TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Jordan-Holder theorem in an abelian category

Statement

If an object $A$ in an abelian category has two composition series, then the two series have the same length and the same composition factors up to permutation and isomorphism.

Facts & Assumptions

Given: Two composition series of the same object $A$ .

[L1]

A composition series is a finite strict subobject chain with simple successive quotients (Composition series and composition factors of an object, Simple object).

[L2]

Any two finite subobject chains admit equivalent refinements (Schreier refinement theorem in an abelian category).

Proof

technique · direct

1.1

By [L2], the two composition series admit equivalent refinements.

givenL2

1.2

A composition series has no proper refinement. Indeed, if $A_{i - 1} < K < A_{i}$ , then the quotient map $A_{i} \to A_{i} / A_{i - 1}$ carries $K$ to a nonzero proper subobject of the simple object $A_{i} / A_{i - 1}$ , contradicting [L1]. So any refinement of a composition series differs from it only by repeated adjacent terms.

L1algebra

2.1

Delete repeated adjacent terms from the equivalent refinements of step 1.1. By step 1.2 this recovers the original two composition series, and the quotient pairing survives. Therefore the original series have the same number of factors, and a permutation matches their factors up to isomorphism.

step 1.1step 1.2∎

Depends on

Used by

Object of finite length Definition
The finite abelian group Z/12 has length three Example
Two composition series of Z/12 refine to the same simple factors Example
FALSE: Jordan-Holder needs finiteness only of the ambient category False statement
The published abelian-group composition-series development is the instance Remark
Length is additive along a subobject Theorem

Dependency tree · two levels

6 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

Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik, Tensor Categories, Section 1.5 (standard reference, not scraped)