CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06‡ 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.

A universal-coefficient splitting cannot in general be chosen naturally

Statement

The noncanonical splitting in the homological UCT cannot in general be chosen naturally in the complex and coefficient group.

Counterexample

Given: the free complex $C_{1} = Z e \oplus Z f$ , $C_{0} = Z g$ , with $d (e) = 2 g$ and $d (f) = 0$ , together with $G = Z / 2$ .

1.1

Since $H_{1} (C) = Z f$ , $H_{0} (C) = Z / 2$ , and $d \otimes G = 0$ , the degree-one UCT sequence is $0 ⟶ (Z / 2) f ⟶ (Z / 2) e \oplus (Z / 2) f \overset{q}{\to} Z / 2 ⟶ 0$ , with $q (a e + b f) = a$ .

givenalgebra

2.1

The chain automorphism $u_{1} (e) = e + f$ , $u_{1} (f) = f$ , $u_{0} (g) = g$ induces the identity on both integral homology groups and hence on both outer UCT terms, while its action on the middle term is $(a, b) \mapsto (a, a + b)$ .

step 1.1algebra

3.1

A section must send $1$ to $(1, t)$ for some $t \in Z / 2$ . Naturality with respect to $u$ would force $(1, t + 1) = (1, t)$ , which is impossible. Therefore a UCT splitting cannot be chosen naturally for all complexes and coefficient groups.

step 2.1contradiction∎

Depends on

The homology universal-coefficient sequence splits nonnaturally

Used by

A nonnatural choice of universal-coefficient splitting Counterexample
The universal coefficient theorem does not always give a natural direct-sum decomposition False statement

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

Weibel, An Introduction to Homological Algebra (standard reference, not scraped)