CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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 gauge of a nonbalanced set need not be a seminorm

Statement refuted

An absorbing convex set need not have a seminorm as its gauge.

Facts & Assumptions

Given: The convex absorbing set $C = (- 1, 2) \subseteq R$ .

[F1]

The gauge is the infimum of positive $t$ for which $x \in t C$ (Minkowski functional of an absorbing set).

Counterexample

technique · direct

1.1

From $x \in t (- 1, 2) ⟺ - t < x < 2 t$ , [F1] gives $p_{C} (x) = x / 2$ for $x \geq 0$ and $p_{C} (x) = - x$ for $x < 0$ .

F1givenalgebra

2.1

Hence $p_{C} (1) = 1 / 2$ while $p_{C} (- 1) = 1$ , contradicting the required equality $p_{C} (- 1) = ∣ - 1 ∣ p_{C} (1)$ . The missing hypothesis is balancedness.

step 1.1∎

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

Gerald Teschl, Topics in Real and Functional Analysis, §5.1 (standard reference, not scraped)