ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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.

The real quarter-turn acquires diagonal Jordan form over $C$

Example

The real quarter-turn $R = (\begin{array}{cc} 0 & - 1 \\ 1 & 0 \end{array})$ has no Jordan form over $R$ , but after scalar extension to $C$ the basis $((1, - i), (1, i))$ gives $[R] = diag (i, - i) = J_{1} (i) \oplus J_{1} (- i) .$

Facts & Assumptions

Given: The displayed real matrix, read over $R$ and then over $C$ .

[L1]

Jordan form over the base field exists exactly when the characteristic polynomial splits over that field (Jordan form over the base field exists exactly when the characteristic polynomial splits).

Verification

technique · computation

1.1L1algebra

Direct computation gives $χ_{R} = x^{2} + 1$ , which has no real root and hence does not split over $R$ ; [L1] rules out real Jordan form.

2.1L1algebra∎

Over $C$ , $R (1, - i) = i (1, - i)$ and $R (1, i) = - i (1, i)$ ; these eigenvectors are independent, so their basis gives the displayed diagonal Jordan form, as [L1] permits.

Depends on

Jordan form over the base field exists exactly when the characteristic polynomial splits

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

S. Axler, Linear Algebra Done Right, 4th ed., Result 5.43 (standard reference, not scraped)

The real quarter-turn acquires diagonal Jordan form over CC