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.
Every continuous group homomorphism is smooth by definition
Statement refuted
Every continuous group homomorphism between Lie groups is smooth by definition.
Facts & Assumptions
Given: The library definition of a Lie-group homomorphism.
A Lie-group homomorphism is required to be both a group homomorphism and a smooth map. Lie-group homomorphism, isomorphism, and automorphism.
Refutation
By [F1], smoothness is an explicit defining hypothesis. Continuity alone is not the same syntactic condition and the definition contains no implication from continuity to smoothness.
The automatic-smoothness assertion for continuous homomorphisms is a substantive theorem requiring proof; it cannot be obtained merely by unpacking [F1]. Therefore the qualification “by definition” is false even though the separate theorem is true.
This is a claim about logical provenance, so dimensions zero and one do not alter it. Lie groups are nonempty; no metric, degeneracy, interval, endpoint, choice, example witness, or biconditional occurs.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
4 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
- Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed. (standard reference, not scraped)
- Alexander Kirillov Jr., An Introduction to Lie Groups and Lie Algebras (standard reference, not scraped)