Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16
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FALSE: Jordan canonical form is a unique literal matrix without fixing block order

Statement

False claim. The Jordan canonical form of an endomorphism is a unique literal matrix even when no ordering convention for its blocks has been fixed.

Facts & Assumptions

Given: The two block diagonal matrices A=J2(0)⊕J1(1) and B=J1(1)⊕J2(0).

[L1]

Shifted-power ranks determine Jordan form uniquely only up to permutation of its blocks (Ranks of shifted powers determine Jordan form up to block order).

Refutation

technique · counterexample
1.1

Both A and B are Jordan matrices with the same block multiset, and a permutation matrix that moves the one-dimensional block past the two-dimensional block conjugates one to the other.

algebra
1.2

Their diagonal sequences are (0,0,1) and (1,0,0), so A≠B as literal matrices.

algebra
2.1

This is precisely the block-order freedom retained in [L1], and it refutes literal uniqueness without an additional ordering convention.

step 1.1step 1.2L1∎

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 10 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