Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26
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.

The same rational function has different Laurent series on different annuli

Example

The rational function

f(z)=1z(z−1)

has different Laurent series on different annuli:

1z(z−1)=−∑n≥0zn−1(0<∣z∣<1),

1z(z−1)=∑n≥0z−n−2(∣z∣>1),

and, with w=z−1,

1z(z−1)=∑n≥0(−1)n(z−1)n−1(0<∣z−1∣<1).

Each is a convergent Laurent series on its stated annulus (Convergent Laurent series on an annulus).

Facts & Assumptions

Given: The function f(z)=1/(z(z−1)).

Verification

technique · direct
1.1

On 0<∣z∣<1, 1z(z−1)=−1z⋅11−z=−∑n≥0zn−1.

algebra
1.2

On ∣z∣>1, 1z(z−1)=1z2⋅11−z−1=∑n≥0z−n−2.

algebra
1.3

Writing w=z−1, on 0<∣w∣<1 one has 1z(z−1)=1w(1+w)=1w⋅11−(−w)=∑n≥0(−1)nwn−1.

algebra
2.1

Since the three right-hand sides are different formal Laurent series attached to different annuli, the same rational function really does have different Laurent expansions on different annuli.

step 1.1step 1.2step 1.3∎

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