TheoremStatement: AI-adaptedProof: 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.

An exact residual check certifies the Padé denominator solve

Statement

If Finite Householder QR solve for a Padé denominator with explicit statuses returns verified-solution with output $X$ for exact complex matrices $M, C$ in its declared exact-controller model, then $M X = C$ exactly. If $M$ is invertible, this solution is unique; in particular, when used with an invertible Padé denominator, it is the exact denominator solve. This includes the original rational-input instance and its finite exact rational audit whenever the adapter encodings are supplied.

Facts & Assumptions

Given: Exact complex matrices $M, C$ and a verified-solution output $X$ .

[F1]

The algorithm returns verified-solution only after exact controller evaluation of $M X - C$ gives the zero matrix (Finite Householder QR solve for a Padé denominator with explicit statuses).

[F2]

The Padé numerator and denominator are $p_{m} (z)$ and $q_{m} (z) = p_{m} (- z)$ , and $r_{m} = p_{m} / q_{m}$ where the denominator is nonzero (Diagonal Padé approximant to the exponential).

Proof

1.1givenF1

By [F1], the exact residual $M X - C$ is zero, so the returned matrix satisfies $M X = C$ .

2.1F2step 1.1algebra∎

If $M$ is invertible, left multiplication by $M^{- 1}$ gives $X = M^{- 1} C$ , which is the unique solution. In the Padé use, take $M = Q_{m} (B) = q_{m} (B)$ and $C = P_{m} (B) = p_{m} (B)$ by [F2]; the conclusion applies whenever this denominator matrix is invertible.

Depends on

Used by

Dependency tree · two levels

8 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

Awad H. Al-Mohy and Nicholas J. Higham, A New Scaling and Squaring Algorithm for the Matrix Exponential (standard reference, not scraped)
Jan Zur, Matrix Functions, TU Berlin full lecture notes (standard reference, not scraped)