CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-28

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 substitution x_i=x^i can erase colour-profile information

Statement refuted

Replacing every cycle-index variable by the one-variable specialization

$x_{i} = x^{i}$

still retains the full colour-profile information of a weighted inventory.

Facts & Assumptions

Given: the trivial action on the one-point set with two colours, blue and red.

[F1]

For the trivial action on one point, the cycle index is the single variable $s_{1}$ , and a blue/red weighting records the two colour orbits by the polynomial $b + r$ .

Counterexample

technique · direct

1.1

Give blue weight $b$ and red weight $r$ . By [F1], the full weighted inventory is $b + r$ , which distinguishes the blue orbit from the red orbit.

2.1

The one-variable specialization named in the statement replaces the cycle-index variables by $s_{i} = x^{i}$ , so here it sends $s_{1}$ to $x$ . The resulting polynomial $x$ no longer distinguishes blue from red, so the specialization has erased colour-profile information. Therefore the displayed claim is false.

step 1.1F1algebra∎

Depends on

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.