LemmaStatement: AI-generatedProof: AI-generated

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.

Gram determinant for the two symmetry directions

Statement

Let $X_{1} = T$ and $X_{2} = Z$ be the generators of the action in Commuting complete flows give the global product action. Define their Gram matrix by $H_{i j} = g (X_{i}, X_{j})$ and their signed orbit-area square by $W = - \det H$ . Under an invertible real change of basis $X_{i}^{'} = A_{i}^{j} X_{j}$ , one has $H^{'} = A H A^{T}$ and $W^{'} = (\det A)^{2} W$ . Consequently invertibility of the orbit Gram matrix and the sign of $W$ do not depend on that basis. Use the metric conventions in Stationary Lorentzian data and tensor conventions. This algebra does not prove that $W$ is positive anywhere.

Facts & Assumptions

[F1]

The stationary flow $ϕ_{t}$ is generated by $T$ , and the normalized axial flow $ψ_{s}$ is generated by $Z$ . The generators are smooth ambient vector fields and the metric is symmetric and bilinear. A basis change here is an invertible two-by-two real matrix at the point under consideration. Commuting complete flows give the global product action Stationary Lorentzian data and tensor conventions Smooth global data for the axisymmetric comparison Periodic isometries and invariant normal averaging

[F2]

The determinant of a real square matrix is its finite Leibniz sum over permutations. We use this definition for each real two-by-two matrix here. For n≥1, the determinant over a commutative ring by the Leibniz formula, and |det A| for a real matrix

[F3]

Matrix multiplication is given by $(B C)_{i j} = \sum_{k} B_{i k} C_{k j}$ , and transpose by $(B^{T})_{i j} = B_{j i}$ . These definitions apply to the real two-by-two matrices here. Entrywise ring-matrix operations, rectangular matrix products, identity matrices and transpose

[F4]

For square matrices $B, C$ of the same positive finite size over a commutative ring, $\det (B C) = \det (B) \det (C)$ and $\det (B^{T}) = \det B$ . Apply these results over $R$ at size two. For same-sized finite square matrices over a commutative ring, det(AB)=det(A)det(B) For every square matrix over a commutative ring, det(A^T)=det(A)

[F5]

A real square matrix of positive finite size is invertible if and only if its determinant is nonzero. A finite square real matrix is invertible if and only if its determinant is nonzero

Proof

**Given:**The generators, metric and change of basis at one point.

Goal: Compute the transformed matrix and signed determinant.

1.1givenF1F3

Bilinearity gives $g (X_{i}^{'}, X_{j}^{'}) = A_{i}^{k} A_{j}^{ℓ} H_{k ℓ}$ . By the definitions of matrix product and transpose this is exactly $H^{'} = A H A^{T}$ .

2.1F2F4step 1.1

Apply determinant multiplicativity twice and transpose invariance to obtain $\det H^{'} = \det (A) \det (H) \det (A^{T}) = (\det A)^{2} \det H$ . Negating this equality gives $W^{'} = (\det A)^{2} W$ .

3.1givenF5step 2.1algebra∎

Since $A$ is invertible, $\det A \neq 0$ by the real-matrix criterion; hence $(\det A)^{2} > 0$ . The determinant identity therefore gives $\det H^{'} \neq 0$ if and only if $\det H \neq 0$ . Applying the same criterion to $H^{'}$ and $H$ proves invertibility invariance, and the positive multiplier preserves the sign of $W$ , including its zero value.

Depends on

Used by · 3

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.