ExampleConstruction: AI-generatedVerification: AI-generatedSession-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 one-sheeted hyperboloid is a regular surface of revolution

Example

The one-sheeted hyperboloid $H = {(x, y, z) \in R^{3} : x^{2} + y^{2} - z^{2} = 1}$ is a regular level set and the surface obtained by rotating the profile $x = \sqrt{1 + z^{2}}$ in the half-plane $y = 0$ , $x > 0$ , about the $z$ -axis.

Facts & Assumptions

Given: The polynomial $F (x, y, z) = x^{2} + y^{2} - z^{2}$ .

[L1]

The power rule and derivative algebra give the continuous Jacobian row $(2 x, 2 y, - 2 z)$ , and the continuous-partials theorem makes $D F (x, y, z) (h) = 2 x h_{1} + 2 y h_{2} - 2 z h_{3}$ (For a natural $n \geq 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $ι (n) x^{n - 1}$ ; for $n = 0$ it is the constant $1$ , with derivative $0$ ; for a natural $n \geq 1$ the function $x \mapsto x^{- n}$ is differentiable at every $x \neq 0$ with derivative $- ι (n) x^{- n - 1}$ ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative).

[L2]

Every positive real has a unique positive square root (Existence and uniqueness of $n$ -th roots: a unique $a^{1 / n} \geq 0$ with $(a^{1 / n})^{n} = a$ ); a regular level is locally a graph with tangent space equal to the derivative kernel (A regular level set is locally a $C^{k}$ graph of dimension $m - n$ , The tangent space to a regular level set).

Verification

technique · direct

1.1

On $F^{- 1} (1)$ the point $(x, y, z)$ cannot be $0$ , so the coefficient vector $(2 x, 2 y, - 2 z)$ in [L1] is nonzero and $D F (x, y, z)$ is surjective onto $R$ .

givenL1

1.2

The level equation is $x^{2} + y^{2} = 1 + z^{2}$ . By [L2], for each $z$ its horizontal section is the circle of positive radius $\sqrt{1 + z^{2}}$ , exactly the rotation of the stated profile.

givenL2algebra

2.1

Hence $1$ is a regular value, and [L2] gives tangent plane ${h : x h_{1} + y h_{2} - z h_{3} = 0}$ at $(x, y, z)$ .

step 1.1L2

3.1

The radius never vanishes, so the rotation has no apex or rank-drop point; steps 1.1 and 1.2 establish both asserted properties.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

53 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

J. M. Lee, Introduction to Smooth Manifolds, regular-level examples (standard reference, not scraped)