Alphabeta MathHow 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.
Convexity conventions, endpoint scope, dyadic approximation, and the exact use of the Axiom of Choice in the midpoint-convex counterexample
The continuity conclusion for a convex function is an interior conclusion: a convex function on a non-open interval need not have an endpoint derivative, and this development does not impose endpoint continuity beyond what a separate hypothesis supplies.
The midpoint-convex counterexample uses the Axiom of Choice exactly through the existence of a Hamel basis (The Axiom of Choice, Assuming the Axiom of Choice, has a Hamel basis over : there is such that every real is a finite -linear combination of elements of in exactly one way, and each basis vector carries a well-defined -linear coefficient map). No choice principle is used in the dyadic induction or in the passage from continuous midpoint convexity to convexity.
Depends on
- The Axiom of Choice
- Assuming the Axiom of Choice, $\mathbb{R}$ has a Hamel basis over $\mathbb{Q}$: there is $B \subseteq \mathbb{R}$ such that every real is a finite $\mathbb{Q}$-linear combination of elements of $B$ in exactly one way, and each basis vector carries a well-defined $\mathbb{Q}$-linear coefficient map
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
30 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
- R. Gardner, Convex Functions, Notes 6.6 (standard reference, not scraped)
- P. Green and W. Gustin, On the gap between convexity and midpoint convexity (standard reference, not scraped)