CounterexampleConstruction: 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 colliding color seed cannot distinguish two Jacobians with different entries

Statement refuted

For any coloring of the columns, one compressed product determines every Jacobian entry allowed by the row support.

Facts & Assumptions

Given: One row with safe support $S_{0} = {0, 1}$ , a coloring that assigns both columns the same color, and seed $s = (1, 1)^{T}$ .

[F1]

The column-interference graph and collision-free coloring for a sparse Jacobian: the compressed observation for seed $s$ is $J s$ .

[F2]

A collision-free column coloring recovers every structurally sparse Jacobian entry: the recovery guarantee requires a proper coloring, and collision-free supports permit reading one entry from each compressed row.

Counterexample

Given: The row pattern and colliding seed in the Facts & Assumptions.

1.1givenF1constructalgebra

The distinct candidate Jacobian rows $J = (1, 2)$ and $K = (3, 0)$ are realized at $(0, 0)$ by the linear maps $F (z_{0}, z_{1}) = z_{0} + 2 z_{1}$ and $G (z_{0}, z_{1}) = 3 z_{0}$ . Both supports lie in $S_{0}$ , but [F1] gives $J s = 1 + 2 = 3$ and $K s = 3 + 0 = 3$ .

2.1givenF2step 1.1algebra∎

Since the same observation is compatible with different entries in columns $0$ and $1$ , this colliding seed does not identify the row. The collision violates the proper-coloring hypothesis of [F2], so the unconditional statement is false.

Depends on

Used by

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Dependency tree · two levels

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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)