Alphabeta Math
PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generatedaudited 2026-09-10
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.

Hodge star squared sign

Statement

On real k-forms, ∗2=(−1)k(n−k)id⁡.

Facts & Assumptions

Given: An oriented Riemannian n-manifold and 0≤k≤n.

[F1]

Hodge star is a smooth bundle isomorphism: The Hodge star exists uniquely and is a smooth bundle isomorphism in every degree 0≤k≤n.

Proof

technique · direct
1.1

On the orthonormal wedge basis of the star construction, ∗2eI=ϵ(I,Ic)ϵ(Ic,I)eI. Switching the ordered blocks of lengths k and n−k takes k(n−k) adjacent transpositions, so the product of these two signs is (−1)k(n−k).

F1given
2.1

Linearity gives the identity for every form. For k=0 or k=n the exponent is zero and the square is the identity. In dimension zero the star multiplies by ε, whose square is one, giving the same formula.

F1step 1.1∎

Source locator

Lee, Problem 16-18(c–e), pp.437–438; block-transposition calculation above fixes the sign.

Depends on

Used by

Dependency tree · two levels

3 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