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 fixed LMNN target triplet can retain a positive margin loss

Example

Take $d = 1$ , $n = 3$ , $C = 2$ , inputs $x_{1} = 0$ , $x_{2} = 1$ , $x_{3} = - 1$ , labels $y_{1} = y_{2} = 1$ , $y_{3} = 2$ , fixed target sets $N_{1} = {2}$ , $N_{2} = {1}$ , $N_{3} = \emptyset$ , and $μ = \frac{1}{2}$ . Every positive-semidefinite matrix is a scalar $M = [m]$ with $m \geq 0$ . For the anchor $i = 1$ , target $j = 2$ , and impostor $l = 3$ , the LMNN hinge term is exactly $[1 + q_{M} (x_{1}, x_{2}) - q_{M} (x_{1}, x_{3})]_{+} = 1$ for every feasible $M$ ; therefore the fixed target triplet cannot attain zero margin loss.

Example references

Fixed target sets and the LMNN push hinge

Facts & Assumptions

Given: The sample, labels, fixed target sets, and weighting parameter in the Example.

[F1]

For $M ⪰ 0$ , every $q_{M} (x, x^{'})$ is nonnegative; the push terms are positive parts, and the full objective is the nonnegative weighted sum of the pull and push terms (Fixed-target large-margin nearest-neighbor objective).

Verification

1.1givenF1algebra

A scalar matrix $M = [m]$ is feasible exactly when $m \geq 0$ . For the designated triplet, $q_{M} (x_{1}, x_{2}) = (0 - 1)^{2} m = m$ and $q_{M} (x_{1}, x_{3}) = (0 - (- 1))^{2} m = m$ , so the hinge argument is $1 + m - m = 1$ and its value is $[1]_{+} = 1$ .

2.1step 1.1F1algebra∎

Every squared distance and every hinge term in $F_{1 / 2}$ is nonnegative. The designated hinge contributes $\frac{1}{2}$ , so $F_{1 / 2} (M) \geq \frac{1}{2}$ for every feasible $M$ , and this triplet's margin loss cannot be zero.

Depends on

Fixed-target large-margin nearest-neighbor objective

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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