ExampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)verified 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.

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L(3,1) correction terms in the stated orientation convention

Example

Use the orientation and cyclic labels for which $d (L (p, 1), i) = ((2 i - p)^{2} - p) / (4 p)$ . For $p = 3$ , the correction terms are $d (L (3, 1), 0) = 1 / 2$ , $d (L (3, 1), 1) = - 1 / 6$ , and $d (L (3, 1), 2) = - 1 / 6$ . Reversing orientation negates all three values.

Example references

The lens-space formula and orientation reversal

Ozsvath and Szabo, Absolutely Graded Floer Homologies and Intersection Forms for Four-Manifolds with Boundary, Proposition 4.8

Hom, Lecture Notes on Heegaard Floer Homology, Section 2.2

Facts & Assumptions

Given: The stated orientation convention and the three labels $i = 0, 1, 2$ .

[F1]

For every $p \geq 1$ and canonical label $i$ , the stated $L (p, 1)$ correction-term formula holds; reversing orientation negates the value at the conjugate label (Lens-space correction-term formula for L(p,1)).

Proof

technique · substitute each label into the formula and simplify

Given: The data above.

1.1F1givenalgebra

For $p = 3, i = 0$ , $(2 i - p)^{2} - p = 9 - 3 = 6$ and $4 p = 12$ , so [F1] gives $d (L (3, 1), 0) = 6 / 12 = 1 / 2$ .

2.1F1step 1.1algebra∎

For $i = 1$ , the numerator is $(- 1)^{2} - 3 = - 2$ ; for $i = 2$ , it is $1^{2} - 3 = - 2$ . Both denominators are $12$ , so the other two values are $- 2 / 12 = - 1 / 6$ . Orientation reversal negates each by [F1].

Depends on

Used by

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Dependency tree · two levels

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Sources

Peter S. Ozsvath and Zoltan Szabo, Absolutely Graded Floer Homologies and Intersection Forms for Four-Manifolds with Boundary (standard reference, not scraped)
Jennifer Hom, Lecture Notes on Heegaard Floer Homology (standard reference, not scraped)