ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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 quarter-turn of $R^{2}$ has characteristic polynomial $x^{2} + 1$ and no real eigenvalue

Example

The real quarter-turn

$A = (\begin{array}{cc} 0 & - 1 \\ 1 & 0 \end{array})$

has $χ_{A} (x) = x^{2} + 1$ and empty real spectrum.

Facts & Assumptions

Given: The displayed matrix, acting on $R^{2}$ .

[L1]

The spectrum over the base field is exactly the root set in that field of the characteristic polynomial (For every finite-dimensional space, $σ_{F} (T)$ is exactly the set of roots in $F$ of $χ_{T}$ ).

[L2]

Every real square is nonnegative (Squares of nonzero elements are positive).

Verification

technique · direct computation

1.1

Directly, $χ_{A} (x) = \det (\begin{array}{cc} x & 1 \\ - 1 & x \end{array}) = x^{2} + 1$ .

algebra

2.1

For every $x \in R$ , [L2] gives $x^{2} + 1 \geq 1 > 0$ , so $χ_{A}$ has no real root.

step 1.1L2algebra

3.1

By [L1], $A$ has no real eigenvalue, despite acting on a nonzero real space.

step 2.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 28 results over 7 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

M. Khovanov, Linear Algebra II notes, §6 (standard reference, not scraped)

A quarter-turn of R2R2 has characteristic polynomial x2+1x2+1 and no real eigenvalue