False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 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.

False: $\mu_P(x,y)$ depends only on the cardinality of $[x,y]$

Statement

If two finite intervals have the same cardinality, then their endpoint Möbius values are equal.

Facts & Assumptions

Given: A four-element chain $c_0<c_1<c_2<c_3$ and the four-element diamond $\{\bot,a,b,\top\}$ with $\bot<a<\top$ , $\bot<b<\top$ , and $a,b$ incomparable.

[L1]

On a finite chain the endpoint value is $0$ whenever the interval contains an intermediate element (On a finite chain, the Möbius function is $1$ on the diagonal, $-1$ on covers and $0$ on longer intervals).

[L2]

The diamond is the Boolean lattice on a two-element set, whose endpoint value is $(-1)^2=1$ (For $A\subseteq B$ in a finite Boolean lattice, $\mu(A,B)=(-1)^{\lvert B\setminus A\rvert}$ ).

Refutation

technique · direct

1.1

The endpoint interval $[c_0,c_3]$ has four elements and Möbius value $0$ by [L1].

1.2

The endpoint interval $[\bot,\top]$ in the diamond also has four elements but has Möbius value $1$ by [L2].

2.1

Equal interval cardinality therefore does not determine the Möbius value; the internal order structure matters.

step 1.1step 1.2∎

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: 72 results over 19 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)

False: μP(x,y)\mu_P(x,y)μP​(x,y) depends only on the cardinality of [x,y][x,y][x,y]