Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30
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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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Linearity alone does not make a tangent vector

Statement

False claim: every linear functional on Cp∞(M) is a tangent vector.

Facts & Assumptions

Given: The evaluation functional E([f]):=f(p).

[F1]

A tangent vector is a linear functional satisfying the Leibniz rule (Derivations at a point and the tangent space).

Refutation

technique · direct
1.1

The map E is linear.

given
1.2

But E([1]⋅[1])=1, while 1 E([1])+1 E([1])=2, so E does not satisfy the Leibniz rule in [F1].

F1given
2.1

Therefore not every linear functional is a tangent vector.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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