CorollaryStatement: AI-adaptedProof: AI-adaptedprecheck passjudge 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.

The normalized integral around a positively oriented circle centred at a is 1

For a positively oriented circle $γ (t) = a + r \exp (i t)$ with $r > 0$ , $\frac{1}{2 π i} \int_{γ} \frac{d z}{z - a} = 1.$

Given: A positively oriented circle of positive radius centred at $a$ .

[L1]

[L2]

The complex numbers form a field containing the embedded real field, every complex number has a unique form $a + b i$ , and every nonzero element has an inverse ( $C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ ).

[L3]

technique · direct

1.1L2L3algebra

By [L3], $2 π \neq 0$ in the embedded real field; the unique complex-coordinate form in [L2] then gives $2 π i \neq 0$ , so division is licensed.

2.1step 1.1L1algebra∎

Substitute [L1] and divide to obtain $1$ . The value is independent of the positive radius, and no winding-number or Cauchy theorem is used.

Depends on