CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-13

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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.

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Degree is an integer and independent of the regular value

Statement

Let $F : M^{n} \to N^{n}$ be a proper smooth map between nonempty connected oriented boundaryless manifolds. Every supplied regular value $y$ gives the same integer $\sum_{p \in F^{- 1} (y)} sgn (d F_{p}) = \deg (F) .$ Thus, if a regular value exists, the compact-support degree is an integer and its signed-count computation is independent of which regular value is used. No existence theorem for regular values is asserted here.

Facts & Assumptions

[F1]

Regular-value formula for compact-support degree identifies the degree with the finite signed sum over any supplied regular fibre, with the empty sum equal to zero.

Proof

Given: The proper smooth map $F$ and any supplied regular value $y$ .

1.1

By [F1], $F^{- 1} (y)$ is finite and $\deg (F) = \sum_{p \in F^{- 1} (y)} sgn (d F_{p})$ . Every summand is $1$ or $- 1$ , so this finite sum is an integer; if the fibre is empty it is the integer $0$ .

F1given

2.1

If $z$ is another regular value, [F1] applied to $z$ gives $\sum_{q \in F^{- 1} (z)} sgn (d F_{q}) = \deg (F)$ . Both signed counts therefore equal the same scalar defined without reference to a regular value. This includes dimension zero, a singleton fibre, cancellation to zero, and two empty regular fibres. The argument applies one theorem to each supplied value and makes no simultaneous choice or appeal to Sard's theorem.

F1step 1.1∎

Depends on

Regular-value formula for compact-support degree

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

Robbin–Salamon, Introduction to Differential Topology (standard reference, not scraped)