ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (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.

A created cancelling pair contributes a $(1 + t) t^{k}$ term

Example

Assume ${A C}_{ω}$ . Let $M$ be a closed smooth $n$ -manifold with a handle presentation, and let $M^{'}$ be the presentation obtained by inserting a geometrically cancelling pair of consecutive indices $k, k + 1$ at an intermediate stage ( $0 \leq k \leq n - 1$ ), transporting the later attaching embeddings across the cancellation diffeomorphism and leaving their indices unchanged. Then $M^{'}$ presents the same manifold $M$ ; if $f$ and $f^{'}$ are adapted Morse functions inducing the two presentations, then $M_{f^{'}} (t) = M_{f} (t) + t^{k} + t^{k + 1} = M_{f} (t) + (1 + t) t^{k}, P_{M^{'}, F} = P_{M, F},$ and the correction polynomials satisfy $Q^{'} = Q + t^{k}$ . In the model case $M = S^{2}$ with the two-critical-point presentation $M_{f} (t) = 1 + t^{2}$ , inserting a cancelling $(0, 1)$ -pair gives $M_{f^{'}} (t) = 2 + t + t^{2}$ and $Q^{'} = 1$ .

Facts & Assumptions

Given: A closed smooth $n$ -manifold $M$ with a finite handle presentation in stages, a geometrically cancelling pair of consecutive indices $k, k + 1$ inserted at an intermediate stage, the resulting presentation $M^{'}$ , and adapted Morse functions $f$ , $f^{'}$ inducing the two presentations (empty-face convention for the closed case).

[F1]

The cancellation theorem deletes any geometrically cancelling pair (Handle cancellation). The creation theorem attaches a $k$ -handle and then a $(k + 1)$ -handle in the standard complementary way on a disc of the outgoing boundary and produces a diffeomorphism of $W \cup h^{k} \cup h^{k + 1}$ with $W$ relative to the incoming boundary, so the modified presentation presents the same manifold; the two new handles are added, and later attaching embeddings are transported across this diffeomorphism without changing their indices (Creation of a cancelling handle pair, Handle decomposition relative to the incoming boundary).

[F2]

Adapted Morse functions on the triad and handle presentations correspond: the Morse numbers of the function inducing a presentation equal the numbers of handles by index (Morse functions and handle decompositions correspond).

[F3]

The Morse polynomial is $M_{f} (t) = \sum_{k} m_{k} (f) t^{k}$ with $m_{k} (f) = # {p : ind (p) = k}$ (Morse numbers and the Morse polynomial); the Poincare polynomial over $F$ is $P_{X, F} (t) = \sum_{k} \dim_{F} H_{k} (X; F) t^{k}$ (Poincare polynomial of a space and of a pair over a field).

[F4]

For every field $F$ there is a unique $Q \in Z [t]$ with nonnegative coefficients and $M_{f} = P_{M, F} + (1 + t) Q$ (Morse polynomial identity).

[F5]

The height function on $S^{2}$ has Morse polynomial $1 + t^{2}$ . (computed below).

[F6]

A homotopy equivalence induces isomorphisms on singular homology with every coefficient group (Homotopy equivalences induce isomorphisms on singular homology); in particular a diffeomorphism does so.

Verification

technique · presentation-comparison

1.1givenalgebra

For $h (x) = x_{3}$ on $S^{2}$ , a point away from the poles has tangent vector $v = e_{3} - x_{3} x$ with $d h (v) = 1 - x_{3}^{2} > 0$ , so it is not critical. In pole charts $h (u) = \pm \sqrt{1 - ∣ u ∣^{2}}$ has Hessian $\mp I_{2}$ at $u = 0$ ; thus the south and north poles have indices $0, 2$ , and $M_{h} = 1 + t^{2}$ . Sphere homology gives $P_{S^{2}, F} = 1 + t^{2}$ , so the correction polynomial is $Q = 0$ .

1.2F1given

Apply cancellation in [F1] to the affected connected component of the intermediate stage, keeping other components fixed; for a $(0, 1)$ -pair its second foot lies on that existing component by the one-intersection condition. It supplies a diffeomorphism of the new presentation's underlying manifold with the old one relative to the incoming face; transport each later attaching embedding across its boundary restriction; hence the modified presentation still presents $M$ , and its handle counts are those of the old presentation increased by one in index $k$ and one in index $k + 1$ .

2.1F2F3step 1.2

Let $f, f^{'}$ be the adapted Morse functions inducing the two presentations by [F2]. Their Morse numbers count the handles by index, so $m_{j}^{'} = m_{j} + δ_{j k} + δ_{j, k + 1}$ for every $j$ , and therefore, by [F3], $M_{f^{'}} (t) = M_{f} (t) + t^{k} + t^{k + 1} = M_{f} (t) + (1 + t) t^{k}$ .

2.2F3F6step 1.2

The diffeomorphism of step 1.2 induces homology isomorphisms by [F6]. Thus the Betti numbers, and hence the Poincare polynomials defined in [F3], agree: $P_{M^{'}, F} (t) = P_{M, F} (t)$ for every field $F$ .

3.1F4step 2.1step 2.2

Apply [F4] to both functions, whose Poincare polynomials coincide by step 2.2: $M_{f} = P_{M, F} + (1 + t) Q$ and $M_{f^{'}} = P_{M, F} + (1 + t) Q^{'}$ . By step 2.1, the nonnegative polynomial $Q + t^{k}$ also satisfies the second identity. Uniqueness in [F4] therefore gives $Q^{'} = Q + t^{k}$ .

4.1F5step 2.1step 3.1∎

Model case: for $M = S^{2}$ the two-critical-point presentation of the height function has Morse polynomial $1 + t^{2}$ by [F5]; inserting a cancelling $(0, 1)$ -pair gives $M_{f^{'}} (t) = 1 + t^{2} + t^{0} + t^{1} = 2 + t + t^{2}$ and $Q^{'} = 0 + 1 = 1$ by steps 2.1 and 3.1. The Euler sum is preserved: $2 - 1 + 1 = 2 = 1 - 0 + 1$ , consistent with the Euler characteristic identity.

Remarks

What the example shows. A birth of a cancelling pair adds $(1 + t) t^{k}$ to the Morse polynomial while leaving the manifold and its homology unchanged, so the correction polynomial of the Morse polynomial identity is exactly the algebraic record of such pairs.
The perfectness defect. The pair is invisible in homology but increases the excess of Morse numbers over Betti numbers; the added term $(1 + t) t^{k}$ has value zero at $t = - 1$ , which is why the Euler identity cannot detect it.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

60 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

Michele Audin and Mihai Damian, Morse Theory and Floer Homology, Part I Chapter 4 Section 4.4, printed pp. 88-91 (PDF pp. 98-100) (standard reference, not scraped)
Liviu Nicolaescu, An Invitation to Morse Theory (2nd ed.), Chapter 2 Section 2.3, printed pp. 46-53 (PDF pp. 56-63) (standard reference, not scraped)

A created cancelling pair contributes a (1+t)tk(1+t)tk term