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.
Positive transverse accessibility between leaves
Definition
Let F be a C² transversely oriented codimension-one foliation on a smooth manifold M without boundary, with its chosen positive transverse direction. A positive transverse segment is a C² map c:[0,1]→M whose transverse derivative is strictly positive in every positively signed foliated chart, including the one-sided endpoint derivatives. It is an immersed curve and need not be embedded. For leaves A,B write when A=B or when such a nonempty segment has c(0)∈A and c(1)∈B. The clause A=B is formal reflexivity, corresponding to Novikov’s empty-segment convention; it does not assert a positive return segment or a closed transversal. This is the direction of Novikov’s A≥B: the positive segment goes from A to B.
Depends on
Used by
Dependency tree · two levels
3 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
- S. P. Novikov, The Topology of Foliations, English translation by J. A. Zilber (standard reference, not scraped)