Alphabeta Math
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.

Two based circle loops are path-homotopic if and only if they have equal degree

Statement

Two based circle loops are path-homotopic if and only if they have equal degree.

Facts & Assumptions

Given: Based loops α and β at [0] in R/Z.

[L1]

Path-homotopic based circle loops have the same degree (Path-homotopic based circle loops have the same degree).

[L2]

Based circle loops of equal degree are path-homotopic (Based circle loops of equal degree are path-homotopic).

Proof

technique · direct
1.1L1

If α and β are path-homotopic, then [L1] gives deg⁡(α)=deg⁡(β).

1.2L2

Conversely, if deg⁡(α)=deg⁡(β), then [L2] gives an endpoint-fixed path homotopy from α to β.

2.1step 1.1step 1.2∎

Steps 1.1 and 1.2 prove the forward and reverse implications, respectively, so the stated biconditional holds.

Depends on

Used by

Dependency tree · two levels

7 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