LemmaStatement: AI-adaptedProof: AI-generatedverified 2026-10-22 (gpt-6.1-sol)

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.

Lorentzian duality sign

Physical scope

Pure mathematics; no physical premises or physical claim.

Statement

For the orientation and signature $η = diag (- 1, 1, 1, 1)$ fixed in Lorentzian Hodge star and periods, the Hodge star satisfies $⋆_{η}^{2} = (- 1)^{k (4 - k) + 1} id$ on $k$ -forms, $0 \leq k \leq 4$ . In particular, on two-forms $⋆_{η}^{2} = - id$ , $⋆_{η} (d x^{0} \land d x^{1}) = - d x^{2} \land d x^{3}$ , and $⋆_{η} (d x^{2} \land d x^{3}) = d x^{0} \land d x^{1}$ .

Statement references

Lorentzian Hodge star and periods

Facts & Assumptions

Given: The coframe, orientation, Lorentzian signs, and Hodge-star definition in the Statement.

[F1]

Lorentzian Hodge star and periods gives $⋆_{η} e^{I} = ϵ_{I} s (I, I^{c}) e^{I^{c}}$ , with exactly one negative $ϵ_{i}$ , and defines the orientation sign $s (I, I^{c})$ .

Proof

1.1F1givenalgebra

For any increasing $I \subseteq {0, 1, 2, 3}$ of size $k$ , apply [F1] twice to obtain $⋆_{η}^{2} e^{I} = ϵ_{I} ϵ_{I^{c}} s (I, I^{c}) s (I^{c}, I) e^{I}$ .

2.1F1step 1.1algebra∎

The product $ϵ_{I} ϵ_{I^{c}}$ is $- 1$ , since the two complementary index sets contain the unique negative metric sign exactly once. Reversing the blocks $I, I^{c}$ uses $k (4 - k)$ transpositions, so $s (I, I^{c}) s (I^{c}, I) = (- 1)^{k (4 - k)}$ . Thus the general square formula follows on the basis $e^{I}$ . For $I = {0, 1}$ , the star is $- e^{23}$ ; for $I = {2, 3}$ , it is $e^{01}$ , which gives the two displayed special cases.

Depends on

Lorentzian Hodge star and periods