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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 0 and 1

Statement refuted

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

Facts & Assumptions

Given: The four-element chain c0<c1<c2<c3 and the four-element diamond with endpoints ⊥<⊤.

[F1]

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

[F2]

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

Counterexample

technique · direct
1.1given

Both endpoint intervals have four elements.

1.2F1F2

Their endpoint Möbius values are nevertheless 0 and 1 by [F1] and [F2].

2.1step 1.1step 1.2∎

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

Remarks

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

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.

Sources