RemarkRemark: AI-adaptedProof: Not applicablejudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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.

Complex contours as planar rectifiable paths: the Euclidean, coordinate-BV, and piecewise-C1 dictionaries

A complex path $γ = x + i y : [a, b] \to C$ is read as the planar path $(x, y) : [a, b] \to R^{2}$ through $C = R [x] / (x^{2} + 1)$ as the Euclidean plane and as a normed real algebra: what the identification preserves. Its polygonal length and rectifiability are therefore those of Paths in $R^{n}$ , inscribed polygonal sums, arc length as their supremum, and rectifiability. By A path in $R^{n}$ is rectifiable exactly when every coordinate has bounded variation, it is rectifiable exactly when both $x$ and $y$ have bounded variation. Every piecewise- $C^{1}$ complex path is rectifiable, and A continuous piecewise- $C^{1}$ path is rectifiable and its length is the sum of the speed integrals over its pieces gives its length as the sum of the speed integrals over its smooth pieces, allowing corners and zero-speed pieces.

Depends on

Used by

Rectifiable complex contours, reversal, concatenation, closedness, and orientation Definition

Dependency tree · two levels

22 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. Ahlfors, Complex Analysis, 3rd ed., Ch. 4 (standard reference, not scraped)