ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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.

An oscillation pattern controls clopen membership

Example

For $ξ < ω_{1}$ , form the disjoint blocks

$a_{ξ} = {3 ξ, 3 ξ + 1}, b_{ξ} = {3 ξ + 2},$

and let $A = {a_{ξ} : ξ < ω_{1}}$ and $B = {b_{ξ} : ξ < ω_{1}}$ . Apply Moore's pattern theorem with $k = 2$ , $l = 1$ , the constant map $π (i) = 0$ , and $χ (0) = 0$ , $χ (1) = 1$ . It returns $a = {a (0) < a (1)} \in A$ and $b = {b (0)} \in B$ with $a < b$ and

$c (a (0), b (0)) = 1, c (a (1), b (0)) = 0.$

Consequently the single point $b (0)$ has the prescribed simultaneous membership pattern

$b (0) \in W_{a (0)} and b (0) \notin W_{a (1)} .$

In Moore's topology on $ω_{1}$ , the finite Boolean combination

$U = W_{a (0)} \cap (ω_{1} ∖ W_{a (1)})$

is therefore a nonempty clopen basic neighborhood of $b (0)$ . This realizes two bits on the graph of one function; it does not claim control of an arbitrary $2 \times 1$ relation beyond those two graph entries (which in this case are all its entries), nor of an arbitrary matrix when $l > 1$ .

Facts & Assumptions

Given: The colouring and topology fixed on the companion page; ordinal multiplication and addition have their usual meanings.

[F1]

The Moore colouring realizes finite binary patterns realizes $o^{*} (a (i), b (π (i))) = χ (i)$ and hence $c (a (i), b (π (i))) = 1 - χ (i)$ for positive finite $k, l$ and uncountable pairwise-disjoint block families.

[F2]

Moore's clopen-generated topology defines $W_{α} = {α} \cup {β > α : c (α, β) = 1}$ and makes every finite Boolean combination of the $W_{α}$ clopen.

Verification

technique · direct application and calculation

1.1

The maps $ξ \mapsto 3 ξ$ , $3 ξ + 1$ , and $3 ξ + 2$ divide $ω_{1}$ into successive three-point blocks: each displayed ordinal is countable, $3 ξ < 3 ξ + 1 < 3 ξ + 2 < 3 (ξ + 1)$ , and different blocks are disjoint. Thus $A$ and $B$ are uncountable pairwise-disjoint families of two- and one-element subsets, respectively.

Givenalgebra

2.1

Use [F1] with $π (0) = π (1) = 0$ and $χ = (0, 1)$ . For the resulting $a < b$ , its parity conclusion gives $c (a (0), b (0)) = 1 - 0 = 1$ and $c (a (1), b (0)) = 1 - 1 = 0$ .

F1step 1.1

3.1

Because $a < b$ , both $a (0)$ and $a (1)$ are strictly below $b (0)$ . The defining endpoint clause in [F2] therefore turns the two equalities of step 2.1 into $b (0) \in W_{a (0)}$ and $b (0) \notin W_{a (1)}$ .

F2step 2.1

4.1

By [F2], $U = W_{a (0)} \cap (ω_{1} ∖ W_{a (1)})$ is clopen and basic, and step 3.1 puts $b (0)$ in it. Hence $U$ is nonempty and is a neighborhood of the stated point. Both bits, the shared column, the strict order, and the complement bit are explicit; the invocation uses positive $k = 2, l = 1$ and makes no assertion beyond the functional graph allowed by [F1].

F1F2step 3.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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