ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

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 half-turn lies on the principal logarithm branch cut

Example

The planar half-turn $R_{π} = - I_{2}$ is invertible, and the principal logarithm algorithm must return a branch-domain failure because its spectrum lies on the removed negative-real axis.

Facts & Assumptions

Given: $R_{π} = - I_{2}$ , the principal logarithm request, and finite complex Francis controls with deflation tolerance $0$ , sweep budget $0$ , stagnation period $1$ , and an empty exceptional-shift table for both Schur calls. The Schur–Parlett threshold is $0$ , its initial partition is the two singletons, and its interval-keyed table contains the exact $1 \times 1$ Jordan certificate $Z = [1]$ , $J = [- 1]$ for each singleton. These block data are not used because the branch test fails first.

[F1]

The holomorphic principal logarithm is defined on $S = C ∖ (- \infty, 0]$ (The principal logarithm is the normalised holomorphic branch on the slit plane).

[F2]

The branch-aware procedure returns branch-domain-failure when an exact Schur diagonal eigenvalue lies in $(- \infty, 0]$ (Principal matrix logarithm and square root by branch-aware Schur–Parlett).

[F3]

With sweep budget $0$ , a complex Francis reduction returns Schur-computed when the initialized matrix is already upper triangular; when $n = 2$ its Hessenberg initialization has no reflector columns (Finite-budget computed Francis Schur reduction with explicit statuses).

Verification

1.1givenF1algebra

The characteristic polynomial of $R_{π}$ is $(z + 1)^{2}$ , so its only eigenvalue is $- 1$ . This lies on the removed ray and hence outside $S$ by [F1].

2.1F2F3step 1.1algebra∎

Here $n = 2$ , so [F3] performs no Hessenberg reflectors. The initialized matrix $- I_{2}$ is already upper triangular, and with zero sweeps the reduction returns the exact pair $Q = I_{2}$ , $T = - I_{2}$ ; this pair passes $Q^{*} Q = I$ and $A = Q T Q^{*}$ . The branch test then sees the diagonal value $- 1$ and returns branch-domain-failure by [F2]. This records only that the principal holomorphic branch is unavailable there.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

37 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

Jan Zur, Matrix Functions, TU Berlin full lecture notes (standard reference, not scraped)
Nicholas J. Higham, Functions of Matrices, Handbook of Linear Algebra, Chapter 17 (standard reference, not scraped)