd,do,oo}-{pp,pm,mp,mm}` enumerate the marked endpoint presentations in the lens `|p|<1`, `|1−p|<1`. They do not enumerate all connected binary attractors or assert that different marked presentations have disjoint images.

The one-dimensional entries `levy`, `heighway`, `heighway-mirror`, and `koch-wiggle` set

```text
p = 1/2 + (i/2)tan θ,  −60° < θ < 60°.
```

The equal contraction ratio is `1/(2cos θ)`. In DD the four traversal signatures are `++`, `+−`, `−+`, and `−−`, respectively. At 45° the first three give the Lévy and Heighway dragon landmarks; the last gives a right-isosceles triangle. At 30° the Koch–wiggle presentation gives the classical Koch curve after a two-level binary refinement.

The Lévy and Koch–wiggle entries also expose their exact DO and OO presentations. For shared sign `s=+1` or `−1`, the traversal signs are `σj=s(−1)^εj`. The endpoint-normalized structural curves agree because the reflection `V(z)=1−conjugate(z)` satisfies `V(h(1−t))=h(t)`. Continuity and endpoint coupling do not imply injectivity.

Calegari's [Wiggle Island](https://arxiv.org/abs/2205.11442) supplies the direct Koch–wiggle comparison. Planar orientation, zipper traversal direction, and parameter-plane conjugation remain separate operations.

## Other exact slices

- `right-triangle`: the OO altitude dissection, `p=cos θ exp(iθ)`, `0°<θ<90°`.
- `real-slice`: all parity charts, connected iff `|a|+|b|≥1`.
- `oo-diagonal`: `f±(z)=±1+λ conjugate(z)`, with the exact Bernoulli-product classification.
- `orthogonal-dd`: `(λ−,λ+)=(ir,−ir)`, connected iff `r≥1/√2`.
- `golden-bee`: an exact mixed-orientation algebraic polygonal tile.

## Normalization and tests

For a general raw pair, let `q` be the fixed point of the average map and `d=F+(q)−F−(q)`. The normalized map is `h(z)=2(z−q)/d`. Reversing multipliers receive the phase `conjugate(d)/d`; direct multipliers are unchanged. An OD input is exchanged into the DO convention, and the result records its raw map order.

Run:

```sh
node tests/family-registry.test.cjs
```

The suite checks independent real-matrix conjugacy, multiplier-coordinate roundtrips, exact integer-polynomial collision identities, original thesis landmarks, strict domain boundaries, cross-parity structural curves, and the distinction between known and unresolved connectivity.