DefinitionDefinition: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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.

Polygonal length and rectifiability of complex paths

Definition

For $a \leq b$ , a complex path is a map $γ : [a, b] \to C$ continuous from the usual real metric to $d_{C} (z, w) = ∣ z - w ∣$ (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, The absolute value makes $R$ a metric space: $d (x, y) = ∣ x - y ∣$ is a metric, its open balls are the intervals $(x - r, x + r)$ , and it is unbounded, Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form, The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane). Its domain and parametrization are part of the path, not merely its trace.

If $a < b$ and $P = (n, t)$ is a partition of $[a, b]$ as in Partition of $[a, b]$ as a finite strictly increasing list $a = t_{0} < t_{1} < \dots < t_{n} = b$ , its subintervals and their lengths, its mesh, refinement, and the common refinement of two partitions, define

$ℓ_{P}^{C} (γ) : = \sum_{j < n} ∣ γ (t_{j + 1}) - γ (t_{j}) ∣, L_{C} (γ) : = \sup_{P} ℓ_{P}^{C} (γ) \in \overline{R} .$

The sum is the ordinary finite real sum of the moduli of the chords. Its supremum is taken in the extended real line (The extended real line $\overline{R} = R \cup {- \infty, + \infty}$ , its order, and the arithmetic that is left undefined, Every subset of $\overline{R}$ has a least upper bound and a greatest lower bound in $\overline{R}$ , agreeing with the real supremum and infimum on nonempty sets bounded in $R$ ). The path is rectifiable when these sums are bounded above by a real number, equivalently when $L_{C} (γ) < + \infty$ . A bounded trace alone does not imply rectifiability; the companion worked example gives a continuous path with bounded trace and unbounded partition sums.

On a singleton interval $[a, a]$ , define $L_{C} (γ) = 0$ and call every path rectifiable. This convention does not invoke a partition, whose definition requires distinct endpoints.

Facts & Assumptions

Given: Reals $a \leq b$ and a continuous map $γ : [a, b] \to C$ .

[F1]

The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane: $d_{C} (z, w) = ∣ z - w ∣$ is a metric with finite nonnegative real values.

[F2]

Partition of $[a, b]$ as a finite strictly increasing list $a = t_{0} < t_{1} < \dots < t_{n} = b$ , its subintervals and their lengths, its mesh, refinement, and the common refinement of two partitions: For $a < b$ , a partition is a finite strictly increasing list $a = t_{0} < \dots < t_{n} = b$ with $n \geq 1$ ; finite sums are as in Finite sums and finite products, by recursion and sums of nonnegative terms are nonnegative by Laws of finite sums and finite products.

[F3]

The extended real line $\overline{R} = R \cup {- \infty, + \infty}$ , its order, and the arithmetic that is left undefined and Every subset of $\overline{R}$ has a least upper bound and a greatest lower bound in $\overline{R}$ , agreeing with the real supremum and infimum on nonempty sets bounded in $R$ : Every subset of $\overline{R}$ has an extended-real supremum; for a nonempty set of real numbers bounded above in $R$ , that supremum is the real least upper bound.

Proof

1.1givenF1F2

If $a < b$ , every partition chord contributes $∣ γ (t_{j + 1}) - γ (t_{j}) ∣ = d_{C} (γ (t_{j + 1}), γ (t_{j}))$ , a finite nonnegative real by [F1]. The finite sum is therefore a finite nonnegative real by [F2].

2.1F2F3step 1.1construct

The set of partition sums is nonempty: the list with $n = 1$ , $t_{0} = a$ , $t_{1} = b$ , and tail $t_{k} = b$ is a partition. Thus [F3] supplies its extended-real supremum. Since every sum is nonnegative, the supremum is not $- \infty$ ; it is finite exactly when some real number bounds all partition sums above. This is the stated rectifiability criterion.

3.1givenF2∎

If $a = b$ , the definition assigns length $0$ and rectifiability directly; no partition is available or used.

Polygonal length and rectifiability of complex paths

Definition

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources