Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31
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.

FALSE: the mapping cylinder by itself supplies model-category data

Statement

The mapping-cylinder factorization by itself specifies a model-category factorization, without first choosing and verifying a model structure.

Facts & Assumptions

Given: The factorization supplied by the mapping cylinder for an arbitrary chain map.

[A1]

The statement refuted is: the mapping-cylinder factorization by itself specifies a model-category factorization without a chosen model structure.

[L1]

The proven corollary gives only a degreewise split inclusion followed by a chain-homotopy equivalence (Every chain map factors as a cofibration-like inclusion followed by a homotopy equivalence).

Refutation

technique · direct
1.1

The result in [L1] does not define a model structure, a class of fibrations, or any lifting axioms. It proves only the chain-level factorization that is actually written.

L1givenalgebra
2.1

A model-category factorization is defined only relative to specified classes of cofibrations, fibrations, and weak equivalences satisfying the model axioms. Since the data in [A1] omit all of that structure, they do not even determine the predicates needed to call the two maps a model-category factorization. The correct conclusion from this page is exactly [L1]: a degreewise split inclusion followed by a homotopy equivalence.

A1L1step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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