Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16
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FALSE: parametrization independence makes orientation reversal leave every contour integral unchanged

Statement

False claim. Parametrization independence implies that reversing a contour leaves every complex line integral unchanged.

Facts & Assumptions

Given: The segment from 0 to 1, its reversal, and the constant integrand 1.

[L1]

Complex and absolute line integrals are invariant under a strictly increasing continuous reparametrization; decreasing reversal is not in that hypothesis (Complex and absolute line integrals are invariant under increasing continuous reparametrization).

[L2]

The integral of a constant is the constant times the endpoint displacement (The contour integral of a constant c is c times the endpoint displacement).

Refutation

technique · direct
1.1

By [L2], the forward segment has integral 1 and the reversed segment has integral −1.

L2algebra
2.1

The values differ. This is consistent with [L1], whose exact hypothesis is increasing reparametrization and therefore does not include orientation reversal.

step 1.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 38 results over 8 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.

Sources