ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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-nearest-neighbor graph can disconnect finite ISOMAP

Example

Let $x_{1} = 0$ , $x_{2} = 1 / 10$ , $x_{3} = 10$ , and $x_{4} = 101 / 10$ in $R$ , use $k = 1$ , and symmetrize by union. The resulting undirected neighbor graph has exactly the two edges ${1, 2}$ and ${3, 4}$ , so it is disconnected. ISOMAP therefore returns its disconnected status and does not form finite all-pairs graph distances.

The diagram shows the two union-graph components; labels are the sample indices.

diagram failed to render:
\begin{tikzpicture}[vertex/.style={draw,circle,minimum size=7mm}]
  \node[vertex] (x1) at (0,0) {$1$};
  \node[vertex] (x2) at (1.4,0) {$2$};
  \node[vertex] (x3) at (4,0) {$3$};
  \node[vertex] (x4) at (5.4,0) {$4$};
  \draw (x1)--node[above] {$1/10$}(x2);
  \draw (x3)--node[above] {$1/10$}(x4);
\end{tikzpicture}

Example references

Hastie, Tibshirani, and Friedman, ESL II, §14.9, printed pp. 572–574

Cameron, Linear and Nonlinear Dimensionality Reduction, §6, PDF pp. 9–13

Facts & Assumptions

Given: The four stated real points, $k = 1$ , union symmetrization, and the ISOMAP connected-graph condition.

[F1]

The nearest-neighbor order is by $(∥ x_{i} - x_{j} ∥_{2}, j)$ , and union edges are included when either endpoint selects the other (Deterministic finite nearest-neighbor lists and graph symmetrizations).

[F2]

ISOMAP returns a disconnected status and does not claim finite all-pairs graph distances when its graph is disconnected (Finite ISOMAP with deterministic neighborhoods and connected-graph shortest paths).

Verification

1.1givenF1algebra

The nearest neighbor of $x_{1}$ and $x_{2}$ is each other at distance $1 / 10$ ; the nearest neighbor of $x_{3}$ and $x_{4}$ is each other at distance $1 / 10$ . Every distance between the two pairs is at least $99 / 10$ , so no cross-pair edge is selected. By [F1], the union graph has exactly the two edges ${1, 2}$ and ${3, 4}$ .

2.1step 1.1F2∎

The graph in step 1.1 has two connected components, so [F2] returns the disconnected status; a path between either cross-component pair does not exist, and the algorithm does not produce a finite all-pairs graph-distance matrix.

Depends on

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

The Elements of Statistical Learning, second edition, corrected 12th printing (standard reference, not scraped)
Maria Cameron, Linear and Nonlinear Dimensionality Reduction, lecture notes (standard reference, not scraped)