Alphabeta Math
CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10
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 matrix with one zero off diagonal is not a gcm

Statement refuted

False claim: every integer matrix with diagonal entries 2 and nonpositive off-diagonal entries is a GCM.

Facts & Assumptions

Given: The witness A with rows (2,0) and (−1,2).

[F1]

GCMs require a symmetric zero pattern. (Generalized cartan matrix).

Counterexample

1.1

Take A=(20−12). All four entries are integers, both diagonal entries are 2, and its off-diagonal entries 0 and −1 are nonpositive. Thus every hypothesis of the false claim holds.

given
2.1

Here a12=0 but a21=−1≠0. This violates the forward implication in the symmetric-zero condition of F1, so A is not a GCM. It is the required counterexample.

F1step 1.1∎

Sources

The displayed finite computation is local.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

1 result within one dependency step 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