ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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.

A rank-deficient Mahalanobis matrix identifies distinct points

Example

On $R^{2}$ , let $M = (\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array})$ , $u = (0, 0)$ , and $v = (0, 1)$ . The matrix is symmetric, positive semidefinite, and singular, while $u \neq v$ . Its Mahalanobis pseudometric satisfies $d_{M} (u, v) = 0$ . Thus the pseudometric need not separate distinct points, whereas a metric requires distance zero only for equal points.

Example references

Definition of the squared Mahalanobis distance and its square root

Pseudometric properties and linear-feature representation

Facts & Assumptions

Given: The matrix $M$ and points $u, v$ in the Example.

[F1]

For $M ⪰ 0$ , $q_{M} (x, x^{'}) = (x - x^{'})^{T} M (x - x^{'})$ and $d_{M} (x, x^{'}) = \sqrt{q_{M} (x, x^{'})}$ . A pseudometric need not separate distinct points (Mahalanobis pseudometric induced by a positive-semidefinite matrix).

[F2]

The square-root distance associated with a positive-semidefinite matrix satisfies the pseudometric axioms (The Mahalanobis distance is a pseudometric and comes from a linear feature map).

Verification

1.1givenalgebra

The matrix $M$ is symmetric, and for every $(a, b) \in R^{2}$ , $(a, b) M (a, b)^{T} = a^{2} \geq 0$ ; its rank is one, so it is singular. Also $u \neq v$ .

1.2givenF1algebra

The difference $u - v = (0, - 1)$ satisfies $(u - v)^{T} M (u - v) = 0$ , hence $q_{M} (u, v) = 0$ and $d_{M} (u, v) = \sqrt{0} = 0$ by [F1].

2.1step 1.2F2∎

Step 1.2 exhibits zero distance between distinct points, so this valid Mahalanobis pseudometric fails the separation property required of a metric; [F2] confirms its other pseudometric axioms.

Depends on

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