ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

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 complex Jacobian and its determinant for $(z_{0} z_{1}, z_{0} + z_{1})$

Example

Let $F : C^{2} \to C^{2}$ be given by

$F (z_{0}, z_{1}) = (z_{0} z_{1}, z_{0} + z_{1}) .$

Then

$J_{C} F (z) = (\begin{array}{cc} z_{1} & z_{0} \\ 1 & 1 \end{array}), \det J_{C} F (z) = z_{1} - z_{0} .$

So the complex Jacobian determinant vanishes exactly on the diagonal ${z_{0} = z_{1}}$ . If $S (w_{0}, w_{1}) = (w_{1}, w_{0})$ is the coordinate swap, then $\det J_{C} S = - 1$ and $\det J_{C} (S \circ F) = - (z_{1} - z_{0})$ , in agreement with the multiplicative chain rule.

Facts & Assumptions

Given: The map $F (z_{0}, z_{1}) = (z_{0} z_{1}, z_{0} + z_{1})$ and the swap $S (w_{0}, w_{1}) = (w_{1}, w_{0})$ .

[L1]

A map is holomorphic exactly when its components are, and for a holomorphic map the complex Jacobian entries are $(J_{C} F)_{j k} = \partial_{z_{k}} F_{j}$ (A map into $C^{n}$ is holomorphic exactly when each of its components is). The coordinate projections $(z_{0}, z_{1}) \mapsto z_{0}$ and $(z_{0}, z_{1}) \mapsto z_{1}$ are complex-linear functionals and hence holomorphic (A real-linear functional on $C^{m}$ is complex linear exactly when its antiholomorphic part vanishes, Holomorphic functions on an open subset of $C^{m}$ ); sums and products of holomorphic functions are holomorphic with the usual derivative rules (Sums, products and nonvanishing quotients of holomorphic functions are holomorphic).

[L2]

The determinant of a $2 \times 2$ matrix is $a d - b c$ (For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix).

[L3]

The composite of holomorphic maps is holomorphic and its complex Jacobian is the product (The composite of holomorphic maps is holomorphic and its complex Jacobian is the product).

[L4]

For equidimensional holomorphic maps, the complex Jacobian determinant of a composite is the product of the determinants (The complex Jacobian determinant of a composite of equidimensional holomorphic maps is the product).

Verification

technique · direct

1.1

By [L1], $J_{C} F (z)$ is $(\begin{array}{cc} z_{1} & z_{0} \\ 1 & 1 \end{array})$ .

1.2

The swap $S$ is linear with matrix $(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array})$ , so $\det J_{C} S = - 1$ by [L2].

2.1

Apply [L2] to that matrix: $\det J_{C} F (z) = z_{1} \cdot 1 - z_{0} \cdot 1 = z_{1} - z_{0}$ , so the determinant vanishes exactly when $z_{0} = z_{1}$ .

step 1.1L2

3.1

By [L3], $J_{C} (S \circ F) = J_{C} S (F (z)) J_{C} F (z)$ , and [L4] gives $\det J_{C} (S \circ F) = (- 1) (z_{1} - z_{0}) = z_{0} - z_{1}$ , exactly as the direct calculation of the swapped matrix would give.

step 2.1step 1.2L3L4∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

46 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

J. Lebl, Tasty Bits of Several Complex Variables, v4.4, §1.3 (standard reference, not scraped)

The complex Jacobian and its determinant for (z0z1,z0+z1)(z0​z1​,z0​+z1​)