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: strict convexity gives a positive-definite Hessian

Statement

False claim: every twice differentiable strictly convex function has a positive-definite Hessian at every point.

Facts & Assumptions

Given: No assumptions beyond the false claim.

[L1]

For every n≥1, the function f:Rn→R given by f(x)=∑j<nxj4 is strictly convex, but its Hessian at the origin is the zero matrix (A strictly convex function can have a singular Hessian).

Refutation

technique · direct
1.1

The function in [L1] satisfies strict convexity globally.

L1
2.1

Its Hessian at the origin is zero and hence not positive definite by [L1], directly refuting the claim.

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