ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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.

An uncertain coordinate interval does not decide orientation

Example

Take $A = (0, 0)$ , $B = (1, 0)$ , and $C = (2, Y)$ where $Y \in [- 1 / 8, 1 / 8]$ . Use an outward-rounding format that represents $0, 1, 2,$ and $\pm 1 / 8$ exactly. The exact orientation determinant is $Y$ . Outward interval evaluation returns $[- 1 / 8, 1 / 8]$ , which contains zero and both signs; the certified result is indeterminate.

Facts & Assumptions

Given: The first two points are exact, and the third point has exact coordinate intervals $C_{x} = [2, 2]$ and $C_{y} = [- 1 / 8, 1 / 8]$ .

[F1]

Dyadic orientation and lifted sphere determinants: orientation is the determinant of ordered rows $(p_{i}, 1)$ .

[F2]

Outward-rounded operations are inclusion-isotone and enclose DAG evaluations: a finite outward arithmetic DAG encloses the exact determinant for every admissible coordinate assignment.

[F3]

Certified geometric predicate for interval-valued coordinates: the interval procedure returns indeterminate when its finite determinant interval contains zero.

[F4]

The interval geometric filter certifies only uniform signs: positive or negative is returned only when that sign holds for every tuple in the box.

[A1]

The selected outward-rounding format represents $0, 1, 2,$ and $\pm 1 / 8$ exactly.

Verification

1.1givenF1algebra

The orientation matrix is $(\begin{array}{ccc} 0 & 0 & 1 \\ 1 & 0 & 1 \\ 2 & Y & 1 \end{array})$ , whose determinant is $Y$ . Since $Y$ ranges over $[- 1 / 8, 1 / 8]$ , admissible inputs realize a negative value, zero, and a positive value.

1.2givenF1F2A1algebra

The Leibniz interval expression has only one nonzero term, namely the interval $C_{y} = [- 1 / 8, 1 / 8]$ ; all other terms contain an exact zero factor. Every intermediate endpoint is among $0, 1, 2, \pm 1 / 8$ , so [A1] makes every outward rounding exact. The returned interval is therefore $[- 1 / 8, 1 / 8]$ , and its inclusion is guaranteed by [F2].

2.1F3F4step 1.1step 1.2∎

The returned interval contains zero, so [F3] requires indeterminate; by [F4] neither positive nor negative could certify all admissible inputs.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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

CGAL Developer Manual, Robustness Issues, §11.1 (standard reference, not scraped)