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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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.

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FALSE: Every complex inner product satisfies $⟨ u, v ⟩ = ⟨ v, u ⟩$

Statement

False claim. Every complex inner product satisfies $⟨ u, v ⟩ = ⟨ v, u ⟩$ .

Facts & Assumptions

Given: Standard $C$ and the vectors $u = i$ , $v = 1$ .

[L1]

Complex inner products are conjugate symmetric: $⟨ u, v ⟩ = \overline{⟨ v, u ⟩}$ (Real and complex inner product spaces, with the inner product linear in the first argument).

[L2]

The standard complex inner product is $⟨ z, w ⟩ = z \overline{w}$ in one dimension (The standard formulas $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and $\sum_{k < n} x_{k} \overline{y_{k}}$ on $C^{n}$ are inner products).

Refutation

technique · counterexample

1.1

By [L2], $⟨ i, 1 ⟩ = i$ , whereas $⟨ 1, i ⟩ = - i$ . These are unequal.

L2algebra

2.1

They are complex conjugates, exactly as [L1] requires. Thus conjugate symmetry, not symmetry, is the correct law.

step 1.1L1∎

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: 32 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

Sheldon Axler, Linear Algebra Done Right, 4th ed., definition 6.2 (standard reference, not scraped)

FALSE: Every complex inner product satisfies ⟨u,v⟩=⟨v,u⟩⟨u,v⟩=⟨v,u⟩