LemmaStatement: AI-adaptedProof: Literature-sourced

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.

Bounded prefix ideals and rank-two closure

Statement

Let $V$ , $A$ , $P$ , and $u$ be as in the positive gallery category: $A$ is a finite central arrangement in the finite-dimensional real vector space $V$ , and every chamber is an open simplicial cone in $V$ . Let $I$ be a set of arrows of $P$ with common source $A$ and bounded length; a prefix of $h$ is an arrow $g$ such that $h = g k$ for some arrow $k$ , with products in traversal order.

A supporting wall hyperplane of a chamber $B$ is a hyperplane $H \in A$ whose wall face $\overline{B} \cap H$ has dimension $\dim V - 1$ . For an intersection $Q$ of supporting wall hyperplanes of $B$ (with empty intersection $V$ ), let $F_{B} (Q)$ be the interior in $Q$ of $\overline{B} \cap Q$ , the unique relatively open face of $\overline{B}$ with linear span $Q$ . The residue at this face is the arrangement ${H / Q : H \in A, Q \subseteq H}$ in $V / Q$ . Let $B_{Q}$ be the unique chamber whose closure contains $F_{B} (Q)$ and whose residue chamber is opposite the residue chamber of $B$ , and put $d_{B} (Q) = u (B, B_{Q})$ .

Then $I$ is the full prefix set of one unique arrow if and only if it contains $1_{A}$ , is closed under taking prefixes, and whenever $g : A \to B$ and distinct supporting wall hyperplanes $M, N$ of $B$ satisfy $g d_{B} (M), g d_{B} (N) \in I$ , also $g d_{B} (M \cap N) \in I$ .

Statement references

Positive gallery category, cancellation and length

Locally finite arrangements and relative face terminology

Facts & Assumptions

[F99]

Import Deligne Proposition 1.14 and Corollary 1.20 for both directions of the bounded-prefix criterion, and (1.5)–(1.6), (1.9.2), (1.13) for its face, residue and $Δ (Q)$ construction, under hypothesis (1.8). No local proof of the substantial criterion or residue correspondence is claimed.

[F1]

The category, minimal class $u$ , traversal-order composition and additive length, with length-zero arrows exactly identities, are those of Positive gallery category, cancellation and length.

[F2]

Use Locally finite arrangements and relative face terminology with $T = V$ for chambers, wall faces, supporting hyperplanes, their intersections, the quotient vector space and open simplicial cones. The objects intersected to form $Q$ , $M \cap N$ and $V / Q$ are linear hyperplanes, not closed wall faces. The imported correspondence in [F99] supplies the unique face and the incident-chamber lift defining $B_{Q}$ .

External input

[E1] Deligne (1.5)–(1.6) identifies chambers incident to the face supported on $Q$ with chambers of the quotient arrangement; (1.9.2) ensures that this face exists for an intersection of wall hyperplanes of $B$ , and (1.13) defines $Δ (Q) = u (B, B_{Q})$ . Under the finite central simplicial hypotheses (1.8), Proposition 1.14 says that the three closure conditions in the Statement make a length-bounded fixed-source set a principal prefix set, with a unique generator. Corollary 1.20 supplies the converse. These are imported inputs, not locally reconstructed proofs.

Proof

Given: The data and hypotheses in the Statement.

Goal: Establish the Statement.

1.1F1F2F99given

The arrangement hypotheses are exactly Deligne (1.8). By the imported correspondence (1.5)–(1.6), (1.9.2), the residue antipode lifts to the chamber $B_{Q}$ in the Statement, and $d_{B} (Q)$ is Deligne's $Δ (Q)$ in (1.13). The gallery congruence agrees with his by Positive gallery category, cancellation and length. Thus the three stated conditions are those of (1.14); the case $M = N$ , if included, adds only a tautology.

2.1F1F99step 1.1∎

Proposition 1.14 gives principality from these conditions, and Corollary 1.20 gives their necessity for a principal prefix set. For uniqueness, if $x$ and $y$ have the same prefix set, write $x = y a$ and $y = x b$ . Additive length gives $ℓ (a) + ℓ (b) = 0$ , so both complements are identities and $x = y$ .

Depends on

Used by · 1

Both gallery divisibility orders are lattices

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

Pierre Deligne, Les immeubles des groupes de tresses généralisés, Inventiones mathematicae 17 (1972), 273–302 · §1, (1.5)–(1.6), (1.8), (1.9.2), (1.13)–(1.14), and Corollary (1.20); printed pp. 275–278, 282

For an intersection Q of supporting wall hyperplanes of a chamber B, the unique face supported on Q identifies its incident chambers with chambers of the quotient arrangement in V/Q; the opposite quotient chamber lifts to B_Q and Δ(Q)=u(B,B_Q). A length-bounded set I of gallery classes of source A is the prefix set of a unique class exactly when it contains the identity, is prefix closed, and contains gΔ(M∩N) whenever it contains both gΔ(M) and gΔ(N), for wall hyperplanes M,N at the target of g.

Hypotheses: A finite arrangement of homogeneous real hyperplanes in a finite-dimensional real vector space V, with every chamber the positive span of a basis of V, as in (1.8). Gallery classes and traversal-order products are those of (1.10)–(1.13); Q is an intersection of supporting wall hyperplanes of B. The set I has fixed source and bounded gallery length.

Reading limits: Checked the PDF text of (1.5)–(1.6), (1.8), (1.9.2), (1.13)–(1.14), and (1.20), and visually inspected printed pp. 276, 278, 282 in the cached Deligne PDF. The quotient correspondence and bounded-prefix criterion are imported; their substantial proofs are not reconstructed. No whole-paper reading or new verification of cancellation is claimed.

Sources

Pierre Deligne, Les immeubles des groupes de tresses généralisés, Inventiones mathematicae 17 (1972), 273–302 · §1, (1.5)–(1.6), (1.8), (1.9.2), (1.13)–(1.14), and Corollary (1.20); printed pp. 275–278, 282