False statementConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

FALSE: unit-ball volume increases with dimension

Statement

False claim: the volume $V_{n} (1)$ of the unit ball increases with the positive integer dimension $n$ .

Facts & Assumptions

Given: The sequence of positive-dimensional unit-ball volumes.

[F1]

Among positive integer dimensions, the unit-ball volume is uniquely maximal at $n = 5$ (The unit-ball volume is maximal in dimension five).

[F2]

$V_{n} (1) \to 0$ as $n \to \infty$ (The volume of the unit $n$ -ball tends to zero with dimension).

Refutation

technique · direct

1.1F1

Fact [F1] gives the explicit strict decrease $V_{6} (1) < V_{5} (1)$ , contradicting monotone increase.

1.2F2

Independently, [F2] says the positive volumes tend to zero, which is incompatible with an increasing positive sequence.

2.1step 1.1step 1.2∎

Either step refutes the universal monotonicity claim.

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

Sheldon Axler, Measure, Integration & Real Analysis, §5C, Exercise 12 (standard reference, not scraped)