ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 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 surface area of a torus is $4 π^{2} a b$

Example

Let $a > b > 0$ . The torus obtained by revolving the circle of radius $b$ whose centre is distance $a$ from the axis has surface area $4 π^{2} a b$ .

Facts & Assumptions

Given: The parametrization $φ (θ, ϕ) = ((a + b \cos ϕ) \cos θ, (a + b \cos ϕ) \sin θ, b \sin ϕ)$ on $[0, 2 π]^{2}$ .

[L1]

The standard trigonometric derivative, Pythagorean, range, and full-period endpoint formulas hold (The derivatives of sine and cosine are cosine and minus sine, Parity and the Pythagorean identity for sine and cosine, Signs, monotonicity intervals, and ranges of sine and cosine, Quarter-turn values and shifts by pi/2 and pi).

[L2]

A regular patch has area $\int_{D} J_{φ}$ ; Jordan-Fubini and the fundamental theorem evaluate the rectangle integral (Regular parametrized surface patches on compact Jordan parameter regions, Surface area and scalar surface integrals on a regular patch, Fubini over a bounded Jordan set when all but a content-zero family of sections are integrable, The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ).

[L3]

The area density satisfies $J_{φ} = ∥ φ_{u} \times φ_{v} ∥_{2}$ at every parameter point, with the cross product given by the coordinate formula (The surface area density is the norm of the cross product of the parameter tangents, The cross product in $R^{3}$ ).

Verification

technique · direct

1.1

Differentiating with [L1] and using the coordinate formula in [L3] gives $∥ φ_{θ} \times φ_{ϕ} ∥_{2} = b (a + b \cos ϕ)$ , which by [L3] is the area density $J_{φ}$ . Since $a > b > 0$ it is positive; the only repeated parameter values occur on the rectangle boundary seams, so [L2] gives a regular patch.

givenL1L2L3algebra

2.1

By [L2] and step 1.1, the area is $\int_{0}^{2 π} \int_{0}^{2 π} b (a + b \cos ϕ) d ϕ d θ$ .

step 1.1L2L3

3.1

The full-period integral of cosine is zero, so the inner integral is $2 π a b$ and the outer integral gives $4 π^{2} a b$ .

step 2.1L1L2algebra

4.1

This is the asserted torus area, with positivity of $a - b$ having discharged the possible degeneracy.

step 1.1step 3.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

46 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

M. E. Taylor, Introduction to Analysis in Several Variables, Exercise 15 (standard reference, not scraped)

The surface area of a torus is 4π2ab4π2ab