DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-12

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.

Differential associated to an exact couple

Definition

For a page- $r$ exact couple, define its associated differential by $d_{p, q} = j_{p - 1, q} k_{p, q} : E_{p, q} ⟶ E_{p - r, q + r - 1} .$ The bidegrees of $k$ and $j$ add to $(- 1, 0) + (1 - r, r - 1) = (- r, r - 1)$ , so total degree decreases by one. The square-zero identity is proved in The exact couple differential squares to zero ↗ before homology is formed. For an initial couple $r = 1$ this map has bidegree $(- 1, 0)$ . The zero couple has zero differential. The composite is specified uniquely; neither a section of $i$ nor a preimage selection is part of this definition.

Differential associated to an exact couple

Definition

Depends on

Dependency tree · 2 levels

Sources