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: contour length depends only on the trace and ignores multiplicity

Statement

False claim. Two contours with the same trace always have the same length.

Facts & Assumptions

Given: A radius $r > 0$ , the paths $γ (t) = r e^{i t}$ and $η (t) = r e^{2 i t}$ for $0 \leq t \leq 2 π$ .

[L1]

A piecewise- $C^{1}$ path has length equal to the integral of its speed (A continuous piecewise- $C^{1}$ path is rectifiable and its length is the sum of the speed integrals over its pieces).

[L2]

Length is invariant under continuous surjective monotone reparametrization; bijective reparametrization does not add multiple coverings (Arc length is invariant under every continuous surjective monotone reparametrization, including pauses and reversal).

Refutation

technique · direct

1.1

Both traces are the same circle of radius $r$ , but their speeds are $r$ and $2 r$ .

algebra

2.1

By [L1], $L (γ) = 2 π r$ and $L (η) = 4 π r$ .

step 1.1L1

3.1

Since $r > 0$ , the lengths differ. This does not contradict [L2], because the double covering is not a bijective reparametrization of the single traversal.

step 2.1L2∎

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: 25 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

Russell Howell and John Mathews, Complex Analysis, §6.2 (standard reference, not scraped)