ExampleConstruction: Literature-sourcedVerification: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Spherical and stereographic pictures of the D4 arrangement

PDF p. 28: Figure 1 and caption

Example

In the $D_{4}$ example with central vertex $2$ and $J = {2}$ , intersect the seven relative planes in $Θ_{J}$ with a unit sphere and project from its north pole to a plane. One coordinate choice gives the following line labels: $H_{1} = H_{α_{1}}$ , $H_{2} = H_{α_{2} + α_{3} + α_{4}}$ , and $H_{3} = H_{α_{1} + α_{2} + α_{3} + α_{4}}$ . The circle labels are $H_{4} = H_{α_{3}}$ , $H_{5} = H_{α_{1} + α_{2} + α_{3}}$ , $H_{6} = H_{α_{4}}$ , and $H_{7} = H_{α_{1} + α_{2} + α_{4}}$ . Here each $H_{β}$ denotes its restriction to $Θ_{J}$ .

Example references

Seven relative D4 hyperplanes and thirty-two chambers

Stereographic projection and its inverse are explicit in coordinates

Facts & Assumptions

[F1]

For $f \in Θ_{J}$ , set $x = f (α_{1})$ , $y = f (α_{3})$ , and $z = f (α_{4})$ . These are coordinates, and the seven relative planes are the kernels of $x, y, z, x + y, x + z, y + z, x + y + z$ , with root labels as in Seven relative D4 hyperplanes and thirty-two chambers.

[F2]

For the unit sphere $X^{2} + Y^{2} + Z^{2} = 1$ , stereographic projection from $(0, 0, 1)$ has inverse $(u, v) \mapsto (2 u, 2 v, u^{2} + v^{2} - 1) / (1 + u^{2} + v^{2})$ , as in Stereographic projection and its inverse are explicit in coordinates.

Verification

Given: The planes and projection in the Example.

Goal: Verify the stated line and circle labels for an explicit coordinate choice.

1.1F1given

Choose coordinates $X = x$ , $Y = (y + z) / \sqrt{2}$ , and $Z = (y - z) / \sqrt{2}$ on $Θ_{J}$ , and use their standard Euclidean unit sphere. This is an invertible coordinate change. In the label order of the Example, the seven plane equations become $X = 0$ , $Y = 0$ , $X + \sqrt{2} Y = 0$ , $Y + Z = 0$ , $\sqrt{2} X + Y + Z = 0$ , $Y - Z = 0$ , and $\sqrt{2} X + Y - Z = 0$ .

2.1F2step 1.1∎

A plane $a X + b Y + c Z = 0$ intersects the sphere in a great circle, meaning the intersection with a plane through the origin. Substitution of the inverse projection gives $2 a u + 2 b v + c (u^{2} + v^{2} - 1) = 0$ . If $c = 0$ , its image is a line; if $c \neq 0$ , completing squares gives $(u + a / c)^{2} + (v + b / c)^{2} = 1 + (a^{2} + b^{2}) / c^{2}$ , a nonempty circle. In step 1.1 the first three equations have $c = 0$ and the last four have $c \neq 0$ . Hence their labels are respectively the three lines and four circles claimed.

Source notes

The article reference attributes the pictured labels. The displayed coordinates are one choice realizing those labels; the verification does not assume that the article specifies this coordinate choice.

Depends on

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

Sources

Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections · PDF p. 28, Figure 1 and caption