TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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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.

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The interval geometric filter certifies only uniform signs

Statement

For every tuple of points represented by the input coordinate intervals, any positive or negative result returned by Certified geometric predicate for interval-valued coordinates is the exact determinant sign for every admissible tuple. An indeterminate result makes no uniform-sign claim.

Facts & Assumptions

Given: A finite box of coordinate intervals, the determinant definition, and the interval evaluation procedure.

[F1]

Dyadic orientation and lifted sphere determinants: each requested predicate is a finite signed determinant expression in the point coordinates.

[F2]

Outward-rounded operations are inclusion-isotone and enclose DAG evaluations: for a finite expression DAG with finite interval outputs, every exact value from every admissible input assignment belongs to its root interval.

[F3]

Certified geometric predicate for interval-valued coordinates: the procedure returns positive only when its final interval has positive lower endpoint, and negative only when its upper endpoint is negative.

Proof

1.1F1F2given

For each admissible choice of coordinates inside the input intervals, the determinant expression in [F1] is an exact evaluation of the finite DAG used by the algorithm. If all DAG operations are finite, [F2] places that exact determinant in the computed interval $J$ .

2.1F2F3step 1.1given∎

If the procedure returns positive, [F3] gives $J = [a, b]$ with $a > 0$ ; every admissible determinant in $J$ is positive. If it returns negative, $b < 0$ and every admissible determinant is negative. When $0 \in J$ or a nonfinite status occurs it returns indeterminate and makes no uniform-sign claim.

Depends on

Used by

An uncertain coordinate interval does not decide orientation Example

Cited to discharge well-definedness by Certified geometric predicate for interval-valued coordinates.

Dependency tree · two levels

11 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)
Mark de Berg et al., Computational Geometry: Algorithms and Applications, 2nd ed., §1.2 (standard reference, not scraped)