ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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.

Nonzero constant series can multiply to zero in $(Z / 4 Z) ⟦ x ⟧$

Example

In $(Z / 4 Z) ⟦ x ⟧$ , the nonzero constant series $2$ satisfies

$2 \cdot 2 = 0.$

Thus exact additivity of formal order cannot be extended from domains to all commutative rings.

Facts & Assumptions

Given: The hypotheses and notation of the statement above.

[F1]

Two residue classes are equal exactly when their representatives are congruent (The congruence class $[a]_{n}$ and the quotient set $Z / n$ ).

[F2]

Multiplication modulo $n$ is defined by $[a]_{n} [b]_{n} = [a b]_{n}$ (Addition and multiplication on $Z / n$ by $[a]_{n} + [b]_{n} = [a + b]_{n}$ and $[a]_{n} [b]_{n} = [a b]_{n}$ ).

[F3]

The product on $R ⟦ x ⟧$ is the Cauchy product (Cauchy multiplication makes $R ⟦ x ⟧$ a commutative ring containing $R [x]$ as the finitely supported subring).

[F4]

Over an integral domain, formal order is additive on products with the $+ \infty$ convention, and the power-series ring is an integral domain (Formal order is non-Archimedean under sums and additive under products over a domain).

Verification

technique · compute the only coefficient

1.1

The residue class of $2$ modulo $4$ is nonzero because $2 \equiv̸ 0 (m o d 4)$ , while its square is $[2]_{4} [2]_{4} = [4]_{4} = [0]_{4}$ . The constant-series embedding preserves multiplication, so the two nonzero constant series multiply to the zero series.

givenF1F2F3

2.1

Each factor has formal order $0$ , whereas the product has order $+ \infty$ . This does not contradict the exact product law because $Z / 4 Z$ is not an integral domain.

step 1.1givenF4∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 59 results over 20 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Nonzero constant series can multiply to zero in (Z/4Z)⟦x⟧(Z/4Z)[[x]]