ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16

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.

The Gram determinant of two vectors is $∥ u ∥^{2} ∥ v ∥^{2} - ∣ ⟨ u, v ⟩ ∣^{2}$ and detects dependence

Example

For two vectors $u, v$ , the Gram determinant is

$\det G (u, v) = ∥ u ∥^{2} ∥ v ∥^{2} - ∣ ⟨ u, v ⟩ ∣^{2} .$

In standard $R^{2}$ , $u = (1, 0)$ and $v = (1, 2)$ give determinant $4$ . Replacing $v$ by $2 u$ gives determinant $0$ .

Facts & Assumptions

Given: Two vectors $u, v$ in an inner product space.

[L1]

A Gram determinant is positive exactly for an independent list and zero exactly for a dependent list (A Gram determinant is nonnegative and is positive exactly when the vector list is linearly independent).

[L2]

Cauchy–Schwarz says $∣ ⟨ u, v ⟩ ∣^{2} \leq ∥ u ∥^{2} ∥ v ∥^{2}$ , with equality exactly for dependent $u, v$ (Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors).

Verification

technique · computation

1.1

Expanding the determinant of $(\begin{array}{cc} ⟨ u, u ⟩ & ⟨ u, v ⟩ \\ ⟨ v, u ⟩ & ⟨ v, v ⟩ \end{array})$ and using conjugate symmetry gives the displayed formula. Its nonnegativity and equality case agree with [L1] and [L2].

L1L2algebra

2.1

For $(1, 0), (1, 2)$ , the Gram matrix is $(\begin{array}{cc} 1 & 1 \\ 1 & 5 \end{array})$ , whose determinant is $4$ ; this is positive, and $(1, 0), (1, 2)$ are independent, as [L1] requires. For $(u, 2 u)$ , the two rows are dependent and the determinant is $0$ .

step 1.1L1algebra∎

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The Gram determinant of two vectors is ∥u∥2∥v∥2−∣⟨u,v⟩∣2∥u∥2∥v∥2−∣⟨u,v⟩∣2 and detects dependence