LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30

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.

An interior configuration loop traces a geometric braid

Statement

For every interior based motion $α : I \to C_{n} (int D^{2})$ at $[Q]$ , the quotient covering $p^{\circ} : F_{n} (int D^{2}) ⟶ C_{n} (int D^{2})$ has a unique lift $\tilde{α}$ starting at $Q$ . Writing $\tilde{α} (t) = (z_{1} (t), \dots, z_{n} (t))$ , the coordinate graphs form a geometric braid based at $Q$ , and its unordered slice at every height is exactly $α (t)$ .

Facts & Assumptions

Given: A natural number $n$ , the geometric base tuple $Q$ , and an interior based motion $α$ at $[Q]$ .

[L1]

$F_{n} (X)$ is the subspace of $X^{n}$ consisting of tuples with pairwise distinct coordinates, and $F_{0} (X)$ is the one-point space (Ordered configuration spaces $F_{n} (X)$ ).

[L2]

$C_{n} (X)$ is the orbit quotient of $F_{n} (X)$ by coordinate permutations; its quotient map $p$ is continuous and surjective, and its fibres are exactly the coordinate-permutation orbits (Unordered configuration spaces $C_{n} (X)$ ).

[L3]

Under the real-complex identification, $D^{\circ}$ is a subspace of the closed unit disk $D^{2}$ ; $D^{2}$ is Hausdorff, and Hausdorffness passes to subspaces (Geometric braids in the disc with setwise endpoints, The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane, The closed disk $D^{2}$ is a connected Hausdorff topological $2$ -manifold with boundary, $T_{0}$ , $T_{1}$ , and Hausdorffness are hereditary).

[L4]

If $X$ is Hausdorff, then every point of $C_{n} (X)$ has an evenly covered neighbourhood under $p : F_{n} (X) \to C_{n} (X)$ ; for $n = 0$ , $p$ is the unique homeomorphism of one-point spaces (Disjoint coordinate neighbourhoods evenly cover the unordered configuration space).

[L5]

For a covering $p : E \to B$ and path $α : I \to B$ with a specified starting lift $e_{0}$ , there is a unique path $\tilde{α}$ starting at $e_{0}$ and satisfying $p \circ \tilde{α} = α$ (Existence and uniqueness of path lifts through a covering map).

[L6]

A geometric braid based at $Q$ is a tuple of continuous point motions in $D^{\circ}$ with pairwise distinct values at every height, bottom tuple $Q$ , and top endpoint set $Q$ (Geometric braids in the disc with setwise endpoints).

[L7]

An interior based motion is a continuous path in $C_{n} (int D^{2})$ whose two endpoints are $[Q]$ (Based motions of an unordered point configuration).

No choice principle is assumed or used: the lift is determined by one prescribed starting tuple and is unique.

Proof

technique · direct

1.1L2L3L4

The quotient is a covering on the open disk. By [L3], $int D^{2}$ is Hausdorff; applying [L4] with $X = int D^{2}$ gives an evenly covered neighbourhood at every unordered configuration, and continuity and surjectivity from [L2] show that $p^{\circ}$ is a covering. For $n = 0$ both spaces are points, so this conclusion still holds.

1.2L1L2L5L7

Lift the motion. Since $α$ is based at $[Q]$ by [L7] and $p^{\circ} (Q) = [Q]$ , [L5] gives a unique continuous lift $\tilde{α} : I \to F_{n} (int D^{2})$ with $\tilde{α} (0) = Q$ and $p^{\circ} \circ \tilde{α} = α$ ; write its coordinates as $z_{1}, \dots, z_{n}$ , taking the empty tuple when $n = 0$ .

1.3L1L2L6L7

Check the strand conditions. By [L1], the coordinate projections of $\tilde{α}$ are continuous; because every lifted tuple lies in $F_{n} (int D^{2})$ , the coordinates remain interior and pairwise distinct, and they start at $Q$ . At the top, $p^{\circ} (\tilde{α} (1)) = α (1) = [Q]$ , so [L2] places the terminal tuple in the coordinate-permutation orbit of $Q$ , giving endpoint set exactly ${q_{1}, \dots, q_{n}}$ ; these conditions are vacuous for $n = 0$ .

1.4L2L5L6

The lift is the required braid and traces the original motion. By [L6], $β = (z_{1}, \dots, z_{n})$ is a geometric braid based at $Q$ : its strands are the graphs $t \mapsto (z_{j} (t), t)$ , so height is their parameter and distinct strands cannot meet. Its unordered slice is $[(z_{1} (t), \dots, z_{n} (t))] = p^{\circ} (\tilde{α} (t)) = α (t)$ for every $t$ , by [L5]; this also holds for the unique empty braid when $n = 0$ .

2.1step 1.2step 1.3step 1.4∎

The unique lift has all the endpoint, collision, continuity and slicing properties required in the statement, establishing the claim.

Depends on

Used by

Dependency tree · two levels

61 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Juan Gonzalez-Meneses, Basic results on braid groups, §§1.1–1.3, printed pp. 3–6 (standard reference, not scraped)
Joan S. Birman and Tara E. Brendle, Braids: A Survey, §1.1, author manuscript pp. 3–5 (standard reference, not scraped)