False statementConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

fs-a-derived-functor-is-canonical-without-supplied-replacement-data.md

Statement

A projective replacement model for a total derived functor is literally canonical without supplied replacement data.

Facts & Assumptions

Given: A projective replacement model for a total derived functor is literally canonical without supplied replacement data.

[F1]

The left total derived construction includes supplied projective models (Left total derived functor on the bounded above derived category).

[F2]

Independence means unique natural isomorphism relative to augmentations (Left total derived functor is independent up to a unique augmentation-compatible natural isomorphism).

[F3]

Contractibility means a nullhomotopic identity, reindexed here to cochains (A contractible complex).

Refutation

1.1

For $F = 1_{A b}$ and $X = 0$ , use either the zero projective complex or $C = (Z \overset{1}{\to} Z)$ in degrees $- 1, 0$ . Both map quasi-isomorphically to zero. The homotopy with degree-zero component $1 : Z \to Z$ and other components zero contracts $C$ . Their literal terms are different.

F1F3algebra

2.1

Applying $F$ leaves these different complexes unchanged, although their derived objects are isomorphic. More generally $P \oplus C \to X$ changes any supplied model $P \to X$ in the same way. Replacement independence gives a unique natural comparison relative to augmentation data, not an equality of all chosen representatives.

F2step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

Boundary check against the licensed construction (standard reference, not scraped)