ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-29

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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.
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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 Hodge star in dimensions two, three, and four

Example

With the standard orientation and inner product on each space: in $R^{2}$ , $⋆ 1 = e_{1} \land e_{2}$ , $⋆ e_{1} = e_{2}$ , $⋆ e_{2} = - e_{1}$ , $⋆ (e_{1} \land e_{2}) = 1$ . In $R^{3}$ , $⋆ 1 = e_{1} \land e_{2} \land e_{3}$ , $⋆ e_{1} = e_{2} \land e_{3}$ , $⋆ e_{2} = e_{3} \land e_{1}$ , $⋆ e_{3} = e_{1} \land e_{2}$ , $⋆ (e_{1} \land e_{2}) = e_{3}$ , $⋆ (e_{2} \land e_{3}) = e_{1}$ , $⋆ (e_{3} \land e_{1}) = e_{2}$ , and $⋆ (e_{1} \land e_{2} \land e_{3}) = 1$ . In $R^{4}$ , $⋆ 1 = ω = e_{1} \land e_{2} \land e_{3} \land e_{4}$ , $⋆ e_{1} = e_{2} \land e_{3} \land e_{4}$ , $⋆ e_{2} = - e_{1} \land e_{3} \land e_{4}$ , $⋆ e_{3} = e_{1} \land e_{2} \land e_{4}$ , $⋆ e_{4} = - e_{1} \land e_{2} \land e_{3}$ , $⋆ (e_{1} \land e_{2}) = e_{3} \land e_{4}$ , $⋆ (e_{1} \land e_{3}) = - e_{2} \land e_{4}$ , $⋆ (e_{3} \land e_{4}) = e_{1} \land e_{2}$ , and $⋆ ω = 1$ .

Facts & Assumptions

Given: The standard oriented orthonormal bases of $R^{2}$ , $R^{3}$ , and $R^{4}$ .

[L1]

In a positively oriented orthonormal basis, $⋆ e_{I} = ε_{I} e_{I^{c}}$ , where $ε_{I}$ is the sign of the permutation listing $I$ followed by its complement (The Hodge star exists uniquely and is given by the complementary-basis formula in an oriented orthonormal basis).

[L2]

$⋆$ is an isometry and satisfies $⋆^{2} = (- 1)^{k (n - k)}$ on $Λ^{k}$ of an $n$ -dimensional space (The Hodge star is an isometry and satisfies $⋆^{2} = (- 1)^{k (n - k)}$ on $Λ^{k} V$ ).

Verification

technique · direct

1.1

In $R^{2}$ , the complementary basis and permutation signs of [L1] give $⋆ 1 = e_{1} \land e_{2}$ , $⋆ e_{1} = e_{2}$ , $⋆ e_{2} = - e_{1}$ , and $⋆ (e_{1} \land e_{2}) = 1$ .

L1algebra

1.2

In $R^{3}$ , the same formula gives the displayed list: for example $⋆ e_{2} = e_{3} \land e_{1}$ because the permutation $(2, 3, 1) \mapsto (1, 2, 3)$ has sign $+ 1$ , and $⋆ (e_{1} \land e_{2}) = e_{3}$ because $(1, 2, 3)$ is already ordered.

L1algebra

1.3

In $R^{4}$ , the formula gives the displayed values, e.g. $⋆ e_{2} = - e_{1} \land e_{3} \land e_{4}$ because $(2, 1, 3, 4)$ has sign $- 1$ , and $⋆ (e_{1} \land e_{3}) = - e_{2} \land e_{4}$ because $(1, 3, 2, 4)$ has sign $- 1$ .

L1algebra

2.1

Square checks with [L2]: in $R^{2}$ , $⋆ ⋆ e_{1} = ⋆ e_{2} = - e_{1} = (- 1)^{1 \cdot 1} e_{1}$ ; in $R^{3}$ , $⋆ ⋆ e_{1} = ⋆ (e_{2} \land e_{3}) = e_{1} = (- 1)^{1 \cdot 2} e_{1}$ ; in $R^{4}$ , $⋆ ⋆ (e_{1} \land e_{2}) = ⋆ (e_{3} \land e_{4}) = e_{1} \land e_{2} = (- 1)^{2 \cdot 2} (e_{1} \land e_{2})$ .

L2step 1.1step 1.2step 1.3algebra

2.2

In each dimension, $⋆$ permutes the orthonormal wedge basis up to sign, so it preserves the Gram norm, as [L2] states.

L2step 1.1step 1.2step 1.3

3.1

Steps 1.1 through 2.2 verify the displayed stars, the square signs, and the isometry in all three dimensions.

step 1.1step 1.2step 1.3step 2.1step 2.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Reyer Sjamaar, Manifolds and Differential Forms, §2.4 (standard reference, not scraped)