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.

There are no nonzero odd-weight level-one modular forms

Example

If $k$ is odd then $M_{k} = {0}$ and $S_{k} = {0}$ .

Facts & Assumptions

Given: An odd integer $k$ and an $f \in M_{k}$ (Level-one modular forms and cusp forms).

[F1]

$- I \in S L_{2} (Z)$ acts on $H$ as the identity and has $c = 0$ , $d = - 1$ , hence factor $(c τ + d)^{k} = (- 1)^{k}$ in the weight- $k$ transformation law (The modular group and its action on the upper half-plane, Level-one modular forms and cusp forms).

Verification

1.1F1givenalgebra

Applying the modular transformation law to $γ = - I$ gives $f (τ) = (- 1)^{k} f (τ) = - f (τ)$ for every $τ \in H$ , since $k$ is odd.

2.1step 1.1givenalgebra∎

Hence $2 f (τ) = 0$ for every $τ$ , so $f = 0$ ; thus $M_{k} = {0}$ for odd $k$ . Since a cusp form of weight $k$ is in particular a modular form of weight $k$ , also $S_{k} = {0}$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

31 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

D. Zagier, Elliptic Modular Forms and Their Applications, in The 1-2-3 of Modular Forms (Universitext, Springer, 2008) (standard reference, not scraped)
J. S. Milne, Modular Functions and Modular Forms (v1.31, 2017) (standard reference, not scraped)