ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 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.

The unit-ball volumes through dimension eight from the Gamma formula

Example

For dimensions $1$ through $8$ , the unit-ball volumes are

$2, π, \frac{4 π}{3}, \frac{π^{2}}{2}, \frac{8 π^{2}}{15}, \frac{π^{3}}{6}, \frac{16 π^{3}}{105}, \frac{π^{4}}{24} .$

Facts & Assumptions

Given: Positive integer dimensions $1 \leq n \leq 8$ .

[F1]

For every $n \geq 1$ , $V_{n} (1) = π^{n / 2} / Γ (n / 2 + 1)$ (The closed form for the volume of the unit $n$ -ball).

[F2]

For every $s > 0$ , $Γ (s + 1) = s Γ (s)$ (The real Gamma functional equation $Γ (s + 1) = s Γ (s)$ ).

[F3]

$Γ (1 / 2) = \sqrt{π}$ ( $Γ (1 / 2) = \sqrt{π}$ from the Gaussian integral).

Verification

technique · direct

1.1

Substitute $n = 1, 2, 3, 4$ into [F1] and use [F2] and [F3]. This gives $2, π, 4 π / 3, π^{2} / 2$ .

F1F2F3algebra

1.2

The same substitution with [F2] and [F3] for $n = 5, 6, 7, 8$ gives $8 π^{2} / 15, π^{3} / 6, 16 π^{3} / 105, π^{4} / 24$ .

F1F2F3algebra

2.1

Steps 1.1 and 1.2 establish every value in the displayed list.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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

University of Toronto MAT237Y1, The Gamma Function and the Beta Function, §2.4 (standard reference, not scraped)
Sheldon Axler, Measure, Integration & Real Analysis, §5C (standard reference, not scraped)