CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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 degenerate two-parameter map can collapse its image to a curve

Statement refuted

A continuous map of a two-dimensional parameter region into $R^{3}$ need not describe a regular surface: its image can be only a curve.

Facts & Assumptions

Given: The map $φ : [0, 1]^{2} \to R^{3}$ , $φ (u, v) = (u, 0, 0)$ .

[L1]

A regular surface patch must be injective in its interior and have nonzero parameter cross product there (Regular parametrized surface patches on compact Jordan parameter regions).

[L2]

The cross product is given by its coordinate formula (The cross product in $R^{3}$ ).

Counterexample

technique · direct

1.1

The parameter derivatives are $φ_{u} = (1, 0, 0)$ and $φ_{v} = (0, 0, 0)$ , so [L2] gives $φ_{u} \times φ_{v} = 0$ at every point.

givenL2algebra

1.2

Also $φ (u, v)$ is independent of $v$ , so the map is not injective on the interior, and its image is exactly the line segment ${(u, 0, 0) : 0 \leq u \leq 1}$ .

givenalgebra

2.1

Both regularity requirements in [L1] fail throughout the interior, and the image is one-dimensional. This supplies the claimed collapse.

step 1.1step 1.2L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

M. E. Taylor, Introduction to Analysis in Several Variables, regular surface charts (standard reference, not scraped)