ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

A one handle joins components or adds a tunnel

Example

A surface $1$ -handle attached along intervals on two different disk components produces a disk. An orientable $1$ -handle attachment along two intervals of the boundary of one disk produces an annulus. A twisted attachment to one disk requires different orientation data.

Facts & Assumptions

[F1]

Attaching a smooth handle with corner rounding: Assume ${A C}_{ω}$ . Let $X$ be a smooth $n$ -manifold with boundary. Attach the handle of def-k-handle-core-cocore-attaching-region-and-belt-sphere by a smooth embedding $h : S^{k - 1} \times D^{n - k} \to \partial X$ that extends to a neighborhood of the disk factor. Form the quotient of $X ⊔ (D^{k} \times D^{n - k})$ identifying $z$ with $h (z)$ in the attaching region. The disk coordinates trivialize the normal bundle of the attaching sphere; this framing is part of the data. Use collars from thm-collar-neighborhood-theorem to give the seam its product smooth charts, then round the compact codimension-two corner. A compatible rounding is a smooth monotone planar profile, transverse to a common diagonal direction, agreeing with the two faces away from a small corner neighborhood. In coordinates along that diagonal it is a graph. This convention fixes the gluing and collar data; changing the attaching embedding is a different question. There is no corner to round when $k = 0$ or $k = n$ .

Verification

Given: The objects and hypotheses in the example.

1.1

A surface handle is a rectangle, attached by its two opposite end edges. Boundary interval parametrizations can be straightened in disk collars: extend their increasing one-dimensional coordinate changes across an annular collar by interpolating a lifted circle coordinate, whose derivative stays positive. If necessary reflect an entire disk or the rectangle to normalize an end orientation. Thus the two-disk attachment is represented by two rectangular disks joined end to end by a rectangular strip. Their union is a longer rectangle before compatible corner rounding, hence a disk afterward.

F1algebra

2.1

For the one-disk orientable attachment, represent the disk as the rectangle $[0, 1] \times [0, 1]$ cut from an annulus along one radial interval. Its two radial edges are the prescribed attaching intervals after the same boundary straightening. Glue in a second rectangle bridging these edges with the orientation-compatible identifications; in coordinates the result is $([0, 2] / (0 \sim 2)) \times [0, 1]$ , an annulus. Reversing just one end identification instead reverses the transverse interval after one circuit, so this coordinate description no longer gives the orientable annulus. The claim explicitly excludes that twist.

F1step 1.1algebra∎

Depends on

Attaching a smooth handle with corner rounding

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

Benedetti, Lectures on Differential Topology (standard reference, not scraped)