Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21
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: a positive-semidefinite Hessian gives strict convexity

Statement

False claim: an everywhere-positive-semidefinite Hessian forces a C2 function on an open convex set to be strictly convex.

Facts & Assumptions

Given: No assumptions beyond the false claim.

[L1]

The Hessian of f(x,y)=x2 is positive semidefinite everywhere, but f is not strictly convex (A positive-semidefinite Hessian need not give strict convexity).

Refutation

technique · direct
1.1

The function in [L1] satisfies the Hessian hypothesis of the false claim.

L1
2.1

It is constant along every vertical line and therefore violates strict convexity by [L1]. Hence the claim is false.

step 1.1L1∎

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