ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

The affine plane and the pair of generic points

Example

For any field $k$ , $A_{k}^{1} \times_{k} A_{k}^{1} ≅ A_{k}^{2}$ functorially. Nevertheless the product has distinct points over the pair of generic points: in $k [x, y]$ , both $(0)$ and $(y - x)$ contract to $(0)$ in each of $k [x]$ and $k [y]$ .

Facts & Assumptions

Given: The objects, hypotheses and conventions in the statement above.

[F1]

Let $A \to B$ and $A \to C$ be maps of commutative unital rings, allowing the zero ring. In the category of all schemes, $Spec B \times_{Spec A} Spec C ≅ Spec (B \otimes_{A} C) .$ The projections correspond to $b \mapsto b \otimes 1$ and $c \mapsto 1 \otimes c$ . (Affine fibre products are spectra of tensor products)

[F2]

For scheme morphisms $f : X \to S$ and $g : Y \to S$ , points of $P = X \times_{S} Y$ are in bijection with quadruples $(x, y, s, r)$ where $f (x) = g (y) = s$ and $r \in Spec (κ (x) \otimes_{κ (s)} κ (y)) .$ The residue field at the corresponding point of $P$ is canonically $κ (r)$ . (Points of a fibre product via residue-field tensors)

Verification

1.1

F1 identifies the product ring with $k [x] \otimes_{k} k [y] ≅ k [x, y]$ : the maps send $x \otimes 1$ to $x$ and $1 \otimes y$ to $y$ , with inverse on every polynomial specified by these images. Consequently morphisms from every $k$ -scheme $T$ into the plane are compatible pairs of morphisms into the two lines. Empty test schemes are allowed.

givenF1algebra

2.1

The ring $k [x, y]$ is a domain, so $(0)$ is prime. The quotient by $(y - x)$ is $k [x]$ , also a domain, so this is another prime, distinct because $y - x \neq 0$ . Substitution $y = x$ is injective on either single-variable subring, giving the asserted contractions. F2 explains these as distinct residue-tensor primes over the same point pair. This works over every field, including characteristic two.

F2step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

Vakil 10.1.2–3 (standard reference, not scraped)