CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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.

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The function 1 / z makes the large-semicircle shortcut fail

Statement refuted

Refuted claim: the pointwise decay of $1 / z$ along larger and larger upper semicircles is enough to force the arc integral to vanish.

Facts & Assumptions

Given: The function $f (z) = 1 / z$ and the upper semicircles $γ_{R}^{+} (t) = R e^{i t}$ .

Counterexample

technique · computation

1.1

On the upper semicircle $γ_{R}^{+} (t) = R e^{i t}$ one has $\int_{γ_{R}^{+}} \frac{d z}{z} = \int_{0}^{π} \frac{i R e^{i t}}{R e^{i t}} d t = i π .$

givenalgebra

2.1

So the arc integral is constant, not vanishing, even though $1 / (R e^{i t})$ is pointwise $O (R^{- 1})$ on the arc. This is the concrete witness behind FALSE: pointwise decay alone makes every large semicircle integral vanish.

step 1.1∎

Depends on

FALSE: pointwise decay alone makes every large semicircle integral vanish

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

L. V. Ahlfors, Complex Analysis, 3rd ed., Ch. 4 §5.3 (standard reference, not scraped)