ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge 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 one-constraint scalar SDP and its dual

Example

Take $n = m = 1$ , $A_{1} = C = [1]$ , and $b_{1} = 1$ . The primal is maximize $x$ subject to $x \geq 0$ and $x \leq 1$ , so its feasible set is $[0, 1]$ and its attained optimum is $1$ . The dual is minimize $y$ subject to $y \geq 0$ and $y - 1 \geq 0$ , so its feasible set is $[1, \infty)$ and its attained optimum is $1$ . Every feasible pair obeys weak duality.

Facts & Assumptions

Given: Scalar positivity is the condition $x \geq 0$ ; the primal and dual signs are those of the finite SDP definition.

[F1]

Finite semidefinite primal-dual pair: the scalar primal constraint is $x \leq 1$ and the dual constraint is $y - 1 \geq 0$ .

[F2]

Weak duality for the finite SDP pair: every primal-feasible/dual-feasible pair satisfies objective $_{P} \leq$ objective $_{D}$ .

Verification

technique · direct

1.1givenF1algebra

The primal feasible interval is $[0, 1]$ , so $x \leq 1$ for every feasible $x$ , and $x = 1$ is feasible; hence the attained primal maximum is $1$ . The dual constraints are $y \geq 0$ and $y \geq 1$ , so every feasible $y$ is at least $1$ , with equality feasible; hence the attained dual minimum is $1$ .

2.1givenF2step 1.1algebra∎

For arbitrary feasible $x, y$ , step 1.1 gives $x \leq 1 \leq y$ , so the objectives satisfy weak duality, in agreement with [F2]. The equality of the two attained values is verified for this example only and is not inferred in general.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

https://arxiv.org/pdf/1705.01843v4 (standard reference, not scraped)