ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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.

The unit ball is Levi pseudoconvex

Example

The unit ball

$B = {z \in C^{m} : ∣ z_{1} ∣^{2} + \dots + ∣ z_{m} ∣^{2} < 1}$

is Levi pseudoconvex.

Facts & Assumptions

Given: The defining function $ρ (z) = ∣ z_{1} ∣^{2} + \dots + ∣ z_{m} ∣^{2} - 1$ of the unit ball.

[L1]

Levi pseudoconvexity is tested by the Levi form of a defining function on complex tangent vectors (Levi pseudoconvex domains).

[L2]

The Levi form is $L_{ρ} (z; v) = \sum_{j, k} \frac{\partial^{2} ρ}{\partial z_{j} \partial {\overline{z}}_{k}} (z) v_{j} \overline{v_{k}}$ (The Levi form and strict plurisubharmonicity).

Verification

technique · direct

1.1

For $ρ (z) = ∣ z_{1} ∣^{2} + \dots + ∣ z_{m} ∣^{2} - 1$ , one has $\frac{\partial^{2} ρ}{\partial z_{j} \partial {\overline{z}}_{k}} = δ_{j k},$ so [L2] gives $L_{ρ} (z; v) = ∣ v_{1} ∣^{2} + \dots + ∣ v_{m} ∣^{2} \geq 0$ for every $z$ and every $v \in C^{m}$ .

L2givenalgebra

2.1

In particular the Levi form is nonnegative on every complex tangent vector at every boundary point of the unit ball. By [L1], the unit ball is Levi pseudoconvex.

L1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

2 results 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

Jiří Lebl, Tasty Bits of Several Complex Variables, §2.3 (standard reference, not scraped)
Harold P. Boas, Lecture Notes on Several Complex Variables, §3.3.1 (standard reference, not scraped)