Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)
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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.

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A connection is c infinity linear in the section being differentiated

Statement

For every connection, ∇X(fs)=f∇Xs for all smooth f, vector fields X and sections s.

Facts & Assumptions

Given: The proposed universal assertion.

[F1]

A connection satisfies ∇X(fs)=X(f)s+f∇Xs (Connection laws in directional form).

[F2]

A prescribed smooth connection matrix in a global frame defines a connection (Local connection forms glue exactly when they obey the transformation law).

Refutation

1.1

On E=R×R over R, take global frame e=1 and zero matrix, giving ∇X(ue)=X(u)e by [F2]. Put X=∂x, f(x)=x, s=e. Then ∇X(fs)=e, while f∇Xs=0.

F1F2given
2.1

At x=0 these are respectively the nonzero unit fibre vector and zero, disproving the assertion even in rank one and at a zero of f. The correct formula has the missing term X(f)s=e. A constant f or a zero section would not witness failure; this explicit nonconstant f and nonzero section do.

F1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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