ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-27

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 square corner carries an explicit power-barrier

Example

For the unit square $Q = (0, 1)^{2}$ , the corner $0$ has the explicit barrier $b (z) = - Re (z^{2 / 3}),$ where the branch of $z^{2 / 3}$ is taken on the first quadrant.

Facts & Assumptions

Given: The unit square $Q = (0, 1)^{2}$ and its corner $0$ .

[L1]

Exterior-cone points are regular because an explicit power-map barrier exists there (Exterior disc points and exterior cone points are regular).

[L2]

A barrier is a negative subharmonic function tending to $0$ at the marked boundary point and staying uniformly below a negative constant away from it (Barriers and regular boundary points).

Verification

technique · direct

1.1givenalgebra

On the first quadrant one may choose the holomorphic branch of $z^{2 / 3}$ . If $z = r e^{i t}$ with $0 < t < π / 2$ , then [given, algebra] $z^{2 / 3} = r^{2 / 3} e^{2 i t / 3}$ has argument in $(0, π / 3)$ , so $Re (z^{2 / 3}) > 0$ . Therefore $b (z) = - Re (z^{2 / 3}) < 0$ on $Q$ near the corner, and $b (z) \to 0$ as $z \to 0$ .

2.1L1L2step 1.1∎

On any set in the square that stays a positive distance from $0$ , the quantity $Re (z^{2 / 3})$ has a positive minimum, so $b$ stays uniformly below a negative constant there. Thus $b$ has exactly the shape required in [L2], and it is the concrete barrier predicted abstractly by [L1].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Sheldon Axler, Paul Bourdon, and Wade Ramey, Harmonic Function Theory, 2nd ed. (standard reference, not scraped)