AlgorithmAlgorithm: Literature-sourcedProof: Not applicablePipeline-generatedjudge 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.

Outward Leibniz determinant sign and nonsingularity certifier

Definition

Input $n \geq 1$ and a nonempty family of real $n \times n$ matrices contained in a box of real intervals. Evaluate the determinant's finite Leibniz expression

\det A = \sum_{σ \in S_{n}} sgn (σ) \prod_{i = 1}^{n} a_{i, σ (i)}

as an outward interval DAG, retaining every permutation term and reporting the final interval $D$ . If any operation returns overflow or another non-finite status, return inconclusive.

If $0 \notin D$ , return nonsingular and the sign of $D$ . If $D = {0}$ , return all-singular. In every other case return inconclusive. This finite procedure has factorially many permutation terms; no efficiency or determinant sign is claimed when zero remains in the enclosure.

Depends on

Used by

The interval determinant sign certifier is sound and failure means unknown Theorem

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

Siegfried M. Rump, Verification methods: Rigorous results using floating-point arithmetic, Acta Numerica 19 (2010), §10.10 (standard reference, not scraped)
Nicholas J. Higham, Numerical Linear Algebra and Matrix Analysis, MIMS EPrint 2015.103, §§5.1 and 5.6 (standard reference, not scraped)