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.

Facet incidence and localized chamber signs

Statement

For the sign facets in Sign facets of a cone arrangement, write $F ⪯ G$ when $F \subseteq \overline{G}$ . This is a partial order, characterized by: every nonzero sign of $F$ equals the corresponding sign of $G$ . Put $A_{F} = {H : F \subset H}$ . This is finite. If $C$ is a chamber, let $C_{F}$ be the chamber of $(I, A_{F})$ with the signs of $C$ on $A_{F}$ . For $F ⪯ G$ , one has $A_{G} \subseteq A_{F}$ . If two chambers $C, D$ both satisfy $F ⪯ C, D$ and $C_{F} = D_{F}$ , then $C = D$ .

Statement references

Sign facets of a cone arrangement

Facts & Assumptions

[F1]

Facets are simultaneous sign fibers, are nonempty and convex, and local finiteness holds, as in Sign facets of a cone arrangement.

Proof

Given: the objects and hypotheses of the Statement.

Use the declared inputs Sign facets of a cone arrangement with the conventions in the Statement and Facts below.

Proof technique: direct.

Goal: prove every assertion in the Statement.

1.1F1

Choose $x \in F$ and $y \in G$ . If $x$ lies in the closure of $G$ , continuity forces every nonzero sign at $x$ to equal that on $G$ . Conversely, under that sign condition, $(1 - t) x + t y$ has precisely the signs of $G$ for every $0 < t \leq 1$ . Convexity places it in $I$ , so it belongs to $G$ and tends to $x$ . This proves the characterization.

2.1F1step 1.1

The sign criterion gives reflexivity and transitivity. Mutual incidence forces equality of every sign, hence equality of fibers, giving antisymmetry. Choose $x \in F$ . All members of $A_{F}$ pass through $x$ , so local finiteness makes this set finite. If $F ⪯ G$ , a zero sign on $G$ cannot be a nonzero sign on $F$ , proving the inclusion.

3.1F1step 1.1∎

The sign fiber of $C$ for the finite arrangement $A_{F}$ is nonempty because it contains $C$ . It is a chamber by the chamber argument of [F1], establishing existence and uniqueness of $C_{F}$ . If $F ⪯ C, D$ , signs of $C, D$ off $A_{F}$ are prescribed by $F$ . Equality $C_{F} = D_{F}$ prescribes equality also on $A_{F}$ . Thus all signs agree and $C = D$ .

Depends on

Sign facets of a cone arrangement

Used by · 5

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.