RemarkRemark: Literature-sourcedProof: Not suppliedSession-authored (Fable 5 assisted)audited 2026-08-21‡ not proved here

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.

‡ Recorded, not proved here. This statement is included so the library can refer to it honestly, with a citation to the literature. It is not proved anywhere in this library: the track that would prove it has not been developed here yet.

The sine-period arc-length integrand has no elementary antiderivative

The reduction in The arc length of one sine period is $4 \sqrt{2} E (1 / \sqrt{2})$ is exact: the sine-period length is $4 \sqrt{2} E (1 / \sqrt{2})$ . What is not proved here is the differential-algebraic statement behind the word elliptic. For a nondegenerate modulus $0 < k < 1$ , the integrand

$\sqrt{1 - k^{2} \sin^{2} t}$

has no elementary antiderivative. In particular $k = 1 / \sqrt{2}$ is nondegenerate, so the displayed arc-length integral is not reducible by an elementary antiderivative.

A local proof would require Liouville's theorem on elementary antiderivatives or equivalent differential algebra. That machinery is not among this development's prerequisites, so the non-elementarity assertion is recorded from Hall's treatment rather than presented as locally proved.

Depends on

The arc length of one sine period is $4\sqrt{2}\,E(1/\sqrt{2})$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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

L. M. Hall, Special Functions, §3.1, Example 3.1.2 (standard reference, not scraped)