# Terminology and notation contract

| Object | Preferred name and notation | Interpretation |
|---|---|---|
| Pair of maps | Binary planar similarity IFS, parameter p | Ordered maps unless an explicit symmetry quotient is being used |
| Multipliers | λ−, λ+ | Nonzero complex coefficients; orientation stored separately |
| Orientation type | Direct (f₀,f₁), mixed (f₀,g₁), reversing (g₀,g₁) | f preserves orientation; g reverses it. Legacy codes DD=(0,0), DO=(0,1), OO=(1,1) are retained in records |
| Ratios and angles | r±=|λ±|, θ±=arg λ± | Intrinsic coefficients shown alongside plotting coordinates |
| Attractor | Kp | Unique compact invariant set |
| Connectedness locus | M=Mᶠᶠ, Mᶠᵍ, Mᵍᵍ; Mε in parity formulas | Parameters with connected Kp, relative to the specified strict contraction domain |
| Equal-contraction slice | r−=r+; w=1/2 | Does not imply equal complex multipliers |
| Homogeneous slice | Common linear part | λ−=λ+ and equal orientation parity; equal modulus alone is insufficient |
| Contact set | Jp=f−(Kp)∩f+(Kp) | Exact first-level intersection |
| Contact difference | Dp=L−Kp−L+Kp, where Lj=λj Jεj | The primary specimen views mark 2; connectedness is equivalent to 2∈Dp |
| Relative difference | Kp−M Kp | An auxiliary real-linear relative similarity state, distinct from the fixed contact-difference notation |
| Exterior coefficients | c₀=1/a=1/λ−, c₁=1/b=1/λ+ | Manuscript/poster indices; legacy c1,c2 fields keep their original meanings. Reversing inverse maps still include conjugation |
| Stable set | SX | Parameters whose full coding kernel equals the relation forced throughout the specified family X |
| Zipper signature | e=(e1,e2) | Endpoint traversal, separate from plane orientation |
| Address tree height | Absolute prefix multiplier | A scale coordinate for a finite 3D scaffold, not an additional spatial dimension of the attractor |
| Symbolic contact | πp(u)=πp(v) | Specify addresses and whether eventually periodic |
| Finite graph | Cylinder-enclosure intersection graph | Candidate contacts of chosen enclosures; topology of Kp not inferred |
| Critical ratio surface | r−²+r+²=1 | Similarity dimension two; not generally the connectedness boundary |
| Angular picture | Toroidal parameter visualization | The periodic coordinate base, not an intrinsic topology claim |
| Radial sampling | Detected transition intervals at depth k | Multiple intervals allowed; thin intervals may be missed |
| Numerical surviving state | Unresolved at depth k | No positive connectedness certificate |
| Rational rejection | Certified disconnection of the exact rational input | Requires replay of the independent rational verifier |

The display coordinates satisfy θ−=α−γ and θ+=−α−γ. The change (α,γ)↦(α+π,γ+π) gives the same pair of multipliers. Consequently this angle torus is a two-fold cover of the multiplier-angle torus, not a globally unique coordinate chart.

Complex conjugation sends (w,α,γ) to (w,−α,−γ). Branch exchange followed by z↦−z sends it to (1−w,−α,γ), exchanging orientations and address symbols. A reflection that uses branch exchange is not a same-weight symmetry at unequal weight.

The laboratory stores α∈[0,π/2], γ∈[0,π], with thirteen weights from 0.35 to 0.65. Mirrored display copies do not enlarge this sampled domain. The live atlas’s preserved direct survey spans the full angular display at weight 1/2.

The four-family manuscript projection uses the canonical coefficient phases, with colour retaining weight. The radial browser atlas’s finite-address hues encode a different quantity. Pixel size, point count, prefix depth, hull depth, density grid and survey resolution remain separate settings.

The [public mathematical guide](./) and [research ledger](../data/research-ledger.json) provide sources and precise interpretation. The [main article and research companion](../publication/) are in preparation with provisional titles. PDFs will be posted only after author review and approval. The ledger preserves the 19 September draft references for traceability. For direct similarities, the square-sum connectedness implication follows from Aoki–Fujimura–Taniguchi (2014, Theorem 1.1 and the binary equivalence in Remark 1.1). Their hypotheses are nonzero holomorphic contractions with distinct fixed points. The manuscript supplies the all-orientation argument; the atlas projection is defined by its own coordinate formula.

## Presentation and compatibility

Use **Direct**, **Mixed**, and **Reversing** for the three orientation types. Write
`f₀(z)=−1+az`, `f₁(z)=1+bz`, `g₀(z)=−1+a conjugate(z)`, and
`g₁(z)=1+b conjugate(z)`, where `a=λ−` and `b=λ+`. The mixed pair has
its direct generator first. These names classify the maps, not the attractor.

Numeric indices `0,1,2`, codes `DD,DO,OO`, parity bits, registry IDs, export
fields and coordinate keys remain unchanged. Presentation labels must not be
used to parse or serialize a parameter. A branch exchange also exchanges
symbolic addresses.

An orientation type has four continuous real parameters. A **slice** imposes
additional equations: equal contraction `w=1/2` leaves three real parameters;
a common linear part has equal coefficients and equal parity. The direct
mirror slice `(c₀,c₁)=(c,conjugate(c))` has two direct generators.

A **marked stable family** keeps the complete coding relation
`Eₚ={(ω,ν):πₚ(ω)=πₚ(ν)}` constant on its specified domain. A persistent
selected address equality is only a contact witness. A complete relation,
its carrier equation, a certified stable patch and ambient-boundary membership
are distinct claims. Zipper traversal signs are independent of planar orientation.

Explore the [mirror-symmetric slice](../../mirror/), [three-dimensional sections](../sections/), and [contact families and certified stable patches](../contacts/) in their dedicated viewers.

## Equivalent contact-coordinate conventions

The primary linked views in the main explorer and direct-similarity laboratory plot `Dp=A−B`, where `A=L−K` and `B=L+K`, with marked target **2**. A legacy laboratory diagnostic plots the translated first-level difference `Δ=f−K−f+K=Dp−2`, with marked target **0**, and labels this convention. Thus `2 ∈ Dp` and `0 ∈ Δ` are the same contact condition. Keep that translation in the comparison record.

For complete length-N prefix centres, the manuscript’s tail estimate bounds the finite hull approximation in exact arithmetic. For a subset of sampled words, truncation bounds describe those evaluated words and do not alone prove Hausdorff coverage of the attractor. Floating-point, binning and sampling errors are separate.