Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16
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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.
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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=(0−110) has no Jordan form over R, but after scalar extension to C the basis ((1,−i),(1,i)) gives [R]=diag⁡(i,−i)=J1(i)⊕J1(−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
1.1

Direct computation gives χR=x2+1, which has no real root and hence does not split over R; [L1] rules out real Jordan form.

L1algebra
2.1

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.

L1algebra∎

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: 19 results over 6 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