DefinitionDefinition: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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.

Bolker condition for a double-fibration transform

Definition

For an analytic double fibration with incidence cotangent relation $C_{Z} \subset (T^{*} G ∖ 0) \times (T^{*} X ∖ 0)$ , let

π_{L} : C_{Z} ⟶ T^{*} G ∖ 0, π_{L} (g, η; x, ξ) = (g, η) .

The double fibration satisfies the global Bolker condition if $π_{L}$ is an injective immersion. The Bolker condition holds at a specified point $c = (g, η; x, ξ) \in C_{Z}$ if the full fiber over its data covector is the singleton $π_{L}^{- 1} (g, η) = {c}$ and $d π_{L} ∣_{c}$ is injective. Thus the pointwise condition has both a fiber-uniqueness clause, which excludes every other relation point over the same data covector even outside a chosen chart, and a local differential-rank clause.

Definition references

Analytic double fibration

Incidence cotangent relation

Facts & Assumptions

Given: An analytic double fibration and its incidence relation $C_{Z}$ .

[F1]

Analytic double-fibration transform and canonical relation: both incidence projections are analytic submersions and $C_{Z}$ is the relation of the transform.

[F2]

Distributions, conormal kernels, and cotangent relations: $C_{Z}$ consists of nonzero conormals with sign convention $(η, - ξ)$ and is independent of coordinates.

Proof

Given: The incidence relation from [F2].

1.1F1F2algebra

The relation is a smooth conic submanifold in local incidence coordinates: the conormal equations are smooth constant-rank linear equations in the covector variables, and the zero section is removed. Thus projection onto its first cotangent factor is a well-defined smooth map.

2.1F2∎

Global injectivity means that every data covector has at most one object covector paired with it; at a specified point $c$ , the corresponding fiber condition is exactly $π_{L}^{- 1} (π_{L} (c)) = {c}$ . Injectivity of the differential means that every tangent vector to $C_{Z}$ mapped to zero is itself zero. Both conditions are invariant under coordinate changes because [F2] makes $C_{Z}$ coordinate independent and cotangent coordinate changes are diffeomorphisms. This proves the two clauses in the definition.

Depends on

Used by

Dependency tree · two levels

17 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

Marco Mazzucchelli, Mikko Salo, and Leo Tzou, A general support theorem for analytic double fibration transforms (standard reference, not scraped)
Johannes Sjöstrand, Singularités analytiques microlocales (standard reference, not scraped)