ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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-dimensional separable hard-margin SVM

Example

Take $X = R$ , $H = R$ , $ϕ (x) = x$ , and $k (x, z) = x z$ . For the sample $(- 1, - 1), (1, + 1)$ , the hard-margin SVM is feasible and its minimum objective value is $1 / 2$ , attained at $w = 1$ , $b = 0$ . The score is $s (x) = x$ , and both training points have signed margin $1$ .

Example references

Hastie, Tibshirani, and Friedman, The Elements of Statistical Learning, §12.2

Hard- and soft-margin support-vector machines with an unpenalized bias

Linear and polynomial kernels as finite feature inner products

Facts & Assumptions

Given: The feature map, kernel, two labeled points, and the hard-margin definition.

[F1]

The hard-margin problem minimizes $\frac{1}{2} ∥ w ∥^{2}$ subject to $y_{i} (⟨ w, ϕ (x_{i}) ⟩ + b) \geq 1$ when the sample is margin-separable (Hard- and soft-margin support-vector machines with an unpenalized bias).

[F2]

The linear kernel $k (x, z) = x z$ on $R$ is positive semidefinite (Linear and polynomial kernels as finite feature inner products).

Verification

1.1givenF1F2algebra

The feature map realizes the stated kernel directly: $⟨ ϕ (x), ϕ (z) ⟩ = x z = k (x, z)$ ; [F2] also verifies that this kernel is positive semidefinite. With $w, b \in R$ , the two hard-margin constraints are $w - b \geq 1$ and $w + b \geq 1$ . Adding them gives $w \geq 1$ .

2.1step 1.1F1algebra

Hence every feasible objective satisfies $\frac{1}{2} w^{2} \geq \frac{1}{2}$ . The candidate $w = 1, b = 0$ satisfies both constraints with equality and has objective $1 / 2$ , so it attains the minimum.

3.1step 2.1givenalgebra∎

For this minimizer, $s (x) = w x + b = x$ . Thus the signed margins at $(- 1, - 1)$ and $(1, + 1)$ are both $1$ , as stated.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

Hastie, Tibshirani, and Friedman, The Elements of Statistical Learning, 2nd ed., corrected 12th printing (standard reference, not scraped)