ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generated

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.

Two contradictory labels make the cyclic perceptron repeat a two-update cycle

Example

Use the ordered one-dimensional sample $(x_{1}, y_{1}) = (0, + 1)$ , $(x_{2}, y_{2}) = (0, - 1)$ . It is not separable by an affine score because both observations have the same augmented input $(0, 1)$ and require opposite strict signs. For the zero-initialized cyclic perceptron, every pass updates on both observations and returns the weight vector to zero, so no no-update pass occurs.

Example references

The cyclic perceptron update procedure

Hastie, Tibshirani, and Friedman, ESL, §4.5.1

Stanford CS229 Spring 2022 lecture notes, §2.2

Facts & Assumptions

Given: The ordered sample and the zero-initialized procedure with augmented input $(x, 1)$ .

[F1]

The procedure updates $w \leftarrow w + y_{i} (x_{i}, 1)$ whenever $y_{i} w^{T} (x_{i}, 1) \leq 0$ , and terminates only after a full pass with no updates. The cyclic perceptron update procedure

Verification

1.1givenF1algebra

Both augmented inputs equal $(0, 1)$ . Starting at $w_{0} = (0, 0)$ , the first observation has margin zero and triggers the update $w_{1} = (0, 1)$ .

2.1F1step 1.1algebra∎

The second observation then has signed margin $(- 1) (0, 1)^{T} (0, 1) = - 1$ , so [F1] updates to $w_{2} = (0, 1) - (0, 1) = (0, 0)$ . Thus each pass repeats the initial state and updates twice, so the stop condition never holds. The sample is nonseparable because a single score at the shared input cannot be strictly positive and strictly negative simultaneously.

Depends on

The cyclic perceptron update procedure

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

1 result within one dependency step 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)
Stanford CS229 Spring 2022 lecture notes (standard reference, not scraped)