False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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.

FALSE: exponential generating functions multiply without the labelled-product hypothesis

Statement

False claim: whenever two labelled classes $A$ and $B$ are combined in any way, the exponential generating function of the result is the product of the EGFs of $A$ and $B$ .

Facts & Assumptions

Given: The labelled product rule of The labelled constructions translate into the usual exponential-generating-function rules.

Proof

technique · direct

1.1

The product rule of The labelled constructions translate into the usual exponential-generating-function rules applies to the labelled product $A ⋆ B$ , where the label set is split into two disjoint parts, one for the $A$ -object and one for the $B$ -object.

given

2.1

If that disjointness requirement is dropped, two one-label structures can be forced to live on the same label. Then the combined object has size $1$ , whereas the EGF product would place it in degree $2$ . So multiplication is not a free rule about arbitrary combinations; it is the rule for the labelled product and depends on that hypothesis.

step 1.1given

3.1

Therefore the claim is false.

step 2.1∎

Depends on

The labelled constructions translate into the usual exponential-generating-function rules

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics — Symbolic Combinatorics (standard reference, not scraped)