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 regression threshold tree

Example

Take a single predictor and the regression sample $(x_{i}, y_{i}) = (0, 0), (1, 0), (2, 1), (3, 1)$ , with maximum depth $1$ . Its candidate thresholds are the midpoints $0.5$ , $1.5$ , and $2.5$ . The child-weighted squared-error score is $2 / 3$ at $0.5$ , $0$ at $1.5$ , and $2 / 3$ at $2.5$ . Thus greedy growth chooses the unique split $x_{1} \leq 1.5$ , producing leaves on feature values ${0, 1}$ and ${2, 3}$ with predictions $0$ and $1$ .

The internal label in the diagram is the split predicate. The left edge means the predicate holds and the right edge means it fails. Each leaf gives the feature values reaching it and its constant prediction.

Facts & Assumptions

Given: The four regression observations and the greedy growth procedure with maximum depth $1$ .

[F1]

Empirical axis-aligned binary decision trees: A nonempty regression leaf predicts its sample mean and has residual sum of squares $\sum_{i \in I} (y_{i} - {\overset{ˉ}{y}}_{I})^{2}$ .

[F2]

Greedy axis-aligned CART growth on a finite sample: Candidate thresholds lie between adjacent distinct coordinate values, and the procedure chooses the candidate minimizing the child score.

[F3]

Greedy CART minimizes the listed split loss and terminates: Every nonempty coordinate-threshold partition occurs among the midpoint candidates, and the chosen split minimizes the child loss on that finite list.

Verification

1.1givenF1F2algebra

The four distinct coordinate values are $0 < 1 < 2 < 3$ , so the midpoint candidates are exactly $0.5$ , $1.5$ , and $2.5$ . At $0.5$ , the right child has responses $0, 1, 1$ , mean $2 / 3$ , and residual sum $2 / 3$ . At $1.5$ , both children have constant responses, so the score is $0$ . At $2.5$ , the left child has responses $0, 0, 1$ , mean $1 / 3$ , and residual sum $2 / 3$ .

2.1F1F3step 1.1algebra

The middle threshold is therefore the unique minimum-scoring candidate. By [F3], it is the best listed split and the procedure chooses it. Its two leaves have means $0$ and $1$ , as displayed.

3.1F2step 2.1algebra∎

The root split has depth $1 = d_{\max}$ , so the procedure stops there and returns the stated two-leaf tree. The diagram merely records the split and leaf predictions; the previously computed scores and the depth cap establish the output independently of the diagram.

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.