LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-13

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.

The two filtrations identify E2 and the composite edge

Statement

For the supplied composite data $A \to I$ , $F (I) \to J$ under the acyclicity and comparison hypotheses of the total-composite lemma, the vertical-first filtration of $Tot (G J)$ collapses to $R^{n} (G F) (A)$ . The other filtration has $E_{2}^{p, q} = R^{p} G (R^{q} F (A)) .$ Its two edges are the canonical maps $R^{n} G (F (A)) \to R^{n} (G F) (A)$ and $R^{n} (G F) (A) \to G (R^{n} F (A))$ .

Facts & Assumptions

Given: The supplied data and exact hypotheses in the statement.

[F1]

The first filtration identifies the target through the augmentation $G F (I) \to Tot (G J)$ (The total Cartan-Eilenberg complex computes the derived composite).

[F2]

The second hypercohomology sequence has $E_{2}^{p, q} = R^{p} G (H^{q} (F (I)))$ and finite resolution-degree filtration (Second hypercohomology spectral sequence).

[F3]

Its edges are inclusion of the bottom horizontal cycles and projection onto resolution degree zero (Hypercohomology edge maps are canonical).

Proof

1.1F1F2

The cohomology of $F (I)$ is $R^{q} F (A)$ , with $H^{0} (F (I)) = F (A)$ . Substitute this into F2 to obtain the displayed $E_{2}$ . F1 identifies the total target with $R^{p + q} (G F) (A)$ through the actual augmentation quasi-isomorphism. Its proof computes the other filtration as one row, so these are two filtrations of the same total complex, not a claimed equality of their second pages.

2.1F1F3step 1.1∎

Set the lower bound to zero in F3. The bottom horizontal-cycle inclusion gives $R^{n} G (H^{0} (F (I))) = R^{n} G (F (A)) \to H^{n} (Tot (G J))$ ; projection gives $H^{n} (Tot (G J)) \to G (H^{n} (F (I))) = G (R^{n} F (A))$ . Transport both along the augmentation isomorphism of step 1.1. This defines the canonical derived-composite edges and agrees with the finite filtration definition. In degree zero both reduce to $G F (A)$ ; vanishing edge terms are permitted. The naturality/choice qualifications are exactly those of F1–F3.

The two filtrations identify E2 and the composite edge

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources