Alphabeta Math
CounterexampleConstruction: AI-generatedVerification: 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.

A four-element chain and a four-element diamond have equal-size endpoint intervals but Möbius values 00 and 11

Statement refuted

The endpoint Möbius value of a finite interval is determined by the number of elements in that interval (False: μP(x,y)\mu_P(x,y) depends only on the cardinality of [x,y][x,y]).

Facts & Assumptions

Given: The four-element chain c0<c1<c2<c3c_0<c_1<c_2<c_3 and the four-element diamond with endpoints ⊥<⊤\bot<\top.

[F1]

The chain computation gives μ(c0,c3)=0\mu(c_0,c_3)=0 (The Möbius table of a four-element chain).

[F2]

The diamond computation gives μ(⊥,⊤)=1\mu(\bot,\top)=1 (The endpoint Möbius value of the four-element diamond is 11).

Counterexample

technique · direct
1.1

Both endpoint intervals have four elements.

given
1.2

Their endpoint Möbius values are nevertheless 00 and 11 by [F1] and [F2].

F1F2
2.1

Hence equal-size intervals can have different Möbius values, and the statement is false.

step 1.1step 1.2∎

Remarks

four-elementchainfour-elementdiamondc0c1c2c3?ab>¹(c0;c3)=0¹(?;>)=1

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 21 results over 9 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