ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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.

The boundary cycle of a round annulus has index $1$ inside the annulus and $0$ on either side

Example

Let $p \in C$ and $0 < r_{1} < r_{2}$ , and let $C_{j} (t) = p + r_{j} \exp (i t)$ on $[0, 2 π]$ for $j = 1, 2$ . Let $Γ$ be the complex chain $((1, C_{2}), (- 1, C_{1}))$ , written $C_{2} - C_{1}$ . Then $Γ$ is a cycle with trace ${∣ z - p ∣ = r_{1}} \cup {∣ z - p ∣ = r_{2}}$ , and

$n (Γ, z) = {\begin{cases} 0, & ∣ z - p ∣ < r_{1}, \\ 1, & r_{1} < ∣ z - p ∣ < r_{2}, \\ 0, & ∣ z - p ∣ > r_{2} . \end{cases}$

Let $s_{1}, s_{2}$ be reals with $0 < s_{1} < r_{1}$ and $r_{2} < s_{2}$ , and put $Ω = {z : s_{1} < ∣ z - p ∣ < s_{2}}$ . Then $Γ$ has trace in the open set $Ω$ and is null-homologous in $Ω$ . The ambient open set is named before the homology because the notion depends on it: the smaller annulus ${r_{1} < ∣ z - p ∣ < r_{2}}$ does not contain the trace of $Γ$ and is therefore not an open set in which $Γ$ is a chain at all.

Facts & Assumptions

Given: A point $p$ , radii $0 < r_{1} < r_{2}$ , the circles $C_{1}, C_{2}$ above, and the chain $Γ = C_{2} - C_{1}$ .

[L1]

The trace of a sum of chains is the union of their traces, the negative of a chain has the same trace, a sum of cycles and the negative of a cycle are cycles, and for $p^{'}$ off the traces involved $n (Γ_{1} + Γ_{2}, p^{'}) = n (Γ_{1}, p^{'}) + n (Γ_{2}, p^{'})$ and $n (- Γ, p^{'}) = - n (Γ, p^{'})$ (Chain integration and the index are additive in the chain, and reverse with it).

[L2]

For $a \in C$ , $r > 0$ and $k \in Z$ , the contour $γ_{k} (t) = a + r \exp (i k t)$ on $[0, 2 π]$ is a closed complex contour with $n (γ_{k}, z) = k$ for $∣ z - a ∣ < r$ and $n (γ_{k}, z) = 0$ for $∣ z - a ∣ > r$ ; for $k \neq 0$ its trace is ${z : ∣ z - a ∣ = r}$ (A circle traversed $k$ times has winding number $k$ inside and $0$ outside).

[L3]

A complex chain is a finite list of pairs $(m_{k}, γ_{k})$ ; a list of closed contours is a cycle; the negative of a chain negates every coefficient; and a single closed contour with coefficient $1$ is a cycle whose trace is that contour's trace (Complex chains, their traces, and cycles).

[L4]

$n (Γ, p^{'}) = (2 π i)^{- 1} \int_{Γ} d z / (z - p^{'})$ , and for a single closed contour with coefficient $1$ this is that contour's winding number (Integration over a complex chain and the index of a chain).

[L5]

A cycle $Γ$ with trace in an open $Ω$ is null-homologous in $Ω$ when $n (Γ, q) = 0$ for every $q \in C ∖ Ω$ (Null-homologous cycles and homologous cycles in an open set).

Verification

technique · direct

1.1

By [L2] with $k = 1$ each $C_{j}$ is a closed complex contour with trace ${∣ z - p ∣ = r_{j}}$ , with $n (C_{j}, z) = 1$ for $∣ z - p ∣ < r_{j}$ and $n (C_{j}, z) = 0$ for $∣ z - p ∣ > r_{j}$ .

givenL2

2.1

By [L1] and [L3] the chain $Γ = C_{2} - C_{1}$ is a cycle and its trace is ${∣ z - p ∣ = r_{1}} \cup {∣ z - p ∣ = r_{2}}$ , and by [L1] and [L4] its index off that trace is $n (C_{2}, z) - n (C_{1}, z)$ .

step 1.1L1L3L4

3.1

Evaluating step 2.1 with step 1.1: for $∣ z - p ∣ < r_{1} < r_{2}$ the value is $1 - 1 = 0$ ; for $r_{1} < ∣ z - p ∣ < r_{2}$ it is $1 - 0 = 1$ ; for $∣ z - p ∣ > r_{2} > r_{1}$ it is $0 - 0 = 0$ .

step 1.1step 2.1

4.1

With $0 < s_{1} < r_{1}$ and $r_{2} < s_{2}$ the trace of $Γ$ lies in $Ω = {s_{1} < ∣ z - p ∣ < s_{2}}$ , and $C ∖ Ω = {∣ z - p ∣ \leq s_{1}} \cup {∣ z - p ∣ \geq s_{2}}$ ; every point of the first set has $∣ z - p ∣ \leq s_{1} < r_{1}$ and every point of the second has $∣ z - p ∣ \geq s_{2} > r_{2}$ , so step 3.1 gives index $0$ at each, and [L5] makes $Γ$ null-homologous in $Ω$ .

step 2.1step 3.1L5∎

Depends on

Used by

Dependency tree · two levels

49 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

J. Lebl, Complex Analysis, Ch. 4 §4.3 (standard reference, not scraped)

The boundary cycle of a round annulus has index 11 inside the annulus and 00 on either side