ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge 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 zero tangent angle makes the bordered corrector singular

Example

For $F (x, y) = x$ , the solution set is the vertical line $x = 0$ . A hyperplane normal perpendicular to its tangent does not define a regular local corrector.

Facts & Assumptions

Given: $F : R^{2} \to R$ , $F (x, y) = x$ , at $(0, 0)$ .

[F1]

Invertibility and a norm bound for the pseudo-arclength bordered matrix: for a rank-one derivative with unit kernel tangent $t$ , the bordered matrix $[D F; τ^{T}]$ is invertible exactly when $τ^{T} t \neq 0$ .

Verification

technique · direct

1.1givenalgebra

The derivative is $D F (0, 0) = [1 0]$ , which has rank $1$ and kernel $span {t}$ for the unit tangent $t = (0, 1)$ .

2.1givenstep 1.1algebra

Take the unit hyperplane normal $τ = (1, 0)$ . Then $τ^{T} t = 0$ and $[\begin{array}{c} D F (0, 0) \\ τ^{T} \end{array}] = [\begin{array}{cc} 1 & 0 \\ 1 & 0 \end{array}],$ whose rows are dependent and whose kernel contains $t \neq 0$ .

3.1F1step 2.1∎

By [F1], the bordered matrix is singular exactly at this failed angle; the inverse bound also cannot hold. This is the tangent-angle failure required by the continuation algorithm.

Depends on

Invertibility and a norm bound for the pseudo-arclength bordered matrix

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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