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.

One color recovers two separated reaction columns of a sparse Jacobian

Example

Consider two independent first-order reactions $A \overset{k_{1}}{\to} B$ and $C \overset{k_{2}}{\to} D$ , with $k_{1}, k_{2} > 0$ and state $y = (a, b, c, d)$ . Their vector field is $F (y) = (- k_{1} a, k_{1} a, - k_{2} c, k_{2} c)$ .

Facts & Assumptions

Given: The vector field and positive rates above.

[F1]

The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case: the Jacobian entries are the coordinate partial derivatives of the vector field.

[F2]

The column-interference graph and collision-free coloring for a sparse Jacobian: columns sharing a row support interfere; a color seed is the indicator vector of its columns.

[F3]

A collision-free column coloring recovers every structurally sparse Jacobian entry: a proper coloring of a safe support pattern recovers each Jacobian entry from the corresponding compressed product.

Verification

1.1givenF1F2algebra

Differentiating the four components in coordinate order $(a, b, c, d)$ , indexed $0, 1, 2, 3$ , gives $J = (\begin{array}{cccc} - k_{1} & 0 & 0 & 0 \\ k_{1} & 0 & 0 & 0 \\ 0 & 0 & - k_{2} & 0 \\ 0 & 0 & k_{2} & 0 \end{array})$ . The row supports are ${0}, {0}, {2}, {2}$ , so no row contains two columns and the coloring that assigns all four columns one color is proper.

2.1givenF2F3step 1.1algebra∎

Its seed is $s = (1, 1, 1, 1)^{T}$ and $J s = (- k_{1}, k_{1}, - k_{2}, k_{2})^{T}$ . Each row has at most one supported column of this color, so [F3] recovers the nonzero entries from these four values; the remaining entries are zero by the declared pattern.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

Thomas F. Coleman and Jorge J. Moré, Estimation of Sparse Jacobian Matrices and Graph Coloring Problems, Argonne National Laboratory Report ANL-81-39 (standard reference, not scraped)
Bradley M. Bell and Kasper Kristensen, Computing Sparse Jacobians and Hessians Using Algorithmic Differentiation (standard reference, not scraped)