CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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.

Two paths can have the same trace and endpoints but different lengths: one traverses $[0, 1]$ once and another traverses it forward, backward, and forward

Counterexample

Let $γ (t) = t$ on $[0, 1]$ . Let $η : [0, 1] \to R$ be the polygonal path with successive values $0, 1, 0, 1$ at parameters $0, 1 / 3, 2 / 3, 1$ . Both traces are $[0, 1]$ , but

$L (γ) = 1, L (η) = 3.$

Thus length belongs to a parametrized path, not to its trace alone. The difference is caused by backtracking, which is excluded by monotone reparametrization invariance.

Facts & Assumptions

Given: The paths $γ$ and $η$ .

[L1]

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

[L2]

The integral of a constant $C$ on $[u, v]$ is $C (v - u)$ (If $m \leq f \leq M$ on $[a, b]$ then $m (b - a) \leq L (f, P) \leq \underline{\int_{a}^{b}} f \leq \overline{\int_{a}^{b}} f \leq U (f, P) \leq M (b - a)$ for every partition $P$ ; in particular every constant function is integrable, with $\int_{a}^{b} c = c (b - a)$ ).

Verification

technique · computation

1.1

The identity path has constant derivative and speed $1$ , so [L1]--[L2] give $L (γ) = 1$ .

givenL1L2algebra

1.2

On the three parameter thirds, $η$ has derivatives $3, - 3, 3$ , hence speed $3$ . Each speed integral is $3 (1 / 3) = 1$ , so [L1]--[L2] give $L (η) = 3$ .

givenL1L2algebra

2.1

Every value of either path lies in $[0, 1]$ , and each path traverses the whole segment, so their traces agree even though steps 1.1--1.2 give different lengths.

step 1.1step 1.2∎

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: 50 results over 16 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

Curve (standard reference, not scraped)

Two paths can have the same trace and endpoints but different lengths: one traverses [0,1][0,1] once and another traverses it forward, backward, and forward