ExampleConstruction: AI-adaptedVerification: Not suppliedSession-authored (Fable 5 assisted)audited 2026-07-31

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 endpoint Möbius value of the four-element diamond is $1$

Example

Let the diamond have bottom $\bot$ , top $\top$ , and incomparable middle elements $a,b$ . It is the Boolean lattice on a two-element set, so For $A\subseteq B$ in a finite Boolean lattice, $\mu(A,B)=(-1)^{\lvert B\setminus A\rvert}$ gives

$\mu(\bot,\bot)=1,\qquad \mu(\bot,a)=\mu(\bot,b)=-1,\qquad \mu(\bot,\top)=1.$

The endpoint recurrence displays the same computation directly:

$1-1-1+\mu(\bot,\top)=0,$

and hence $\mu(\bot,\top)=1$ (The Möbius recurrence: $\mu_P(x,x)=1$ and both interval sums of $\mu_P$ vanish when $x<y$ ).

Depends on

Used by

A four-element chain and a four-element diamond have equal-size endpoint intervals but Möbius values 0 and 1 Counterexample

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 68 results over 16 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

R. Stanley, Enumerative Combinatorics, Volume 1, §§3.6–3.8 (standard reference, not scraped)

The endpoint Möbius value of the four-element diamond is 111