LemmaStatement: AI-generatedProof: 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.

Deleting relative root hyperplanes removes all cone-boundary points

Statement

Let $(W, S)$ be a finite-rank Coxeter system in its canonical real representation $V = R^{S}$ , with basis $(α_{s})$ , dual $Θ$ , root system $Φ$ , contragredient action, and Tits cone $U$ . For $I \subseteq S$ put $E_{I} = {f : f (α_{i}) = 0 (i \in I)}$ and $C_{I} = {f \in E_{I} : f (α_{s}) > 0 (s \notin I)}$ . Fix $J \subseteq S$ and put $E = E_{J}$ , $T = U \cap E$ , and $T^{\circ} = {int}_{E} T$ . With $Φ^{J}$ and $A$ as in the restricted-root-trace lemma, put $D = ⋃_{K \in A} K$ . Then $T ∖ D = T^{\circ} ∖ D$ .

Statement references

The relative root deletion set is the restricted-hyperplane union

Full-dimensional J-chambers lie in the relative interior

Interior, closure, boundary, exterior, derived set and isolated point in a topological space

External input

Iyama–Wemyss, the sentence immediately after Definition 1.6, states that the $J$ -chambers $x C_{I} \subseteq E_{J}$ with $∣ I ∣ = ∣ J ∣$ are the connected components of $(U \cap E_{J}) ∖ ⋃_{α \in Φ^{J}} H_{α}$ .

Facts & Assumptions

[F1]

The restricted deletion set agrees with the root deletion set by The relative root deletion set is the restricted-hyperplane union.

[F2]

Every source $J$ -chamber is contained in $T^{\circ}$ by Full-dimensional J-chambers lie in the relative interior.

[F3]

Every point of a topological space belongs to its connected component, including when that space is a subspace, as defined and justified in Connected components, quasicomponents, and totally disconnected spaces.

[F4]

In the topological space $E$ , the interior of $T$ is the union of open subsets of $E$ contained in $T$ ; in particular $T^{\circ} \subseteq T$ , by Interior, closure, boundary, exterior, derived set and isolated point in a topological space.

Proof

Given: The objects and hypotheses of the statement. Goal: Establish its conclusion.

1.1F1F2F3given

By [F3], every point $p \in T ∖ D$ belongs to a connected component of that space. By [F1] and the external input, that component is a source $J$ -chamber. By [F2] it is contained in $T^{\circ}$ , so $p \in T^{\circ} ∖ D$ .

2.1F4step 1.1given∎

Conversely $T^{\circ} \subseteq T$ by the definition of interior in [F4], so $T^{\circ} ∖ D \subseteq T ∖ D$ . Together with the first inclusion this proves equality, also for the zero-dimensional case.

Source notes

The external statement is used only as explicitly recorded above. It is an independent prerequisite source, not the article under study. The local topology and linear-algebra argument is authored here; the source proof has not been reviewed.

Depends on

Used by · 2

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.

External dependencies

Osamu Iyama and Michael Wemyss, Tits Cone Intersections and Applications, author-hosted draft · Sentence following Definition 1.6, printed p. 6 (PDF p. 30)

Iyama–Wemyss, the sentence immediately after Definition 1.6, states that the $J$-chambers $xC_I\subseteq E_J$ with $|I|=|J|$ are the connected components of $(U\cap E_J)\setminus\bigcup_{\alpha\in\Phi^J}H_\alpha$.

Hypotheses: Finite Coxeter graph, canonical real reflection representation and dual action; arbitrary vertex subset J, not required spherical. Our I is the source secondary subset, and our J is the source fixed subset.

Reading limits: Read the extracted source text of §1.1 and §§1.3–1.4 (printed pp. 3–6), including Definitions 1.3–1.6 and the following component assertion. Checked the finite-rank setup and notation correspondence. No source proof review or local reconstruction; no infinite-rank topology or local-finiteness result imported.

Sources

Osamu Iyama and Michael Wemyss, Tits Cone Intersections and Applications, author-hosted draft · Sentence following Definition 1.6, printed p. 6 (PDF p. 30)