TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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.

Path-homotopic based circle loops have the same degree

Statement

Path-homotopic based circle loops have the same degree.

Facts & Assumptions

Given: Based loops $α, β : I \to R / Z$ at $[0]$ and an endpoint-fixed path homotopy from $α$ to $β$ .

[L1]

Every based circle loop $γ$ has a unique lift $\tilde{γ}$ beginning at zero, and $\deg (γ) = \tilde{γ} (1)$ (The degree of a based circle loop).

[L2]

Endpoint-fixed homotopic paths in the base have lifts with the same endpoint whenever their lifts begin at the same point (The endpoint of a lifted path depends only on its endpoint-fixed homotopy class).

Proof

technique · direct

1.1givenL1

Let $\tilde{α}$ and $\tilde{β}$ be the unique lifts used in [L1]. Both begin at the common point $0$ .

2.1step 1.1L2

Since the base loops are endpoint-fixed homotopic and the lifts have the same initial point, [L2] gives $\tilde{α} (1) = \tilde{β} (1)$ .

3.1step 2.1L1∎

Reading these terminal values through [L1] yields $\deg (α) = \deg (β)$ .

Depends on

Used by

Dependency tree · two levels

9 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

Allen Hatcher, Algebraic Topology, Ch. 1, Section 1.1 (standard reference, not scraped)
J. Peter May, A Concise Course in Algebraic Topology, Ch. 1, Section 5 (standard reference, not scraped)