ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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.

Binary necklaces of length $6$ from both $CYC (2 Z)$ and Burnside's lemma

Example

There are $14$ binary necklaces of length $6$ .

Facts & Assumptions

Given: The necklace count $\frac{1}{n} \sum_{d ∣ n} φ (d) m^{n / d}$ (The number of necklaces of length $n$ on an $m$ -letter alphabet is $\frac{1}{n} \sum_{d ∣ n} φ (d) m^{n / d}$ ).

[L1]

Burnside's lemma counts orbits by averaging fixed points (Cauchy-Frobenius orbit counting: $∣ G ∣ ∣ X / G ∣ = \sum_{g \in G} ∣ X^{g} ∣$ for a finite group action).

Verification

technique · direct

1.1

The corollary with $m = 2$ and $n = 6$ gives $(1 / 6) (φ (1) 2^{6} + φ (2) 2^{3} + φ (3) 2^{2} + φ (6) 2^{1}) = (1 / 6) (64 + 8 + 8 + 4) = 14$ .

givenalgebra

1.2

Burnside gives the same value. The six rotations of a $6$ -bead necklace fix $64, 2, 4, 8, 4, 2$ colourings respectively, since the numbers of position-orbits are $6, 1, 2, 3, 2, 1$ . Their average is $(64 + 2 + 4 + 8 + 4 + 2) / 6 = 14$ .

L1algebra

2.1

The cycle-construction count and the direct Burnside count therefore agree at length $6$ .

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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 (standard reference, not scraped)

Binary necklaces of length 66 from both CYC⁡(2Z)CYC(2Z) and Burnside's lemma